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Contents
Chapter 1 Introduction to the Structural Biology of Membrane
Proteins
Mary Luckey
1.1
1.2
1.3
1.4
1
Introduction
Membrane Features
Lipid Polymorphism
Classes of Membrane Proteins
1.4.1 a-Helical Bundles
1.4.2 b-Barrels
1.5 Functions of Membrane Proteins
1.5.1 Channels
1.5.2 Transporters
1.5.3 Enzymes
1.5.4 Receptors
1.6 Membrane Protein Complexes
1.7 Conclusions
References
1
2
5
7
7
8
9
10
11
14
14
15
17
17
Chapter 2 Molecular Dynamics Simulations: Principles and
Applications for the Study of Membrane Proteins
Victoria Oakes and Carmen Domene
19
2.1
2.2
Introduction
Classical Molecular Dynamics
2.2.1 Additive Force Fields
2.2.2 Polarisable Force Fields
RSC Theoretical and Computational Chemistry Series No. 10
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2.2.3 Practical and Technical Considerations
2.2.4 Applications
2.3 Coarse-grained Molecular Dynamics
Simulations
2.4 Ab initio Molecular Dynamics
2.5 Enhanced Sampling Techniques and Free Energy
Methods
2.6 Conclusions
References
Chapter 3 Free Energy Calculations for Understanding Membrane
Receptors
Andrew Pohorille
3.1
3.2
Introduction
The Basics of Free Energy Calculations
3.2.1 The Parametric Formulation of Free Energy
Calculations
3.2.2 Ergodicity, Variance Reduction Strategies,
and the Transition Coordinate
3.3 Free Energy Perturbation Methods
3.3.1 Theoretical Background
3.3.2 Alchemical Transformations
3.4 Probability Distribution Methods
3.5 Thermodynamic Integration
3.5.1 Theoretical Background
3.5.2 Adaptive Biasing Force Method
3.6 Replica Exchange for Enhanced Sampling in
Configurational Space
3.7 Applications of Free Energy Calculations:
Case Studies
3.7.1 Binding of Anesthetic Ligands to
Receptors
3.7.2 Free Energies of Ions across Channels
3.7.3 Conformational Transitions in Receptors
3.8 Non-equilibrium Properties from Free Energy
Calculations
3.8.1 Theoretical Background
3.8.2 Example – the Leucine–Serine Channel
3.9 Summary and Conclusions
References
25
29
30
33
34
40
40
59
59
60
60
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Chapter 4 Non-atomistic Simulations of Ion Channels
Claudio Berti and Simone Furini
4.1
4.2
Introduction
Methods Based on Continuum Distributions
of Ions
4.2.1 Poisson–Boltzmann
4.2.2 Poisson–Nernst–Planck
4.2.3 Improvements of Classical Continuum
Theories of Electrolytes
4.3 Particle-based Methods
4.3.1 Brownian Dynamics
4.3.2 Monte Carlo
4.4 Methods to Include Atomic Detail in
Non-atomistic Models
4.4.1 Atomic Detail in Brownian Dynamics
4.4.2 Atomic Detail in Continuum Models
4.5 Concluding Remarks
References
Chapter 5 Experimental and Computational Approaches to Study
Membranes and Lipid–Protein Interactions
Durba Sengupta, G. Aditya Kumar, Xavier Prasanna and
Amitabha Chattopadhyay
5.1
5.2
5.3
5.4
5.5
Introduction
5.1.1 Membrane Components
Role of Membrane Lipids in Membrane Protein
Organization and Function
Mechanisms for Lipid Regulation of Membrane
Proteins
5.3.1 Specific Membrane Effects
5.3.2 Non-specific Membrane Effects
Range of Time Scales Exhibited by Membranes
Lipid–Protein Interactions: Insights from
Experimental Approaches
5.5.1 Determining Near-neighbor Relationships in
Membranes: Interaction of Melittin with
Membrane Cholesterol utilizing FRET
5.5.2 Interaction of the Actin Cytoskeleton with
GPCRs: Application of FRAP
107
107
111
112
117
119
122
124
127
128
129
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133
133
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5.6
Computational Approaches to Study Membrane
Organization and Lipid–Protein Interactions
5.6.1 Simulating Single Component and
Multi-component Bilayers
5.6.2 Atomistic Simulations Elucidating
Lipid–Protein Interactions
5.6.3 Coarse-grain Methods to Analyze Membrane
Protein Interactions
5.6.4 Enhanced Sampling Methods
5.7 Future Perspectives: The Road Ahead
Acknowledgements
References
Chapter 6 Computer Simulation of Ion Channels
Ben Corry
6.1
6.2
Introduction to Ion Channels
Questions that can be Addressed and Associated
Timescales
6.3 Ion Permeation
6.4 Ion Selectivity
6.4.1 Na1/Ca21 Selection
6.4.2 Na1/K1 Selection
6.5 Channel Gating
6.6 Interactions of Channels with Drugs
and Toxins
6.6.1 Toxin–Channel Interactions
6.6.2 Channel Blockage by Small Molecules
6.7 Conclusions
Acknowledgements
References
Chapter 7 Computational Characterization of Molecular
Mechanisms of Membrane Transporter Function
Noah Trebesch, Josh V. Vermaas and Emad Tajkhorshid
7.1
7.2
Membrane Transport – A Fundamental Biological
Process
Substrate Binding and Unbinding
7.2.1 Spontaneous Binding Simulations Revealing
a Binding Mechanism and Site
149
151
151
153
154
155
155
155
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161
165
169
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174
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7.2.2
Proposing Substrate Binding Sites through
Molecular Docking
7.2.3 Unraveling Substrate Release Pathways
7.3 Capturing Localized Transporter Motions with
Equilibrium Molecular Dynamics
7.3.1 Substrate-induced Structural Changes of an
Antiporter
7.3.2 Gating Elements in a Neurotransmitter
Transporter
7.4 Computational Description of Global Structural
Transitions in Membrane Transporters
7.4.1 Nonequilibrium Simulation of Structural
Changes
7.4.2 Application to an ABC Transporter
7.5 Water within Transporters
7.5.1 Water Leaks in Transporters
7.5.2 Water in Proton Pathways
7.6 The Lipid Frontier
7.6.1 Why Now? Initial Barriers to Simulating
Lipid–Protein Interactions
7.6.2 Computational Probes of Lipid–Protein
Interactions
7.7 Concluding Remarks
Acknowledgements
References
Chapter 8 Computational Studies of Receptors
Maria Musgaard and Philip C. Biggin
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Introduction
Network Models Can Provide Insight into
Large-scale Conformational Changes
Network Models to Examine Gating
Network Models to Compare Dynamics
Network Models to Suggest Novel Mechanisms
for Modulation
Molecular Dynamics to Aid Crystallographic
Interpretation
Molecular Dynamics to Move between States
Molecular Dynamics to Refine Working
Models
202
204
205
206
206
209
210
212
215
216
216
219
219
220
221
222
223
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237
239
241
242
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8.9
Molecular Dynamics to Explain the Effects of
Ions and Water
8.10 Molecular Dynamics to Quantify Free Energy
Requirements
8.11 Conclusions
References
Subject Index
251
253
255
255
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CHAPTER 1
Introduction to the Structural
Biology of Membrane Proteins
MARY LUCKEY
Department of Chemistry and Biochemistry, San Francisco State
University, 1600 Holloway Ave., San Francisco CA 94132, USA
Email: luckey@sfsu.edu
1.1 Introduction
The biomembrane is an essential feature of life, defining the interior of cells
and controlling the entry and exit of molecules and ions.1 Lipids in the
membrane make a hydrophobic barrier for aqueous solutes; this barrier is
semipermeable due to the presence of membrane proteins. In addition to
transport functions, membrane proteins have roles as receptors, enzymes,
and structural components. Biomembranes also contain glycolipids and
glycoproteins, whose carbohydrate constituents are external to the cell
and are important in recognition. This introduction focuses on membrane
proteins in the context of their lipid environment.
Membrane proteins share some features with soluble proteins, for
example the protein interior (excluding channel walls) is tightly packed with
mostly nonpolar residues. However, the surface of membrane proteins is
amphiphilic, allowing interactions with both the polar exterior and the
nonpolar interior of the membrane. The need to maintain this amphiphilic
structure increases the difficulty of purifying and handling membrane proteins. In addition, their localization in the membrane often impedes their
overexpression. These difficulties hampered their structural characterization
for many years, so that X-ray crystallography of membrane proteins lagged
RSC Theoretical and Computational Chemistry Series No. 10
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Chapter 1
25 years behind that of soluble proteins. In 1988, Harmut Michel, Johann
Deisenhoffer, and Robert Huber received a Nobel Prize for the first crystal
structures of membrane proteins, those from the photosynthetic reaction
center of Rhodopseudomonas viridis.2 (Rhodospeudomas viridis has been
renamed Blastochloris viridis.) Over the next two decades, new membrane
protein crystal structures were sufficiently rare to grace the covers of Nature
and Science when they did appear. However, their number rose exponentially; the data base now contains more than 650 unique structures of
membrane proteins determined with X-ray crystallography or NMR
spectroscopy,3 with good overall agreement in structures that have been
solved by both. The availability of detailed membrane protein structures
allows researchers to further probe their function and dynamics using
genetic, biochemical, biophysical, and computational tools.
1.2 Membrane Features
The native environment of membrane proteins is dynamic and asymmetric,
described by Singer and Nicolson as fluid and mosaic in their 1972 model
that became the paradigm for membrane structure (Figure 1.1).4 Lipids
consisting of polar head groups and nonpolar acyl chains form a twodimensional fluid bilayer (Figure 1.2). Lipid composition is diverse, with
most of the lipids randomly distributed in the bulk phase of the bilayer while
Figure 1.1
The Fluid Mosaic Model of Singer and Nicolson views the membrane as a
fluid lipid bilayer with a mosaic of intrinsic proteins. The fatty acyl chains
from each leaflet form the nonpolar interior, while the aqueous periphery
contains extrinsic proteins and carbohydrate chains (not shown).
From S. J. Singer and G. L. Nicolson, Science 1972, 175, 720. r 1972,
American Association for the Advancement of Science. Redrawn with
permission from AAAS.
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Introduction to the Structural Biology of Membrane Proteins
Figure 1.2
3
Phospholipid bilayers may exist in the liquid disordered phase (Ld, left) or
the liquid ordered phase (Lo, right). Ld and Lo phases are depicted with
representative snapshots from coarse-grained simulations using the Martini
force-field. DPPC (dipalmitoyl phosphatidylcholine) is shown in green,
DOPC (dioleoyl phosphatidylcholine) in orange, cholesterol in purple,
with portions of headgroups visible as beads. Water is not shown for clarity.
Kindly provided by Dr Svetlana Baoukina, University of Calgary.
Undulations
(10-6-1s)
Flip-flop
(10-3-104 s)
cis-trans isomerization
(10-10s)
Bond vibrations
(10-12s)
Rotational diffusion wobble
(10-8s)
Protrusion
(10-9 s)
Lateral diffusion
(10-7s)
Figure 1.3
A variety of lipid motions create disorder in the fluid lipid bilayer. Several
kinds of lipid motions are shown here with their approximate correlation
times.
Kindly provided by Prof. Carmen Domene, King’s College London.
some are localized by specific interactions with proteins and/or other lipids,
often in regions of ordered lipids called rafts (see below).
In the lipid disordered phase (Ld, also called La), fluidity of the bilayer
results from the constant and varied motion of lipids (Figure 1.3). Although
the lateral diffusion rate in pure lipid bilayers is very fast, the measured
mobility of bulk lipids on the surface of cells is much slower. This difference
has been explained by single-particle tracking on the surface of cells containing cytoskeletons: single-molecule trajectories are fast within small
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Chapter 1
regions, where they are confined until they hop to a contiguous region,
producing a slower overall progression.5
The mosaic distribution of membrane proteins results from wide variations in lateral mobility, from those that diffuse rapidly on the surface to
those anchored by the cytoskeleton. A large proportion of membrane
proteins function in protein assemblies, which themselves have varying
lifetimes. Some complexes of membrane proteins are very stable, such as the
respiratory complexes involved in energy transduction (see below), while
others are the result of transient interactions such as those involved in signal
transduction. Many lipid-anchored proteins are observed in membrane rafts
enriched in sphingomyelin and cholesterol in the lipid-ordered (Lo) state
(see Figure 1.2). The presence of rafts varying in size (diameters from 10 to
200 nm) and in duration (from o1 ms to fairly stable lifetimes) enhances
nonrandom distributions and dynamic interactions in the membrane.6
Lipid–lipid interactions probably drive raft formation, given that even
simple lipid mixtures reveal fluid immiscibility (Figure 1.4). The fused
Figure 1.4
A ternary phase diagram showing that concentrations of three lipid
components can reveal regions with lipid compositions that produce
immiscible fluid phases. Using giant vesicles of POPC (palmitoyl oleoyl
phosphatidylcholine)/PSM (palmitoyl sphingomyelin)/cholesterol tagged
with fluorescent markers, separation into two distinct liquid phases is
observed at 25 1C for compositions within the gray region of this phase
diagram. Several representative vesicles are shown with compositions
near the edges of the phase boundary, with the dark phase being highly
enriched in PSM and moderately enriched in cholesterol, while the
bright phase is strongly enriched in POPC. Scale bars are 20 mm and
the vesicles were imaged before the domains fully coalesced.
Kindly provided by Prof. Sarah Veatch, University of Michigan.
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hydrocarbon rings of cholesterol are nearly rigid, allowing the sterol to align
with lipids containing saturated acyl chains, especially sphingomyelins, and
promote the tight packing of the Lo phase.
1.3 Lipid Polymorphism
Of the possible physical states for lipid aggregates, only a few are present at
biological conditions, while others are useful in the laboratory. A pure lipid
bilayer (constituted by only one lipid species) can be in lamellar crystalline
(Lc), lamellar gel (Lb), or lamellar liquid crystalline (called fluid, La, or
disordered, Ld) states, depending on the temperature and pressure. The shape
of the lipid molecule may introduce significant curvature to the bilayer, an
important force during some membrane processes.7 Since excessive curvature
disrupts the bilayer, biological membranes contain a significant fraction of
non-bilayer lipids (lipids such as phosphatidylethanolamine that do not form
bilayers in the pure state) to minimize the curvature tension. Non-bilayer
states include hexagonal phases (hexagonally packed arrays in long cylinders)
and cubic phases (cubic packing of rod-like elements, Figure 1.5). Hexagonal
phases may be important in biological transitions, such as pore formation
and membrane fusion. The lipidic cubic phase (LCP) has become an
important tool in the crystallization of membrane proteins.8
In contrast to the simplified mixtures employed for physical studies, the
lipid composition of most membranes is very complex: a typical membrane
has about a hundred lipid species, and over 1000 different species of lipids
have been identified in biological samples.9 Prokaryotic organisms vary
the fatty acid content of their membranes to maintain fluidity at extreme
Figure 1.5
Lipid polymorphism results in lamellar, hexagonal, and cubic phases. The
lipid bilayers shown in Figure 1.2 are of lamellar phase. The hexagonal
phase may be normal, HI (not shown) with nonpolar regions inside the
tubes, or inverted, HII (left) with polar groups and water inside. In cubic
phases, a variety of three-dimensional systems of lipid channels or
networks are interpenetrated by water channels, represented here by the
bicontinuous type (right).
Kindly provided by Dr Simone Aleandri and Prof. Ehud Landau, University
of Zurich.
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Chapter 1
temperatures (longer and more saturated fatty acids have higher melting
points than shorter and unsaturated fatty acids). Eukaryotic organisms have
markedly different lipid contents of their envelope and organellar membranes due to the sites of lipid synthesis and patterns of lipid trafficking.
Prokaryotes lack cholesterol and other steroids, while some eukaryotic
membranes possess very high sterol contents (cholesterol in animals,
ergosterol in yeast and fungi, and stigmasterol and sitosterol in plants.)
Clearly, the membrane composition is mosaic in the plane of the bilayer,
with dynamic mixing of its lipid species. The membrane also presents
significant variations normal to the bilayer plane. The lipid headgroups
make up the polar exterior of the bilayer, with the acyl chains extending
across the nonpolar interior. Elegant probability maps for the positions of
each constituent have been achieved by joint refinement of X-ray and
neutron diffraction data (Figure 1.6). The penetration of water molecules
defines the two interfacial regions of the membrane; water is completely
absent from the hydrocarbon core. In addition to their functional importance, such as exclusion of charged molecules from the interior, these
variations are important considerations for the structures of proteins that
span the bilayer.
Figure 1.6
The joint refinement of X-ray and neutron diffraction data provides the
structure of a DOPC bilayer (indicated normal to the bilayer) as density
peaks for portions of the DOPC molecule [methyl (CH3), methylene
(CH2), double-bonds (C¼C), carbonyls, choline, glycerol, and phosphate]
and water. The hydrocarbon core of the bilayer, which lacks water, is
sandwiched between two interfacial regions.
From S. H. White et al., J. Biol. Chem. 2001, 276, 32395. r 2001 by the
American Society for Biochemistry and Molecular Biology. Reprinted by
permission of the ASBMB.
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1.4 Classes of Membrane Proteins
The amphiphilic membrane proteins described above are classed as integral
membrane proteins (also called intrinsic proteins) because they are
embedded in the membrane: they interact extensively with membrane
lipids10 and cannot be purified without disrupting the membrane using
detergents or mechanical rupture. In contrast, peripheral membrane proteins (also called extrinsic proteins) can be washed off the membrane
without disrupting it. They bind to membrane lipids, other membrane
proteins, or both by a combination of electrostatic and nonpolar interactions. Some proteins insert into the membrane or bind it transiently, with
binding being enhanced by calcium ions, lipid anchors, and/or repeats of
small peptide domains. Amphitropic proteins are proteins whose activities
are regulated by binding to the membrane, such as some phospholipases
and blood clotting factors. Some phospholipases undergo significant conformational changes in order to bind their lipid substrates, while others
simply trap them by dimerization.
Integral membrane proteins are adapted to an environment that varies
from the cytosol in viscosity, dielectric constant, and isotropy, and contains
gradients of pH, redox potential, and pressure. They are described by their
number of transmembrane segments (TMSs). Bitopic membrane proteins
with one TMS tend to form dimers (such as glycophorin A) or associate with
larger, polytopic membrane proteins (such as the H protein in the photosynthetic reaction center). Monotopic membrane proteins do not cross the
bilayer but insert into one leaflet; an example is prostaglandin H2 synthase, an
important target of nonsteroidal anti-inflammatory drugs. In addition, proteins may bind to the membrane via one or more lipid anchors (myristoyl,
palmitoyl, farnesyl, geranylgeranyl, and glycosylphosphatidylinositol groups)
added by post-translational modifications.
Integral membrane proteins are classified by their overall structure into
two dominant classes: a-helical bundles and b-barrels, which are discussed
below. Other structural patterns can occur, such as the Wza protein that
spans the bilayer with a barrel of amphipathic a-helices.11 A governing
principle is the need for secondary structure in the TMS to satisfy the
hydrogen-bonding of peptide groups, since in the low dielectric bilayer the
cost of disrupting these hydrogen bonds is B4 kcal per residue, producing a
prohibitive DG of B80 kcal to unfold a twenty-residue TM a-helix.12
1.4.1
a-Helical Bundles
In the majority of integral membrane proteins each TMS is an a-helix of
twenty or more predominantly nonpolar residues. The presence of one or
two polar residues is tolerated within the overall hydrophobicity needed.
Charged amino acids are positioned for functional purposes, they may form
ion pairs, or they may be present in their uncharged states with altered pKas
due to low dielectric environments. Alternatively, charged or polar side
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Chapter 1
chains can fold towards the membrane interface, snorkeling to orient their
polar groups beyond the nonpolar core of the bilayer. Snorkeling is most
commonly observed with lysine residues, but has also been seen with other
amino acids (Arg, Tyr, Asp, Glu, Asn, and Gln). Another role for polar side
chains near the TMS ends is the formation of hydrogen bonds with peptide
bond groups to cap the ends of the helices.
Tryptophan and tyrosine residues play a special role at the interface of the
polar and nonpolar domains due to electrostatic interactions of their aromatic rings with the hydrocarbon core; often their presence in a ring around
the protein structure helps to define the location of the interface. Another
informative aspect of the primary structure derives from the ‘positive inside
rule’: positively charged amino acids predominate on the cytoplasmic
portions of integral membrane proteins due to some restrictions in their
insertion during biogenesis. Numerous algorithms utilize the positive inside
rule combined with hydropathy plots to predict the structure of a-helical
membrane proteins.13
The first membrane protein observed (by electron microscopy) to consist
of an a-helical core was bacteriorhodopsin (bR), a light-driven proton pump
from Halobacteria. bR contains seven roughly parallel TM helices with a
covalently bound retinal cofactor near the center (Figure 1.7). Characterization of bR intermediates at different stages of its photocycle has revealed
small structural shifts accompanying the electronic transitions.14 Sophisticated folding studies of bR have defined folding transitions as well as the
effect of curvature stress.15
bR is a bacterial homolog of rhodopsin, a photoreceptor in the retina.
Rhodopsin belongs to the large class of G protein-coupled receptors (GPCRs)
that respond to a variety of stimulants by binding to G proteins (see below)
and triggering signaling cascades. The TM a-helices in many GPCRs and in
bR are quite regular and oriented fairly perpendicular to the bilayer. However, most a-helical bundle membrane proteins contain some tilted a-helices
as well as TM helices with distortions due to kinks and bulges, short unwound portions, or stretches of 310 helices, in addition to half helices that
stack to span the bilayer.
1.4.2
b-Barrels
Membrane-spanning b-barrels are found in the outer membranes of Gram
negative bacteria as well as mitochondria and chloroplasts, where they
function in transport, phage binding, catalysis, and adhesion.16 Varying
from monomers to oligomers and from open barrels to tightly packed interiors, most form pores that dissipate ion gradients. b-barrels typically
consist of an even number (8 to 26) of amphiphilic b-strands crossing the
bilayer at a tilt of B451, each containing 9 to 11 residues hydrogen-bonded to
the adjacent strands. Exceptions include VDAC (voltage-dependent anion
channel), a mitochondrial b-barrel with an odd number of strands
(Figure 1.8), and OmpX, an adhesion protein with several long b-strands
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Introduction to the Structural Biology of Membrane Proteins
Figure 1.7
The fold of a typical a-helical membrane protein is represented by the
high resolution structure of bacteriorhodopsin, showing its seven transmembrane helices [labeled A to G and colored from the N terminus
(blue) to the C terminus (red)] and the bound cofactor retinal.
Kindly provided by Prof. Eva Pebay-Peyroula, University Grenoble Alpes.
extending past the bilayer. Porins, a family of trimers with three open pores,
vary in their selectivity towards ions and solutes due to their pore architecture. Larger b-barrels that facilitate diffusion of specific large solutes
contain internal plug domains that may be displaced in conformational
changes driven by an energized complex linked to the inner membrane.
1.5 Functions of Membrane Proteins
The tremendous progress in the structural biology of membrane proteins
supported by countless genetic, biochemical, and biophysical studies now
provides exciting details of their different molecular functions. Specificity of
channels is illuminated by their architecture and gating mechanisms.
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Chapter 1
Figure 1.8
The fold of a b-barrel membrane protein is represented by the X-ray
structure of the mitochondrial porin VDAC, again colored from the N
terminus (blue) to the C terminus (red). This b-barrel is unusual in that it
has an odd number of b-strands (19, numbered in the figure), which
makes the N- and C-termini b-strands parallel.
From M. Bayrhuber et al., Proc. Natl. Acad. Sci. U. S. 2008, 105, 15370.
r 2008, National Academy of Sciences, U.S.A. Reprinted with permission.
Fundamental transport mechanisms are shared among transporters of
widely varying families. Capturing intermediates in the reaction cycles of
membrane enzymes provides details of their mechanism. An atomic picture
of a receptor interaction with signaling transducers provides vital clues to
the molecular transmission of information. Before looking at examples of
these membrane functions, it is worth remembering that the classes overlap:
for example, many receptors form ion channels and some enzymes also
transport substrates or ions.
1.5.1
Channels
Channels carry out passive diffusion (down concentration gradients), hence
passage of solutes does not require energy-driven conformational changes;
however, channel proteins may undergo conformational changes when
gated by voltage, pH, calcium ions, or other ligands. Although the selectivity
of channels varies, some exhibit exquisite specificity while maintaining very
fast permeation rates. The first X-ray crystal structure of an ion channel, that
of a truncated potassium channel called KcsA, revealed how residues lining
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Introduction to the Structural Biology of Membrane Proteins
Figure 1.9
11
Channels may be highly selective, as first observed at high resolution in
the selectivity filter of the potassium channel KcsA. In this cutaway view
of the filter with the extracellular surface at the top, the electron density
(blue mesh) observed for dehydrated K1 ions is shown within the
channel in addition to a hydrated K1 ion in the central cavity.
Reprinted from, R. MacKinnon, Potassium Channels, FEBS Lett., 555,
62–65, Copyright 2003 with permission from Elsevier.
the channel transfer dehydrated potassium ions while excluding sodium and
other ions (Figure 1.9).17 Structures of aquaporins, such as the glycerol
channel GlpF, similarly revealed how water channels can prevent the
passage of protons: hydrogen bonding to channel-lining residues requires
the reorientation of each water molecule as it passes through the center of
the channel.18 Other elegant channel structures include those for connexin
found in gap junctions, TRP (transient receptor potential) channels important in neurological sensing, and mechanosensitive channels of both
vertebrates and prokaryotics.
1.5.2
Transporters
Transporters are generally considered a separate class from passive channels, even though the evolution of highly similar chloride transporters and
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19
chloride channels erodes this distinction. Active transporters are classified
as primary, those that rely on exergonic chemical reactions such as the
hydrolysis of ATP, and secondary, those that are driven by ion gradients.
The prototype for secondary transport is the extremely well characterized
lactose permease, the Lac carrier called LacY.20 The LacY protein can
accumulate lactose up to 100-fold inside Escherichia coli when coupled with
the obligatory uptake of protons. The LacY structure exhibits a two-fold
pseudosymmetry with the N-terminal and C-terminal domains forming two
distinct but similar lobes. Each domain spans the bilayer with six a-helices
and a central sugar-binding site open to the cytoplasm (Figure 1.10). Despite
the low sequence homology between the two domains of LacY, they likely
resulted from gene duplication since inverted topology repeats are seen in
the structures of many transporters.
Other transporters in the same family have been crystallized in conformations open to the periplasm. Much evidence has established that the
change between outward-facing (co) and inward-facing (ci) conformations
effects solute uptake. This alternating access model for the mechanism of
transport is a general model for transporter action, employed by exporters as
well as importers.21 In addition to the two states ci and co, intermediate
states that are closed (occluded) or partially open (gated) are often observed.
While the LacY structure makes it easy to visualize the transition as a rocker
Figure 1.10
The first high resolution structure of LacY protein (a C154G mutant)
clearly shows the transporter open to the cytoplasm, with the substrate
analog thiodigalactoside bound between the two domains. The twelve
a-helices are colored from the N terminus (blue) to the C terminus (red).
From J. Abramson et al., Science 2003, 301, 610. r 2003, American
Association for the Advancement of Science. Reprinted with permission
from AAAS.
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Introduction to the Structural Biology of Membrane Proteins
13
switch («), alternating access can involve quite dissimilar shape changes, such as the up-and-down motion of an interior domain in the ‘elevator
mode’ of a glutamate transporter, GltPh.22
Alternating access occurs in larger, more complex proteins that carry out
primary transport, such as the maltose transporter from Escherichia coli.
Crystal structures of the entire complex of five subunits (MalF, MalG, MalK2,
and MalE, and the periplasmic maltose-binding protein) have been obtained
in both ci and co conformations, showing the role of the MalF and MalG
TMSs in alternating access (Figure 1.11).23 The structures reveal key interactions between the subunits and indicate that the maltose-binding protein
covers the opening on the periplasmic side in co. These and additional
conformations also reveal the effect of binding analogs of ATP to the MalK
subunits (on the cytoplasmic side) and suggest how the binding and
hydrolysis of ATP drive the conformational changes that activate the complex to enable maltose uptake. The maltose transporter is representative of a
large family of ABC transporters that include the human protein CFTR
Figure 1.11
High resolution structures of the maltose transporter in ci (left) and co
(right) conformations show how it utilizes the alternating access mechanism of transport, with the transition between states dependent on
ATP binding and hydrolysis. Details of the subunit interactions reveal
critical links involved in the conformational changes (not shown). The
internal cavities are shaded (grey) and the MalK (red and green), MalF
(blue), MalG ( yellow), and MalE (MBP, pink) subunits are labeled.
Kindly provided by Dr Michael Oldham and Prof. Jue Chen, Rockefeller
University.
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(the cystic fibrosis transmembrane conductance regulator) and the drug
exporter P-glycoprotein.
1.5.3
Enzymes
Membrane enzymes are wonderfully diverse, from the phospholipases
mentioned above and other enzymes such as diacylglycerol kinase that act on
lipid soluble substrates, to proteases including those involved in processing
amyloid precursor proteins, to redox complexes that carry out electron
transfers and generate proton gradients (see below). Structural biology is now
able to characterize the various steps of the enzymatic reaction cycle, beautifully done with a P-type ATPase called SERCA (the sarcoplasmic reticulum
calcium pump), which uses the hydrolysis of ATP to transport calcium ions.24
SERCA is a very large protein with ten TMSs and three large cytoplasmic
domains involved in major conformational changes (Figure 1.12). The two
major states of SERCA are called E1, with two high affinity Ca sites exposed to
the cytoplasm, and E2, with two low affinity Ca sites exposed to the lumen.
The reaction cycle involves the binding of calcium and ATP to the E1 enzyme,
the hydrolysis of ATP with phosphorylation of the enzyme as a high-energy
state with the calcium sites occluded, a conformational change to E2 releasing the calcium, and the release of the bound phosphate allowing the
enzyme to revert to E1. Over 20 crystal structures of SERCA bound to different
ligands have contributed to the understanding of the detailed reaction cycle.
This successful approach has now been applied to the Na, K-ATPase and
other membrane enzymes.
1.5.4
Receptors
The first GPCR to have a high resolution crystal structure was rhodopsin (see
above), which became an elegant prototype for other GPCRs. Now, the crystal
structures of many GPCRs allow the comparison of detailed ligand-binding
sites to provide fundamental understanding of their physiological roles as well
as valuable information for pharmaceutics.25 Once a ligand has docked, conformational changes allow a GPCR to bind to its G protein(s) at the other side
of the membrane. In a biological tour de force, this initial interaction was
captured in crystals of the b2-adrenergic receptor (b2AR) with its G-protein, Gs,
and the inverse agonist carazolol (Figure 1.13).26 The structure shows the
conformational changes in b2AR as well as in Gs that are the first steps of the
signal transmission.
The mechanism of signal transduction is also targeted in structural work
on neurotransmitter receptors, such as the beautiful crystal structures of
glutamate, serotonin, and GABA (g-aminobutyric acid) receptors.27,28 Intense
studies of these and other receptors (such as those for epidermal growth
factor, dopamine, and acetyl choline) are laying the foundation for the
fundamental understanding of brain functions along with insight into
neurological disorders and bases for rational drug design.29
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Introduction to the Structural Biology of Membrane Proteins
Figure 1.12
15
Key intermediates of the reaction cycle of SERCA have been captured in
high resolution structures using different ligands. The structures of E1
bound to phosphate, two Ca21 ions, and an ADP analog; E2 bound to
the phosphate analog BeF3; and E2 bound to an ATP analog, in
addition to either AlF4 or the inhibitor thapsigargen (TG), are shown
in the cycle. Significant shifts are evident in the domains of SERCA
(A domain, yellow; N domain, red; P domain, blue; TM segments M1–2,
purple, M3–4 green, M5–6 wheat, and M7–10 grey.)
Reproduced by permission from Macmillan Publishers Ltd: Nature,
Copyright 2007.
1.6 Membrane Protein Complexes
Many membrane proteins contain multiple subunits or form complexes to
carry out their functions. This is especially true for energy transducing
complexes. The photosynthetic complexes PS1 and PS2 are the largest
complexes to have crystal structures. The high resolution crystal structures
of the respiratory chain complexes provide astonishing mechanistic insights
that reveal striking differences in the modes of coupling electron transport
and proton transport. Complex I couples the flow of electrons through its
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16
Figure 1.13
Chapter 1
The interaction of a G protein-coupled receptor (the b2-adrenergic receptor) with its G protein is captured in the high resolution structure of the
ternary complex of b2AR (green), the agonist BI-167107 (yellow spheres),
and the Gs heterotrimer. Only the aRas domain (gold) of Gs makes
extensive contact with the receptor as it swings away from aAH (also
gold) and the other subunits of the Gs protein, Gb (cyan) and Gg (blue).
Reproduced by permission from Macmillan Publishers Ltd: Nature,
Copyright 2011.
hydrophilic domain to a concerted piston-like movement in its membrane
domain that allows the passage of protons (Figure 1.14).30 In contrast, Complex III utilizes the separation of two centers of electron carriers that oxidize
ubiquinol in a cycle called the Q cycle concomitant with the ejection of protons. Complex IV (cytochrome oxidase) accepts the electrons from cytochrome
c and protons from the cytosol to reduce O2 to H2O, while ejecting additional
protons.31 Finally, the protons reenter on a rotating ring of subunits of the ATP
synthase, whose rotor is mechanically coupled to a stator causing rotation of
the catalytic a and b subunits, changing their conformation to effect the
catalysis.32 Membrane structural biology has contributed greatly to understanding each of these very large molecular machines.
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Introduction to the Structural Biology of Membrane Proteins
Figure 1.14
17
A membrane protein complex composed of B2000 amino acids in 14
polypeptide chains is shown in the high resolution structure of complex
I from T. thermophilus. The overall shape and subunit interactions
suggest a coupling mechanism in which the lateral helix acts as a piston
to drive the conformational changes that open proton channels in
response to the reduction of the bound quinone (Q).
Reproduced by permission from Macmillan Publishers Ltd: Nature,
Copyright 2013.
1.7 Conclusions
Building on this remarkable progress, the field of membrane protein
structural biology is moving to address many more challenges. Researchers
are studying even larger complexes, such as the nuclear pore with as many as
a thousand proteins; more proteins from eukaryotes, especially humans,
such as CFTR; more dynamic processes, such as fusion. The rest of this
volume illustrates the contribution of biophysical and computational research to these studies. It is axiomatic that the interplay between simulations and experiments enhances both.
References
1. M. Luckey, Membrane Structural Biology, Cambridge University Press,
2nd edn, 2014.
2. J. Deisenhofer and H. Michel, Annu. Rev. Cell Biol., 1991, 7, 1.
3. http://blanco.biomol.uci.edu/mpstruc/listAll/list.
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Chapter 1
4. S. J. Singer and G. L. Nicolson, Science, 1972, 175(4023), 720.
5. A. Kusumi, T. K. Fujiwara, R. Chadda, M. Xie, T. A. Tsunoyama, Z. Kalay,
R. S. Kasai and K. G. N. Suzuki, Annu. Rev. Cell Dev. Biol., 2012, 28, 215.
6. D. Lingwood and K. Simons, Science, 2010, 327(5961), 46.
7. R. Lipowsky, Biol Chem, 2014, 395(3), 253.
8. (a) V. Cherezov, Curr. Opin. Struct. Biol., 2011, 21(4), 559; (b) P. Nollert,
J. Navarro and E. M. Landau, Methods Enzymol., 2002, 343, 183.
9. B. Brügger, Annu. Rev. Biochem., 2014, 83, 79.
10. A. G. Lee, Trends Biochem. Sci., 2011, 36(9), 493.
11. R. F. Collins and J. P. Derrick, Trends Microbiol., 2007, 15(3), 96.
12. F. Cymer, G. von Heijne and S. H. White, J. Mol. Biol., 2015, 427(5), 999.
13. K. D. Tsirigos, A. Hennerdal, L. Kall and A. Elofsson, Proteomics, 2012,
12(14), 2282.
14. M. Andersson, E. Malmerberg, S. Westenhoff, G. Katona, M. Cammarata,
A. B. Wohri, L. C. Johansson, F. Ewald, M. Eklund, M. Wulff,
J. Davidsson and R. Neutze, Structure, 2009, 17(9), 1265.
15. P. J. Booth, Curr. Opin. Struct. Biol., 2012, 22(4), 1.
16. J. W. Fairman, N. Noinaj and S. K. Buchanan, Curr. Opin. Struct. Biol.,
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17. D. A. Doyle, J. Morais Cabral, R. A. Pfuetzner, A. Kuo, J. M. Gulbis,
S. L. Cohen, B. T. Chait and R. MacKinnon, Science, 1998, 280(5360), 69.
18. R. M. Stroud, L. J. Miercke, J. O’Connell, S. Khademi, J. K. Lee, J. Remis,
W. Harries, Y. Robles and D. Akhavan, Curr. Opin. Struct. Biol., 2003,
13(4), 424.
19. C. Miller, Nature, 2006, 440(7083), 484.
20. L. Guan and H. R. Kaback, Annu. Rev. Biophys. Biomol. Struct., 2006, 35, 67.
21. L. R. Forrest, R. Krämer and C. Ziegler, Biochim. Biophys. Acta, 2011,
1807(2), 167.
22. O. Boudker and G. Verdon, Trends Pharmacol. Sci., 2010, 31(9), 418.
23. D. Khare, M. L. Oldham, C. Orelle, A. L. Davidson and J. Chen, Mol. Cell,
2009, 33(4), 528.
24. M. G. Palmgren and P. Nissen, Annu. Rev. Biophys., 2011, 40, 243.
25. V. Katritch, V. Cherezov and R. C. Stevens, Annu. Rev. Pharmacol. Toxicol.,
2013, 53, 531.
26. S. G. F. Rasmussen, B. T. DeVree, Y. Zou, A. C. Kruse, K. Y. Chung,
T. S. Kobilka, F. S. Thian, P. S. Chae, E. Pardon, D. Calinski,
J. M. Mathiesen, S. T. A. Shah, J. A. Lyons, M. Caffrey, S. H. Gellman,
J. Steyaert, G. Skiniotis, W. I. Weis, R. K. Sunahara and B. K. Kobilka,
Nature, 2011, 477(7366), 549.
27. P. J. Focke, X. Wang and H. P. Larsson, Structure, 2013, 21(5), 694.
28. A. Penmatsa, K. H. Wang and E. Gouaux, Nature, 2013, 503(7474), 85;
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32. M. Nakanishi-Matsui, M. Sekiya, R. K. Nakimoto and M. Futai, Biochim.
Biophys. Acta, 2010, 1797(8), 1343.
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CHAPTER 2
Molecular Dynamics
Simulations: Principles and
Applications for the Study of
Membrane Proteins
VICTORIA OAKESa AND CARMEN DOMENE*a,b
a
Department of Chemistry, Britannia House, 7 Trinity Street,
King’s College London, London SE1 1DB, UK; b Chemistry Research
Laboratory, Mansfield Road, University of Oxford, Oxford OX1 3TA, UK
*Email: carmen.domene@kcl.ac.uk
2.1 Introduction
Research into novel chemical problems is now commonly facilitated by
computational methods in order to supplement experimental data, instruct
future work and, in many cases, provide details for which no experimental
methods are applicable. This is of particular importance in the context
of membrane proteins, where the complexity of the membrane prevents
established techniques to be utilised in order to understand structure–
function relationships for soluble proteins.1 Three-dimensional structures of
membrane proteins have only began to emerge in the last two decades, due
to the difficulty in their expression and purification, and their instability
when removed from their native environment.2 Progress in this field3 and
the emergence of new techniques for structure determination, such as cryoelectron microscopy,4 have led to a wealth of available structures from many
RSC Theoretical and Computational Chemistry Series No. 10
Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
Published by the Royal Society of Chemistry, www.rsc.org
19
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4–13
membrane protein families.
Molecular dynamics (MD) simulations have
become a powerful technique to describe the structure and dynamics of
these systems in atomistic detail, resulting in appreciable contributions to
the molecular understanding of membrane protein function.
Ion channels (Chapter 4 and 6), transporters (Chapter 7) and membrane
receptors (Chapter 8) facilitate the communication and transport between the
cell interior and exterior; the elucidation of such mechanisms and how they
are influenced by their environment (Chapter 5) is key to understanding
fundamental sensations related to light, taste, smell, sound, heat, pain
and hormones, for which computational approaches have presented considerable insights. It is also imperative to understand when the behaviour of
membrane proteins veers from normal physiological function. Incorrect
modulation of membrane proteins has been linked to a wide range of
diseases that include cardiac disorders, neurological indications, muscle
afflictions and kidney failure; hence, they are arguably the most important
pharmaceutical drug targets.14–17 Computational algorithms are becoming
increasingly more popular in the drug discovery context; the identification of
putative binding sites provides an optimal starting-point for targeted design,
resulting in the discovery of novel effector molecules to be used as lead
compounds.18 MD simulations have proven particularly fruitful in this
respect, with numerous studies emerging in recent years.19–22 With the
increasing speed and availability of supercomputers, and the development of
new algorithms, the scope for novel discoveries is continuously expanding.
In this chapter, we provide an overview of the theoretical background of
MD simulations on biological macromolecules, some practical considerations when performing such calculations, and their application to membrane proteins. This review is intended as a precursor to the following
chapters, which provide an extensive survey of dynamical insights obtained
from computational simulations, related to the structure and function of
membrane proteins, and additional techniques to study these features.
2.2 Classical Molecular Dynamics
In MD simulations, the motion of interacting particles is calculated by the
integration of Newton’s equations of motion (EOM). The potential energy of
the system and the force, derived from the negative gradient of potential
with respect to the displacement in a specified direction, can be used to
calculate the acceleration, and hence forecast the time evolution of the
system, in the form of a trajectory. The potential energy can be obtained by
classical or quantum mechanical methods, with the former predominant
due to its reduced computational expense by utilising empirical force fields
and associated parameters (Section 2.2.1).
In systems with a couple of atoms, the solutions to the EOM can be gained
analytically, resulting in a continuous trajectory over time. However, in larger
systems, the subsistence of a continuous potential instigates a many body
problem for force evaluations, rendering analytic solutions unattainable.
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Molecular Dynamics Simulations
21
In these circumstances, finite difference methods can be used; forces are
assessed at discrete intervals, and considered constant in the hiatus. Positions and velocities at the next timestep, as these intervals are known, are
computed using force evaluations for each atom combined with current
positions and velocities. Forces are then recalculated and this procedure is
repeated, propagating a trajectory describing the flux of the atomic coordinates over time in a given equilibrium state, which can then be analysed
for the properties of interest. An overview of this procedure is given in
Figure 2.1. The timestep suitable for stable dynamics is dictated by the
Figure 2.1
An overview of the molecular dynamics procedure, using the CHARMM
force field and the Velocity-Verlet integration algorithm.
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highest vibrational frequency of the molecule, typically 1 fs corresponding to
a fraction of the vibrational motion of a C–H bond (10 fs), and the integration
algorithm used. Details on this can be found in Section 2.2.3.
2.2.1
Additive Force Fields
Molecular mechanics (MM) is a mathematical model utilising the
approximation that atoms can be considered as balls and, subsequently,
nuclear motion can be evaluated using the EOM from classical physics.
The consideration of only nuclear coordinates is based on the Born–
Oppenheimer approximation; this states that electronic and nuclear
wave functions can be treated separately due to the vast difference in
mass, and hence velocity, of electrons and nuclei. Electrons are said to
adjust ‘instantaneously’ to changes in nuclear position, thus they can be
ignored when calculating motion. For this reason, the analytic expression
for the energy of a system described by MM, known as the force field,
is composed solely of inter- and intramolecular energetic contributions,
including bond stretching, angle bending, bond rotations and non-bonded
terms. This architecture neglects the electronic properties, such as dipole
moments and vibrational frequencies, but allows for the evaluation of
molecular motion in (biological) systems with a large number of degrees
of freedom. A wide variety of force fields for biological molecules are
available, including, but not limited to, CHARMM (Chemistry at Harvard
Molecular Mechanics),23 AMBER (Assisted Model Building with Energy
Requirement)24 and OPLS (Optimized Potentials for Liquid Simulations).25
Each one varies in their functional form and the parameters therein,
which are generally obtained to provide a suitable reproduction of
experimental and/or quantum mechanical data. The CHARMM force field
will be used as a representative example to demonstrate these aspects,
with a recent comparison of the aforementioned force field available in
ref. 26.
In the CHARMM force field, the individual terms for bond lengths
and angles are based on simple harmonic potentials on the basis of an
energetic penalty associated with a deviation from the equilibrium value.23
A torsional angle potential function is also used to model the steric barriers
(between 1, 4 pairs) associated with the rotation of atoms. The inclusion
of a Urey–Bradley term, defined by the distance separated by two bonds,
has provided notable improvements for the representation of in-plane
deformations and the separation of symmetric and asymmetric bond
stretching modes. Furthermore, improper terms assist the replication of
out-of-plane bending modes. For the reproduction of non-bonding interactions, Coulomb and Lennard-Jones (LJ) potentials are used to express
electrostatic and van der Waals forces, respectively. Cross terms are also
included to account for the interdependence of internal coordinates, which
are ordinarily a function of two internal coordinates, such as stretchstretch or stretch-bend.
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The functional form of the potential energy function in the current
CHARMM force field is as follows:
Uð~
RÞ ¼
X
Kb ðb b0 Þ2 þ
X
Ky ðy y0 Þ2 þ
angle
þ
KUB ðS S0 Þ2
UB
bonds
þ
X
X
impropers
X
Kw ð1 þ cosðnw dÞÞ
dihedrals
Kimp ðj j0 Þ2 þ
X
nonbond
e
Rminij
rij
12 Rminij 6
qi qj
þ
rij
e1 rij
where K is representative of the force constant of the respective terms, b is
the bond length, S is the Urey–Bradley 1,3 distance, y is the bond angle, n is
the multiplicity, w is the dihedral angle, d is the phase factor, j is the
improper angle, e is the LJ well depth, Rmin is the distance at the LJ minimum,
q is the partial atomic charge, e1 is the effective dielectric constant and rij is the
interatomic distance between atoms i and j. Symbols throughout associated
with the subscript zero denote the equilibrium values. The generation of such
parameters is generally based on the reproduction of vibrational and
crystallographic data supplemented by ab initio calculations.
The CHARMM force field encompasses parameters for proteins,27–29
lipids,30–32 sterols,32,33 nucleic acids34–36 and carbohydrates.37–40 The
accurate parameterisation of protein and lipid molecules is crucial for
the investigation of membrane proteins via computational methods.
CHARMM22 provided the first all-atom parameter set for proteins developed
to accurately produce condensed-phase properties.27 Optimisation of the f,
c dihedral parameters lead to the inclusion of grid-based energy correction
maps (CMAP).28 The current release includes additional refinements to
overcome issues with force field bias for a-helices.29 The development of
lipid parameters has been hindered due to the complex phase behaviour of
lipid bilayers and the difficulties in gaining detailed structural information
of this phenomenon. The publication of the CHARMM36 lipid parameters
showed a significant improvement in the reproduction of bilayer surface
areas, density profiles and deuterium order parameters over previous releases.32 Further sets of parameters are required for ionic species. Ions are
ordinarily used in solution to neutralise the simulation system in molecular
simulations. However, the situation is complicated by the presence of additional interaction sites localised on the proteins. Traditionally, LJ parameters for ionic interactions are optimised to emulate the free energies of
ions in bulk water.41 Nonetheless, specific pairwise interactions can be
substituted corresponding to the free energy of solvation of an ion in liquid
N-methylacetamide (NMA), indistinguishable to those in bulk water to
emulate cation–protein interactions.42,43 Finally, the CHARMM General FF
(CGenFF)44 was developed to account for the diverse range of compounds of
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pharmacological interest, particularly in drug design, where MD simulations
are growing in popularity.
2.2.2
Polarisable Force Fields
The veracity of insights gained from MD simulations is inherently reliant on
the quality of the underlying force fields. Force fields are generally parameterised under certain conditions, such as temperature, pressure and pH,
which are specific to each individual force field; when the simulation
conditions diverge from those in the parameterisation process, the accuracy
of the data can suffer. A significant limitation inherent to all additive force
fields is the absence of polarisation, leading to much deprecation of the
treatment of electrostatic interactions. The term polarisation appertains to
the fluctuation of the charge distribution in response to an external electric
field, induced by the presence of additional atoms or molecules.45 The use of
fixed-point charges assigned at the nuclei and the evaluation of the Coulomb
formula to appertain the magnitude of the electrostatic forces ignore this
phenomenon, which can account for up to 30% of the interaction energy.46
Despite initial efforts to include polarisation effects,47 significant efforts to
develop polarisable force fields for biological molecules were not realised
until the turn of the century, and are now considered particularly beneficial
where ion channels are concerned. All-atom MD simulations of lipid bilayers
significantly underestimate the dielectric constant of lipid hydrocarbons,
and accordingly overestimate the energetic barriers facing permeating
species.48 In addition, the behaviour of ions interacting with protein atoms
may considerably deviate from that in a solvated environment. The selectivity
filter of K1 channels is a prime example where dehydrated ions accommodate
a narrow channel lined with carbonyl oxygens, for which specialised
parameters have been derived. Furthermore, substantial inaccuracies of
simulations with multivalent cations, such as Mg21 and Ca21 have been noted
as a consequence of their high charge density and the polarisation of surrounding molecules that is not captured in typical simulations.49 Polarisable
force fields, therefore, potentially provide a uniform solution to such issues.
Three schemes have been proposed, namely the fluctuating charge model,
the induced dipole model and the Drude oscillator approach.50 In the
fluctuating charge model, molecular charges remain constant throughout
the simulation with individual point charges readjusted in consonance with
the electronegativity.51 Deficiencies concerning out-of-plane polarisation in
conjugated systems limit its applicability to some extent, despite some
successes within the CHeq force field52,53 in recent biomolecular studies.54
AMOEBA is the most noteworthy force field utilising an induced dipole
model, where atomic multipoles are used explicitly to represent electrostatics, calculated via a self-consistent field procedure, heightening the
computational expense of this procedure.55
Finally, in the Drude oscillator approach, a subsidiary charged particle is
attached to the nucleus by a harmonic spring, and treated as an extra degree
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of freedom. This protocol is easily executable in established MD platforms
with the computational expense, relative to a non-polarizable FF, increasing
by a maximum factor of two. Studies concerning protein secondary structure
formation57 and structural perturbations of DNA–ion interactions,58 for example, have emerged since the release of the Drude-2013 force field.59,60
Recent parameterisation efforts have shown significant improvements for
the accurate representation of ion–protein interactions in comparison to
additive force fields, reproducing energetics obtained from QM calculations
of 30 unique ion-bound enzymatic proteins.61 The ability of such force fields
to replicate such quantum mechanical methods at a fraction of the computational expense will have significant implications for the modelling and
understanding of metalloproteins, including membrane transport proteins
such as ion channels. The reader is directed to ref. 62 for an extensive review
of the field, with details on the mathematical background, accompanying
parameterisation protocols and recent applications.
2.2.3
Practical and Technical Considerations
Simulations of membrane protein systems are now extremely accessible to
any researcher in an academic or industrial institution; force fields are
widely available for all the components required,27–32 with many MD algorithms available on open-source platforms such as NAMD,63 AMBER,64 or
GROMACS,65 for example. In recent years, web-servers have also been developed that set up the initial system and provide the necessary input to use
such software, with the CHARMM-GUI service leading this field.66,67 A
conscientious practitioner, however, will understand the inner workings that
allow a successful simulation to be performed. On these grounds, the basic
principles will be outlined in the following section.
Since the first simulation of a biological macromolecule, bovine pancreatic trypsin inhibitor (BPTI), was published, the complexity and accuracy
has significantly increased.68 For example, the initial simulations of BPTI
were performed in vacuo68 with subsequent simulations incorporating
an implicit water model by the inclusion of a dielectric constant in the
electrostatic energy term.69 Advanced implicit70 and explicit71 water models
are currently available for MD simulations; the latter are arguably more
accurate72–74 and are therefore predominant in contemporary membrane
protein studies, despite the heightened expense.75–77 The present-day
availability of computational resources and efficient models to calculate
interaction potentials renders such computations feasible.78 TIP3P,79 for
example, is a ‘simple’ pairwise model where water molecules are assigned
partial atomic charges at three sites. In this model, electrostatic interactions
are calculated between balanced anionic oxygen and cationic hydrogen
atoms, and van der Waals interactions are determined by a single oxygen
centred point charge, providing a popular model for MD due to its level of
accuracy and affordability. Concurrently, octane slabs originally served as
lipid bilayers for embedded proteins,80 with a full atomistic description of
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lipid molecules now standard. Implicit membrane models are still utilised
for protein-folding investigations, where higher-order timescales are required.81 Overall, simulation systems are continuously transitioning to more
realistic descriptions of biological systems, with membrane protein simulations generally constituting explicit protein atoms, lipid bilayers, water
molecules and ions.82
In any respect, simulation systems must be of finite size and peripheral
regions encounter an anomalous environment, introducing practical difficulties for performing MD simulations. Periodic boundary conditions
(PBC) are generally used to mimic an infinite environment, with a finite
number of atoms, to account for the system size and deleterious boundary
effects.83 Any particles that leave the simulation box throughout the simulation are simultaneously replaced by an identical particle on the opposite
side. Furthermore, each particle is liable to interactions with particles in
adjacent boxes removing unphysical interactions with the system boundary
(Figure 2.2). Any geometry that can fill an infinite space by translational
operations alone is a suitable choice.84 However, the utmost efficiency can
be achieved when the simulation box reflects the geometry of the system in
question. The use of the cube/parallelepiped is the most widely recorded,
and suitable for proteins embedded in a lipid bilayer.82 When simulating
proteins that require a concentration gradient akin to that experienced in
ion channels, for example, PBC can be troublesome as the ion concentration
is inherently equal.85 The most straightforward method to overcome this is
to simulate the ion channel in the presence of an electrochemical driving
force, by the application of a constant electric field in the direction
Figure 2.2
(a) Side and (b) top view of a membrane protein (the 5-HT1B G-protein
coupled receptor)88 simulation system, with the atoms explicitly
present coloured, and the atoms replicated as a consequence of
cubic periodic boundary conditions in grey. Water molecules are
excluded in the latter for clarity.
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orthogonal to the membrane. Alternatively, PBC can be exploited. The
simulation system can be separated into two chambers by parallel model
bilayers, with the ionic concentration controlled in such a manner to induce
an electrochemical gradient to observe permeation.87
Further approximations are, however, generally required to realise feasible
computational costs when utilising PBC. Theoretically, non-bonded interactions (van der Waals and electrostatic interactions) should be calculated
for all atoms present in the system, including those in neighbouring boxes.
Therefore, a minimum image convention is employed in which each atom
interacts with only one image of the recurrent atoms.89 Furthermore, on the
basis of the underlying LJ potential, which decays proportionally to r6,
short-range non-bonded (van der Waals) interactions are truncated above a
specified interatomic distance.90 A non-bonded neighbour list is generated
at regular intervals to indicate which atom pairs should be considered in this
calculation, avoiding the calculation of all interatomic distances at each
timestep.91 The Verlet neighbour list stores such information, using a distance criterion slightly larger than the cut-off to extend the applicability of
the list in future timesteps in which atoms may enter the calculation
threshold.92 Switching functions are also required to diminish discontinuities in force and energy calculations. These can be applied throughout the
whole potential range or between lower and upper cut-off distances to
moderately abate the potential in this range. The latter is recommended to
avoid perturbing the equilibrium structures.93
Electrostatic interactions must also be considered to ensure efficient
molecular simulations. Interactions of single-point charges decay proportionally to r1; hence, switching functions are arguably unsuitable in this
case.94–96 A numerical solution to the Ewald summation, an infinite sum of
electrostatic interactions for a charge neutral system, can be derived using
the Particle Mesh Ewald (PME) method as an alternative. This procedure
reduces the calculation expense from the order of N 2 to N log N, where N is
the total number of atoms in the system.97–100
By utilising all of the above, calculations can feasibly be performed;
thereafter, it is important to consider the relationship between the resulting
trajectory and the experimental data. Microscopic properties, identified
within the simulations, are connected to macroscopic properties of
the system via statistical mechanics.101 An ensemble can be defined as a
collection of independent microstates that displays indistinguishable
macroscopic conditions. This provides a key concept in this field and is
fundamental in the accurate reproduction of experimental data throughout
simulations. Isothermal-isobaric conditions are most widely used experimentally; therefore, simulations require regulation, by thermostats and
barostats, to sample the appropriate ensemble.102,103
Firstly, to achieve a given temperature, the velocities can be scaled by a
constant.104 Furthermore, the temperature can be coupled to an external
heat bath, which can supply or remove heat where required. There are a
variety of schemes designed to generate a statistical ensemble at a constant
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temperature using these means. In the Berendson thermostat, for example,
velocities are scaled at each step according to the rate of change at each
step.105 This is often used to achieve the initial temperature, but influences
the fluctuations in the kinetic energy throughout simulations, failing to
sample the correct canonical ensemble.106 Extensions have been proposed
where a properly constructed random force is added, controlling the kinetic
energy and removing such shortcomings.104 The Langevin thermostat is an
alternative stochastic method to achieve temperature control via heat bath
coupling.107 All particles simultaneously obtain random and frictional
forces, the balance of which generates the correct canonical ensemble. In
extended system methods, such as the Nosé–Hoover, a thermal reservoir is
considered as an extra degree of freedom in the equations of motions.108
Both Langevin and Nosé–Hoover thermostats are widely used in MD
simulations.102
Methods for pressure control are generally analogous to those of
temperature, replacing the velocity term with volume. The hybrid Nosé–
Hoover103 Langevin piston109 barostat is an extended system method, using
a compressing piston as an additional degree of freedom described by a
Langevin equation. The combination with Langevin dynamics overcomes
the earlier bias caused by the fictitious mass of the piston. This approach
also allows isotropic, anisotropic or semi-isotropic coupling of volume
fluctuations; the latter is recommended in membrane simulations to allow
proportional scaling of the x and y dimensions of the lipid bilayer.82
Finally, the time evolution of the energy of the system should be of
paramount importance within MD simulations.110 To truly abide by classical
Newtonian dynamics, the trajectory must be time-reversible with the total
energy of the closed system conserved. Discretisation of variables, introduced by the use of a finite timestep in MD simulations, can cause an energy
drift, an unphysical phenomenon where the energy of the system gradually
changes over time.
Integration algorithms are generally optimised to accommodate discrete
timesteps and maintain the conservation of energy, alongside considerations for computational efficiency.111 Taylor expansions are used to approximate the key terms (positions, velocities and acceleration) in all
algorithms, differing in their interpretation. The Verlet algorithm provides
an inexpensive method, in terms of both computational time and storage,
using positions at previous timesteps (t, and (t dt)) to calculate those at
future (t þ dt) timesteps.112 Nonetheless, the inclusion of an explicit velocity
term in both the Leap-Frog113 and Velocity-Verlet114 algorithms demonstrated paramount accuracy; the latter is somewhat preferred as all terms are
evaluated at the same timestep, whereas velocities and positions sequentially ‘leap’ over each other in the so-called Leap-Frog method. The complexity of the velocity expression is increased in Beeman’s algorithm
providing a more accurate, yet intensive, method.115 Higher-order methods,
such as the Runge–Kutta family,116 are generally unsuitable for biomolecular simulations, as a consequence of their inefficiency and inability to
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maintain the fundamental properties of a simulation system, such as energy
conservation.117 Overall, the Velocity-Verlet algorithm provides a symplectic,
time reversible method that conserves momentum and energy, with an
excellent compromise between accuracy and expense, therefore finding a
place in most MD software of common usage. It is worth noting that supplementary algorithms in these packages are included to increase the
timestep; the SETTLE118 algorithm, an analytical version of the SHAKE
function,119 is used, for example, to restrain bonds involving hydrogen
atoms, and increase the timestep from 1 to 2 fs.
2.2.4
Applications
Despite the universal availability of MD algorithms and force fields applicable to biological macromolecules, simulations of intrinsic membrane proteins, whose system size regularly exceeds 100 000 atoms, are inherently
limited by computing resources. However, expansions in computer hardware
and high-performance computing facilities and improved software means
that MD simulations on the nanosecond timescale are now routine, with
microsecond simulations attained in recent years.120 Equilibrium MD
simulations are therefore relevant to study a wide range of biological phenomena; substrate binding, for example, is a fundamental process for which
a nanosecond timescale is appropriate. Ligand binding is integral for activation, and hence basic function, of many integral membrane proteins. MD
simulations have provided considerable insight into such mechanisms. In
membrane transporters, ATP or ions are required as a driving force for largescale conformational changes between the inward facing and outward facing
edges that are functional to transport species against the electrochemical
gradient. In Na1 coupled secondary transporters, for example, the location
of ion binding sites, the progression of binding events and the following
structural perturbations have been unearthed to some extent by MD
simulations.121–124
In a pharmaceutical environment, classical MD has become an established tool to identify putative binding sites and consequentially establish
how drugs function on a molecular level. The microscopic mechanisms of
anaesthesia, for example, remain elusive despite its ubiquity in modern
medicine. Isoflurane, a common general anaesthetic, is a known inhibitor of
proton-gated ion channels125 and mammalian nicotinic acetylcholine
receptors,126 for which MD simulations have enabled the identification
of multiple pore and subunit interface binding sites127,128 before crystallographic data was available,129,130 and advocating a multi-site model for
anaesthetic action in the Cys-loop receptor family. Modulation of voltagegated sodium channels by anaesthesia has also been pensively studied by
computational methodologies,131–133 with particular focus on the role of
lateral fenestrations between the central pore and the surrounding bilayer as
hydrophobic drug access pathways.134 In-depth understanding of drug entry
and binding mechanisms is, in general, essential to guide future drug
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discovery advances, in a continued effort to develop subtype-selective
blockers with optimal response properties in an affordable manner.
Unfortunately, the high computational expense of atomistic MD simulations for membrane proteins remains a significant weakness for the investigation of other aspects of protein structure and function. Many
biological phenomena that occur on extended timescales, such as protein
folding, complex association and conformational changes associated with
gating, are generally unattainable by atomistic equilibrium MD without the
use of tailor-made software. The Shaw group has pioneered the production
of millisecond long unbiased simulations by the development of the
Supercomputer Anton, optimised for use with in-house MD software.135
However, such technology is not widely available, leading to the development of alternative approaches to accelerate sampling.
2.3 Coarse-grained Molecular Dynamics Simulations
Using a reduced representation is one such approach, utilising classical MD
simulations, to increase the speed and hence the timescales obtainable.
Coarse-grained (CG) molecular dynamics, as this is known, reduces the
number of degrees of freedom in a simulation system by treating a group of
atoms as a single entity, significantly curtailing the computational expense
of each iteration. To convert an all-atom structure to a coarse-grained model,
hydrogen atoms are not considered, with a number of heavy atoms (typically
three or four) grouped into a single interaction site, known as a ‘‘bead’’.
Interaction potentials are then characterised dependent on the CG model,
with required parameters generally developed to reproduce microscopic
properties recorded in atomistic simulations and thermodynamic data
derived by experimental means.
Klein and co-workers developed an initial CG model for phospholipids,
using a distance dependent model for pairwise potentials and utilising an LJ
style potential to calculate non-bonded interactions and reproduce the
biophysical properties of the membrane.136,137 Parameterisation efforts
subsequently focused towards lipid and surfactant molecules.138,139 With
regards to the Klein model for amino acids, backbone and side chain atoms
are generally classified as a single site, with exceptions for bulky constituents.140 Interaction potentials are optimised to reproduce the surface tension and density of a representative side-chain analogue in bulk solution,
leading to unique parameters for most amino acids with high accuracy and
excluding the use of protein structural data during parameterisation.
Marrink and colleagues undertook similar work to construct CG
representations of lipid and surfactant molecules.141,142 The MARTINI force
field, as this is known, was quickly expanded to include proteins,143–146
sterols,143,147 carbohydrates,148,149 polymers,150–152 dendrimers,153,154 nanoparticles155,156 and DNA157 providing a versatile, ever-expanding toolkit for
bio-molecular simulations, the most popular CG model to date. In this
method, individual sites are classified to maintain the chemical nature of
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the region: polar, non-polar, apolar (a mixture of polar and non-polar
groups) and charged, with further sub-groups representing the hydrogen
bonding capabilities, polarity, and alternative grouping schemes for specific
cases, such as rings, where the overall symmetry defines the number of sites
required (Figure 2.3).
Short-range non-bonded interaction energies are calculated according to
the standard LJ 12-6 and Coulomb potentials, with parameters developed to
reproduce the free energy of hydration, free energy of vaporisation and oil/
water partitioning coefficients through extensive validation against experimental data concerning the physical properties of the bilayer.141,143,158 The
deficiencies of the LJ potential have been noted in this case, demonstrating
the diminished stability of liquid phases159 and the inaccurate representation of the surface tension at the air–water interface160 and water–oil
interfaces,161 leading to possible corrections in a future release.162
Overall, the implementation of both the Klein and Marrink models,
alongside other CG schemes,163 reduces the number of degrees of freedom
in the system, curtailing the number of required calculations in each iteration, as well as increasing the timesteps (20–40 fs) whilst maintaining
a level of chemical specificity without fine chemical details, thereby
providing a powerful tool to generate molecular simulations on an extended
timescale.
Figure 2.3
Representation of a coarse-grain map of a short polypeptide sequence
(Phe-Lys-Ser), with coloured circles representing the interaction sites,
known as beads. The yellow, blue and red circles correspond to N-type
(non-polar), P-type (polar) and Q-type (charged) beads, respectively.
The polypeptide backbone atoms (green circles) are a unique case,
where their classification is dependent on the secondary structure;
a P-type bead defines a helix or a bend, with polarity considerably reduced
in a-helices or b-strands due to inter-helical hydrogen bonding.
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The limitations of CGMD must also be evaluated when considering such
approaches.162 CG models generally have difficulties reproducing physical
properties for which they are not parameterised, arising from the reduced
representations.164 The chemical resolution is restricted; the subtle differences between lipid molecules, for example, cannot be captured accurately
in this manner.162 Specific bonded parameters or an elastic network are
often required to constrain the protein in the appropriate secondary structure, which is inherently maintained in atomistic simulations by the explicit
representation of hydrogen bonds. In addition, long-range non-bonded
interactions are generally neglected throughout. Finally, the speed-up of
CGMD is not universal across all molecule types;162 therefore, it is often
difficult to assess the acceleration versus the loss of detail and hence the
utility of CGMD in some cases.
However, CGMD methodologies are in common usage today, tackling a
wide range of biophysical scenarios. The MARTINI force field has been at the
forefront of developments in numerous areas of biochemical research, with
an exhaustive review of such applications provided by Marrink and Tieleman
in ref. 162. Investigations concerning membrane proteins, in particular,
have had notable success in elucidating lipid–protein and protein–protein
association mechanisms, as well as the conformational changes in proteins
in biological membranes due to the extended timescales the techniques have
enabled.165
Identification of specific lipid166–168 and cholesterol165,168 binding sites
in G-protein coupled receptor (GPCR) proteins is an area of research where
both atomistic and CGMD simulations are employed; the advancements by
both experimental and computational means are reviewed in Chapter 5.
Furthermore, the available timescales are appropriate to observe protein
self-assembly. Therefore, CGMD is an integral tool to observe GPCR oligomerisation. In the case of rhodopsin, transmembrane helices 1, 2 and
amphipathic helix 8 were identified to be the primary dimer interface.169
Oligomerisation properties of b2-adrenergic receptor were also elucidated by
this method, with transmembrane helices 1, 4 and 7 being the preferable
interfacial region for energetic reasons concerning the hydrophobic
mismatch of the monomer in the membrane.170
Lipid self-assembly is also within the bounds of CGMD, which can be used
to optimise the setup of atomistic membrane simulations. Enveloping
proteins by self-assembled lipid bilayers ensures the most energetically
favourable orientation of the proteins within the membrane, demonstrating
notable benefits to manual insertion. CG models can then be transformed
into atomistic representations to perform MD simulations.
It is also worth noting that dual-resolution approaches (denoted MM/CG)
have been developed, where atoms in regions of chemical interest are treated
explicitly and the remaining system coarse-grained, to exploit the chemical
detail and extended timescales of the respective methods.171–176 The method
was applied to the b2-adrenergic receptor as a means of predicting ligand
poses at a high-throughput, with key attributes of ligand–receptor
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interactions conserved between atomistic and multi-scale simulations.
Subsequently, this method was applied to bitter taste receptors TAS2R38 and
TAS2R46, with the binding predictions consistent with site-directed mutagenesis and functional assays.178–180 These initial results suggest that CGMD
can provide an accurate yet economical method to obtain GPCR/ligand
complexes and aid drug design.181
2.4 Ab initio Molecular Dynamics
Advanced techniques are also required at the other end of the spectrum;
where complex chemical phenomena occur on a sub-nanosecond timescale
yet require the knowledge of molecular electronic properties. Bond making
and breaking in chemical reactions and excited state dynamics are two such
examples that cannot be described by empirical force fields alone. A full
electronic description of a system can be derived from first principles via
quantum mechanical (QM) methods; the theoretical and mathematical
background of the QM treatment can be found, for example, in ref. 182.
Density functional theory (DFT) is a popular QM method, providing an attractive compromise between the accuracy of the exchange-correlation
functionals and the scaling properties with respect to the number of atoms
in the system.183,184
Car and Parrinello spearheaded the integration of DFT within an
MD framework, leading to the emergence of the ab initio method, CPMD.185
In this scheme, electronic degrees of freedom are included explicitly
and coupled to the motion of the nuclei by dynamic variables. Protocols
based on Born–Oppenheimer MD,186 which employs the time-independent
Schrödinger equation to evaluate the electronic structure, and combinatory
approaches have since been proposed.187 These approaches have been
applied to investigate the mechanism of proton transport in aqueous
solution,188–191 at hydrophobic interfaces,192 in enzymes and in the cationselective membrane channel gramicidin A (gA).193,194 In a study using
ab initio metadynamics, Dreyer et al. identified a free energy barrier at the
mouth of the gA channel,194 implicating a membrane dipole potential in
the relationship between the proton transfer rate and the transmembrane
voltage observed experimentally.195 The authors note that these conclusions
are somewhat speculative, as a result of limited timescales and hence partially
converged trajectories, as well as the potential boundary effects induced by
the relatively small system size (B2000 atoms). Such limitations arise from the
substantial computational requirements of performing ab initio MD, meaning
it is generally unsuited to large-scale biological assemblies.
As a means to reduce such expense and broaden the range of biological
processes that can be studied, hybrid quantum mechanics/molecular
mechanics (QM/MM) methods have been developed.47 In QM/MM simulations, the bulk of the system is treated using classical mechanics, with the
specific region of biochemical interest evaluated by quantum mechanical
methods and a coupling scheme to account for the boundary region.196,197
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As with both classical and quantum MD simulations, the timescales
and phenomena that can be sampled by unbiased QM/MM simulations are
limited by the computational resources, and enhanced sampling techniques as described in the following section are regularly used. These approaches have provided extensive insight into the mechanism of enzymatic
reactions,198,199 as well as phenomena in several classes of membrane
proteins.200–206
QM/MM simulations have found obvious applications in the investigation
of proton migration in membrane proteins. For example, a mechanism for
proton transport across aquaporin GlpF has been proposed.200 In the lightsensitive GPCR bacteriorhodopsin, QM/MM has been used to determine
the structure and IR spectra of protonated local water networks to provide
information on the photosynthetic cycle.201,202
QM/MM simulations have also been applied to study the structure and
energetics of binding sites in ion channels. Analysis of the prototypical K1
channel, KcsA, suggested that ion-binding energetics are manipulated by
polarisation and charge transfer effects between the carbonyls and cations,
potentially screening the ion-ion repulsion integral, but maintaining the
canonical structure throughout.203 Proton transfer was also observed between two residues constituting part of the selectivity filter, imposing additional energetic consequences on the permeation profile.204 Furthermore,
the coordination numbers of K1 and Na1 were found to be similar in the
filter and water, but overestimated in atomistic simulations.205 The conduction of anionic Cl ions through the NanC porin has also been reinvestigated by QM/MM methods; this study supported extensive solvation
in the binding sites proposed from atomistic simulations, yet highlighted
distinct differences concerning the average water–Cl distance and polarisation effects resulting from proximal protein residues.206 The observed
differences in the binding of charged species indicate that conduction and
selectivity mechanisms cannot be truly quantified without the inclusion of
polarisation effects.207
2.5 Enhanced Sampling Techniques and Free Energy
Methods
Additional techniques are often required to sample phenomena where
standard atomistic, coarse-grain and QM methodologies are not appropriate. In order to obtain statistically significant estimates of conductance in
ion channels, for example, a copious amount of conduction events should be
observed and, therefore, atomistic MD simulations must be performed
at length, i.e. trajectories of tens of microseconds, which are generally at
odds with current computational capabilities. Furthermore, large-scale
conformational changes, integral to the function of membrane channels,
receptors and transporters, can occur on the second time-scale, and involve
high-energy barriers between stable structures which cannot be overcome by
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thermal fluctuations in equilibrium simulations. Accordingly, advanced
techniques are required to calculate the underlying free energy of what is
called a ‘rare event’, as well as accelerate sampling.
In the archetypal method, known as free energy perturbation (FEP), the
relative free energy differences of multiple processes can be calculated
by using alchemical transformations via a thermodynamic cycle.208,209
Figure 2.4(a) demonstrates a generalised cycle to calculate the relative free
binding energies and assess the affinity of related molecules to a known
binding site.
Instead of attempting to overcome energetic barriers and extended timescales directly to observe binding events, the ligands can be artificially
transformed in the bound and unbound states to calculate the relative free
energy by an alternative route:
D(DG) ¼ DG2 DG2 ¼ DG4 DG3
Practically, intermediate (non-physical) steps are required to perform this
conversion in a stepwise manner, for which a coupling parameter (l), scaled
between 0 and 1, is used to dictate the contributions of the initial (U1) and
final (U2) states to the potential energy of the whole system (U).210
U ¼ lU2 þ (1 l)U1
FEP simulations provided the earliest insight into the conduction
mechanism in K1 channels.211 The primary permeation axis, known as the
selectivity filter, is constituted of a conserved amino acid sequence arranged
to form rings of oxygen atoms, which are able to bind dehydrated ions in a
cage-like structure, in four adjoining sites (S1–S4), with intermediary binding sites in the central cavity (SC), and at the extracellular mouth (S0 and
SEXT) of the protein, displayed in Figure 2.4(b).5 These initial simulations
Figure 2.4
(a) General thermodynamic cycle to calculate the relative free binding
energies of two molecules (L1 and L2) to an identical binding site. (b)
Structure of the selectivity filter in K1 channels.5 (c) An example
thermodynamic cycle to assess the relative affinity of K1 and Na1 ions
(orange and purple spheres, respectively) towards the S2 and S4 sites of a
K1 channel selectivity filter (black trace).
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provided evidence that S1/S3 and S2/S4 ion configurations play a crucial role
in conduction, with the latter being the most thermodynamically favourable
loading state.211 FEP computations have also been utilised to understand the
molecular origins of selectivity in the K1 channel filter. An exemplar thermodynamic cycle to elucidate the relative free energies of K1 binding vs. Na1
binding in the S2–S4 configuration is shown in Figure 2.4(c). Initial calculations proposed an energetic penalty of 4.5 0.5 kcal mol1 for this
transformation;211 a general consensus has since emerged that Na1 occupancy of K1 binding sites is less favourable, with the selectivity of the S2 site
being the most pronounced.43,211–213 Alternative binding sites for Na1 have
also been proposed in the plane of the four carbonyl atoms, which displayed
paramount stability.214
FEP calculations are also becoming increasingly useful in modern day
drug discovery for the characterisation of the ligand affinity ratios of drug–
protein complexes.215 By obtaining the relative binding energies of different
ligands to an identical binding site216 or, conversely, of one ligand to wild
type and mutated sites,217–219 FEP provides computational analogues to
competition assays220 and site-directed mutagenesis studies,221 respectively,
which play a major role in industry to elucidate the key determinants of
ligand binding and identify and optimise lead compounds.
Several algorithms also exist to enhance sampling along a pre-defined
set of reaction coordinates and estimate the potential of mean force (PMF),
such as umbrella sampling,222 metadynamics,223 steered MD224 or adaptive
biasing force.225 These reaction coordinates, known as collective variables
(CVs), are chosen to investigate a specific transition. An estimate of the free
energy profile as a function of such CVs can be recovered, in addition to the
equilibrium properties of the system, to provide a wealth of information
about the simulation system at a fraction of the expense of unbiased
simulations. An overview of how simulation algorithms explore the potential
energy surface is given in Figure 2.5, with the theoretical background and
relevant applications of each method described from this point forward. To
the best of our knowledge, few studies using QM approaches with rare-event
techniques to investigate the behaviour of membrane proteins have been
published to date.195,226 Therefore, it is noted that the selected applications
described have been performed by entirely classical means.
Umbrella sampling (US)222 is perhaps one of the most popular enhanced
sampling methods in this field, where a bias potential along user-defined CVs
provokes the conversion between stable thermodynamic states. Independent
MD simulations are performed at intermediary steps (Figure 2.5(b)), known as
windows, which can be combined using the weight histogram analysis
method (WHAM)227,228 or umbrella integration.229 Each window represents
equilibrium sampling of energetically distinct locales; accordingly, such
evaluation estimates the consolidated equilibrium free-energy surface.
US simulations have also proven popular when deciphering the conduction mechanism in K1 channels.42,214,230–233 In order to obtain a free
energy profile of this pathway, the windows are defined by different ionic
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Molecular Dynamics Simulations
Figure 2.5
37
Simplified representation of how enhanced sampling methods explore
the potential energy surface in comparison to unbiased MD simulations.
Different colours denote independent simulations, with the progression
from dark to light representing the evolution of the trajectory. (a) In
classical molecular dynamics, the trajectory represents an ensemble of
microstates that generally remain well within an energy level unless
thermal fluctuations can overcome the surrounding barriers, something
usually unlikely. (b) In umbrella sampling, a biased potential is used to
perform independent simulations in different but overlapping regions
on the potential energy surface, which can be combined to obtain the
potential of mean force. (c) In steered molecular dynamics, an external
force is applied to an atom, or group of atoms, to enforce an alternative
stable thermodynamic state, regardless of the barrier. (d) In metadynamics, biasing potentials (Gaussians) are added to promote sampling
of less probable regions of the energy surface. (e) In adaptive biasing
force, the instantaneous force along the reaction coordinate is evaluated
directly and counteracted by an external biasing force of equal and
opposite magnitude, effectively flattening the surface.
configurations in the selectivity filter. Initial umbrella sampling calculations
suggested a conduction mechanism whereby ions cycle between occupying
the S1/S3 and S2/S4 positions, with the remaining sites containing a single
water molecule throughout, prompted by approaching ions in the central
cavity (Figure 2.6(a)),42 in line with FEP computations.211 Subsequent umbrella sampling calculations demonstrated a mechanism in which direct
ion–ion contacts and vacant sites could feasibly compete with those previously calculated by the inclusion of additional ionic configurations
(Figure 2.6(b)).230 The expanse of the estimated energy landscape is inherently limited by the input of such configurations, highlighted further by a
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38
Figure 2.6
Chapter 2
Proposed mechanisms of ion conduction in K1 channels: (a) chains of
alternating K1 ions and water molecules cross the selectivity filter,42 or
(b) cation pairs move through the selectivity filter without intervening
water molecules.230 Coloured spheres represent individual K1 ions. The
approach of an ion from the central cavity prompts a simultaneous ion
movement throughout the selectivity filter, resulting in the exit of one
ion that rapidly diffuses into the extracellular solution.
systematic comparison with metadynamics simulations that identified further possible iterations.234
Steered molecular dynamics (SMD) is a further enhanced sampling
method, with applications for a wide range of membrane proteins
(Figure 2.5(c)).224 SMD are akin to well-established experimental techniques,
such as atomic force microscopy or optical tweezers, where an external force
is applied to an atom or group of atoms to overcome the barriers and sample
a specific process.224 Relative free energies can then be obtained by
Jarzynki’s equality.235 This method can be used to quantify protein–ligand
binding energies, for example, with understanding of the transport, binding
and unbinding mechanisms observed on-the-fly, gaining notable insights
relevant to the drug discovery process.236 Potential inhibitors of the M2
channel protein have been discovered in this manner.237 SMD simulations
of the 5HT2A have also elucidated specific driving forces for GPCR activation,238 in addition to key translocation properties in substrate-specific
channel proteins239 and ATP binding cassette transporters.240
Metadynamics is another enhanced sampling technique employed in this
field to sample rare events and obtain the free energy underlying the
events.223 In this case, a biased potential is utilised to advance the sampling
along suitable CVs, diverting from the configurational space previously inhabited. The biasing potential is adjusted by the addition of a Gaussian
function, augmenting the energy of the system and departing from the local
free energy minima, allowing the exploration of alternative thermodynamic
states separated by energetic barriers. Once convergence has been achieved,
effectively when the entire free energy profile has been flattened, it can be
easily reconstructed to provide an unbiased estimate of the landscape as a
function of the CVs, hence elucidating the underlying properties of the
system. The CV definition, therefore, is of paramount importance in the
accuracy of such profiles, which should include all the degrees of freedom
relevant to the process in question. In practice, however, only a minimal set
is used to alleviate the computational cost, which increases exponentially
with each CV. This can be problematic when considering systems with
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39
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241
complex conformational transitions, such as ion channels,
and thus
specialised methods are often preferred.
Extensive CV selection can be avoided by replica-exchange methods. In
parallel tempering metadynamics, metadynamics runs are performed at
different temperatures and configurations are exchanged at regular intervals.242 Runs at elevated temperatures are more likely to overcome barriers,
and consequentially introduce uncharted transitions via exchange. In biasexchange metadynamics, multiple replicas are undertaken with different
CVs, instead of temperatures.243 Configurations from random runs are
exchanged, so numerous CVs and high dimensional transitions can be
explored in a more efficient manner than traditional metadynamics.
The latter has recently been proposed as an efficient method to study
conduction in bacterial voltage-gated sodium (NaV) channels.244 In NaVAb,
for example, an arrangement of four glutamates, the so-called ‘EEEE ring’,
forms a highly charged region at the extracellular mouth of the selectivity
filter known as binding site SHFS.6 Deeper sites formed of rings of carbonyl
atoms, denoted SCEN and SIN, are proposed to form the remainder of the
permeation pathway, demonstrating a notably wider and shorter selectivity
filter than in K1 channels, able to accommodate hydrated ions throughout.
In order to investigate permeation in NaVAb by US, the number and species
of ions involved must be defined at the outset, yet this is unknown.
Therefore, multiple independent simulations are required to assess the
feasibility of single and multi-ion conduction events, none of which are
representative of the entire energy landscape.245,246 When considering
traditional metadynamics, a single CV along the channel axis is insufficient
to characterise the conduction pathway in Na1 channels, as in K1 channels; an additional CV describing radial displacement from the axis is
required to properly accommodate on- and off-axis sites, requiring a
4-dimensional energy profile (when two ions are involved).247 Metadynamics is more efficient than classical US simulations at sampling this
4-dimensional space. Furthermore, when coupled with replica-exchange
methods, like in bias-exchange metadynamics, this technique can be used
to investigate permeation events that involve a variable number of ions or
different ion species.244 Consequently, this method is likely to gain traction
in non-selective channels such as TRPV, for which crystal structures are
now available but equivalent properties are unknown. Further examination
of the role of metadynamics in the analysis of ion channels can be found in
ref. 248.
Metadynamics simulations can also be likened to the adaptive biasing
force method (ABF),225 an additional rare-event technique which aims to
provide homogenous sampling of the potential energy surface. This approach is largely based on thermodynamic integration,249 whereby the instantaneous force along the reaction coordinate is evaluated directly and
counteracted by an external biasing force of equal and opposite magnitude.
This effectively provides a smooth energy landscape and uniform sampling
irrespective of the energetic barriers allowing for accelerated dynamics.
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40
Chapter 2
The application of ABF to the study of membrane protein structural organisation was established in its infancy, examining the reversible folding
processes of alanine-rich a-helical peptides250–252 to probe the secondary
structure formation prior to membrane insertion, in line with the ‘two-stage’
model proposed by Popot and Engelman.253 Subsequently, the transmembrane region of Glycophorin A (GpA) embedded in a membrane mimetic
environment was used to chronicle the in situ a-helical recognition and association mechanisms.254 Independent GpA a-helices were found to exhibit
stable dynamics oriented perpendicular to the membrane interface, prior
to the formation of native contacts and tilting to generate the canonical
structure, lending further evidence to the ‘two-stage’ scheme.
ABF has also been applied to understand selective transport phenomena in
membrane proteins, in a similar vein to other enhanced sampling methods.
Ivanov et al. investigated members of the Cys-loop receptor family, the nicotinic acetylcholine (nAChR) and glycine (GlyR) receptors, in order to explore
the origin of cation and anion selectivity in the respective channels.255 Rings
of oppositely charged residues at the extracellular mouth of the channels were
found to stabilise the translocating ions, notably contributing to their selectivity. A barrier in the central region of the nAChR pore also supported
evidence of a hydrophobic gating mechanism and the proposed closed state
of the crystal structure.256 ABF has since been used in multiple studies concerning ammonium,257 glycerol,258 CLC,259 and urea260 transporters.
2.6 Conclusions
The accessibility of three-dimensional crystal structures, advanced empirical
force fields and efficient setup/simulation protocols has enabled MD to
become an established tool to probe structure–function relationships of
membrane proteins. With the continual expansion of enhanced sampling
algorithms and high-performance computing resources, as well as the
development of dual-resolution techniques, MD simulations are now able
to illustrate an immense range of biological phenomena. Computational
investigations of this nature have provided considerable insights in the area
of biophysics, contributing to our fundamental understanding of transport
and communication across the cell membrane on a molecular level. The
applicability of such techniques to membrane proteins of pharmaceutical
interest can also aid the affordable, rational design of drug molecules,
setting a precedent for the next generation of drug discovery. The continuous
increase of high-resolution structures from a diverse range of protein
families and organisms will significantly advance this field in the future.
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00059
CHAPTER 3
Free Energy Calculations for
Understanding Membrane
Receptors
ANDREW POHORILLE
NASA Ames Research Center, Exobiology Branch, MS 239–4, Moffett Field,
California 94035–1000, USA
Email: andrew.pohorille@nasa.gov
3.1 Introduction
Membrane receptors are dynamic protein complexes that mediate transport
or signal transduction across membranes in response to environmental
signals. Because of their fundamental importance to many physiological
processes, they have been subjects of extensive studies with basic biology,
medicine, pharmacology, and biotechnology coming to mind. How do they
bind ligands, such as neurotransmitters, channel blockers, or anesthetics?
What conformational states do they adopt in response to signals? What is
their efficiency and selectivity in transporting ions and small molecules
across membranes? These are examples of questions that are at the heart of
efforts to understand and control the mechanisms of receptor action. Some
of them address equilibrium properties of a receptor system, whereas others
deal with dynamic processes. In all cases, finding the answers can be greatly
aided by the knowledge of free energy differences between different states of
the system. For example, free energy of ligand binding to a target site on a
receptor determines how effective is a ligand in modulating receptor action.
The free energy of an ion along a pore of an ion channel informs us about
RSC Theoretical and Computational Chemistry Series No. 10
Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
Published by the Royal Society of Chemistry, www.rsc.org
59
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Chapter 3
the efficiency and selectivity of ion transport. For some problems, free
energies can be measured but for many others they cannot. This is where
computational methods shine; in principle, free energies can be calculated
from computer simulations of properly designed systems. Even more importantly perhaps, the results allow for the interpretation of the mechanisms
of receptor action in both thermodynamic and structural terms.
The determination of free energy differences from numerical simulations of
a molecular system is a mature field, the foundations for which were already
laid in the first half of the last century.1–4 However, free energy calculations
have risen to the level of a reliable theoretical tool only due to both methodological developments and the continuous increases in computational
power in the last 40 years. At present, there is a remarkable range of techniques available for this purpose. Their apparent conceptual and methodological complexity and diversity may easily confuse not only a novice to this
subject matter, but even a seasoned practitioner. The purpose of this chapter
is to provide the conceptual basis for the classes of methods that are most
frequently applied to membrane systems, provide guidelines on how these
methods should be properly applied, and illustrate their use in relevant examples. This is done at the expense of some mathematical rigor and detailed,
tutorial-level explanations. Those interested in expanding their knowledge of
different methods for free energy calculations may wish to reach for a
monograph on theory and applications,5 an exposition of the subject matter
from a mathematical perspective,6 or several recent reviews.7–11
In the next section, the basics of free energy calculations are presented to
underscore the connection between different, broadly used classes of methods,
which are discussed in more detail in the next three sections. We also describe
briefly a specific class of techniques to improve sampling configurations of the
system. In the subsequent section, several applications of free energy calculations to membrane receptors are reviewed. The emphases are on the diversity
of problems and methods, and on information that can be gauged from
simulation results. The last section is devoted to the connection between free
energy and dynamic properties and, more specifically, on how free energy
calculations can be used to determine currents through ion channels.
3.2 The Basics of Free Energy Calculations
3.2.1
The Parametric Formulation of Free Energy
Calculations
Consider a Hamiltonian, H, that depends on a parameter l, which is
often called ‘‘order parameter’’, ‘‘general extent parameter’’, ‘‘coupling
parameter’’, ‘‘reaction coordinate’’, or ‘‘transition coordinate’’,
HðG; lÞ ¼
N
X
p2i
þ Uðx1 ; . . . ; xN ; lÞ;
2mi
i¼1
(3:1)
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Free Energy Calculations for Understanding Membrane Receptors
61
where N is the number of particles in the system, U is potential energy and xi,
pi, and mi abbreviate, respectively, the Cartesian coordinates, momenta, and
mass of particle i. If the system of interest is in the equilibrium state at a
fixed value of l and temperature T, the probability of finding it in a microstate defined by a point in phase space, G ¼ (x1, . . . , xN, p1, . . . , pN), is given
by the Boltzmann distribution
PðG; lÞ ¼ Ð
exp½bHðG; lÞ
1
exp½bHðG; lÞ;
Ql
dG0 exp½bHðG0 ; lÞ
(3:2)
where b ¼ 1/kBT, kB is the Boltzmann constant, and Ql is the partition
function at l.
The difference in the Helmholtz free energy, D A, between two states at
l ¼ 1 and 0, which is the quantity that we wish to calculate, is given as the
ratio of the corresponding partition functions.
1 Q1
D A ¼ ln :
b Q0
(3:3)
In some instances, we are interested not only in D A, but also in the full
free energy dependence along l, D A(l), between the endpoints l ¼ 0 and 1.
In the parametric formulation, this dependence can be usually readily
obtained as
1 Ql
D AðlÞ ¼ ln :
b Q0
(3:4)
The contribution from the kinetic term in the Hamiltonian can be integrated analytically and usually is not considered in calculations of free energy. Then, D A is expressed in terms of the ratio of configurational integrals,
which are analogous to partition functions but no longer depend on particle
momenta. This quantity is formally called excess free energy. In most cases,
we will consider it without explicit reference to its full name. Then, G involves only coordinates and U is substituted for H in the Boltzmann
distribution.
The purpose of introducing a parameter is to avoid calculating individual
partition functions, which is usually quite difficult, and instead focus directly on estimating their ratio, a task that is computationally much more
tractable. Many problems that require knowledge of free energy differences
can be formulated in this framework. For example, l might be an external,
macroscopic parameter, such as temperature. As shown in Figure 3.1, l may
also be a physical coordinate that describes, for example, conformational
changes in a protein, transitions between bound and unbound states of a
protein–ligand complex, or the position of an ion that moves through a
channel. In ‘‘computational alchemy’’, l is used to interpolate between two
different Hamiltonians, which represent different physical systems. These
might be a wild type and mutated protein or two different ligands interacting
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62
Figure 3.1
Chapter 3
Different applications of parameter l in free energy calculations. The
parameter may represent (a) the distance between atoms, (b) and (c)
single or coupled torsional degrees of freedom, (d) the mutation of a
group of atoms to a different chemical group, (e) the creation of a
molecule in solution, and (f) the transition between the ordered and
disordered structure of a polymer.5
with their target. Although calculating D A in these different cases requires
slightly different theoretical treatments, the conceptual basis remain the
same. For this reason, we will draw a distinction between different types of
problems that involve l only when it is necessary.
The most frequently used methods for calculating D A can be derived
through simple manipulations of the original eqn (3.3). Yet, remarkably,
they lead to approaches that are quite different at a technical level. For
specific problems, they might offer different degrees of accuracy and convenience. One simple transformation is to multiply the integrand in the
numerator of eqn (3.3) by the identity exp(bU0)exp(bU0). This yields:
Ð
expðbU1 ÞdG
D A ¼ b1 ln Ð
expðbU0 ÞdG
ð
¼ b1 ln expf½bðU1 U0 ÞgPðG; 0ÞdG
(3:5)
¼ b1 lnhbDUi0 ;
where DU ¼ U1 U0 U(G, 1) U(G, 0) and the angular brackets denote an
equilibrium average at the given l. An interesting feature of this formula is
that D A is estimated from sampling system 0 only. Methods based on eqn
(3.5) are called free energy perturbation (FEP) methods because system 1 is
treated as a perturbation of system 0. These methods are discussed in more
detail in Section 3.3.
Another approach relies on unconstrained simulations in which the full
range of l is sampled. For example, l might be a linear measure of the
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distance between a protein and a ligand, and states 0 and 1 would correspond to the bound and unbound state, respectively. In analogy to eqn (3.2),
we can define probability of finding the system at a given value of l, Pl, as:
Pl ¼
Ql
;
Q
(3:6)
Ql dl:
(3:7)
where Q is defined as:
ð1
Q¼
0
Then, the free energy along l, D A(l), is given as
D AðlÞ ¼ b1 ln
Pl
P0
(3:8)
and D A is obtained from the values of D A(l) at the endpoints. Pl is obtained
from computer simulations, usually as a histogram, by way of recording how
often the system is observed at different values of l. This generic idea has
been implemented in a number of creative ways, discussed further in Section 3.4, giving rise to a class of techniques called probability distribution or
histogram methods.
In the third approach, the quantity that is being estimated is the derivative, dA/dl, rather than D A. From the definition of free energy it follows that
ð
@H
expðbHÞdG dA
@H
@l
¼
¼ ð
:
(3:9)
dl
@l l
expðbHÞdG
Then, D A is obtained by way of integrating the average derivative of the
Hamiltonian with respect to l in the range between 0 and 1. This quantity
has units of force and is often called ‘‘thermodynamic force’’. The approaches that rely on estimating D A or, more generally, D A(l) from the
thermodynamic force are called thermodynamic integration (TI) methods.
These approaches and their more formal derivation are discussed in
Section 3.5.
3.2.2
Ergodicity, Variance Reduction Strategies, and the
Transition Coordinate
Central to the foundations of statistical mechanics, and to calculations of
free energy in particular, is the concept of ergodicity. Stated somewhat informally, a physical system is ergodic if it assumes, in the long run, all
conceivable microstates that are compatible with the conservation laws. For
nonergodic systems, computer simulations initiated from different points in
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Chapter 3
phase space would not yield, in general, the same values of free energy,
independently of their length.
Even though condensed phase chemical and biological systems are believed to be formally ergodic, their behavior during computer simulations
may appear to be nonergodic. This means that the system does not properly
explore the phase space and, thus, the calculated statistical averages may
exhibit strong dependence on the initial conditions. This, in turn, means
that they may be burdened with substantial errors. This phenomenon is
called quasi-nonergodicity. This might occur if a system diffuses slowly
compared to the simulation time and, therefore, covers a volume in the
phase space that is insufficient to yield reliable statistical averages. More
commonly, quasi-nonergodicity is due to the existence of high energy barriers separating different volumes in the phase space. Then, transitions
across these barriers constitute rare events that might never happen during
simulations or occur sufficiently infrequently that statistical averages cannot
be reliably estimated. To combat quasi-nonergodicity and improve these
estimates, advanced strategies that improve sampling of rare events are
needed. These strategies are not unique to statistical mechanics. In other
fields of science they are known as variance reduction techniques.
Variance reduction strategies are essential for successful free energy calculations. A large majority of them are based on two ideas. One is stratification, also known as multistage sampling or ‘‘window’’ method. In general,
it relies on partitioning the relevant region of the phase space into parts to
ensure that all of them are adequately sampled. Specifically in free energy
calculations, Pl may vary considerably with l. In unstratified simulations,
low probability regions are sampled very infrequently and, as a result, precision in estimating D A suffers. If l is divided into intervals, the variability of
Pl in each stratum is reduced compared to the variability in the full range.
Furthermore, every stratum is sampled, even if it is associated with low
values of Pl.
The second, commonly used strategy is enhanced sampling, sometimes
called, depending on the context, importance sampling, non-Boltzmann
sampling, biased sampling, or histogram reweighting. The idea behind this
variance reduction technique is to sample from a distribution P 0 (G, l) chosen
such that it is more uniform along l than P(G, l), given by eqn (3.2). If the
connection between these two distributions is known, the sampling from P 0
yields improved estimates of D A.
A variety of ingenious stratifications and enhanced sampling techniques
that greatly improve the efficiency and accuracy of free energy calculations
have been developed. Some of them will be discussed further in this chapter.
The final issue to be discussed in this section is the choice of l. Considering that the free energy difference between two macrostates of the
system is a state function, and thus does not depend on the path taken to
evaluate it, this issue appears to be only of peripheral importance. In practice, however, the efficiency of free energy calculations may depend critically
on the choice of l. If this choice is not made properly, free energy
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calculations may be plagued with quasi-nonergodicity. Furthermore, if one
wants to segue from calculating free energies to kinetic properties, such as
rates, the results are likely to be erroneous if l is selected poorly.
The transition between two macrostates can be formally described by a onedimensional transition coordinate, the committor probability, which is defined as the probability of proceeding to the target state rather than returning
to the initial state. In most cases, this coordinate is difficult to calculate and
offers only limited insight to the process of interest. For these reasons, it may
be advantageous to choose a l that only approximates the committor probability, but that is easier to calculate, can be efficiently sampled, and carries a
clear physical meaning. Sometimes, it might be helpful to extend the representation of the transition path beyond one dimension and consider l as a
collection of several parameters rather than a single parameter. It should be,
however, kept in mind that kinetic properties can be reliably calculated only if
l is the committor probability or its close approximation.
Central to the parametric formulation of free energy calculations is the idea
of the separation of timescales. In general, different degrees of freedom may
evolve at different timescales. If separation of timescales exists in the system,
all slow degrees of freedom should be captured in the transition coordinate.
Then, precise estimates of D A, which require calculating appropriate statistical averages over all degrees of freedom orthogonal to l, can be carried out
relatively fast. Conversely, if evolution of the system along some degrees of
freedom in the hyperspace orthogonal to l is slow, and in particular if equilibria between metastable states are involved, calculating the required statistical averages becomes inefficient. As a result, the efficiency and reliability of
the free energy estimates suffer. Standard stratification and enhanced sampling techniques are not helpful in this respect, as they are aimed at improving sampling and removing metastability along l, but not other degrees
of freedom. If equilibration along these coordinates is a concern, different
techniques that will be discussed in Section 3.7 have to be brought to bear.
In systems of biological interest, there is usually a multiplicity of time
scales that may involve collective motions and cannot be cleanly separated
into slow and fast. For example, the slowest degree of freedom in channel
opening or closing may involve conformational transitions in the whole
protein assembly accompanied by relaxation of the surrounding lipid molecules. Consequently, there is no simple recipe for selecting the transition
coordinate in such systems that would be theoretically justified and practical
to implement. Instead, we have to be guided by experience, physical intuition and our understanding of the problem at hand.
3.3 Free Energy Perturbation Methods
3.3.1
Theoretical Background
Free energy perturbation4,5,12 is one of the oldest and most frequently used
methods for free energy calculations. For example, it is often a method of
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66
Chapter 3
choice for calculating free energies of binding small ligands to proteins and
protein assemblies.13–17 The utility of FEP relies largely on the concept of the
thermodynamic cycle, a series of reversible transformations connecting two
states of interest (see Figure 3.2). Since free energy is a state function, values
of D A associated with each transformation in the cycle sum up to zero. This
allows for replacing calculations of D A in the transformation of interest,
which might be computationally difficult, with the evaluation of free energy
differences for the remaining transformations in the cycle. An example of a
problem in which the concept of a thermodynamic cycle combined with FEP
is particularly useful is the calculation of the free energies of binding for a
series of ligands.
The starting point for FEP is eqn (3.5). Since the right side of this equation
depends only on the difference, DU between energies of the system in states
1 and 0, it can be expressed in terms of P0(DU), the probability distribution of
values of DU sampled from state 0.
ð
1
DA ¼ ln expðbDUÞP0 ðDUÞdDU:
(3:10)
b
The roles of state 0 and 1 can be, of course, reversed. If sampling is carried
out from state 1, D A is given as
ð
1
DA ¼ ln expðbDUÞP1 ðDUÞdDU;
(3:11)
b
where P1(DU) has the same meaning as P0(DU), but for values of DU sampled
from state 1. These two probability distributions are not independent.
Instead, they are connected via the relation
exp(bDU)P0(DU) ¼ exp(bD A)P1(DU),
(3.12)
as follows from eqn (3.2) and (3.3).
From this relation, it follows that its left hand side, which is also the integrand in eqn (3.10) is proportional to the probability distribution of DU
sampled from state 1. The proportionality constant is defined by D A. Since
the integrand is typically shifted with respect to P0(DU) toward low values of
DU, it can be accurately evaluated only if the low-DU tail of P0(DU) is
adequately sampled. The shift depends largely on the variance of the probability distribution, although the variance is the sole determinant of the shift
only if P0(DU) is Gaussian. If the probability distribution is broad, its relevant
tail is sampled quite rarely and D A cannot be estimated with acceptable
accuracy in simulations of typical length. This is illustrated in Figure 3.3. For
this reason, it is highly recommended to plot exp(bDU)P0(DU) as a function
of DU. If this function is noisy or is missing the low-DU tail, the estimate of
the free energy difference will be poor.
The problem is likely to occur in most simulations of practical interest.
Fortunately, it can be readily remedied by way of stratification. In its
Published on 30 November 2016 on http://pubs.rsc.org |
Thermodynamic cycles for free energy calculations. Left: Calculations of free energy, DAbinding, of forming a host–guest (e.g.
receptor–ligand) complex. Instead of proceeding directly along the top horizontal line, it may be preferred to carry out
calculations along the vertical lines. This may require imposing restrains on the position of the ligand and the associated free
energy, DArestraint, has to be estimated and accounted for calculating DAbinding. Right: Calculations of differences in free energy,
DAsolvation, of solvating two solutes, 0 and 1. Again, calculations along the vertical lines may be preferred to calculations along
the horizontal lines.
Reproduced with permission from A. Pohorille, C. Jarzynski and C. Chipot, Good practices in free-energy calculations. J. Phys.
Chem. B, 2010, 114(32), 10235–10253. Copyright (2010) American Chemical Society.
Free Energy Calculations for Understanding Membrane Receptors
Figure 3.2
67
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68
Figure 3.3
Chapter 3
Histograms of P0(DU), P1(DU), and the corresponding integrands in eqn
(3.10) and (3.11). If sampling were satisfactory, P0(DU) and exp(bDU)P1(DU)
would differ by a constant. The same applies to P1(DU) and
exp(bDU)P0(DU). This is clearly not the case. In particular, systematic
error was introduced due to missing samples in the shaded areas.
Reproduced with permission from A. Pohorille, C. Jarzynski and C.
Chipot, Good practices in free-energy calculations. J. Phys. Chem. B,
2010, 114(32), 10235–10253. Copyright (2010) American Chemical
Society.
simplest implementation, one defines state M that corresponds to a value of
l between 0 and 1. Then
1
Q1
1
QM Q1
(3:13)
DA ¼ ln
¼ ln
¼ DA0;M þ DAM;1 ;
b
b
Q0
Q0 QM
where D A0,M and D AM,1 are the free energy differences between states M and
0, and 1 and M, respectively. Eqn (3.13) can be generalized to n strata by
applying this relation serially
DA ¼ n1
n1
X
1X
lnhexpðbDUi Þii ¼
DAi;iþ1 ;
b i¼0
i¼0
(3:14)
where DUi ¼ U(G, li11) U(G, li), Ai,i11 is the free energy differences between
states i þ 1 and i and, for convenience of notation, index n corresponds to
l ¼ 1. Thus, it is possible to estimate the free energy difference between the
final and initial states as a sum of the free energy differences between states
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with intermediate values of l. Even though stratification involves n1
simulations instead of a single one, it is almost always more efficient because the probability distributions have smaller variances and, thus, the
corresponding free energy differences can be estimated from substantially
smaller samples.
From the discussion above, it follows that errors in FEP calculations are
not only of statistical but also of systematic nature. If sampling is incomplete, which in practice is always the case, missing samples are not
randomly distributed, but instead are located primarily in the tails of P(DU).
Stratification is a powerful tool for reducing systematic errors. When the
probability distributions are narrow and overlap well, the tails give only
negligible contribution to the calculated D A.
Considering that the same free energy difference can be estimated from
‘‘forward’’ and ‘‘backward’’ simulations, as given in eqn (3.10) and (3.11),
respectively, it is usually advantageous to combine both simulations such
that the variance of D A is minimized. Conceptually, there are several ways of
doing so,11,18,19 but all lead to the same result.11 Here, we briefly outline the
original approach, known as the Bennett Acceptance Ratio (BAR) method.18
It relies on introducing a function w(G, l) that weights the contributions
from configurations sampled from states 0 and 1. It is done as follows:
expðbDAÞ ¼
Q1
Q0
Ð
Q1 w exp½bðU0 þ U1 Þ=2dG
Ð
¼
Q0 w exp½bðU0 þ U1 Þ=2dG
Ð
Q1 w expðbDU=2ÞexpðbU0 ÞdG
Ð
¼
Q0 w expðbDU=2ÞexpðbU1 ÞdG
¼
(3:15)
hw expðbDU=2Þi0
;
hw expðbDU=2Þi1
where the arguments of w are suppressed for clarity. The optimal w that
minimizes the variance of the estimated D A is a hyperbolic secant function.18 An important advantage of BAR is that it does not require modifications in the simulation protocol, as it is implemented only after all the
samples from states 0 and 1 have been drawn and the corresponding values
of DU have been stored. Hyperbolic secant w, which is a positive symmetric
function, reaches the maximum when P0(DU) ¼ P1(DU) if samples from 0 and
1 are of equal size. This means that reliable estimates of D A require good
sampling in the overlap region between these two distributions. In contrast,
in unidirectional simulations, say from state 0, a similar reliability is
achieved if the region near the peak of the integrand in eqn (3.10) or,
equivalently, P1(DU) is adequately sampled. Since this region is located
further in the low-DU tail of P0(DU), the latter condition is more demanding.
In fact, it can be proven that, given fixed computational resources, BAR
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70
Chapter 3
almost always yields more reliable estimates of D A than unidirectional
simulations.11
Clearly, BAR can be combined with stratification. Then, in each strata, one
combines samples generated from forward and backward simulations in
states i and i þ 1, respectively. For multiple strata, one can generalize BAR to
the Multistate Bennet Acceptance Ratio (MBAR)20 and use samples from all
strata to improve the estimates of D Ai,i11. Although samples from other
strata may fall into the relevant overlap region only infrequently, the reliability of the calculated free energy difference will always get better, even if
only slightly. This is why it is recommended to use MBAR, especially since
the computational overhead associated with this procedure, which, as BAR,
is carried out in post-processing, remains small.
From eqn (3.12), it immediately follows that
1
1
ln P1 ðDUÞ ln P0 ðDUÞ þ DU ¼ D A
b
b
(3:16)
for all values of DU. This means that the left hand side of this equation
plotted as a function of DU in the overlap region between P0(DU) and P1(DU)
should be a constant equal to D A.18 This offers a valuable consistency check;
if the right hand side is not approximately constant, and especially if there is
a clear drift in this quantity (see Figure 3.4), the estimate of D A should not be
trusted, probably due to insufficient sampling of configurations relevant to
state 0 and 1 or persistent quasi-nonergodicity.
3.3.2
Alchemical Transformations
In most applications of FEP, l is a parameter in the Hamiltonian that transforms interatomic interactions rather than a physical coordinate. Two types of
transformations can be distinguished: creation/annihilation and mutations.
In the former, atoms, molecules, or groups of atoms progressively appear or
vanish. The latter type can be considered a combination of creation and annihilation – a group of atoms is substituted by a different group of atoms.
Together, these transformations are called alchemical transformations.
Consider transformation A-B, where A and B are, for example, two solutes dissolved in bulk solvent, two ligands bound to a macromolecule, or two
amino acid side chains. With these transformations is associated, respectively, change in the solvation free energy, the relative binding free energy,
and the mutation free energy. The transformations can be carried out in two
distinct ways, called the single-topology and double-topology paradigm (see
Figure 3.5). They share the feature that the Hamiltonian corresponding to
l equal to 0 and 1 describes systems A and B, respectively. In the singletopology paradigm,21,22 a common geometry is built that contains the
minimal set of atoms needed to build A and B. This common geometry,
which may not correspond to any molecule that exists in nature, is usually
based on the more complex of the two molecules or groups of atoms. During
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Figure 3.4
71
bDA from eqn (3.16). The calculated value (solid circles) is nearly
independent of DU and fluctuates slightly around the exact value (horizontal line). The corresponding probability distributions are shown as
solid curves in the inset. However, if P1(DU) is sampled slightly inaccurately (the dashed curve in the inset), bDA (open circles) is no longer
constant as a function of DU.
Reproduced with permission from A. Pohorille, C. Jarzynski and C.
Chipot, Good practices in free-energy calculations, J. Phys. Chem. B,
2010, 114(32), 10235–10253. Copyright (2010) American Chemical
Society.
the transformation, atoms disappear, appear, or change their identity. For
example, during mutation of ethane to methanol, the carbon atom of one
methyl group is transformed to oxygen, two hydrogen atoms attached to this
carbon are annihilated, and the third one is altered from methyl to hydroxyl
hydrogen. In the double-topology paradigm,22,23 all atoms of both A and B
always exist in the system, but never interact between themselves. The
transformation from l ¼ 0 to l ¼ 1 can be considered as a simultaneous
annihilation of A and creation of B. The total Hamiltonian of the combined
system, Ht(Gt, l), can be written as:
Ht(Gt, l) ¼ H 0 (G 0 ) þ HB(GB, l) þ HA(GA, l)
where H 0 (G 0 ) is the part of the Hamiltonian describing particles unaffected by
the transformation, whereas HA(GA, l) and HB(GB, l) are the terms associated
with A and B, respectively. In the simplest, although not necessarily the most
efficient case of linear scaling, the total Hamiltonian takes the form:
Ht(Gt, l) ¼ H 0 (G 0 ) þ lHB(GB) þ (1 l)HA(GA)
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72
Figure 3.5
Chapter 3
Single (a) and double (b) topology paradigms for free energy calculations
of the serine-to-glycine mutation. In the single-topology paradigm, serine
serves as the common geometry. As l is switched off from 1 to 0, the
hydrogen atoms and the hydroxyl group on Cb turn to dummy atoms (DH
and DO), Cb is modified to H, and the Ca–Cb bond becomes a Ca–H bond.
In double topology, the CH2OH of serine and H of glycine coexist, but do
not interact. As l changes from 1 to 0, the interactions of the serine side
chain with the rest of the system are progressively switched off and
interactions of the glycine hydrogen are switched on. This is indicated by
the change of color between red and yellow.5
Due to their non-physical nature, alchemical transformations are associated with a number of difficulties. For example, as atoms are annihilated,
their van der Waals radii are reduced. This allows residual charges on
these atoms to approach other atoms in the system sufficiently close so that
electrostatic forces become quite large, thus creating instabilities in integrating the equations of motion. For this reason, it is often required to decouple or at least delay the reduction of atomic radii from the reduction
of charges. Another difficulty, frequently encountered in dual topology, is
related to possible collisions between incoming atoms and atoms in
the reminder of the system. This usually occurs at the tail end of the
transformation. One way to remedy this difficulty is via non-linear scaling of
van der Waals energy with l, such that the original potential is modified to a
soft-core form.24,25 This could markedly improve smoothness of the calculated D A(l). Also, special care should be taken to account correctly for
changes in non-bonded contributions to free energy associated with bonds,
planar angles, and dihedral angles involving atoms that are created or
annihilated.
Additional difficulties emerge during FEP calculations of free energies of
binding. In these calculations, the ligand is annihilated or, more precisely,
decoupled from the rest of the system. As its interactions with the host become weaker during this transformation, the ligand tends to dissociate from
the binding pocket in the host. This could have an adverse effect on the
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efficiency of free energy calculations as it greatly increases the configurational space available to the ligand, which is usually quite rigid, while
interacting with the host. Accurate sampling of the translational and rotational degrees of freedom associated with the motion of the ligand no
longer bound to the host might be a daunting task, yet it is necessary to
account correctly for the primarily entropic contributions to the free energy.
This problem can be dispensed with by way of restrain forces imposed to
keep the ligand bound to the host. Subsequently, the contribution of these
artificial restrains to the calculated free energy of binding has to be carefully
subtracted.16 Detailed discussion of these and other important technical
issues associated with alchemical transformations and calculating binding
energies is beyond the scope of this chapter, but several excellent treatments
of these issues are available.14,16,26–28
3.4 Probability Distribution Methods
In its generic application, the probability distribution method is perhaps the
simplest approach to calculating D A(l), both conceptually and technically. If
l is a physical coordinate, i.e. it can be expressed in terms of coordinates of
selected particles in the system, all that needs to be done is to run standard
computer simulations, such as molecular dynamics, and keep a tally, usually
in the form of a histogram, of how many times the system was in states
corresponding to different values of l. Once the histogram has converged,
D A(l) can be obtained from eqn (3.8). In many problems of practical interest, however, this simple scheme turns out to be quite inefficient because l is
sampled highly non-uniformly if the free energy changes markedly along the
parameter. This, in turn, will produce an undesired effect of non-uniform
statistical error. For example, if the free energy corresponding to two values
of l differs by 5 kBT, these values will be sampled at a ratio of 7 : 1000. The
problem becomes particularly acute if two macrostates of the system that
exist along l are separated by a high barrier. Then, transitions between these
macrostates are rare and obtaining reliable estimates of the ratio of the
corresponding partition functions, which is at the heart of all free energy
calculations (see eqn (3.3)) becomes very difficult. Fortunately, variance reduction techniques are often quite successful in remedying this problem.
A number of enhanced sampling techniques can be used in combination
with probability distribution methods.5 Here, we limit the discussion to only
one that is most frequently used and also most relevant to problems involving membrane proteins. This technique is called importance sampling,
and in its simplest and most common implementation, ‘‘umbrella sampling’’.29 For clarity, the underlying idea will be presented here in the specific and most frequently used formulation rather than its most general
form. Consider a system with the potential energy U þ Ub(l) instead of the
original U. As we will see, the role of the additional biasing potential, Ub,
which in this case depends only on the parameter, is to make sampling
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along l more uniform. As follows from eqn (3.2) and (3.8), the free energy,
D Ab(l), of this biased system is:
Ð
exp½bðUðG; lÞ þ Ub ðlÞÞdG
D Ab ðlÞ ¼ b ln Ð
exp½bðUðG; 0Þ þ Ub ð0ÞÞdG
Ð
exp½bUb ðlÞ exp½bUðG; lÞdG
1
Ð
¼ b ln
exp½bUb ð0Þ exp½bUðG; 0ÞdG
1
(3:17)
¼ D AðlÞ þ Ub ðlÞ Ub ð0Þ;
where, in the second line, we used the fact that Ub does not depend on G.
This means that D A(l) is given as:
D A(l) ¼ D Ab(l) [Ub(l) Ub(0)].
(3.18)
This equation not only provides the prescription for how to recover D A(l)
from biased simulations but also contains clues to an efficient choice of the
biasing potential. If Ub(l) Ub(0) is chosen equal to D A(l) in the whole
range of l, then D Ab(l) and, equivalently, the probability of sampling along l
calculated from the biased simulation would be constant. In most cases, it
corresponds to optimal or nearly optimal sampling of l. In practice, however, such a choice is, of course, impossible, as it implies that we can accurately predict D A(l) without simulations. For many problems, however, we
can make a good guess, which will in turn markedly improve the uniformity,
and thus the efficiency, of sampling along l. Conversely, a poor guess
will produce negligible gains or even a loss in accuracy. Both scenarios are
illustrated in Figure 3.6.
Figure 3.6
Umbrella sampling. The solid black curve represents the actual D A(l),
the dashed curve is the biasing potential, and the red solid curve is the
sampled, biased free energy. In the left panel, the biasing potential has
been properly chosen, as the sampled probability distribution is more
uniform than the original one. However, when the biasing potential is
slightly shifted along l (right panel), the resulting biased distribution is
even more difficult to sample than the original one.5
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There are both conceptual and mathematical connections between FEP
and importance sampling probability distribution methods. Both methods
are based on the idea of sampling from one system to gain information
about another one; however, in FEP we seek the difference in free
energy between these systems, whereas in importance sampling this difference is known and, instead, the free energy of the system of interest is
obtained from knowledge of the free energy of the biased system and this
difference.
Enhanced sampling can be readily combined with stratification. The full
range of l is divided into a number of strata, often called ‘‘windows’’. The
system is kept in each stratum by way of a restraining potential that acts
outside this stratum. Then, stratified calculations yield a probability distribution in each window. From these distributions, the probability distribution in the full range of l needs to be reconstructed. This can be simply
done by way of matching the appropriate endpoints, but this procedure
would be quite inaccurate. A better approach is to construct strata such that
two consecutive windows overlap and match the probability distributions in
the overlapping region. The method for optimal matching, known as the
weighted histogram analysis method (WHAM),30 is essentially a version of
BAR,20 which again underscores the connection between FEP and probability distribution methods. In general, stratification is a good idea even
when D A(l) is a weakly changing function of l because it improves diffusive
sampling of the full parameter range.31,32
Probability distribution methods and TI, discussed in the next section, are
alternatives to FEP for calculating free energies of binding. Even though
different parts of the thermodynamic cycle are explored in these methods, it
has been recently shown that, most encouragingly, the calculated values of
D A are of similar accuracy.28 Which of these methods is chosen in practice
can be dictated by several considerations. Many of them are related to
technical issues. In some circumstances, however, probability distribution
methods and TI are clearly preferable to FEP. One such case is when there
are strong electrostatic host–ligand interactions. The binding energy in
water may still be small, but the free energy of annihilating the ligand both
bound to the host and free in aqueous solution would be quite large. Then,
FEP is not an optimal route to estimating D A because calculations of a small
number as the difference between two large numbers are usually quite difficult to carry out reliably.
3.5 Thermodynamic Integration
3.5.1
Theoretical Background
If l is a parameter in the potential energy that scales some interatomic
interactions, then TI, eqn (3.9), takes a particularly simple form. All that
needs to be done is to carry out simulations at a series of different values of
l between 0 and 1. At each l, the average value of the derivative of the
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Chapter 3
potential energy with respect to l, which usually can be readily evaluated,
needs to be obtained. The values of l do not have to be evenly spaced; in fact
this is rarely the optimal choice. However, they have to cover the range of l
sufficiently densely to ensure subsequent, accurate numerical integration. In
this type of applications, TI is similar to FEP. Not surprisingly, many practical issues and concerns, previously discussed in the context of FEP, such as
single and dual topology paradigms or diffusion of the ligand from the
binding site apply to both methods. TI might appear to provide an easier
route to estimating D A than FEP, as there are no concerns about overlap
between probability distributions or systematic errors. Upon closer analysis,
however, this impression may be misleading. Generating n integration
points along l in TI is analogous to using n strata in FEP. The difference is
that in the latter method, D A is estimated via MBAR, which is the optimal
procedure given the data, whereas in TI D A is obtained via numerical integration. It can be shown that the choice of integration algorithm is equivalent to implicitly selecting a model for P(DU), which is not optimally based
on the data. If the overlap between probability distributions in consecutive
strata is poor and, consequently, a significant portion of relevant samples is
missing, TI may yield more accurate estimates of D A than FEP. However, if
FEP is properly executed, such an outcome is unlikely.
More common applications of TI are to problems in which l is a physical
coordinate. Then, TI is an alternative to probability distribution methods.
The derivative of the free energy is expressed as
ð
@H bH
e
dðl l*ÞdG
dA @l
ð
¼
;
(3:19)
dl l*
ebH dðl l*ÞdG
where the delta distribution selects a specific value of l equal to l* and |l*
denotes that the derivative is taken at this value.
Direct use of this equation is inconvenient because it requires expressing
the Hamiltonian in generalized coordinates, which may be quite tedious.
Instead, a commonly used approach is to keep l fixed at l* by way of the
applied force, Lrl, where L is a Lagrange multiplier associated with the
constraint l ¼ l*.
To carry out the integration in eqn (3.19), we need to account for the
change in volume elements associated with the transformation from the
Cartesian to a generalized coordinate system. This yields the formula:33
dA
1 @ ln j J j
¼ Lþ
;
(3:20)
dl
2b @l
where | J| is the Jacobian (the determinant of the Jacobian matrix) for the
transformation between the coordinate systems and the derivatives are
taken at l ¼ l*.
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Combining TI with constrained dynamics carried out at specific quadrature points, {l*}, allows for sampling states corresponding to different values
of l, including low-probability transition regions. This approach also has its
drawbacks. In particular, restricting sampling to the hypersurface l ¼ l*
limits paths connecting metastable states, thus enhancing possible quasinonergodicities in the system. A similar problem arises in other fields. For
this reason, in some optimization techniques of multidimensional functions,
additional dimensions are artificially introduced to improve the search efficiency. Another drawback is that evaluating the derivative of free energy by
way of eqn (3.20) requires calculating the second derivatives of l with respect
to the Cartesian coordinates, which could be complicated in some cases.
In an alternative approach, dA/dl can be calculated from unconstrained
simulations. Analogously to the probability distribution method, values of
the derivative are histogrammed and subsequently averaged in bins along l
rather than calculated at discrete points. Generally, dA/dl can be calculated
from the formula
dA
¼ hrU w r wi
dl
(3:21)
for any vector w, providing that it satisfies the condition w rl ¼ 1.34–36 In
particular, the choice of w ¼ @x/@l yields
dA
@U 1 @ ln j J j
¼
þ
;
(3:22)
dl
@l b @l
which is the exact equivalent of eqn (3.20).36 Note that the factor 1/2 in
the second term on the right hand side is no longer present. It accounts for
the strict constraint imposed on the momentum along l, which does not
exist in unconstrained simulations. Other choices of w are also possible. In
P 1
particular, if w ¼ mlM1rl, m1
mk ð@l=@xk Þ2 , where M is the mass
l ¼
k
matrix and ml is the generalized mass associated with l, then36
dA
d
dl
¼
ml
:
dl
dt
dt
(3:23)
An interesting feature of this formula is that second order space derivatives are no longer needed, as they have been replaced with first order derivatives over time and spacial coordinates, which are easier to evaluate. In
this form, the equation formally resembles Newton’s equation of motion,
but involves statistical averages instead of instantaneous quantities, an
interesting result in its own right.36 In general, different implementations of
eqn (3.20) yield different formulas for the derivative of the free energy.32 In
the limit of infinitely long simulations, estimates of dA/dl obtained from
these formulae are identical, but the corresponding variance will, in general,
depend on w. Thus, in an efficient implementation of TI in unconstrained
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simulations, it is desirable to choose w such that the variance is as small as
possible.
3.5.2
Adaptive Biasing Force Method
In comparison to constrained simulations, unconstrained simulations improve the ergodic behavior, but suffer from the same difficulties as the
generic implementation of the probability distribution method – crossing
high free energy barriers between metastable states along l would be rare.
This, in turn, would result in inefficient equilibration along the transition
coordinate. One solution could be to employ an equivalent of umbrella
sampling – guess dA/dl and apply this quantity as a biasing force. There is,
however, a better way to approach the problem, which has been implemented in the Adaptive Biasing Force (ABF) method.32,36,37
The underlying idea of ABF and other adaptive algorithms38–42 is an
adaptive adjustment of some quantity carried out during simulations that
eventually leads to a uniform sampling of chosen transition coordinates.
This quantity could be a force, a potential, a probability distribution or a
transition probability matrix. An important advantage of well-designed
adaptive algorithms is that, in contrast to traditional enhanced sampling
techniques, such as umbrella sampling, no prior guess about the dependence of D A on l is needed. This is particularly important when little is
known a priori about the shape of D A.
In ABF, the instantaneous force acting along l and calculated, for example,
from eqn (3.22) or (3.23), is considered as a sum of the average force, which
depends only on l and a random force with zero average that reflects
fluctuations of all other degrees of freedom. In many instances, the random
force can be satisfactorily approximated as diffusive, leading to a simple
physical picture in which the system diffuses along l in the potential of the
mean force. To preserve most of the characteristics of the dynamics, the
adaptive algorithm is applied to the systematic average force, but not to
the random force. At each time step, the running average of the force for the
current value of l is updated. The biasing force applied along l is equal to
this quantity. Over time, as the estimate converges to the average force at
equilibrium, the total, biased force becomes approximately equal to zero for
all values of l. This means that all values of the transition coordinate are
sampled with equal probability. Reaching perfect convergence in which the
biasing force is exactly equal to the average force is not a requirement. It is
sufficient that the biasing force allows for efficient sampling of all values of l.
As in probability distribution methods, stratification is very useful for
improving efficiency of free energy calculations.28 It is also useful to monitor
the total biased force, as forpsufficiently
long times, t should converge to zero
ffiffi
at the rate proportional to t, as shown in Figure 3.7. If this is not the case,
quasi-nonergodicity is most likely at play. It is also always essential to estimate statistical errors associated with the calculated D A. How to do it has
been described by Rodrı́guez-Goméz et al.43
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Figure 3.7
79
Convergence of the average force in ABF simulations in which imidazole
is transferred across the water–carbon tetrachloride interface. Upper
panel: convergence of the biased force, Fb, averaged over l, to zero. Lower
panel: decay of the deviations in Fb from the constant
pffiffi value as a function
of time. The red line corresponds to t0.6, close to t expected for strictly
statistical errors.
Other adaptive algorithms are also being used in biomolecular simulations. One is the Local Elevation technique,38 further incorporated in
metadynamics.40 The basic idea behind this approach is to add to the
current potential a small, repulsive function in the neighborhood of the
sampled value of l in order to increase the probability of exploring regions
of the configurational space that have not been yet sufficiently sampled.
A common form of the biasing potential is the weighed sum of Gaussians
of different width.44 This procedure converges to a solution for which the
biased free energy, D Ab(l), is constant or, equivalently, the biasing potential is equal to D A(l).45 The disadvantage of this procedure is that the
width and weight of the Gaussian function has to be carefully tuned, especially since it is often not natural to approximate D A(l) as a sum of
Gaussians. In another adaptive algorithm, the adaptive biasing potential
method,39 the biasing potential is periodically updated to yield a uniform
probability distribution along l. In general, properly executed adaptive
methods improve the efficiency of free energy calculations, sometimes in a
dramatic fashion. It does matter, however, what is being adapted. The
superior efficiency of ABF is due to the fact that force, in contrast to
potential or probability distribution, is a local property. This means that no
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information about the shape of the free energy in the whole stratum is
needed to carry out efficient adaptation.
3.6 Replica Exchange for Enhanced Sampling
in Configurational Space
The enhanced sampling techniques discussed in the previous sections can
markedly improve the efficiency of sampling along l, but they are not designed
to combat quasi-nonergodicities in the configurational hyperspace perpendicular to l. In other words, they are not helpful in equilibrating degrees of
freedom orthogonal to l. A family of techniques that is aimed at enhancing
sampling along all degrees of freedom is called parallel tempering, replica
exchange, or multiple walker methods.46–52 As recently reviewed,53 they rely on
generating a number of copies of the system of interest and simulating them
in parallel at different temperatures or with different Hamiltonians. One copy,
or replica, is the target for which we wish to compute the free energy. The
purpose of other replicas is to supply configurations that are structurally more
diverse than those in the target system. For example, increased temperatures
make it more likely that the system will traverse energy barriers separating
metastable states. Occasionally, an attempt is made to swap configurations
between two different replicas. If the swap involving the target replica is successful, chances are that the new configuration belongs to the region in configurational space that would be difficult to access in sequential simulations
without swaps. It is, of course, critical to ensure that samples generated with
swaps are still drawn from the Boltzmann distribution. This is guaranteed if
the acceptance of swaps is governed by the Metropolis–Hastings criterion. If
two replicas, i and j, differ only in the temperature (TioTj), the exchange of
configurations between the replicas is accepted with the probability, p:
p ¼ min(1, e(bibj)[Ui (l)Uj (l)]).
(3.24)
This means that the exchange of configurations is always accepted if the
energy, Uj, of the replica simulated at the higher temperature Tj is lower than
the energy, Ui, of the replica at Ti. Even if the energy of replica j is higher than
the energy of replica i, there is a chance that the swap will be accepted, but
the probability of acceptance decreases exponentially with the difference in
energies and temperatures. It follows that the difference in temperature
between replicas i and j should be sufficiently small to ensure that the acceptance ratio is large enough to allow for the efficient mixing of configurations. For this reason, in most applications, replicas are ordered according
to the increasing temperature and exchanges are attempted only between
consecutive replicas. Considerable efforts have been expended to optimize
the temperature distribution among the replicas.54–56
Even though temperature is, in principle, an excellent, general parameter
to reduce barriers, applications of temperature-based replica exchange suffer
from several serious problems. One is size consistency. As the system size
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81
increases so does the average energy difference between replicas. This means
that to keep the acceptance ratio constant, the temperature difference between replicas has to decrease, which in turn adversely influences the efficiency of free energy calculations. Additionally, increasing the temperature
could cause havoc in membrane systems, generating large fluctuations in
the structure of the bilayer, water penetration into its hydrophobic core, or
even mixing of the water and membrane components. This pushes the
system to regions of the configurational space that are usually not of interest. Furthermore, the increased temperature may induce transitions in the
protein structure, for example from the open to the closed state of a channel,
that would be unwanted in simulations of ion transport. Taken together,
this means that temperature-based replica exchange should be applied to
membrane systems with great care.
Alternatively, the potential energy rather than temperature differs between
replicas. Usually, these differences are parametrized by way of a parameter, z.
Then, the Metropolis–Hastings criterion takes the form:
p ¼ min(1, eb[Ui (l,zi)Uj (l,zj)]),
(3.25)
where zi and zj are the values of z for replicas i and j, respectively.
Often, z scales a subset of interatomic interactions, for example
solute–solute or solute–solvent interactions.57–60 This is equivalent to a
non-physical procedure of scaling temperature for these but not other
interactions. In another version, torsional potentials of a solute, e.g. proteins,
are scaled to induce its increased flexibility.61 Using Hamiltonian replica
exchange reduces or even eliminates the problem of size consistency because
the number of degrees of freedom affected by scaling is substantially reduced, However, it might still lead to disappointing outcomes. If interactions
that involve the membrane are scaled, problems with preserving its integrity
will persist. If these interactions are not scaled, membrane relaxation that
often accompanies conformational transitions in receptors will not improve.
It is also permissible and perhaps advantageous to modify the
Hamiltonian in ways other than simple scaling of selected interactions. In
general, it is desirable if modifications affect the degrees of freedom involved in the process of interest, but not others. However, defining such
modifications that accommodate constraints on replica exchange techniques for membrane systems is often difficult. Even though replica exchange
techniques have been combined with all basic free-energy calculation
methods,49,51,62,63 no version has been specifically designed with membrane
systems in mind. This is an interesting area for future theoretical studies.
3.7 Applications of Free Energy Calculations:
Case Studies
How useful are free energy calculations for understanding membrane receptors? Instead of reviewing the rich literature on this subject, we focus on
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Chapter 3
several examples selected to illustrate the diversity of methods and applications, ranging from ligand binding and ion transport to conformational
changes accompanying receptor action. We wish to demonstrate that all
techniques outlined in the previous sections can be fruitfully applied to
provide valuable insight into membrane proteins that is otherwise very
difficult to obtain.
3.7.1
Binding of Anesthetic Ligands to Receptors
A quantity of great interest in studies of membrane receptors is the binding
free energy of ligands that modulate receptor action. Among them are anesthetics. Despite persistent efforts spanning over a century, the mechanism
of anesthesia remains enigmatic. There is now a consensus that inhaled
anesthetics work by modulating the function of ion channels, with the
pentameric ligand-gated ion channel (pLGIC) receptor family that includes
glycine, acetylcholine, and gamma aminobutyric acid type A receptors being
the most important targets.64–67 A wide variety of molecules can act as anesthetics,68,69 which raises a puzzling question: how can so many dissimilar
compounds that differ in size, shape, electronic structure, and conformational flexibility modulate the same receptors? The work of many investigators points to two likely answers to this question: (1) there is a binding site
with unusual properties not yet characterized that can accommodate many
anesthetics, or (2) there are multiple anesthetic binding sites, possibly on a
single receptor. Over the years a considerable body of evidence has accumulated in support of both of these mechanisms.
Two recent advances allowed for probing these mechanisms in ways that
were not previously possible. First, the atomic-scale structure of an anesthetic
binding site on GLIC, a newly discovered member of the pLGIC family70,71 (see
Figure 3.8) was identified.72 Second, it has been demonstrated that GLIC is
modulated at clinically relevant concentrations of anesthetics.73 These findings provide a unique model system for addressing questions about
anesthetic binding on the basis of the three-dimensional structure of both the
protein and the sites that bind more than one anesthetic. Along these lines,
the free energy of binding propofol and isoflurane, two anesthetics known to
modulate GLIC, to the crystallographically identified allosteric binding site
and a site inside the pore were calculated with the aid of FEP, applying a
number of features discussed in Section 3.3.74 To ensure good overlap between consecutive probability distributions, all calculations were stratified
with l changing by 0.05 in the range 0.1olo0.9, and even less near its extremal values. Soft core potential and electrostatic decoupling were used to
prevent unphysical electrostatic repulsion near the end values of l. D A was
reconstructed from stratified calculations by way of BAR. Flat-bottom spherical restraints were applied to keep the anesthetics in the binding site.
To improve the accuracy of the calculations, a force-field model of isoflurane was developed and tested.75 In particular, both FEP and ABF calculations of the hydration free energy were carried out. In FEP, l was a
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Figure 3.8
83
Top (left panel) and side (right panel) view of GLIC. The X-ray structure is
in blue and the structure from MD simulations initialized from the X-ray
structure is in gold.
parameter that decoupled solute–solvent interactions. In ABF, l was a
transition coordinate for the transfer of the solute from water to the gas
phase across the liquid–vapor interface. The results from both methods
agreed to within statistical error and reproduced the free energy of hydration
with chemical accuracy. This demonstrates that different methods, if properly applied, yield consistent free energy estimates.
For both anesthetics, it was found that interactions with the pore site were
stronger than interactions with the allosteric site (see Figure 3.9). The pore
binding site was not identified from X-ray data because it was occupied by a
detergent molecule, which very likely was an artifact of the crystallization
procedure. The anesthetics were found to bind to both the open and closed
state of the channel, with preference for the latter. The calculated binding
affinities were in broad agreement with the experiment. In contrast to
propofol and isoflurane, ethanol was found to bind only very weakly to the
receptor. This result is also in accord with experimental findings that
ethanol does not inhibit the current through GLIC.73
A number of ambiguities persist, however. Molecular-level studies of
anesthetic binding to ion channels require the knowledge of both open and
closed states of the receptor. However, only the open form of GLIC has been,
so far, crystallized, and it was found not to be stable in MD simulations.76,77
Thus, the precise structure of both states, which may importantly affect the
calculated binding affinities, remains uncertain. Considering that anesthetics inhibit GLIC, it is expected that they would preferentially bind to the
open state. It is, therefore, somewhat worrisome that the anesthetics are
predicted to bind stronger to the closed state than to the open state. Also, the
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84
Chapter 3
Figure 3.9
Left: View of the GLIC channel with two propofol molecules blocking a
pore restrained to be open (shown in red and orange), and one bound
in the crystallographic binding site (purple). Center: The two propofol
molecules bound to the pore (gray). Right: Analogous magnification
of two isoflurane molecules in the pore. Isoleucines forming the hydrophobic gate are shown in cyan.
Reproduced from ref. 74.
FEP results contradict a generally acceptable notion that anesthetics act
allosterically, which is based on the observation that there appears to be no
competitive binding between them and channel blockers. Despite these
concerns, the study on the interactions of anesthetics with GLIC are a good
example of how free energy calculations can be used to generate testable
hypotheses regarding the mechanism of action of membrane proteins.
3.7.2
Free Energies of Ions across Channels
Another important common application of free energy calculations is to
obtain free energy profiles of ions along the pores of ion channels. These
profiles provide valuable information about the mechanism and selectivity
of ion transport, help to identify binding sites in the channel if they exist
and, as will be discussed in the next section, enable the estimation of
channel conductance.
Some of the simplest transmembrane channels are built of a-helical
bundles of antibacterial peptaibols, antiamoebin and trichotoxin, which
contain, respectively, only 16 and 18 amino acids per monomer. The former
is thought to be a hexamer,78 whereas the latter is a heptamer.79 For both,
the free energy profiles of K1 and Cl have been calculated by way of ABF in
several strata 5 to 8 Å wide.78,80 In both cases, the barrier to transport K1 was
found to be lower than the barrier to transport Cl, reflecting the observed
selectivity for cations over anions.
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Another simple antibiotic channel that is synthesized non-ribosomally is
gramicidin A. The channel is a tail-to-tail dimer of a peptide, 15 residues in
length, that is built of alternating L- and D-amino acids.81,82 This unusual
sequence allows the peptide to adopt a cylindrical structure. Calculating the
free energy profile turns out to be a considerable challenge because the
narrow pore accommodates only a single file of water molecules.83 Because
water molecules in a single-file arrangement have a specific orientation, the
large dipole moment of the water column creates significant electronic polarization effects that have to be accounted for explicitly rather than in an
average fashion, as is usually done in standard force fields. Only then it is
possible to achieve semi-quantitative agreement between simulations and
experimental results.84 The two-dimensional free energy profile of K1 along
the pore axis and the radial coordinate was obtained with the aid of WHAM
from a series of umbrella samplings in windows 0.5 Å wide. Using small
windows makes methods based on probability distributions nearly equivalent to constrained TI, which has the advantage of not requiring a biasing
potential. As for peptaibol channels, the calculated barrier to the transport of
K1 is located in the middle of the bilayers, but is substantially higher, equal
to 7.2 kcal mol1 compared to the ion in bulk aqueous solution. An independent estimate of the barrier from FEP calculations in which the ion at
the center of the pore was alchemically interchanged with a water molecule
in the bulk yielded 8.6 kcal mol1, in good agreement with the umbrella
sampling results. This underscores the consistency between results obtained
by way of different free-energy calculation methods.
The free energy profiles for Na1 and Cl were also calculated in the pore of
GLIC.77,85,86 Although different simulations differed in a number of ways,
the profiles obtained with the aid of ABF85 and umbrella sampling86 were
qualitatively consistent. They yielded a high free energy barrier for Cl and a
small or no barrier for Na1 in the transmembrane pore. On the intracellular
side, Na1 was stabilized in a well that acts as a reservoir for cations. The
calculations also revealed that E222 may act as an electrostatic gate to
chloride transport.85,86
Even more complicated were calculations of the free energy profiles for
ions transported through the bacterial voltage-gated potassium channels
KcsA,87,88 which is one of the structurally best studied ion channels.89–92 A
remarkable characteristic of KcsA is its high selectivity for K1 over Na1,
enforced mainly by the selectivity filter, a highly conserved region in all
potassium channels. The origin of ion selectivity has been the subject of
extensive debate.87,90,93–97 It is broadly believed that there are five binding
sites, S0–S4, along the filter occupied by three ions separated by water molecules, as seen in Figure 3.10. Because multiple ions are involved in transport, a three dimensional free energy profile, with the energetics of an ion
described on each dimension, was calculated by way of umbrella sampling.88
Considering that ions move toward the water-filled cavity near the center
of the channel in a concerted way, only a small part of the reduced, threedimensional configurational space needs to be sampled, as the rest
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86
Chapter 3
Figure 3.10
Schematic of the selectivity filter in KcsA. Five binding sites, S0–S4, are
occupied either by ions or by water molecules. K1 ions are green and
the Na1 ion is purple.
Reproduced from B. Egwolf et al., Ion Selectivity of the KcsA Channel: A
Perspective from Multi-Ion Free Energy Landscapes, J. Mol. Biol., 401,
831–842, Copyright 2010 with permission from Elsevier.88
comprises high energy regions. Two combinations of ions were considered:
K1/K1/K1 and Na1/K1/K1. The results yielded a number of interesting results. They revealed that binding to sites S0 and S1 is similar for K1 and Na1.
The selectivity appears to be associated mainly with the interactions of ions
with S2. For K1, there is a free energy minimum at this site, whereas Na1
faces a steep free energy increase.
3.7.3
Conformational Transitions in Receptors
A quantity that is particularly relevant to understanding the mechanisms of
action of membrane channels and receptors is the free energy associated
with transitions between the active and inactive forms of a channel.
Calculating this quantity, however, is usually quite challenging because
conformational changes associated with this process are often quite slow
and a suitable transition coordinate is difficult to identify. Nevertheless,
several simulations along these lines have been carried out, yielding interesting results.77,98–100
The energetics of the transition between the open and closed states was
studied for the transmembrane domain of GLIC.77 The free energy of this
transition was calculated by way of the string method.101,102 This approach
belongs to a class of methods that do not require defining a transition
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coordinate, only the initial and final state need to be known. The initial
pathway connecting these two states is then modified to obtain the pathway
that corresponds to the free energy minimum or a Boltzmann-averaged
ensemble of low free-energy pathways. In the future, these methods, not
discussed in this chapter, may become valuable, but they are not free of
problems with quasi-nonergodicity; these problems appear in the space of
pathways rather than the configurational space. The calculations reveal that
only the closed state corresponds to a free energy minimum. The free energy
increases monotonically and substantially during the transition to the open
state. This result indicates that the transmembrane domain is constitutively
closed and opens only through interactions with the extracellular domain.
Another system that has been the subject of free energy calculations is the
ionotropic glutamate receptor, a tetrameric ligand-gated ion channel that
transduces signals carried by neurotransmitters into electrical impulses.
Once attached to the ligand-binding domain (LBD), the neurotransmitter
triggers its closing. This transition drives the opening of the transmembrane
domain allowing cations to flow through the channel.103–105 Computational
studies98,99 focused on the first step in this process – the closing of the LBD
upon binding the ligand. According to a purely ‘‘structural’’ hypothesis,
ligand efficacy is directly correlated with the degree of LBD closure, i.e. full
agonists induce tighter closure than partial agonists. An alternative
‘‘dynamic’’ view is that agonists keep the LBD closed, whereas partial
agonists are unable to prevent occasional partial opening of the domain. To
test these two hypotheses, the process of interest was divided into two steps:
ligand docking to LBD and LBD closing in response to the presence of the
bound ligand.99 The free energies associated with each step were calculated
from appropriately designed thermodynamic cycles by way of umbrella
sampling for three agonists, three partial agonists, and three antagonists.
The distance between the centers of mass (COM) of the ligand and the
residues in the binding cleft was used as the transition coordinate for
docking. Additional restraining potentials were added to keep the LBD in the
open conformation and, subsequently, the calculated free energies were
corrected for contributions from these potentials. To measure the LBD
closure, a two-dimensional transition coordinate was chosen. Each dimension was the distance between groups of atoms on the opposite flaps of the
LBD that come together upon ligand binding. In the open state, these distances are approximately 14 Å, whereas in the fully closed state there are only
8 Å. The total calculated free energies of binding that included both docking
and closure correlated very well with the measured binding affinities, but the
free energies from each step exhibited no or poor correlation. This indicates
that both steps contribute importantly to the activation of the receptor. For
most agonists and partial agonists the minimum free energy corresponds to
the fully closed conformation of the LBD, in contrast to the antagonists,
which are bound to the open structure (see Figure 3.11). Agonists and
partial agonists differ, however, in their binding affinity, which supports the
‘‘dynamic’’ hypothesis.
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Figure 3.11
Chapter 3
(a) Two-dimensional maps of the free energy of interaction between the
LBD and its agonists (top row), partial agonists (middle row), and
antagonists (bottom row). Contour lines are separated by 1 kcal mol1,
with the darker colors corresponding to lower free energies. Lower and
upper dotted lines point, respectively, to the positions of the closed and
open conformations of the LBD. The agonists and partial agonists, with
the exception of kainate, induce the closed conformation, whereas
the antagonists bind to a partially open conformation. The fully
open conformation corresponds to the apo form (lower right corner).
(b) Order parameters that describe the closure of the LBD.
Reprinted by permission from Macmillan Publishers Ltd: Nat. Struct.
Mol. Biol.,99 Copyright 2011.
3.8 Non-equilibrium Properties from Free Energy
Calculations
The primary function of ion channels is to facilitate passive ion transport
across cell membranes through a water-filled pore. Although recent advances in high-resolution structure determination of membrane proteins
have greatly improved our understanding of how ion channels function, the
detailed description of their mechanism of action still remains a major
challenge because it cannot be fully achieved from the analysis of ‘‘frozenframe’’ structures, even at atomic resolution. Computer simulations, especially at the atomistic level, can in principle provide the required level of
detail. However, they suffer from a number of problems, mainly related to
insufficient time scales of simulations and inaccurate force fields. How can
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we assess the reliability of simulation results? One way to do so is to compare
the calculated and experimentally measured electrophysiological properties
of ion channels, such as ionic conductance, ion selectivity, or reversal
potential. Satisfactory agreement constitutes a strong argument that simulations provide a good description of ion transport through the channel.
Conversely, if such agreement is lacking, then conclusions drawn from
simulations are uncertain.
By definition, ion transport is a non-equilibrium process and, therefore,
has to be described with the aid of non-equilibrium statistical mechanics.
The most direct computational method for this purpose is molecular
dynamics, in which an electric field is applied across the membrane,106
although other methods, such as Poisson–Nernst–Planck or Brownian dynamics simulations are also used.107–111 Even though the process of interest
is non-equilibrium in nature, it will be shown below that, in many instances,
equilibrium quantities, such as changes in free energy of an ion being
transported through the channel, can be quite helpful both conceptually and
practically. Conversely, non-equilibrium simulations provide a path to reconstructing these free energy changes. This underscores a deep connection
between equilibrium and non-equilibrium dynamics that is valid under
many circumstances of practical interest.
Consider ionic conductance, defined as the ratio of ionic current to applied voltage. This quantity, which is probably the most comprehensive
electrophysiological characteristic of ion channels, is routinely measured
in single channel recording experiments. In non-equilibrium molecular
dynamics simulations, it can be calculated through the integration of instantaneous currents,106 which is in practice equivalent to counting the net
number of ions that traverse the channel. If the current is proportional to the
voltage, the channel is called Ohmic. Then, conductance can be fully characterized from a single measurement or simulation at one applied voltage.
In practice, establishing the Ohmic relation requires data at several voltages.
For many channels, however, this relation does not hold and the full characterization of conductance requires measuring the current–voltage (I–V)
dependence over a broad range of electric fields. Then, the agreement between the calculated and measured I–V dependence is a very good indicator
of the reliability of simulations. Determining such a dependence usually
requires substantial, and frequently prohibitively large, computer resources.
Even if one is willing to settle for only a single simulation at a physiologically
relevant applied field, calculating conductance might turn out to be a demanding task. Under these conditions, ionic currents through many channels are quite low, and very long simulations are required to determine them
with sufficient statistical precision. To increase the current, calculations are
often carried out at high voltages, markedly larger than those observed
physiologically, and then extrapolated to lower voltages with the aid
of procedures of unknown accuracy.112 These difficulties motivate efforts
to combine computer simulations with theory of transport in order to
improve the efficiency of calculating elecrophysiological properties without
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Chapter 3
sacrificing accuracy. The free energy of ions as a function of their position in
the channel is an essential ingredient of this approach.
This research agenda can be realized in a number of ways, but the general
goal remains the same – to calculate the I–V dependence at satisfactory accuracy on the basis of a single equilibrium or non-equilibrium simulation.
In this section, we describe its implementation, in which molecular
dynamics simulations are combined with an electrodiffusion equation. For
many years, this equation has been broadly used by electrophysiologists. In
its simple forms, it is known as the steady-state Nernst–Planck or Goldman–
Hodgkin–Katz (GHK) equations.113 We will outline the theory behind this
approach, illustrate its application in an example, and briefly discuss its
limitations and alternative approaches.
3.8.1
Theoretical Background
Assume that transport of ions across a transmembrane channel can be described as a diffusive process in the presence of the intrinsic potential of
mean force, A(z), which is a function of ion position z along the channel, and
the applied voltage, V(z). If the concentrations of ions on both sides of the
membrane and the external electric field remain constant in time, the
system is in a steady state and can be described by a one-dimensional
electrodiffusion equation, which is a stationary form of the Smoluchowski
equation:
drðzÞ rðzÞ dEðzÞ
J ¼ DðzÞ
þ
:
dz
kB T dz
(3:26)
In this equation, J is the net current across the channel, D(z) is the
position-dependent diffusivity, r(z) is the number density of ions per unit
length along z in the channel, which is proportional to the probability
density of ions in the channel, and
E(z) ¼ A(z) þ qV(z),
(3.27)
where q is the ionic charge. Here, V(z) is assumed to change linearly with z.
Since water is a conducting phase, the electric field acts only in the membrane region.
Several other conditions have to be fulfilled for eqn (3.26) to be valid. Ion
diffusion is assumed to be Fickian rather than, for example, single file, as it
would be in very narrow channels. Ion crossing events should be uncorrelated. Also, the free energy profile of an ion in the absence of the field,
A(z), which is due to all other components of the system, including not only
the membrane and water, but also other ions, is assumed to be independent
of the applied voltage. How well these assumptions hold will be discussed
briefly at the end of this section.
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To obtain the integrated form of the electrodiffusion equation, we
multiply both sides of eqn (3.26) by a function g(z). This yields
gðzÞ
drðzÞ
rðzÞ dEðzÞ
J
¼ gðzÞ
þ gðzÞ
DðzÞ
dz
kB T dz
and integrating over z in the limits between z1 and z2:
ð z2
drðzÞ
dz ¼ rðz2 Þgðz2 Þ rðz1 Þgðz1 Þ
gðzÞ
dz
z1
ð z2
dgðzÞ
dz:
rðzÞ
dz
z1
The first integral in the numerator can be carried out by parts. Then,
we obtain
ð z2
1
d
EðzÞ
J ¼ ð z2
lnðgðzÞÞ rðz1 Þgðz1 Þ rðz2 Þgðz2 Þþ rðzÞgðzÞ
dz :
gðzÞdz
dz
kB T
z1
z1 DðzÞ
(3:28)
EðzÞ
If we choose gðzÞ ¼ exp
, which is the integrating factor for
kB T
eqn (3.26), the integral in the numerator vanishes and we obtain
J¼
rðz1 ÞexpðEðz1 Þ=kB TÞ rðz2 ÞexpðEðz2 Þ=kB TÞ
ð z2
:
expðEðzÞ=kB TÞ
dz
DðzÞ
z1
(3:29)
This equation requires knowledge of the full free energy profile, A(z), the
position-dependent diffusivity, D(z), and the ion line densities, but only at
the endpoints of integration. Since the system is assumed to be in the
steady state, the calculated J should not depend, in principle, on the choice
of z1 and z2. In practice, however, this is not strictly the case, as both A(z)
and D(z) are burdened with errors. Error analysis80 indicates that the best
accuracy is achieved when the limits of integration are chosen close to the
ends of the channel, but in the region in which there are still no ambiguities regarding channel integrity due to, for example, fraying or lipid
penetration.
A(z) is an equilibrium property and therefore can be calculated from
equilibrium simulations in the absence of the electric field. This can be done
by way of one of the approaches outlined in the previous sections, choosing
positions of the ion along the channel axis z as l. The probability density
method with a properly designed umbrella potential would be suitable for
this purpose. However, the method that is expected to be particularly efficient for the problem at hand is ABF. In fact, this method has been
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Chapter 3
successfully used to estimate A(z) for calculations of ionic conductance of
several channels.78,80,85
Alternatively, A(z) can be estimated from simulations with applied voltage.
To see how it works, we return to eqn (3.28) and note that g(z) could be an
arbitrary function that is non-zero everywhere in the interval between z1 and
1
, the integral in the numerator can be carried out
z2. If g(z) is chosen as
rðzÞ
analytically. This yields
1
½Aðz20 Þ Aðz1 Þ þ qV ðz20 z1 Þ
lnrðz1 Þ lnrðz20 Þ kB T
J¼
:
ð z0
2
1
dz
z1 rðzÞDðzÞ
This formula can be rearranged as an equation for the free energy, given the
current, applied field, diffusivity, and the non-equilibrium ion density
profile.
"
Aðz20 Þ
Aðz1 Þ ¼ kB T lnrðz1 Þ
lnrðz20 ÞJ
#
1
dz qV ðz20 z1 Þ:
z1 rðzÞDðzÞ
ð z0
2
(3:30)
Changing z20 between z1 and z2 allows for reconstructing the full free energy
profile.
Utilizing eqn (3.30) is not the only way to obtain A(z) from nonequilibrium simulations.114–118 In particular, ‘‘milestoning’’, which in its
most recent general version118 can be considered as an efficient numerical
solution of the Fokker–Plank equation, has been successfully used for this
purpose in a related problem of unassisted ion permeation through
membranes.117
D(z), which is another equilibrium property of the system, can be calculated from a series of short molecular dynamics trajectories by way of the
fluctuation–dissipation theorem, from the Einstein relation, or fitted to
some quantities obtained from simulations.78,119–121 A detailed discussion of
the advantages and disadvantages of each of these approaches is beyond the
scope of this chapter. Here, it suffices to observe that the consistency
between the results obtained by way of different approaches is a strong
argument for the reliability of the estimated diffusivity.
Once A(z) and D(z) have been determined from equilibrium or nonequilibrium simulations, the current at any applied voltage can be obtained
from eqn (3.29), providing that the assumptions underlying this equation
remain in force. All that is needed are relatively short simulations needed to
estimate the boundary density terms r(z1) and r(z2) at the required voltage.
This does not require observing any ion crossing events. Taken together, it
means that knowledge of A(z) allows the calculation of the I–V dependence
only from a series of relatively short, additional simulations.
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With an additional assumption, the I–V dependence can be estimated
without any reference to the boundary densities. A(z) is related to the equilibrium density profile r0(z) through
r0(z) ¼ rref exp(A(z)/kBT)
where rref is the reference density to ensure the correct density unit conversion. Substituting this relation to the numerator of eqn (3.26) yields
rðz1 Þ
rðz2 Þ
expðqV ðz1 Þ=kB TÞ expðqV ðz2 Þ=kB TÞ
r ðz Þ
r0 ðz2 Þ
J¼ 0 1
:
ð z0
2
1
dz
z1 rðzÞDðzÞ
Near the edges of the channel, the effects of the electric field are small and
the densities at different voltages are constrained by the bulk ion concentrations. Then, it is often assumed r(z)Er0(z) or, equivalently, r(z)/r0(z) ¼ 1
at the boundaries. This leads to:
J¼
expðqV ðz1 Þ=kB TÞ expðqV ðz2 Þ=kB TÞ
:
ð z0
2
1
dz
z1 rðzÞDðzÞ
(3:31)
With these approximations, the I–V dependence can be obtained from A(z)
and D(z) alone.
If it is further assumed that A(z) and D(z) are constant across the membrane, further simplifications follow. Similar assumptions are made in the
solubility–diffusion model of solute permeation across membranes.122 With
these approximations, the electrodiffusion equation takes a form
J ¼D
qeel
;
kB T
where eel is the electric field and D is the diffusion constant. This is a form
of the Nernst–Planck equation frequently used in electrophysiology. In this
equation, the current is a linear function of the voltage, which means that
the channel is Ohmic. For a number of channels, it has been determined
that diffusivity is a weak function of z78,80 and, therefore, substituting it with
a single diffusion coefficient is justified. In contrast, the assumption that
A(z) is constant is highly questionable on both theoretical and empirical
grounds, greatly limiting the applicability of the Nernst–Planck equation.
3.8.2
Example – the Leucine–Serine Channel
We will illustrate how the approach outlined above works in the example of
LS3, a small, hexameric channel. LS3 is formed in the presence of an electric
field from a synthetic peptide that contains only two amino acids – leucine
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123,124
and serine – in a heptad repeat (LSSLLSL)3.
When folded into an ahelix, the peptide contains a hydrophobic and a hydrophilic face, consisting,
respectively, of leucine and serine residues. In the channel, hydrophilic faces
form the water-filled pore whereas hydrophobic faces are exposed to the
membrane. The electrophysiological properties of the LS3 channel and its
several mutants have been extensively studied.124 It has been determined
that the channel is not Ohmic and, in particular, is rectifying, which means
that the current is not symmetric with respect to the direction of the applied
electric field.
Extensive non-equilibrium molecular dynamics simulations were carried
out for the LS3 channel embedded in the 1-palmitoyl-2-oleoyl-sn-glycero-3phosphocholine membrane in the presence of 1 M aqueous solution of KCl
at three applied voltages: 200, 100, and 100 mV. In addition, an equilibrium simulation in the absence of an electric field was also performed.
The structure of the channel, which was quite stable throughout the
simulation, is shown in Figure 3.12. The radius of the pore is equal to
approximately 5 Å in a broad range of 15 Å near the center of the bilayer,
but widens toward the ends. At all times, the pore was filled with water.
A(z) for both K1 and Cl was calculated either from the equilibrium ion
density profiles as kBT ln(r0(z)) or reconstructed from eqn (3.30) using the
currents and non-equilibrium density profiles obtained from simulations
with the applied voltage. The free energy profiles for both ions have similar
shapes, exhibiting a barrier in the center of the bilayer. This is known as the
Born barrier and is due to the transfer of a charge from an aqueous medium
of high dielectric constant to a low dielectric constant environment in
the interior of the membrane. Similar free energy profiles, but characterized
by markedly higher barriers, exist for unassisted ion permeation through
membranes. For example, for Na1, this barrier was estimated at 27 kcal mol1.125
In contrast, the barrier to transport K1 and Cl through the LS3 channel is
Figure 3.12
Side (left panel) and top (right panel) view of the synthetic hexameric
LS3 ion channel. Each subunit is in a different color.
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only 2.5–3.0 kcal mol . This difference highlights the significance of waterfilled pores in the facilitation of ion translocation through membranes.126
Remarkably, A(z) obtained from the equilibrium and non-equilibrium
simulations at different applied voltages are quite similar, usually within
statistical error, indicating that both approaches are valid. The only significant difference is observed for the profile reconstructed for Cl from
simulations at 200 mV. The diffusivities needed for calculating A(z) from
eqn (3.30) were obtained by way of the Einstein relation. For both ions, they
were found to be nearly independent of z and, thus, were replaced by a single
diffusion constant, D, in each case. Nearly identical diffusion constants were
found from solving eqn (3.30) self-consistently for A(z) and D to match the
equilibrium free energy profile.
Once A(z) and D are available, the I–V dependence can be calculated from
eqn (3.29). The results are shown in Figure 3.13, including the currents estimated for two additional voltages, 50 mV and 200 mV with the aid of
short molecular dynamics simulations aimed at determining the boundary
density terms at these voltages. In agreement with the experiment,123,124 the
channel was found to exhibit non-Ohmic behavior in which the currents
at positive voltages are markedly smaller than the currents at the corresponding negative voltages. The currents at voltages 100 mV predicted from
the electrodiffusion model are in close agreement with those calculated
directly from simulations, but for 200 mV they are overestimated.
It has been determined experimentally that the LS3 channel is selective for
K1 over Cl by a factor of 10. In simulations, however, the selectivity is only
Figure 3.13
Current–voltage curves for the LS3 channel. The total and K1 current
from MD simulations are in green and red, respectively. Experimentally
measured current is in blue. Triangles abbreviate the measurement
points.
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Chapter 3
about 2 at 200 mV, decreasing to 1.6 at 100 mV. At 100 mV there appears
to be almost no selectivity. It is possible that this is caused by a slight imbalance in the potentials describing the interactions of Cl with other
components of the system. If this were the case, the Cl current should be
scaled down to restore the experimentally observed selectivity. This would
turn a qualitative agreement between the calculated and measured currents
at different voltages to the remarkably good quantitative agreement.
The case of LS3 channel represents an example of how free energy
calculations combined with simple theory of ion transport yield electrophysiological information that otherwise can be obtained only from a series
of long molecular dynamics simulations at considerable gains of efficiency,
but without an appreciable loss of accuracy. A similar approach, however,
does not have to be equally successful for all channels. In a number of cases,
some of the assumptions underlying the electrodiffusion equation might not
hold. For example, ion transport through the narrow pore of gramicidin A
appears to follow a single file rather than Fickian diffusion.127 In order to
describe such process, a different diffusion equation is required.128 Whether
it can be effectively combined with molecular simulations to yield the I–V
dependence is not known. In the potassium gated ion channel KcsA, the
assumption of independent ion movement through the channel does not
hold. Instead, ions appear to move through the channel in a concerted
fashion.88 However, this does not necessarily mean that the electrodiffusion
model breaks down, as the motion of ions might still be diffusive along a
collective coordinate. Another difficulty is related to the validity of the assumption about the independence of A(z) and D(z) of the external electric
field at high applied voltages. In general, efficient and accurate methods for
determining electrophysiological properties of ion channels with the aid of
free energy profiles of ions are both important and an interesting area of
theoretical research on membrane proteins.
3.9 Summary and Conclusions
Free energy calculations have already reached the level of maturity at which
it is possible, at least in principle, to treat complex systems of biological
interest, such as membrane receptors. The theory underlying several basic
methods for estimating free energies has been carefully worked out. A wide
array of enhanced sampling techniques and adaptive algorithms has led to
significant improvements in efficiency. As a result, many problems that
would have been impossible to tackle with basic methods have become quite
feasible. Equally importantly, a number of theoretical tools are available to
control and estimate statistical errors and some systematic biases. For example, applying BAR or, more generally, MBAR, improves the reliability of
FEP calculations. A number of techniques exist to identify and reduce quasinonergodicities in simulations that otherwise might go unnoticed. Several of
them have been mentioned in this chapter. The effects of artificial forces
introduced to the system, for example to restrain its parts or encourage
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improved sampling, should be carefully monitored and accounted for in
estimating the free energy. It should be kept in mind that introducing nonphysical interactions to the system could be a double-edged sword. It could
markedly accelerate the convergence of free energy calculations, but also
could push the system into undesirable regions of the configurational space.
Examples of the latter, such as membrane deformations caused by temperature replica exchange or unwanted conformational transitions in receptors have been mentioned in this chapter. For all these reasons, it is
essential to follow ‘‘good practices’’ in free energy calculations, which are
not unlike established protocols in experimental studies.
In practice, membrane receptors remain among the most challenging
systems to which free energy calculations are applied. In part, it is due to the
sheer size of systems under study, as membrane receptors are among the
largest protein assemblies in a cell. Moreover, their mechanisms of action
typically involve highly concerted motions in response to the external
stimulus that extend over many atomic units. Capturing or even understanding these motions at a qualitative level is quite difficult for both theorists and experimentalists. An inherent feature of many receptors is that
effects of the stimulus, which could be a ligand, electric field, pH change, or
mechanical force, are relatively weak. This implies that barriers separating
active and inactive forms of receptors are relatively low, typically of the order
of a few kcal mol1. This, in turn, means that potential functions describing
complex systems containing not only a protein receptor, but also the
membrane, water molecules, and ions, have to be quite accurate. This is by
no means guaranteed, especially that most force fields are calibrated on
water-soluble proteins or their building blocks. If potential functions are not
sufficiently accurate, the results will be unreliable, no matter how carefully
the methods for free energy calculations are applied. An additional complication is due to the fact that a number of high-resolution structures of
membrane receptors have been shown or are suspected to be non-native.
Simulating such structures is unlikely to provide informative results.
The complex nature of motions in membrane proteins brings to light one
of the main difficulties in calculating free energies in these system: identification of the appropriate parameter l or, more generally, finding a lowdimensional configurational hyperspace. In some cases, intuition clearly,
and often correctly, points to a proper transition coordinate. For example,
the pore axis is usually used as the coordinate for describing transport of
ions through channels. In other instances, intuition gives us no clues, as is
the case of signal transfer between extra-membrane and transmembrane
domains in response to ligand binding. All we know is that it has to be a
collective variable, but identifying it is a major challenge. The problem is not
unique to membrane receptors. The same difficulty is encountered, for
example, in studies of folding/unfolding transitions in proteins or protein–
protein and protein–nucleic acid interactions. How to deal with this difficulty is a subject of active research and a number of creative ideas are being
examined, but general, practical solutions are yet to be found.
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One additional limitation of current free energy calculations is that their
vast majority is carried out in the classical mechanics framework. For many
problems this is sufficient, but there are systems for which this is not the
case. Two well-studied membrane receptors, bacteriorhodopsin and the M2
channel from the influenza virus, serve as representative examples. Both
involve transfer of protons that cannot be satisfactorily described with
classical approaches. Although purely quantum or mixed quantum/classical
approaches to calculating free energies have been used for a variety of
chemical and biological systems, their application to membrane receptors is
rare and their reliability for these systems is not documented.
Despite several important limitations, free energy calculations are becoming an important tool for understanding membrane receptors. So far,
most applications have concentrated on the binding of ligand, ion transport,
and conformational transitions between different states of receptors. A few
illustrative examples have been discussed in this chapter. Studies have also
been carried out on protein insertion into membranes and channel assembly inside membranes. There is, however, no question that the area is
not nearly as developed as free energy calculations for water-soluble
proteins. This is in part due the paucity of structures available for study.
As structures of membrane receptors become more readily available and
reliable, theoretical tools specifically designed for membrane proteins improve and computational capabilities increase, the number of interesting
applications of free energy calculations to membrane receptors is likely to
grow rapidly.
Some readers might be disappointed that the chapter concludes without a
recommendation of the most suitable methods for free energy calculations
for membrane receptors. This is on purpose. When thoughtfully applied, all
methods have their place in our toolbox. So far, much has been gain through
cross-pollination between different, yet conceptually related, methods, and it
is likely to remain so.
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00107
CHAPTER 4
Non-atomistic Simulations
of Ion Channels
CLAUDIO BERTI AND SIMONE FURINI*
Department of Medical Biotechnologies, University of Siena,
Viale Mario Bracci 16, I-53100 Siena, Italy
*Email: simone.furini@unisi.it
4.1 Introduction
The study of electricity in biological systems has been an important area of
biophysics since the eighteenth-century and the first experimental evidence
that dissected muscular tissues contracted in response to electrical
stimuli (Luigi Galvani, De viribus electricitatis in motu muscolari, 1792).
However, the connection between these processes and ion currents across
cell membranes was firmly established only in the 1950’s by a series of experimental and theoretical studies that culminated in the Hodgkin and
Huxley model. The H&H model explains the action potential of excitable
cells by two voltage- and time-dependent components of the membrane
current: an inward transient current transported by Na1 ions, and an outward sustained current transported by K1 ions.1 At the molecular level, these
membrane currents are due to ionic fluxes across separate ion channels,
respectively selective for K1 and Na1 ions. Since the late 70’s, with the development of the patch–clamp technique,2 it has been possible to measure
the current through a single ion channel experimentally and single-molecule
electrophysiological experiments have been performed for almost any ion
channel with an important biological function. The result is that today
ion channels are likely the class of protein with more single-molecule
RSC Theoretical and Computational Chemistry Series No. 10
Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
Published by the Royal Society of Chemistry, www.rsc.org
107
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Chapter 4
experimental data available. Mathematical modeling has been extremely
successful in interpreting these experimental data,3 with many details about
the microscopic mechanisms of conduction and selectivity revealed before
any experimental data on the atomic structures of the ion channels was
available. The field of ion channel research changed abruptly in 1998 with the
release of the first structure of a K1 selective channel at atomic resolution.4
With the availability of experimental atomic structures of ion channels, the
focus of channel modeling changed from predicting the structural characteristics from experimental measurements of ionic currents to understanding
the relationship between atomic structure and channel function.
The analysis of the structure–function relation in ion channels is primarily
an analysis of their conduction properties, as this is the most important
functional characteristic of these membrane proteins. Ion channels conduct
ions at an extremely fast pace and with exquisite selectivity. Understanding
the atomic details of these processes represents a major goal of biophysics
since the discovery of independent Na1 and K1 conductance through the
cell membrane. Moreover, the human genome codes for more than 400 ion
channels, with these proteins playing a role in almost all biological processes, such as muscular contraction, transmission of nerve impulses, and
regulation of cell homeostasis. Thus, not surprisingly, several hereditary
diseases (e.g., cystic fibrosis and long-QT syndrome) are due to impaired
fluxes across ion channels,5 and these proteins are also the targets of many
pharmaceutical compounds (e.g., anesthetics and antiarrhythmics). Therefore, computational methods for simulating conduction through ion channels have important applications, both in basic and applied research.
Nowadays, Molecular Dynamics (MD) is accepted as the most accurate
computational method to investigate the structure–function relation of
complex biological molecules at the atomic level.6 In an MD simulation, the
entire system (membrane, protein, water molecules, and ions) is described
with atomic detail (Figure 4.1), and the atomic trajectories are simulated by
numerical integration of the classical equations of motion, with the forces
among atoms calculated by means of an empirical function known as the
force field. The physical description of the system is entirely contained
within this empirical function. Therefore, a correct definition of the force
field is crucial for a satisfactory agreement between MD simulations and
experimental behavior. Today, the most common force fields in simulations
of biological molecules are AMBER,7 CHARMM,8 and OPLS.9 Some physical
processes, such as electronic polarization, are still not properly described by
classical force fields,10 and improved force fields are certainly to be expected
in the following years.11 However, the accuracy of the force fields in use
today is corroborated by more than 20 years of MD simulations of membrane
proteins, with many atomic details about conduction and selectivity through
ion channels revealed by atomistic simulations.12
Today, one of the main shortcomings concerning computational studies of
the structure–function relation in ion channels is certainly the computational
cost of MD simulations. The number of atoms needed to simulate an ion
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Non-atomistic Simulations of Ion Channels
Figure 4.1
109
Atomistic description of an ion channel used in MD simulations.
A snapshot from a typical MD trajectory of a K1 channel is shown. The
channel is shown in blue (cartoon representation), lipid molecules are in
white, K1 and Cl ions in green and yellow, respectively, and water
molecules in red.
channel embedded in a lipid membrane ranges from 100 000 to more than
500 000, depending on the size of the protein. Thanks to highly optimized
codes13,14 and specialized hardware,15 it is now possible to simulate MD
trajectories for atomic systems of this size at a pace of a few microseconds per
day. In ion channels with high-conductance (10–100 pS), the movement of an
ion between the intracellular and the extracellular compartment takes as little
as a few nanoseconds. Therefore, under these circumstances, MD simulations
are a workable strategy to investigate the conduction properties of ion channels. In MD simulations, it is possible to mimic the presence of a membrane
potential by applying a constant electric field along the axis of the channel.16
Moreover, recent methods have been proposed to control the concentration
of ions in the extracellular and intracellular compartments.17 Once the
membrane potential and ion concentrations are set to experimental values
and sufficient computational resources are available, the analysis of ion
conduction by MD simulations is just a matter of counting the ions moving
across the channel in the simulated trajectory. In cases where the current–
voltage characteristic of the channel is linear, a single MD simulation at a
fixed membrane potential is sufficient to estimate the functional characteristics of the protein. However, many ion channels deviate from this ohmic
behavior, with current–voltage characteristics that could be highly asymmetric. For instance, inward rectifier K1 channels have a much higher
conductance for currents entering inside the cell than for outward currents,
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and this characteristic is of primary importance for their biological function.3
Whenever channels with non-ohmic current–voltage characteristics need to be
analyzed, MD simulations need to be replicated at different membrane
potentials. Moreover, MD simulations of conduction through ion channels are
usually done using high membrane potentials (4100 mV, but in some cases
even close to 1 V). In ohmic channels, the functional characteristics at
physiological potentials could easily be extrapolated from these simulations at
higher membrane potentials. However, this extrapolation is not possible for
non-ohmic channels, which need to be simulated at the working (usually low)
potentials. The consequent increase in the computational cost of the simulations makes the problem practically intractable, if not for some test cases.
Most of the atoms in MD simulations of biological systems are oxygen and
hydrogen atoms from water molecules. Therefore, using a continuum model
for the solvent could drastically reduce the computational cost of atomistic
simulations. In Brownian Dynamics (BD) simulations, random forces are
used to model the effects of the water molecules. In classical BD simulations,
only the dynamics of the ions are described explicitly, while the rest of the
system (water, protein, and membrane) is described as static continuum
structures. The explicit treatment of water molecules reduces the computational cost to a fraction of the cost of MD simulations. Thus, within the BD
framework, the simulation of hundreds of nanoseconds per day can be
achieved on a standard desktop. A further reduction of the computational cost
could be obtained using a continuum description also for the ionic species.
This is the approach adopted in computational analyses based on the
Poisson–Boltzmann (PB) or the Poisson–Nernst–Planck (PNP) theory. When
ion distributions are described by continuum functions, physical effects
related to the finite-size of mobile particles are neglected. In narrow ion
channels, or in the presence of ion binding sites with high affinity, neglecting
the finite size of ions might be a severe approximation, and methods have
been proposed to include these size-effects in PB or PNP models.
This chapter describes the computational methods mentioned in the
previous paragraph, and how they could be used to analyze ion channels.
The common characteristic of these methods is that they are based on
non-atomistic descriptions of ion channels. As discussed in the next
sections, these methods can be used to get insight into how channels work at
the molecular and atomic level. Here, the term non-atomistic is used only to
differentiate these approaches from MD simulations, where the system is
described in full atomic detail. In Section 4.2, methods based on a continuum description of the entire system are discussed. Then, in Section 4.3,
methods that use a particle-based description for ions are introduced. The
chapter concludes with a brief discussion about how atomistic and nonatomistic simulations could be combined together, taking the best out of
each of these strategies. Multiscale models that use the results of atomistic
simulations in less computationally demanding approaches for reproducing
experimental behavior might be the key for unveiling the structure–function
relation in complex biological molecules such as ion channels.
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4.2 Methods Based on Continuum Distributions
of Ions
Continuum theories of electrolytes are widely used to describe physical
processes in biological systems. For instance, the equilibrium potential of an
ion across the cell membrane is approximated by the Nernst potential of that
particular ion species, which is derived considering the drift and diffusion of
a continuum distribution of ions across a semi-permeable membrane, and
the Goldman–Hodgkin–Katz equation that approximates the equilibrium
potential of the cell membrane is based on the same continuum description
of electrolytes. In these macroscopic situations, continuum theories of
electrolytes are firmly established. Instead, this section discusses the
application of such theories to a microscopic situation: the analysis of ion
conduction through a single membrane protein. The description of conduction in ion channels by continuum theories of electrolytes entails two
main approximations. The first obvious one is that the distribution of ions is
described by continuum functions ci(r), where r is the position in threedimensional space (i ¼ 1, .., N ionic species are considered). The second
approximation is that the protein and the membrane are treated as static
continuum structures. Since the presence of charged residues is usually
critical for the functional properties of membrane proteins, it is common to
include protein charges in the continuum models of ion channels. The
distribution of electrical charge in the protein is usually defined as a set of
point charges, placed at positions that reproduce the charged residues
important for the functional properties of the channel (i.e., the one that
plays a relevant role in conduction processes). The surface of the protein and
the position of these point charges are based on the experimental knowledge
available for the particular ion channel. If the atomic structure of the
channel is known, the position of the point charges is immediately defined
by the atomic coordinates, and standard algorithms for contour tracing can
be used to define the protein boundaries. However, for the continuum
models treated here, only a description of the protein surface and of the
main charged moieties are needed. Thus, it is not strictly necessary to know
the protein structure at the atomic level for continuum calculations, as long
as a reasonable model of the channel surface can be defined. Indeed, for
several ion channels, analyses based on continuum theories of electrolytes
preceded the experimental resolution of the atomic structure, and they were
functional to reveal how channels operate at the molecular level before the
protein structure was solved.18
The description of ion channels as static continuum structures has an
obvious consequence: if dynamical movements of the protein at the atomic
level are crucial for its functional properties, the model cannot provide an
accurate description of the system. This shortcoming is likely to have a
severe impact in channels where direct or water-mediated interactions
between the permeating ions and the protein atoms play an important role
for conduction processes. Therefore, methods based on a static description
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of the protein structure are likely to fail in pores with size comparable to the
size of the permeating ions. Instead, in pores with diameter wider than that
of hydrated ions, atomic movements are likely to have a minor effect on the
functional properties, and methods based on continuum theories of electrolytes might provide an accurate description of the physical process of ion
conduction.
4.2.1
Poisson–Boltzmann
The Poisson–Boltzmann (PB) equation is based on the hypothesis that,
under the effect of an electrostatic potential, f(r), the mobile charges distribute according to Boltzmann statistics:
zi efðrÞ
ci ðrÞ ¼ c0i exp (4:1)
kB T
where kB is the Boltzmann constant, T is the temperature, e is the elementary
charge, and zi and c0i are, respectively, the valence and the reference concentration of the i-th ionic species. The PB equation results from the combination of eqn (4.1) with Poisson’s equation:
e0 r ½eðrÞrfðrÞ ¼4prfixed ðrÞ 4pxðrÞ
N
X
zi efðrÞ
kB T
zi ec0i exp
i¼1
(4:2)
In this equation, e0 is the vacuum permittivity, e(r) is the dielectric constant
at position r, and rfixed(r) is the charge distribution in the region occupied by
the protein and membrane (which differently from the distribution of ion
charges is fixed in space). The multiplicative factor x(r) is equal to 1 in space
regions accessible by mobile charges, while it is equal to 0 inside the protein
and membrane. The region occupied by the water molecules is highly polarizable, and it is usually described by a dielectric constant close to 80.
Lower values of the dielectric constant (around 60) have also been adopted
for the analysis of ion channels in order to take into account the lower
mobility of water molecules in confined environments.19 The dielectric
constant is much lower for the region occupied by the protein and membrane. Common values of the protein dielectric constant in PB calculations
of ion channels range between 2 and 6, with higher values better suited for
ion channels that present flexible polar groups close to the protein surface.
The Taylor’s expansion of eqn (4.2), truncated to the first term, gives the
Linearized PB equation:
e0 r ½eðrÞrfðrÞ ¼ 4prfixed ðrÞ þ 4pxðrÞ
N
X
c0 ðzi eÞ2
i
i¼1
kB T
fðrÞ
(4:3)
which was introduced by Debye and Hückel in 1923. It is instructive to
consider the analytical solution of the Linearized PB equation in the simple
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case of a point charge Q in a uniform dielectric medium with relative
dielectric constant e. In this ideal system, eqn (4.3) is reduced to:
r2 fðrÞ ¼ N
X
4pc0i ðzi eÞ2
4pQ
dðrÞ þ
fðrÞ
e0 e
kB Te0 e
i¼1
(4:4)
Where d is a Dirac’s delta, and r is the distance from the point charge Q.
The solution of eqn (4.4) is:
fðrÞ ¼
Q
r
exp 4pe0 er
lD
(4:5)
This electrostatic potential corresponds to the Coulomb potential at distance
r from a point-charge Q in a uniform dielectric medium multiplied by an
exponentially decreasing factor, which decays to zero with space-constant lD
equal to:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
kB Te0 e
lD ¼ u
u
N
P
t
4p
c0i ðzi eÞ2
(4:6)
i¼1
This space-constant is known as the Debye length, and it represents how fast
the mobile charges in solution screen the point charge Q. At a distance equal
to lD, the electrostatic potential is around 40% of the corresponding value in
the absence of mobile charges and, at a distance of 3lD, the point charge Q is
almost completely screened with a residual potential lower than 5% of the
potential of the naked charge. As an example, in a water solution (e ¼ 80) of
symmetric monovalent electrolytes at 150 mM, the Debye length is around
0.8 nm, which means that the charge of a mobile ion is almost completely
screened after 2–3 nm. The Debye length is a useful quantity for computational analysis of ion channels, as it discriminates between situations
where classical continuum theories of electrolytes are in good agreement
with experiments, from situations where correction terms are needed. In
summary, when the geometrical characteristics of the system exceed the
Debye length, charges are well screened, and continuum theories of
electrolytes give an accurate description of the physical processes, while
correction terms are likely to be necessary for channels with diameters close
to the Debye length (this point is further discussed in Section 4.2.3).
The PB equation has been widely used to analyze complex biological
molecules, and several codes for its numerical solution are available, with
Delphi20 and APBS21 being two popular choices. In the context of ion
channels, applications of the PB equation include: (i) analysis of the
electrostatic potential along the pore, (ii) pKa calculations to identify the
protonation states of protein residues, and (iii) estimates of binding energies
for protein–ligand interactions. The electrostatic potential, calculated by the
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Chapter 4
numerical solution of the PB equation, might provide important clues on the
functional state of the channel and on the molecular determinants of its
conduction properties. This strategy has been used to investigate the conduction properties of different kinds of ion channels, including among
others K1-channels,22–24 aquaporins,25 P2X receptors,26 the acetylcholine
receptor,27 and voltage dependent anion channels.28 The general idea of this
sort of calculations is to estimate the electrostatic energy of an ion moving
between the intracellular and the extracellular space. If the channel is highly
permeable by a particular ion species, the energy profile for ions of that
species is expected to show local energy minima, corresponding to the ion
binding sites, separated by low energy barriers. By changing the characteristics of the protein, i.e. the protein surface and the position/values of
the charged moieties, it is possible to gain insight into the molecular
determinants of the channel conductance. K1 channels offer a possible
example of how PB calculations have been used to analyze the relation
between the protein structure and the functional characteristics of conduction. As mentioned in Section 4.1, the atomic structure of K1 channels
was solved for the first time in 1998, for the bacterial channel KcsA.4 The
pore domain of KcsA is made of four identical subunits, which are symmetrically arranged around the axis of the channel. The architecture of
this region is conserved in the entire family of K1 channels. Each subunit
contains three a-helices, known as S5, P-loop, and S6. S5 and S6 are transmembrane helices; they constitute the scaffold of the pore, with S6 on the
internal side. The P-loop helices of the four subunits form a funnel at
the extracellular entrance of the channel. The region responsible for the
selective conduction of K1 ions, known as the selectivity filter, is at the
center of this funnel. The sequence of the selectivity filter is shared, with a
few conservative mutations, by the entire family of K1 channels, as TVGYG.
The selectivity filter is B1.2 nm long, with an internal diameter close to the
size of a K1 ion. This region presents five binding sites for dehydrated K1
ions.29 Binding of K1 ions in the selectivity filter is mediated by direct
atomic interactions between the dehydrated ions and oxygen atoms from the
protein. In this region, a description based on the PB equation is obviously
inappropriate. Instead, analyses based on the PB equation have been used to
investigate ion conduction at the intracellular side of the selectivity filter,
where the pore of the K1 channel opens into a water-filled cavity. Continuum
electrostatic calculations proved that a K1 ion is energetically stable inside
this water-filled cavity, and that the electrical dipole associated with
the P-loop a-helices is responsible for the electrostatic attraction at this
binding site.30 The presence of the intracellular cavity is crucial for the
efficient conduction of ions across K1 channels, as it shorten the actual
distance between the extracellular and the intracellular compartment,
focusing the transmembrane potential across the selectivity filter. This
focusing effect had been hypothesized long before the atomic structure of
K1 channels was revealed, and it was later confirmed by PB calculations.
When the X-ray structure of KcsA is used as a model for protein surface, the
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electrostatic potential calculated by the PB equation exhibits a high barrier
between the intracellular space and the pore-cavity.23 This energy barrier is
due to the S6 helices. Indeed, in the X-ray structure of KcsA, the helices S6
form a bundle at the intracellular entrance of the pore that blocks the passage of ions. The C-terminal end of helices S6 might rotate around a conserved glycine residue, increasing the radius of the pore at the intracellular
entrance. This open-state was observed for the first time for the MthK
channel31 and, later, a similar structure was observed in other K1 channels.32,33 PB calculations confirmed that the outward rotation of the S6 helices
reduces the energy barrier at the intracellular entrance of the channel.23
A comparison between the structures of different K1 channels in the openstate reveals that the degree of opening is different for different channels,
being maximal in MthK and around 0.5 nm lower in the voltage gated K1
channel Kv1.2.33 As MthK is a high-conductance channel, it is tempting to
establish a relation between the degree of opening at the intracellular
entrance and the channel conductance. However, PB calculations proved
that two effects modify the electrostatic potential when the helices S6 rotate.
The outward movement of helices S6 obviously decreases the energy barrier
at the intracellular entrance but, at the same time, it destabilizes the binding
site for cations inside the intracellular cavity. As these two are expected to
modify the channel conductance in opposite directions, the different degree
of opening of the intracellular entrance in the open-state is not likely to play a
dominant role on the channel conductance. Instead, PB calculations proved a
strong correlation between the concentration of cations in the cavity and the
presence of negative residues in helices S6. The presence of negative residues
at the C-terminal of S6 might explain the high-conductance observed experimentally for BK (big conductance) K1-channels.34 Similar analyses based on
the PB equation have been performed for different families of ion channels.
As a general rule, whenever long-range electrostatic forces dominate the ion–
channel interactions, the PB equation is a suitable strategy to get insight into
the relation between the structural and functional characteristics.
The electrostatic force exerted by protein atoms on permeating ions
depends on the protonation state of the ionizable residues. Thus, determining
the pKa of these residues is crucial for any analysis on the structure–function
relation of ion channels. The PB equation represents a computationally
efficient way to estimate changes in pKa values.35,36 The change in pKa of an
ionizable group in a protein is due to changes in the protonation energy
when the residue is moved from a water solution to its position inside the
protein. Both inside the protein and for the isolated residue, the protonation
energy is equal to DGU DGP, where DGU and DGP are the energies of the
unprotonated and protonated states, respectively. If the electrostatic terms
are the dominating ones, these energies (both in the protonated and
unprotonated states) can be estimated as:
X
DG ¼
qi Fi
(4:7)
i
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Chapter 4
Where the sum is over all the electrical charges in the system and Fi is the
electrostatic potential at the position of the charge qi. The electrostatic
potential, F, can be calculated through the numerical solution of the PB
equation. In order to estimate the change in pKa, the PB equation needs to
be solved under four different conditions, corresponding to the protonated/
unprotonated residue inside the protein and in water solution. Then, the
change in pKa is calculated as:
DpKa ¼
DDG
2:303kB T
(4:8)
with DDG being the difference in protonation energy between the residue
embedded in the protein and one isolated in the water solution. Since this
strategy to calculate shifts in pKa is based on the PB equation, the value of
DpKa is sensible to the dielectric constants adopted for the water solution
and the protein interior. Moreover, the method is based on a static representation of the protein structure. If the protonation state modifies the local
structure of the protein, a static protein surface does not describe the system
accurately, and the estimated DpKa is likely to be wrong. In these situations,
an alternative strategy for the calculation of DpKa values is to estimate the
change in energy, DDG, by methods based on MD simulations. However, it is
important to note that, despite the inherent approximations, DpKa values
calculated by the BD equation are in good agreement with the experimental
results for several ion channels.37 Electrostatic calculations based on the PB
equation are also widely used in combination with MD simulations to estimate protein–ligand binding energies. The free energy of binding between a
protein and a generic ligand, DGbind, is given by:
DGbind ¼ Gcomplex Gprotein Gligand
(4.9)
where Gcomplex, Gprotein, and Gligand are, respectively, the free energies of the
complex formed by the two molecules, and of the protein and the ligand
isolated in water solution. In the MM/PBSA approach (Molecular Mechanics
Poisson Boltzmann Surface Area), each free energy term in eqn (4.9) is
calculated as:
G ¼ Ebond þ EVdW þ Eelec þ GPB þ GSA TS
(4.10)
The first three energy terms in eqn (4.10) are directly calculated from MD
trajectories: GPB is the polar contribution to the solvation energy, GSA is an
energy term proportional to the surface area of the molecule, and TS is the
contribution to the energy of entropic terms, which can also be estimated
from the oscillations of the solute molecules in MD trajectories. The polar
contribution to the solvation energy, GPB, is calculated by solving the PB
equation for different snapshot along the MD trajectories and taking the
average of these values. Thanks to this average, oscillations of the structure
around its local energy minimum are taken into account in the calculation of
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binding energies within the MM/PBSA approach. Moreover, if three independent trajectories are used to calculate Gcomplex, Gprotein, and Gligand, the
structural changes induced by the formation of the complex are also properly included in the model. Energy calculations with the MMPB/SA method
have been widely used to estimate the binding energy of drugs, toxins, or
other molecules to ion channels. The interest in this sort of calculations is
motivated by the fact that a better understanding of the molecular determinants of channel–ligand interactions might drive the design of better
pharmaceutical compounds with higher selectivity and fewer side effects.
4.2.2
Poisson–Nernst–Planck
The numerical solution of the PB equation provides the distribution of ions
around a protein at equilibrium, i.e. with ionic fluxes equal to zero in the
whole simulation domain. Instead, for the analysis of ion channels, it is
usually important to calculate the ion fluxes across the protein. These ionic
fluxes can be estimated by the numerical solution of the Poisson–
Nernst–Planck (PNP) differential equations. The Nernst–Planck equation
defines the flux of the i-th ion species, Ji, as:
1
Ji ðrÞ ¼ Di ðrÞ rci ðrÞ þ
ci ðrÞrWi ðrÞ
(4:11)
kB T
Where Di(r) is the ion diffusion coefficient. The first term represents the flux
due to diffusive processes (Fick’s law), while the second term is the flux due
to the potential energy Wi(r) acting on the ion species. As a result of mass
conservation, ionic fluxes at steady-state satisfy the differential equation
(continuity equation):
r Ji(r) ¼ 0
(4.12)
In order to solve the set of differential eqn (4.12), it is necessary to define the
relation between the potential energy, Wi(r), and the distribution of ions in
the system. In the classical PNP model, the potential energy of each ion
species is assumed equal to the electrostatic energy:
Wi(r) ¼ zief(r)
(4.13)
Then, the electrostatic potential is related to the distribution of mobile ions
in the system by the Poisson’s equation:
e0 r ½eðrÞrfðrÞ ¼ 4prfixed ðrÞ 4p
N
X
ci ðrÞzi e
(4:14)
i¼1
Eqn (4.12) and (4.14) define a complete set of differential equations (PNP
equations). The numerical solution of the PNP equations provides the
electrostatic potential and the concentration of the different ion species in
the entire simulation domain (Figure 4.2). Then, eqn (4.11) can be used to
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Figure 4.2
Chapter 4
Schematic representation of a model-system for PNP calculations.
The results of PNP calculations for a K1 channel are shown on two
orthogonal surfaces crossing at the axis of the channel. A schematic
representation of the atomic structure of the channel is shown in
transparency. (A) Electrostatic potential, with red/blue corresponding
respectively to the positive/negative potential. Concentration of K1 (B)
and Cl (C) ions, with the color ranging from blue (0 mM) to red (1 M).
calculate the ionic fluxes. PNP equations are based on the same continuum
description of the system used for the PB equation and, consequently, they
entail the same approximations. More precisely, PNP equations might be
considered as a generalization of the PB equation to the case of non-zero
fluxes, as they reduce to the PB equation in the case of ionic fluxes identical
to zero.
Numerical simulations of ion channels based on the PNP equations have
been widely used to investigate the relation between structural features and
functional characteristics. Moreover, in many cases, theoretical studies of
ion conduction based on the PNP methods preceded the experimental
resolution of the channel structure, and they were a powerful approach to
make hypothesis about the structure–function relation before the highresolution structure of the channel was available (for a review, see ref. 18).
For instance, PNP models proved that rectification properties of ion channels could be explained by the asymmetric distribution of charged residues
at the extra/intracellular entrances of the channel, and they explained how
charged residues, or the composition of the bathing solutions, might modify
the conduction and selectivity properties of ion channels.38,39 With the
advent of high-resolution structures of ion channels, numerical solvers of
the PNP equations in three-dimensions were developed.40–42 The input
parameters for a PNP simulation are: (i) the surface of the protein, which can
be derived from the experimental structure; (ii) the space-dependent
dielectric constant; (iii) the diffusion coefficient of the different ion species;
and (iv) the boundary conditions for the electrostatic potential and ion
concentrations. As for the numerical solution of the PB equation, the
simulation domain is usually divided into two regions: one with low
dielectric constant, representing the protein and the membrane; and one
with high dielectric constant, representing the water solution. Diffusion
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coefficients are set to zero inside the protein and the membrane, which
prevents ions from entering these regions. In the rest of the system, where
mobile ions can diffuse, the diffusion coefficients are a primary determinant
of the conductance values estimated by the PNP model (see eqn (4.11)). The
diffusion coefficient could be used as a fitting parameter, adjusted to reproduce the experimental ion currents. An alternative strategy is to estimate the
diffusion coefficient by atomistic simulations, as described in Section 4.4.
Three-dimensional PNP simulations, with the channel profile estimated
from its experimental atomic structure, have been performed for several ion
channels.41,43,44 In narrow pores, as K1 channels, the PNP theory deviates
from the experimental data, as discussed in the next section. However, even
in these cases, the PNP approach could still be useful to analyze how the ions
approach the (large) entrances of the channel. As an example, PNP calculations confirmed the predictions discussed in the previous section in the
context of the PB equation about how the outward rotation of helices S6
impacts on the conductance of K1 channels. As hypothesized on the base of
PB calculations, the different degrees of opening of the intracellular entrance does not have an impact on the channel conductance, which instead
is modulated by the presence of negative residues at specific positions along
helices S6,45 a result that was confirmed by experimental measurements.34
As the calculation of ion currents by the PNP theory is extremely efficient
from a computational point of view, this method is perfectly suited for a
rapid evaluation of the functional characteristics of structural models. For
instance, this approach has been used to compare different structural
models of the alpha-hemolysin channel.46 This pore-forming toxin is a
homo-oligomer that might exist both in heptameric and hexameric states.
The atomic structure of the heptameric state was solved experimentally
by X-ray crystallography. Instead, the structure of the hexameric state was
experimentally observed only at low-resolution by atomic force microscopy.
Different models of the hexameric state were defined on the base of geometrical constraints. Then, PNP simulations were used to estimate the
current–voltage characteristics, and the model with better agreement with
the experimental data was identified. More recently, a similar approach was
used to assess structural models of K1 channels.47 The distance between
predicted and experimental functional characteristics is correlated with the
root mean square distance from the reference structure, which suggests that
this approach might be useful for the fast-screening of structural models.
4.2.3
Improvements of Classical Continuum Theories
of Electrolytes
PB and PNP are mean-field theories, and the forces on mobile ions are
calculated as purely electrostatic forces. Microscopic effects, as van der
Waals forces or interactions mediated by water molecules, are neglected.
Moreover, as described in the next paragraphs, the classical PB and PNP
equations underestimate the dielectric self-energy, and they completely
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Chapter 4
neglect the effects related to the finite size of ions. At distances larger than
2–3 Debye lengths, mobile charges are screened by the corresponding
counter ions. Therefore, in macroscopic systems, the approximations
described above are perfectly legitimate, and the predictions of PB/PNP
calculations are in agreement with the experimental results. Instead, it is not
clear if these continuum theories can work in small systems whose dimensions are comparable to or smaller than the Debye length.
The dielectric self-energy results from the interaction between the mobile
ions and the electrical charge that they induce at dielectric boundaries in
non-homogenous systems. At the interface between two media with different
dielectric constant, the electric field exhibits a discontinuity. Instead, the
electrical displacement, defined as the product between electric field and
dielectric constant, is conserved. The effect of this discontinuity on the local
electric field can be described by a distribution of the electric charge
(induced charge) at the interface between the two media. In particular, when
an ion is embedded in a high-dielectric medium (water solution, eE80), it
induces a charge of the same sign at the boundary with a low-dielectric
medium (protein, e ¼ 2–6). The interaction with this induced charge pushes
the ion away from the boundary surface. Thus, in ion channels, the dielectric
self-energy hampers the entrance of ions inside the pore, and this effect does
not depend on the sign of the ion charge: it is repulsive both for anions and
cations. The potential energy that was used in eqn (4.1) to define the
Boltzmann distribution of mobile charges cannot describe this process, as
the electrostatic potential is the same for all the ions. The potential energy in
eqn (4.1) has opposite sign for cations and anions and, consequently, a
repulsive term with the characteristics of the dielectric self-energy is
certainly missing. The expected result is that the PB equation overestimates
ion concentrations inside the channel. These qualitative predictions were
confirmed by a comparison between PB calculations and particle-based
simulations based on Brownian Dynamics in model channels (see Section
4.3 for a description of Brownian Dynamics).48 The force acting on the
mobile ions calculated by the numerical solution of the PB equation is lower
than the force estimated by particle-based simulations when the radius of
the channel is close to 1 Debye length, while the two models converge for
channels wider than 2 Debye lengths. The underestimation of the force and
the consequent overestimation of the ion concentrations inside the channel
have also an effect on the conduction properties estimated from the PNP
equations. When PNP calculations are compared to particle-based simulations, the currents estimated by PNP are higher.49 Again, the difference
between the two methods vanishes for channels wider than 2 Debye lengths.
Intuitively, the overestimation of the current by the PNP theory is due to the
fact that, in a particle-based simulation, an ion enters the channel in all-ornone way. In other words, a position inside a narrow channel is empty or
occupied by an ion of a specific sign, and this ion is repelled by the electrical
charge it induces at the dielectric boundary with the protein surface.
Instead, in PNP, fractional charges are possible, and a position along the
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channel could have concentrations different from zero for both cations and
anions. The consequence is an overestimation of the shielding effect that
produces an overestimation of the ionic currents through the channel.
The dielectric self-energy can be included in continuum theories of electrolytes by modifying the potential energy perceived by mobile ions, with a
term describing the interactions with the electrical charge induced at the
dielectric boundary. This energetic term is crucial for explaining conduction
and selectivity in ion channels. For instance, in the gramicidin channel, the
dielectric self-energy and electrostatic interactions with the charged residues
of the protein almost cancel each other out for monovalent cations, providing an energy profile with low-energy barriers.50 Instead, for anions or
divalent cations, the same two energy terms give an energy profile with high
barriers. This difference in energy between monovalent cations and ions
with different charges might explain why the latter are excluded from permeation events. PB and PNP equations modified to include the dielectric
self-energy were compared to particle-based simulations in simplified
models of ion channels.51,52 The results obtained with these modified versions of PB and PNP equations were in agreement with particle-based
simulations, even for channels with radius close to one Debye length.
A second problem of the PB and PNP equations emerges in the presence of
high-affinity binding sites for ions inside the channel. These high-affinity
binding sites are usually the result of charged residues of the opposite sign
close to the pore walls. For instance, a ring of four glutamate residues is
present in bacterial Na1 channels,53 Ca21 selective channels present two
rings of negatively charged residues,54 and six rings of positive and negative
residues have been observed in calcium-release-activated channels.55 In the
classical PB and PNP equations, there is no energy term that discourages the
accumulation of mobile charges. Therefore, a high concentration of protein
charges might cause the accumulation of mobile charges of the opposite
sign up to concentrations that are physically impossible (i.e., corresponding
to more than one ion in the volume occupied by the ion itself). This shortcoming of continuum electrolyte theories is well known in biology from the
study of highly charged molecules, such as nucleotide sequences, and it is
due to the fact that in PB and PNP equations the mobile ions are treated as
dimensionless particles. While this approximation might be valid at low
concentrations, it is certainly not in the crowded environment of an ion
channel with a diameter comparable to the size of the permeating ions. The
effects related to the finite-size of ion particles can be included in the
framework of the PNP theory using a chemical potential (mi) calculated with
the Density Functional Theory (DFT) of hard-sphere fluids.56,57 In this context, the classical PNP equations are obtained when the chemical potential is
approximated by its ideal components while, using DFT, it is possible to
include the effect of interactions between finite-sized particles in the
chemical potential. The model was extremely successful in reproducing, and
even predicting, the functional characteristics of the ryanodine receptors,58
and it explained anomalous mole fraction effects59 or the selectivity to
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60,61
divalent ions in model channels.
The success of the PNP equations
modified to include finite-size effects clearly demonstrates the importance of
ion size for conduction properties. Ion channels are crowded environments,
where different kinds of ions, water molecules, and protein atoms compete
for a limited space. Selectivity is the result of this competition.
It is important to remark that PB and PNP are mean-field theories and,
consequently, they neglect ion–ion correlations, even when the equations are
modified to include self-dielectric energy and size-effects. The mean-field
approximation might fail in channels where conduction is a multi-ion process. As an example, the selectivity filter of K1 channels host between two
and four ions in the conductive state and conduction of these ions proceeds
by a knock-on mechanism, where an incoming ion pushes the ions already
inside the selectivity filter.29 The movements of the ions inside the selectivity
filter are highly correlated, a process that cannot be captured by mean-field
models.
4.3 Particle-based Methods
The main function of ion channels is to move ions between the intracellular
and the extracellular compartments. Thus, methods based on particle-based
descriptions of ionic species are naturally suited to describe how these
proteins operate. Treating ions as particles immediately solves the shortcomings described in Section 4.2.3. Indeed, with finite-size particles, the
over-accumulation of ions is naturally avoided, and ion–ion correlations
are explicitly described. In general terms, particle-based simulations are
more computationally demanding than calculations based on continuum
descriptions of ions, as the numerical solution of differential equations is
replaced by the need to sample a statistically relevant number of conduction
events, which could be a daunting computational task, in particular for
channels with low conductance. Moreover, in particle-based simulations,
special attention is needed for the treatment of boundary conditions. The
electrostatic difference across an ion channel is due to an imbalance of
charges between the two sides of the membrane. In experimental conditions,
the difference in the number of charges between the two compartments is
negligible with respect to the total number of ions in the system, and ion
concentrations are maintained through exchanges with reservoirs, i.e.
intracellular and extracellular compartments that contain a number of ions
comparable to Avogadro’s number. The situation is necessarily different in
numerical simulations, as the computational cost of particle-based simulations increases dramatically with the number of particles. The consequence is that the number of ions in the simulation of ion channels rarely
exceeds a few hundreds and, in order to reproduce the experimental conditions, it is necessary to control this (low) number of ions during
the simulation. The aim of the algorithms for the control of boundary
conditions is to reproduce, in the simulation domain and for each ion
species, the statistical properties of ions in contact with reservoirs at the
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corresponding chemical potentials. This condition can be achieved by
coupling the simulation domain with reservoirs of ions using the Grand
Canonical Monte Carlo (GCMC) algorithm.62 In a simulation with the GCMC
method, particles are created/destroyed with a Metropolis scheme, based on
the difference between the change in free energy and the target chemical
potential. In order to separately control the ion concentrations at the two
sides of the membrane, the simulation domain is divided into an extracellular and an intracellular compartment, and the creation and destruction
of particles in both compartments are accepted with probabilities, Pcreate and
Pdestroy, equal to:
c0i
DW mi
exp ci þ 1
kB T
Pcreate ¼
c0i
DW mi
exp 1þ
ci þ 1
kB T
Pdestroy ¼
1
DW mi
1 þ exp ci
kB T
c0i
(4:15)
(4:16)
where ci is the concentration of the i-th ion species in the corresponding
compartment (measured in number of ions), c0i is the reference concentration in that compartment, DW is the change in free energy if the change in
the number of particles is accepted, and mi is the chemical potential. The
chemical potential of the i-th ion species may be estimated from theoretical
equations62 or it can be calculated using iterative procedures, where an
initial guess of the chemical potential is refined until the target ion concentrations are established in the two compartments.63,64 Whenever the
GCMC scheme is adopted, it is necessary that the creation/destruction of
ions by the algorithm does not interfere with the physical processes of ion
conduction. In order to satisfy this requirement, the simulation domain is
usually divided into three regions (Figure 4.3). Ions are not created/destroyed
inside the pore or close to its intracellular and extracellular entrances.
Instead, this region is flanked by two boundary-reservoirs, where the
GCMC algorithm is used to control the ion concentration. As ions are free
to move between the three compartments, this scheme reproduces the
experimental conditions of an ion channel, with an accuracy that improves
for boundary-reservoirs further from the ion channel.
It is important to remark that in MD simulations, ions, as any other
atoms, are described as particles. Thus, the method to control the boundary
conditions described above could be used also in MD simulations.
However, this section deals with coarser methods, where only the ions are
treated as particles. The solvent is still described as a structure-less continuum
medium, and the proteins and membrane are described as static continuum
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Chapter 4
Figure 4.3
Simulation domain with reservoir-boundaries for particle-based simulations. The simulation domain of a typical BD simulation is shown (blue
box). The red lines delimit the regions where ions are created/destroyed
by the GCMC algorithm. The shape of the channel is shown by
black lines. K1 and Cl ions are shown as green and yellow spheres,
respectively. These are the only particles explicitly described in BD
simulations.
structures. As a consequence, these models are still going to fail
if protein movements are crucial for the functional properties of the channel.
Section 4.4 discusses possible strategies to include these effects in nonatomistic simulations.
4.3.1
Brownian Dynamics
In classical BD simulations of electrolytes, only the ions are modeled
explicitly. The trajectory of an ion, x(t), of mass m in a viscous medium is
described by Langevin’s equation:
::
mx (t) ¼ mg(x)x_(t) þ R(t) þ F(t)
(4.17)
The three terms in eqn (4.17) correspond to the frictional, random, and
systematic forces acting on the ion. The random and frictional forces
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represent the effects of collisions of ions with the surrounding water molecules. The systematic force is the sum of all forces other than those caused
by the surrounding solvent molecules. The magnitude of the first two terms
depends on the frictional coefficient g, which is related to the diffusion
coefficient of the ion species through Einstein’s relation. For an ion in bulk
solution, the frictional term of the force is prevailing over the inertial one. In
this case, Langevin’s equation is reduced to:
mg(x)x_ (t) ¼ R(t) þ F(t)
(4.18)
Eqn (4.18) is the commonly used equation to describe the Brownian motion
of particles suspended in a fluid. In this chapter, the term Brownian
Dynamics is used indifferently for the systems described by eqn (4.17) or
(4.18). The reason is that, inside ion channels, the force due to ion–protein
interactions might dominate the stochastic force, and the inertial term in eqn
(4.17) might be comparable to the other forces. Thus, numerical simulators
based on both eqn (4.17) and (4.18) have been proposed. The Langevin’s
equation of motion can be numerically solved by the method developed by van
Gunsteren and Berendsen.65 The time step for numerical integration depends
on the highest frequency movements in the system. In bulk solution, a time
step of 100 fs is perfectly suitable while, inside ion channels, where strong
systematic forces might dominate, a time step as low as 10 fs might be
necessary.
In classical BD implementations, the systematic force is calculated as the
sum of two terms: a short-range repulsive force and an electrostatic force.
The short-range force prevents ion–ion overlap or the overlap of ions with
the region occupied by the protein or the membrane. Lennard-Jones
potentials are commonly used to describe short-range repulsive forces, but
other expressions of the short-range force have also been used in BD
simulations.48,62 The electrostatic force is due to all the contributions of
electric fields in the system. These include all the other ions, the electric
charges induced at the dielectric surface separating the region where ions
diffuse (high dielectric medium) from the region occupied by the protein
and membrane (low dielectric medium), the electric charges embedded in
the protein or in the membrane, and any externally applied electric field
(introduced to mimic the presence of a membrane potential). The electrostatic potential is calculated by solving the Poisson’s equation in the simulation domain. As the electrostatic potential depends on the ion positions,
Poisson’s equation needs to be solved at run-time. In order to simplify the
numerical solution of Poisson’s equation, the electrostatic potential can be
divided into four terms: (i) the potential due to the external electric field,
(ii) the potential due to interactions with protein charges, (iii) the Coulomb
potential due to other mobile ions in a homogenous dielectric medium, and
(iv) the force due to charges induced at the dielectric boundaries by mobile
ions. The first two terms do not depend on the particular configuration of
the ions. Thus, it is possible to calculate their contribution to the
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electrostatic potential by solving Poisson’s equation once for all at the
beginning of the simulation. The Coulomb potential can be easily calculated
at the run-time. The only term that requires the numerical solution of
Poisson’s equation at each time step is the last one. A possible strategy to
avoid the computational cost associated with the numerical solution of
Poisson’s equation at run-time is to preliminary calculate the electrostatic
potential due to the charges induced at the dielectric boundary for all
the possible ion positions in the simulation domain. Then, the stored
values could be used to estimate the electrostatic potential at each time
step. The efficiency of this strategy is strictly related to the number of values
that need to be calculated for the electrostatic potential and, consequently,
on the size of the memory that needs to be accessed at run-time. In order
to limit the number of pre-calculated potentials, a common strategy is to
impose cylindrical symmetry on the system. However, the structure of
ion channels is not always symmetrical. Moreover, the usage of tabulated
electrostatic potentials necessarily introduces an interpolating error related
to the size of the grid. An alternative strategy is to solve Poisson’s equation at
run-time at the exact positions occupied by the ions. The run-time solution
to Poisson’s equation is possible thanks to the Induced Charged Computation (ICC) method,66 which is around two orders of magnitude faster than
iterative boundary element methods for the solution of Poisson’s equation
for a given level of accuracy and for the same number of discretization
elements.67 Moreover, with this method, the electrostatic potential is
calculated exactly at the current position of the ions. Thus, lookup tables
are not required and asymmetric systems can be simulated as easily as
symmetric ones.68
BD has one major advantage over MD simulations, which is the reduced
computational cost. As only ions are modeled, trajectories of hundreds of
nanoseconds per day can be produced on a standard desktop. Therefore, it is
possible to use the BD approach to estimate the currents through an ion
channel at different membrane potentials and ion concentrations with
limited computational effort. The classical BD model has proved to be extremely successful to describe ion conduction through membrane proteins.
The low computational cost of the BD approach comes at the price of a loss
in atomic detail. When the radius of the channel is comparable to the radius
of the permeating ions, intimate atomic interactions are likely to play a
crucial role in the conduction and selectivity mechanisms and, since these
interactions are not properly described, classical BD simulations are unable
to reproduce ion fluxes through these channels. For instance, in the
selectivity filter of K1 channels, K1 ions directly interact with carbonyl
oxygens from the protein residues TVGYG. The energy barriers experienced
by permeating ions increase by several kcal mol1 if any carbonyl oxygen of
the selectivity filter moves away from the pore, with dramatic effects on the
conduction properties.69 In a classical BD simulator, the protein is represented as a static continuum structure and, as a consequence, this sort of
events cannot be reproduced. In order to use a BD simulator in pores with
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diameters comparable to the size of the ions, it is necessary to include in the
model some details of the atomic interactions between the permeating ions
and the surrounding environment. A possible strategy to solve this problem
is to extend the set of particles described by eqn (4.17) to include some
particles that mimic the mobile elements of the protein. Chung and Corry
proposed a model of K1 channels based on this idea, where the mobile
particles represented the carbonyl oxygens of the selectivity filter.70 These
mobile particles were restrained by harmonic potentials around the average
position occupied by the carbonyl oxygens in the experimental structure of
K1 channels. By these and similar strategies, it is possible to analyze the
mechanisms of conduction in simplified systems, and in this way to unveil
which characteristics of an ion channel are responsible for its functional
properties. Section 4.4 describes a more systematic strategy to include
atomic details in BD simulations.
4.3.2
Monte Carlo
Monte Carlo (MC) simulations are an alternative to the numerical integration of the equation of motion for sampling the configurational space at
equilibrium. In brief, in an MC simulation, a new configuration of the system is generated by random changes of the current configuration. The new
randomly generated configuration is accepted as the next configuration of
the system with a probability equal to:
DW
P ¼ min 1; exp (4:19)
kB T
with DW being the change in free energy if the new configuration is
accepted. MC simulations have been used to estimate the distribution of ions
inside an ion channel at equilibrium. Then, in the case of an ohmic channel,
the ion flux between two compartments at the same ion concentration can
be estimated by integrating the Nernst–Planck equation. This approach was
used to analyze conduction through a simple model of Ca21 selective
channels,59 proving that the presence of binding sites with different affinities for the various ion species might explain the emergence of anomalous
mole fraction effects (i.e., a decrease in channel conductance when new ion
species are added to the medium). In more general cases, in order to estimate the ion fluxes, it is necessary to calculate the concentration and the
chemical potential for all the ion species in the simulation domain. The
continuity equations define a first set of relations between concentrations
and chemical potentials that need to be satisfied at the steady state. In the
PNP theory, the mathematical model is completed with Poisson’s equation,
which links ion concentrations to the electrostatic potential and consequently to the chemical potential if this is approximated by its ideal
components. Instead, in the Local Equilibrium Monte Carlo (LEMC)
method, a numerical simulation based on the MC algorithm is used to
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estimate the relation between chemical activity and ion concentration.
Traditionally, MC simulations are used to sample the configurational space
of a system at equilibrium while, in LEMC, Monte Carlo steps are used to
sample non-equilibrium states.71,72 The method is based on the hypothesis
that, at a local level, the system does not deviate significantly from an
equilibrium state. In practice, the simulation domain is divided into smaller
cells, and it is assumed that each cell is at equilibrium. Under this
hypothesis, the state of each cell samples a Grand Canonical ensemble with
the volume, temperature, and chemical potentials fixed. Thus, MC steps
corresponding to creation/annihilation (as described in Section 4.3 for
GCMC simulations in the reservoir-boundaries) or particle movements
between cells can be simulated once the chemical potential of the different
ion species is known. The LEMC algorithm estimate ion concentrations and
chemical potentials by an iterative procedure. Starting from an initial guess
of the chemical potentials, the ion concentrations are updated by MC steps.
Then, these ion concentrations are used in the continuity equations to
calculate the chemical potentials, and the two phases are repeated until
convergence. The accuracy of LEMC was proved by comparison with PNPDFT simulations for test cases.72 An alternative method that can be used to
simulate non-equilibrium conditions by MC sampling is the Dynamic Monte
Carlo (DMC) method. In DMC, the positions of ion particles are sampled by a
classical MC scheme, eqn (4.19), with the further constraint that the
maximum displacement of a particle is limited by an upper boundary rmax.73
The method is based on the idea that the configurations sampled by MC steps
can be interpreted as a dynamic trajectory of the system. The parameter of
the algorithm rmax is critical for this interpretation of MC sampling. The
maximum displacement in MC steps needs to be shorter than the mean free
path in the corresponding dynamic trajectory.74 It is important to remark that,
if DMC simulations are used to sample ion movements in water solution, the
mean free path is the average path between particle collisions when the
system is described in full atomic detail, not for ions moving in the gas phase.
Compared to BD simulations, DMC simulations do not give direct access to
dynamic trajectories. However, a direct proportionality exists between the
time intervals and the number of steps in DMC simulations and, consequently, dynamical features and ionic fluxes can be quantitatively estimated
once this coefficient of proportionality has been established.
4.4 Methods to Include Atomic Detail in
Non-atomistic Models
Computational methods based on an implicit description of the solvent,
such as the ones described in Sections 4.2 and 4.3, guarantee high computational efficiency, but this comes at the price of a loss in atomic detail. At
the other side of the spectrum, MD simulations use a full atomistic
description of the system, but analysis of the experimental behavior by this
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approach is hampered by a high computational cost. The perfect method
should combine the strengths of the two approaches, i.e. the atomic detail of
MD simulations and the computational efficiency of the methods based on
implicit descriptions of the solvent. A general strategy to combine the
two methods is to estimate the parameters of the coarser models by
atomistic simulations. This multiscale strategy, where a detailed (computationally expensive) model is used to estimate the parameters of a
coarser (computationally efficient) model, is common to many fields of
biology. An example, close to the topic of this book, is the modeling of the
electrical activity of the heart, where Markov’s models of ion channels are
used inside models of cardiac cells, and models of cardiac cells into models
of the full organ.75 A similar approach could be used to include information
estimated from atomistic simulations into more efficient descriptions of ion
conduction.
4.4.1
Atomic Detail in Brownian Dynamics
The fundamental equation of BD simulations, eqn (4.17), incorporates the
atomic detail of ion–protein and ion–water interactions into two parameters:
the diffusion coefficient and the systematic force. Therefore, a straightforward strategy to include atomistic details in BD simulations is to estimate
these two parameters by MD simulations.
In classical BD simulations, the ion diffusion coefficients are set to
experimental values or they are used as fitting parameters to reproduce the
experimental ionic fluxes. Moreover, a constant value is usually used for the
entire simulation domain. However, atomic interactions with the protein
channel or with water molecules might modify the ion diffusion coefficient
inside the channel, and MD simulations are a possible method to estimate
these changes in diffusivity. The root mean square displacement (RMSD) of a
diffusing particle increases linearly with time, with a slope equal to six times
the diffusion coefficient. Therefore, a direct method to estimate diffusion
coefficients is to calculate the RMSD of ions in MD trajectories, but
alternative strategies based on autocorrelation functions have also been
proposed. The diffusion coefficient summarizes into a single parameter the
effect of the random collisions with solvent molecules. When this parameter
is estimated from MD simulations, a direct link is established between the
implicit treatment of the solvent molecules used in BD simulations and
atomistic processes.
The second parameter in Langevin’s equation related to atomistic
processes is the systematic force F, which represents the entire force acting
on the diffusing particles that is not due to random collisions. This systematic force is the result of interactions with other ions and protein
atoms, and it also encapsulates the drifting effect of a difference in the
electrostatic potential across the channel. Therefore, a link between BD
simulations and the atomic detail of ion–protein interactions necessarily
involves the calculation of this force, or of the corresponding potential energy,
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by MD simulations. In order to estimate the potential energy acting on the i-th
ion, Wi, by MD simulations, it is convenient to re-formulate this potential
energy as:
Wi(r) ¼ Weq(r) þ Wdrift(r)
(4.20)
where Weq is the potential energy at equilibrium and Wdrift represents the
effect on the potential energy of an external electric field forcing ion
movement across the channel. Eqn (4.20) is based on the hypothesis that the
membrane potential does not modify how ions interact with protein atoms.
This hypothesis needs to be verified on a case-by-case basis. It is likely to
be true if the conductive structure of the channel is stable. For instance, the
conductive state of the selectivity filter of K1 channels is stabilized by a
network of h-bonds with residues in the P-loop helices. As long as the
channel is stable in this open-conductive state, an external electric field is
not expected to modify the ion–protein interactions and, consequently,
eqn (4.20) provides an accurate description of the energy of permeating ions.
The situation is different in Na1 selective channels. The selectivity filter of
the bacterial Na1 channels has a ring of four glutamate residues at its
extracellular entrance.53 The side chains of these residues are highly mobile,
and they can easily switch between two configurations. In the upper configuration, the charged moieties are directed towards the extracellular
solution, but the side chains of these glutamate residues might also assume
a downward configuration, where they point to the pore lumen. Rapid
switches between these two states have been observed in MD simulations,
and an external electric field is likely to modify the relative stability of
the two configurations.76 As a consequence, an ion at the extracellular
entrance of the channel is going to interact differently with the protein at a
different electric field, a situation that cannot be properly described by
eqn (4.20).
In channels where the effect of the electric field on ion–protein interaction is minimal, eqn (4.20) offers a method to include details from MD
trajectories in BD simulations. The potential of mean force at equilibrium,
Weq, can be estimated by several methods. Indeed, numerous insights into
the mechanisms of conduction and selectivity at the atomic level have been
revealed by energy calculations based on MD simulations.77 The important
points for the current discussion are that: (i) it is possible to calculate the
potential of mean force for ion–protein interactions by MD simulations
and (ii) this calculation usually requires a fraction of the time needed to
analyze the same processes in classical MD simulations. The potential of
mean force calculated by MD simulations is, by definition, the average
potential acting on the ions, where the average is taken over all the degrees
of freedom not explicitly considered in the definition of Weq. In other
words, if Weq is calculated as a function of the coordinates of the ions inside the channel, the average is taken over all the possible configurations of
the protein atoms, water molecules, and other ions that are not explicitly
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included in the definition of the energy function. Therefore, by Weq, the
effect of atomic interactions with parts of the systems described as continuum structures (protein and membrane) is correctly included in the BD
framework.
The effect of the external electric field could be included in the model by
solving the PB equation, with the correct boundary conditions in the extracellular and intracellular environment and with no screening changes inside
the channel. Bernèche and Roux successfully adopted this approach to simulate the conduction of K1 ions across the conductive state of the selectivity filter of the KcsA K1 channel.78 These BD simulations, with forces
calculated by MD trajectories, provided an estimate of the conduction
properties of the channel in qualitative agreement with the experimental
results. In the BD simulations of K1 channels by Bernèche and Roux, only
the ions inside the channel were explicitly described, with entrance and exit
rates from this region defined to guarantee an instantaneous equilibrium
with bulk concentrations at both openings. A similar framework could be
used to trace the dynamics of ions both inside and outside the channel. In
principle, MD simulations could be used to calculate Weq in the entire system. However, the computational cost of calculating Weq by MD simulations
increases exponentially with the number of variables and so, the number of
ions involved prevents the calculation of Weq in the entire system. This
shortcoming could be easily circumvented by the observation that, outside
the channel, the atomic details of ion–protein interactions are not crucial for
the conduction properties. As discussed in previous sections, when the
geometrical features of the simulated system are coarser than a hydrated
ion, ion–ion and ion–protein interactions can be correctly described as the
sum of electrostatic and steric terms. This approximation is certainly valid
outside the channel. Therefore, it is always possible to divide the simulated
system in two regions: inside the channel, where forces are dictated by
atomic interactions; and outside the channel, where forces can be calculated
as in the classical BD simulations described in Section 4.3.1. Comer and
Aksimentiev used this approach to simulate ion conduction through a pore
partially occluded by a DNA fragment.79 In this study, the potential of mean
force at equilibrium, Weq, was calculated as the sum of ion–ion interactions
and interactions between ions and other molecular components (protein
and DNA). At short distances, the ion–ion interactions were calculated by
MD simulations, while at longer distances (41.4 nm) they were estimated as
Coulomb forces. At short distances, the force between two ions is affected by
interactions mediated by water molecules of the corresponding hydration
shells, and these effects are not properly described by the electrostatic
and steric terms commonly used in classical BD simulations. Therefore, the
approach proposed by Comer and Aksimentiev introduced the atomic
details of water-mediated ion–ion interactions and of ion–protein interactions in the framework of BD simulations. Thus, this strategy combines
the atomic detail of MD simulations with the computational efficiency of BD
simulations.
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4.4.2
Chapter 4
Atomic Detail in Continuum Models
In the previous section, the link between BD simulations and atomic detail
was established by diffusion coefficients and potential energies calculated by
MD simulations. The same parameters appear in the Nernst–Planck equation for ion fluxes, eqn (4.11). Therefore, similar arguments could be used to
include atomic detail in continuum theories of electrolytes. In the case of ion
channels, eqn (4.11) can usually be reduced to a one-dimensional equation
along the axis of the channel. The reason is that the highest energy barriers
for ion conduction are usually in the direction orthogonal to the membrane,
while ion movements parallel to the membrane are expected to equilibrate
rapidly. For a one-dimensional problem, eqn (4.11) can easily be integrated
between two arbitrary boundaries, z1 and z2, giving:
ci ðz1 Þexp
Ji ¼
Wi ðz1 Þ
Wi ðz2 Þ
ci ðz2 Þexp
k T
kB T
ð z2 B
dz
Wi ðzÞ
exp
kB T
z1 Di ðzÞ
(4:21)
This equation offers a link between MD simulations and continuum theories
of electrolytes. Indeed, as mentioned in the previous section, diffusion
coefficients and potentials of mean force can be estimated by MD trajectories. In eqn (4.21), Di(z) describes one-dimensional diffusion of ions along
the axis of the channel. Thus, as the surface area orthogonal to the membrane usually changes along the channel, this one-dimensional diffusion
coefficient also changes, even in the case of constant three-dimensional
diffusion coefficients. Once Di and Wi are known (by MD simulations), in
order to calculate ion fluxes by eqn (4.21), it is necessary to estimate the ion
densities at two z-values along the axis of the channel. MD simulations can
be used to estimate ion densities at the intracellular and extracellular entrances of the channel, and these calculations are much less computationally demanding than the simulations required to estimate ion fluxes.
Therefore, the current–voltage characteristics of an ion channel can be
obtained by a series of short MD simulations at different voltages. The most
time-consuming step of the procedure is the estimate of the energy Wi(z).
However, this potential of mean force is calculated only once, which represents a great advantage over the estimate of the entire current–voltage
characteristics by a set of independent (long) MD simulations. As for the
methods described in the previous section, this method is based on the
hypothesis that the external electric field does not modify the ion–protein
interactions. Moreover, it is important to remark that eqn (4.21) is based on
a continuum description of ion distributions and on a mean-field approximation. Thus, correlations between ion movements are not considered. If
ion movements are strongly correlated, the conductance of the channel
cannot be estimated by a mean-field theory. The method has been tested in a
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model of the trichotoxin channel, where it provided results in good agreement with MD simulations at the corresponding membrane potentials and
ion concentrations.80
4.5 Concluding Remarks
Mathematical modeling and numerical simulations have played a crucial
role in the analysis of ion channels, and simplified descriptions of conduction processes, as the ones described in this chapter, were functional for
revealing how ion channels operate at the microscopic level. Thanks to the
availability of high-resolution structures of ion channels, it is now possible
to analyze conduction processes in full atomic detail by techniques as MD
simulations. However, the computation cost of atomistic simulations is still
prohibitively high for a systematic analysis of ion channels and the situation
is not likely to change in the near future, at least not for many of the
low-conductance channels with important biological functions. For these
systems, the simulation by simplified models is still the best (if not the only)
option for accessing experimental behavior. When combined with MD
simulations, simplified theories could also represent an efficient strategy to
simulate experimental properties while, at the same time, providing a direct
relation with the structural features of the protein at the atomic level. It is
also important to remember that simplified theories are more than a mere
way to increase the computational efficiency. MD trajectories include many
details that are not essential for the description of the simulated processes.
In order to get physical insight into how channels operate from MD simulations, it is necessary to isolate the important events from an overwhelming
number of random atomic movements. In simplified theories, only a wellreasoned set of features is included in the model. Identifying these features
and how they impact on the functional properties offer a more profound
understanding of the structure–function relation in complex biological
molecules such as ion channels.
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00137
CHAPTER 5
Experimental and
Computational Approaches to
Study Membranes and Lipid–
Protein Interactions
DURBA SENGUPTA,*a G. ADITYA KUMAR,b XAVIER PRASANNAa
AND AMITABHA CHATTOPADHYAY*b
a
CSIR-National Chemical Laboratory, Dr. Homi Bhabha Road,
Pune 411 008, India; b CSIR-Centre for Cellular and Molecular Biology,
Uppal Road, Hyderabad 500 007, India
*Email: d.sengupta@ncl.res.in; amit@ccmb.res.in
5.1 Introduction
The cellular membrane is a complex combination of lipids and proteins,
whose composition is organelle-, tissue- and age-dependent.1–4 Membrane
proteins are implicated in a wide variety of cellular functions, and comprise
B30% of the human proteome5 and B50% of the current drug targets.6 They
also represent important disease biomarkers.7 Recent breakthroughs in our
understanding of membranes have revealed a highly dynamic, anisotropic,
and heterogeneous lipid environment.8 This complex environment of the
membrane has been shown to play an important role in determining the
interactions of membrane lipids with proteins. Membrane organization and
dynamics, and lipid–protein interactions are studied using a wide range of
experimental approaches. Concomitantly, molecular details of membranes
RSC Theoretical and Computational Chemistry Series No. 10
Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
Published by the Royal Society of Chemistry, www.rsc.org
137
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Chapter 5
and its interactions are being increasingly probed by computational
methods, thanks to the recent advances in computational power. In this
chapter, we discuss representative experimental and computational methods to analyze membranes in general, with special focus on lipid–protein
interactions. Our aim is to provide a comprehensive overview of our current
understanding of this field by combining the outcomes of experimental
and computational approaches. The information obtained by exploring
the critical lipid–protein interactions would form an important step in
our overall understanding of membrane protein function in health and
disease.
5.1.1
Membrane Components
The predominant lipid components of cellular membranes are phospholipids, cholesterol, and sphingolipids (see Figure 5.1). In the plasma membrane of several cell types, lipids with the zwitterionic phosphatidylcholine
(PC) headgroup form the largest component.3 Phospholipids with charged
headgroups such as phosphatidylserine (PS) or phosphatidylinositol (PI) are
Figure 5.1
Chemical structures of representative classes of lipids found in eukaryotic cells: (a) glycerophospholipid, (b) cholesterol, and (c) sphingolipid.
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Approaches to Study Membranes and Lipid–Protein Interactions
139
relatively less abundant. The degree of saturation of phospholipid acyl
chains also varies across cellular membranes. Sphingolipids are crucial
constituents of the cell membrane, although they are minor components in
terms of abundance. Cholesterol is an integral part of eukaryotic membranes
and cholesterol content can vary a lot depending on the nature of the cell.
For example, cholesterol content is much higher in cells of neural origin, but
much lower in non-neural cells.9
Membrane proteins interact with the membrane in a number of ways,
including those that span the bilayer completely, interact with one leaflet, or
are attached by lipid anchors. They are involved in a wide variety of cellular
processes, such as signaling and transport. G protein-coupled receptors
(GPCRs) constitute one of the largest families of membrane proteins that
initiate cellular signaling in response to a variety of ligands. Ion channels are
another class of important membrane proteins that transport ions across the
cell membrane. Yet another important class of membrane proteins are those
that interact with the membrane from the outside. The structural characterization of membrane proteins is challenging, although there have been
some successes in the last few years.10 Both a-helical and b-sheet membrane
protein structures have been resolved. Recent reports have highlighted several complex structures in a combination of different structural elements
together with large disordered regions.
5.2 Role of Membrane Lipids in Membrane Protein
Organization and Function
The function of several membrane proteins has been shown to be dependent
on membrane lipids. Several facets of membrane proteins, such as ligand
binding, protein–protein association, and conformational dynamics have
been related to membrane lipid composition. The molecular details of how
membrane lipids affect membrane protein function are beginning to emerge.
Several membrane receptors that transmit signals across the membrane
have been shown to bind ligands in a manner that is dependent on the
membrane lipid composition. The most striking examples are those belonging to the GPCR family.11–15 A number of experimental findings have
demonstrated that membrane cholesterol modulates the ligand binding
characteristics of the serotonin1A receptor.16,17 In fact, a stereospecific
requirement of cholesterol has been recently reported to be necessary for
receptor function.18 Modulation of ligand binding by cholesterol has been
demonstrated in related receptors, such as the a and b-adrenergic receptors.12,19 Phospholipids, but not cholesterol, have been shown to modulate
ligand binding and G-protein coupling in the neurotensin receptor, another
important member of the GPCR family.20,21 Phospholipids have recently
been suggested to act as allosteric modulators of the b2-adrenergic receptor,
with specific lipid headgroups facilitating agonist or antagonist binding.22
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Chapter 5
Interestingly, although sphingolipids have a low abundance in the plasma
membranes, they play an important role in GPCR function.15
The modulatory effect of the membrane is not limited to the GPCR family.
Membrane composition has been demonstrated to have functional effects
on ion channels and modulates the transport of ions across the cell membrane. The ion channel gramicidin has been demonstrated to be uniquely
sensitive to membrane properties such as bilayer thickness and membrane
mechanical properties.23 The function of the nicotinic acetylcholine receptor, a synaptic ion channel, is known to be modulated by the membrane
thickness.24 Similarly, the function of potassium channels has been
demonstrated to be cholesterol dependent.25,26
In addition, the structure and conformational dynamics of several
membrane proteins have been suggested to depend on membrane lipid
composition. Due to the experimental challenges involved in membrane
protein crystallization, only few lipid-dependent structural changes have
been reported. Interestingly, the conformational details of the ammonia
channel have been demonstrated to be dependent on the membrane lipid
composition.27 Additionally, functional protein domains, such as those that
interact with the membrane surface, have been shown to adopt a membranedependent conformation. The N-terminal domain of the GPCR chemokine
receptor 1 (CXCR1) has been shown to be involved in ligand binding and
adopts membrane-dependent structural characteristics.28,29
5.3 Mechanisms for Lipid Regulation of Membrane
Proteins
The mechanism underlying the effect of membrane lipids on the organization, function, and dynamics of membrane proteins is complex, arising
due to the various degrees of spatiotemporal heterogeneity displayed by
membranes. It has been proposed that membrane lipids can exert their
effects on membrane proteins in several ways: (i) by direct (specific) interaction with the protein, (ii) through an indirect (non-specific) modulation of
the membrane physical properties (thickness, order) in which the protein is
embedded, or (iii) a combination of both.30
5.3.1
Specific Membrane Effects
Membranes with embedded proteins are believed to contain several classes
of lipids, depending on their interaction with membrane proteins. These are
termed bulk lipids, annular lipids, and nonannular lipids.31,32 In crowded
membranes, i.e., in membranes with high protein/lipid ratio, the number of
bulk lipids would be low. The annular lipids are those that form an ‘annulus’
or a shell around the protein. The nonannular lipids could be present at
inter-helical or inter-receptor sites where they remain associated with
the protein for longer time scales. Interestingly, the effect of the protein
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Approaches to Study Membranes and Lipid–Protein Interactions
141
could extend beyond the first annular lipid shell, due to long-range
interactions.
Recent crystallographic results have provided evidence for specific lipid
interactions, since it has become possible in the last few years to resolve
membrane lipids in the high resolution crystal structure of a membrane
protein.33 A large number of membrane protein structures have been reported with bound lipid molecules.12,34 One of the first examples of ‘bound’
lipid molecules in membrane proteins was bacteriorhodopsin35 and, subsequently, aquaporin.36 Figure 5.2(a) depicts the crystal structure of
aquaporin with nonannular lipid molecules bound at the inter-monomer
sites. In the GPCR family, ‘bound’ cholesterol molecules were observed
between monomers37 as well as at an inter-helical site in the b2-adrenergic
receptor.38 Subsequently, closely associated cholesterol molecules have been
resolved in several GPCR structures.12,14 In addition, phospholipids in close
association with GPCRs have been resolved from crystallographic studies.
A number of phospholipid binding sites have been distinguished in the
crystal structure of the A2A adenosine receptor, which form almost a complete annulus around the receptor, in addition to bound cholesterol (see
Figure 5.2(b)).39 Several of these sites have been predicted (or validated) by
computational studies and will be discussed below. Interestingly, a recent
NMR spectroscopy study reported two distinct time scales of cholesterol–
GPCR interactions,40 in line with the predictions from simulations.41 Specific
Figure 5.2
Closely bound lipid molecules in membrane protein structures. Representative snapshots of (a) aquaporin (PDB: 3M9I) and (b) A2A adenosine
receptor (PDB: 4EIY) with crystallographically resolved bound lipids
(phospholipid and cholesterol, respectively). The protein is represented
in ice-blue, cholesterol molecules in pink, and phospholipids in cyan.
The protein surface is rendered in light gray.
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Chapter 5
lipid interactions have been observed in the ion channel family as well, such
as in the voltage-dependent K1 channel Kv1.2 chimera42 and the voltage-gated
Na1 channel (NavAb).43 Interestingly, a lipid acyl chain was shown to directly
block the channel conductance in the non-conductive conformation of the
TRAAK ion channel (K2P channel family), moving away in the conductive
state.44
5.3.2
Non-specific Membrane Effects
The complex environment of the cellular membrane is dynamic and anisotropic, both laterally and transversally,45 and can indirectly modulate protein
function through a variety of non-specific effects. It is important to understand the properties of the membrane bilayer itself, in order to analyze its
indirect effect on proteins. Membrane thickness, defined as the distance
between the lipid headgroups in the two leaflets, is dependent on several
factors such as the lipid acyl chain length, saturation, membrane phase, and
cholesterol content. Difference in membrane thickness and transmembrane
helix length, known as the hydrophobic mismatch, has been shown to
directly modulate channel properties and conductance in gramicidin.46
Interestingly, modulation of the GPCR structure and function by hydrophobic mismatch has been previously reported.47 Another important
membrane property, namely elasticity, has been shown to affect the ligand
binding property of the serotonin1A receptor.48 The membrane dipole
potential, the potential arising due to non-random dipolar organization of
the membrane components, i.e., phospholipids and interfacial water, could
have implications on membrane protein function and organization.49
Membrane curvature is another parameter that could affect the membrane
protein function and distribution.50
5.4 Range of Time Scales Exhibited by
Membranes
Membrane dynamics and lipid–protein interactions span a large range of
time scales (see Figure 5.3). At the sub-nm and sub-ns scale, molecular
interactions play a role in the direct association of lipids with membrane
proteins. Local lipid chain dynamics and protein segmental conformational
dynamics occur at the ms time scale. Membrane topological and curvature
changes can extend over mm and display dynamics at longer (seconds) time
scales. Due to this reason, the study of membrane dynamics and organization requires experimental and computational methods spanning a wide
range of time scales. The corresponding experimental and computational
tools used to probe various time scales are shown in Figure 5.3. A popular
experimental strategy extensively used to analyze membrane organization,
dynamics, and lipid–protein interactions is based on fluorescence spectroscopy. This includes tools such as fluorescence resonance energy transfer
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Figure 5.3
143
Range of time scales displayed by lipids and proteins in biological
membranes. The range of time scales in which membrane phenomena
take place is truly remarkable and can span more than ten orders of
magnitude. The corresponding experimental and computational tools
used to probe various time scales are also shown. It is obvious that it is
not possible to address these time scales simultaneously using any single
technique (either experimental or theoretical). The judicious choice of an
experimental or theoretical tool with matching time scales is crucial for
addressing problems in membrane biology.
(FRET), fluorescence recovery after photobleaching (FRAP), fluorescence
correlation spectroscopy (FCS), and monitoring solvent relaxation rates
using fluorescence.51–55 These techniques probe membrane phenomena at
varying time scales (see Figure 5.3). The advantages of using fluorescencebased techniques include enhanced sensitivity, minimal perturbation,
multiplicity of measurable parameters, and suitable time scales that allow
the analysis of several relevant molecular processes in membranes. Likewise,
computational approaches, such as atomistic and coarse-grain simulations,
have been very successful in analyzing various aspects of the lipid–protein
interaction and membrane dynamics.41,56 A few representative examples of
experimental and computational approaches to analyze lipid–protein interactions and membrane dynamics are discussed below, with focus on
approaches used by us. It should be noted here that this is not an exhaustive
description of all available methods to study membrane organization and
dynamics.
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5.5 Lipid–Protein Interactions: Insights from
Experimental Approaches
We discuss below the application of two representative fluorescence-based
experimental approaches, FRET and FRAP, in deriving molecular details of
lipid–protein and protein–cytoskeleton interactions in the membrane, from
previous work carried out in our laboratory.
5.5.1
Determining Near-neighbor Relationships in
Membranes: Interaction of Melittin with Membrane
Cholesterol utilizing FRET
FRET is a powerful biophysical tool for determining proximity relationships
between fluorophores in membranes. FRET requires a donor and an acceptor fluorophore in such a way that the emission spectrum of the donor
has substantial overlap with the excitation spectrum of the acceptor.57,58 The
photophysical consequences of FRET are well understood and discussed in
the literature. These include: (i) quenching of the donor emission and donor
excited state lifetimes, and (ii) increase in the sensitized emission from the
acceptor. These changes in photophysics can be quantitatively converted
into energy transfer efficiency, which is related to the proximity between the
donor and acceptor probes typically in the 1–10 nm scale.
Melittin, a cationic hemolytic peptide, is the principal toxic component in
the venom of the European honey bee, Apis mellifera. It is a small linear
peptide composed of 26 amino acids in which the amino-terminal region is
predominantly hydrophobic whereas the carboxy-terminal region is hydrophilic due to the presence of a stretch of positively charged amino acids. This
amphiphilic property of melittin makes it water soluble and yet it spontaneously associates with natural and artificial membranes. Such a sequence
of amino acids, coupled with its amphiphilic nature, is characteristic of
many membrane-bound peptides and putative transmembrane helices of
membrane proteins. This has resulted in melittin being used as a convenient
model for monitoring lipid–protein interactions in membranes.59 Melittin is
intrinsically fluorescent due to the presence of a single tryptophan residue at
the 19th position, which makes it a sensitive probe to study the interaction of
melittin with membranes and membrane-mimetic systems.60,61
Since melittin is a hemolytic peptide, its natural target is the erythrocyte
membrane, which contains a high amount of cholesterol. Interestingly, the
presence of cholesterol in the membrane is known to inhibit the lytic activity
of melittin, both in model membranes62 and in erythrocytes.63 This brings
up the possibility that membrane cholesterol could specifically interact with
melittin, thereby giving rise to these effects. In order to test whether melittin
specifically interacts with cholesterol, i.e., whether there is any clustering of
cholesterol molecules around melittin in the membrane, we carried out a
quantitative near-neighbor relationship assay by FRET in two dimensions
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62,64
(as in a membrane bilayer) utilizing the Fung and Stryer formalism.
For
this, melittin acted as the donor and dehydroergosterol (DHE) as the acceptor for energy transfer. DHE is a naturally occurring fluorescent analogue
of cholesterol which is found in yeast and differs from cholesterol in having
Figure 5.4
(a) Spectral overlap (shown as striped area) between donor (melittin,
solid line) and acceptor (DHE, dashed line). Considerable spectral overlap is an important criterion for FRET to occur. The inset shows the
chemical structure of DHE. (b) Mapping out the distance of interaction
between donor and acceptor in the membrane from experimental FRET
data and simulated energy transfer plots (see text and ref. 62 for more
details). Adapted and modified from ref. 62.
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three additional double bonds and a methyl group (see inset in Figure 5.4(a)).
A number of reports have shown that DHE faithfully mimics natural
cholesterol in biophysical, biochemical, and cell biological studies.65,66
Figure 5.4(a) shows that there is considerable spectral overlap between
the emission spectrum of membrane-bound melittin with the absorption
spectrum of membrane-bound DHE, an essential criterion for efficient
energy transfer. A complication of FRET measurements, where donors and
acceptors are localized in the plane of the membrane, is that there could be
a large number of donor–acceptor pairs (and therefore donor–acceptor
distances) for a random distribution of donors and acceptors. Analyzing
the energy transfer results for such a system requires an analytical solution.
Results of the energy transfer measurements between membrane-bound
melittin and DHE were analyzed using the Fung and Stryer formalism64 for
the energy transfer efficiency of randomly distributed donors and acceptors
in membranes. The distance (typically in Å) between the donor and acceptor that results in 50% energy transfer efficiency, termed as Förster
distance (R0), is an important parameter in the analysis of energy transfer
data and usually ranges between 10–100 Å.58 R0 for the melittin–DHE pair
was calculated to be 16 Å.62 The extent of energy transfer was quantitated
from the extent of quenching of the donor (melittin) fluorescence. The
dependence of the efficiency of energy transfer (E) on the surface density of
the acceptor (DHE) for various R0 was calculated using the Fung and Stryer
framework and is shown in Figure 5.4(b). The experimentally obtained
energy transfer efficiencies were then compared to calculated efficiencies of
energy transfer for a random distribution of donors and acceptors in a twodimensional plane using this formalism. The series of calculated plots of
energy transfer efficiency as a function of the acceptor surface density in
the membrane for a range of R0, and for randomly distributed donors and
acceptors in the plane of the bilayer, were generated by the numerical
integration:
ð1
E ¼ 1 ð1=t0 Þ
½FðtÞ=Fð0Þdt
(5:1)
0
where t0 is the excited state lifetime of the donor in the absence of the
acceptor, and F (t) is the fluorescence intensity of the donor in an infinite
plane at time t and is given by:
F (t) ¼ F (0) exp(t/t0) exp(sS(t))
(5.2)
where exp(sS(t)) is the energy transfer term, F (0) is the initial fluorescence
intensity, s is the surface density of the acceptor (number of acceptors per
phospholipid headgroup area), and S(t) is given by:
ð1
SðtÞ ¼
a
½1 expfð t=t0 ÞðR0 =rÞ6 g2prdr
(5:3)
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where a is the distance of the closest approach of donor and acceptor, the
expression 2prdr represents the probability of finding an acceptor within a
distance r from the donor in two dimensions, and R0 is the Förster distance
between the donor and acceptor. The dependence of the energy transfer
efficiency on the surface density of the acceptor for a range of values of R0,
calculated by numerical integration of eqn (5.2) and (5.3) is shown in
Figure 5.4(b). The experimental data points were superimposed on the
simulated curves and we found that the experimental data fitted best to
an R0 value of 24 Å. This means that the measured energy transfer efficiency
considerably exceeded the actual value of R0 (16 Å). This implies a preferential association of the donor (melittin) and acceptor (DHE) in the membrane.
These results therefore suggest that there is a close molecular interaction
between melittin and DHE, and their distribution in the membrane is not
random, even at low sterol concentrations. Since DHE is a naturally occurring fluorescent cholesterol analogue, this result indicates that there is a
specific interaction of melittin with membrane cholesterol.62
5.5.2
Interaction of the Actin Cytoskeleton with GPCRs:
Application of FRAP
FRAP is a popular approach used to measure the lateral diffusion of lipids
and proteins in membranes (for a recent review, see ref. 55). FRAP involves
the generation of a concentration gradient of fluorescent molecules by
irreversibly photobleaching a fraction of fluorophores in the region of
interest. The dissipation of this concentration gradient with time due to
diffusion of the fluorophores into the bleached region from the unbleached
regions of the membrane provides an indicator of the mobility of the
fluorescently tagged lipid or protein in the membrane. The recovery of
fluorescence into the bleached area in FRAP experiments is represented by
an apparent diffusion coefficient (D) and mobile fraction (Mf). The rate of
fluorescence recovery provides an estimate of the lateral diffusion coefficient
of the diffusing molecules, whereas the extent of fluorescence recovery
provides an estimate of the mobile fraction, i.e., the fraction of molecules
that are mobile in this time scale. Figure 5.5 shows the underlying principles
of FRAP measurements.
The dynamic heterogeneity observed in cell membranes could be attributed to the differential confinement of diffusion experienced by the membrane components. When viewed from this perspective, cellular signaling
mediated by proteins could be interpreted as a consequence of differential
mobility of the various interacting partners. This forms the basis of the
‘mobile receptor’ hypothesis, which proposes that receptor–effector interactions at the plasma membrane are controlled by lateral mobility of the
interacting components.68,69 The confinement to diffusion is provided by
the intricate network of the cortical actin cytoskeleton that lies immediately
below the plasma membrane.70 The boundaries of confinement are defined
by transmembrane proteins anchored to the cytoskeleton, thereby acting as
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pickets. This model of the plasma membrane is called the ‘membrane
picket-fence model’.71 The relation between membrane heterogeneity and
differential mobility of the membrane components and their role in regulating cellular signaling represent an interesting problem in contemporary
cellular biophysics.
In our work, we have explored the role of the actin cytoskeleton in the
dynamics of a representative GPCR, the serotonin1A receptor, and its
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72
implications in signaling. Lateral diffusion of membrane lipids and
proteins is known to be influenced by cytoskeletal proteins. We destabilized
the actin cytoskeleton using cytochalasin D, a potent inhibitor of actin
polymerization. Upon destabilization of the actin cytoskeleton by increasing
concentrations of cytochalasin D, the mobile fraction of the receptor showed
a significant increase (the diffusion coefficient remained constant) (see
Figure 5.6). This was accompanied by an increase in signaling by the
receptor, as measured by a reduction in cAMP (Figure 5.6(b)).72 The fact that
the change in signaling was correlated with the change in receptor dynamics
was supported by a positive correlation of B0.95 obtained from a plot of
these two parameters (Figure 5.6(b)). Such a tight correlation between the
mobile fraction of the receptor and its signaling is supportive of the mobile
receptor hypothesis. These results imply that the actin cytoskeleton could
play a regulatory role in signaling by membrane proteins.
5.6 Computational Approaches to Study Membrane
Organization and Lipid–Protein Interactions
In order to analyze the various aspects of membrane organization and lipid–
protein dynamics, computational approaches provide powerful tools to
sample time scales ranging from ns to ms (see Figure 5.3) and length scales
from sub-nm to mm. Atomistic simulations have been successful in analyzing
the conformational dynamics of lipids and proteins. With increasing
Figure 5.5
Application of FRAP to monitor the diffusion of lipids and proteins in
membranes. Panel (a) shows a schematic representation of a membrane
surface with fluorescently tagged proteins (shown in yellow). The dotted
line represents the region of interest (ROI), where the laser beam is
focused. Fi denotes the total fluorescence intensity in the ROI prior to
photobleaching (see panel (b)). Upon irradiation by a strong laser beam,
a population of fluorescently tagged molecules is photobleached, thereby
creating a concentration gradient of fluorescent molecules. The total
fluorescence intensity in the ROI immediately after photobleaching is
termed as F0. The gradient in concentration of fluorescent molecules
created this way will dissipate as time progresses, since unbleached
fluorescent molecules from outside the ROI will move into the ROI due
to lateral diffusion. FRAP is extensively used to measure diffusion
coefficients of fluorescently tagged proteins and lipids in membranes.
The total fluorescence intensity in the ROI at a given time t after
photobleaching (at photobleaching, t ¼ 0) is shown as Ft. Analysis of
the rate of fluorescence recovery (from F0 to Ft in time t) provides a
measure of the lateral diffusion coefficient (D). Since membranes are
quasi-two-dimensional, diffusion coefficients measured this way represent two-dimensional diffusion coefficients. The extent of fluorescence
recovery (Mf), on the other hand, offers information on the fraction of
fluorescent molecules that are mobile in the time scale of FRAP. The
interpretation of D and Mf becomes complicated if the dimensions of the
membrane micro-heterogeneities (domains) happen to be of the order of
or smaller than the dimensions of the ROI (see ref. 67 for details).
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Figure 5.6
(a) A schematic representation of the destabilization of the actin cytoskeleton (shown as maroon rods) in cellular membranes using cytochalasin D (CD). (b) Close correlation between the number of diffusing
molecules and cellular signaling of the serotonin1A receptor, indicating
the fundamental role of cell membrane dynamics in signaling. The
dashed lines denote the 95% confidence band. Data for panel (b) are
taken from ref. 72.
computational resources, the dynamic interactions between membrane
lipids and proteins have been analyzed with greater detail. Coarse-grain
simulations have been able to reproduce both specific interactions and nonspecific effects, such as changes in the local membrane thickness. Longer
scale phenomena such as lipid flip–flop, or energetics of membranedependent protein association have been probed by increased sampling
methods and free energy calculations. An integrated approach connecting
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various strategies would be instrumental in understanding membrane lipid
and protein dynamics across the hierarchical levels.
5.6.1
Simulating Single Component and Multi-component
Bilayers
Atomistic simulations of membranes have been able to reproduce several
aspects of their structure and dynamics. Atomistic simulations have been
used to study the self-assembly of lipids into bilayers73–75 and their phasedependent behavior has been characterized.76 Similarly, coarse-grain (CG)
simulations have been successful in reproducing the self-assembly of
bilayers.77 Both atomistic and CG simulations have been used in conjunction
with X-ray scattering or neutron scattering experiments on lipid bilayer structure determination.78–81 Current efforts are focused on improving the available
force-fields and toward a closer match with detailed experimental data.82,83 It
has been observed that the acyl chain region structure and dynamics, but not
the interfacial dynamics, are generally well described by these methods. In
addition, changes in temperature, dehydration, and cholesterol content are
predicted correctly in a qualitative fashion.84 Multi-component bilayers are
being increasingly used to study biologically relevant membranes. However,
atomistic simulations of heterogeneous membrane compositions are difficult,
especially due to limited sampling of lipid diffusion.85 To overcome time
scale issues, coarse-grain simulations of multi-component bilayers have been
performed. For example, coarse-grain simulations have been used to capture
lipid dynamics in bilayers closely representing cellular membranes.86,87 The
improvements in the force-field parameters of various membrane components
pave the way toward studying lipid–protein interactions.
5.6.2
Atomistic Simulations Elucidating Lipid–Protein
Interactions
One of the biggest breakthroughs in computational biology has been the
analysis of ms time scale membrane protein dynamics with atomic resolution. These simulations have allowed us to analyze the details of lipid–
protein interactions with improved accuracy. Additionally, lipid-dependent
conformational dynamics are currently being explored. One of the major
challenges is to minimize artifacts in atomistic simulations of membrane
proteins, whose crystal structures are heavily engineered.88 In general,
membrane protein simulations have been shown to accurately reproduce
membrane protein dynamics.
Atomistic molecular dynamics simulations have been successful in
demonstrating the preferential interaction of membrane cholesterol with
certain sites on GPCRs, such as the serotonin1A receptor,89 the b2-adrenergic
receptor,90 and the A2A adenosine receptor.91 Multiple sites have been
identified, a few of which correspond to crystallographically resolved sites.
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A unique motif corresponding to a putative cholesterol binding site, the
cholesterol recognition/interaction amino acid consensus (CRAC) site, has
been highlighted in the serotonin1A receptor.89 A representative snapshot of
a cholesterol molecule associated at the CRAC site on transmembrane helix
V is shown in Figure 5.7. The cholesterol molecule is associated at a ms time
scale to that site. Several other interaction sites have been observed, but due
to the limited time scale of the simulations, they could not be resolved in
detail. These multiple cholesterol interaction sites have been suggested to be
of comparable energy and in competition with other bilayer components.41
These factors contribute to a large stochasticity in the interaction sites, and
the relative strengths of binding remain unclear.
Phospholipid interaction sites have also been identified in the serotonin1A
receptor by ms time scale simulations. A phospholipid molecule was
observed to be associated with the receptor during the time scale of the
simulations.89 However, no unbinding events were observed. Recently,
specific phospholipid association at the cytosolic leaflet has been suggested
to stabilize an active state structure of the b2-adrenergic receptor.92 Interestingly, the simulations suggested that the receptor embedded in an
anionic membrane shows increased lipid binding, providing a molecular
mechanism for the experimental observation that anionic lipids can
Figure 5.7
A representative snapshot of the serotonin1A receptor embedded in a
1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine/cholesterol bilayer, highlighting closely associated cholesterol molecules. The cholesterol
molecule at the CRAC site (shown in cyan) of the receptor is shown in
red. The remaining cholesterol molecules are shown in dark salmon. The
phospholipid headgroups are represented in orange, the acyl chains in
gray, and the receptor in ice-blue. The surrounding water has been
omitted for clarity. The figure is based on results from ref. 89.
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enhance the receptor activity. Although several facets of GPCR function have
been elucidated by atomistic simulations, the molecular details of how the
membrane tunes activation are not known.
In addition, lipid effects have been demonstrated in ion channels and
other related membrane proteins using atomistic simulations. The stability
of gramicidin A in lipid bilayers has been successfully probed with atomistic
simulations. The effect of hydrophobic mismatch has been elucidated and
shown to affect peptide orientation and function.93 It has been shown that,
although the channel conformation of gramicidin A is the most stable
structure, it is possible for gramicidin A to change from channel to nonchannel conformation, depending on the local environment of the host
bilayers. In more complex ion channels, such as in Kir2.2, an inwardly rectifying potassium channel, phosphatidylinositol 4,5-bisphosphate binding
sites have been predicted from multi-scale simulations that show good
agreement with experimental results.94 Additionally, in the potassium
channel KcsA, several nonannular lipids close to the channel’s selectivity
filter were identified from simulations that appear to have a functional
role.95 Atomistic simulations therefore have been able to probe membrane–
protein interactions in several membrane proteins at the sub-ms time scale.
5.6.3
Coarse-grain Methods to Analyze Membrane Protein
Interactions
Recent improvements in coarse-grain methods have made it possible
to analyze the ms time scale association of membrane components with
embedded proteins. The association of cholesterol with membrane proteins
in general, and GPCRs in particular, is currently being extensively explored
by coarse-grain simulations.41 One of the first studies was able to identify
several interaction sites on the serotonin1A receptor that are reminiscent
of high occupancy sites.96 Additionally, the cholesterol interaction sites
identified in coarse-grain simulations were similar to those identified in
atomistic simulations.97 Taken together, the atomistic and coarse-grain
simulations have been able to suggest a general picture of cholesterol
interaction sites, although no consensus model exists. They can be thought
to represent hot-spots instead of binding sites that exhibit ms time scale
lifetimes and fast exchange with bulk lipids. Phospholipid interaction sites
have been identified in the b2-adrenergic receptor98 that are at the same
site as that observed in atomistic simulations of a related receptor, the
serotonin1A receptor.89 Coarse-grain simulations are emerging as an important tool to analyze lipid–protein interactions in GPCRs.
An important contribution of coarse-grain simulation studies has been the
elucidation of GPCR association. One of the first studies, focusing on
rhodopsin organization in bilayers of varying thickness, suggested that
receptor association is influenced by bilayer perturbations around the
receptor.99 A similar trend was reported for b1- and b2-adrenergic receptors, in
which the mismatch was observed to correlate with receptor association.100
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The influence of bilayer perturbations on receptor association can be
considered to be an indirect or non-specific membrane effect. On the other
hand, direct effects arising from the cholesterol interaction sites have been
reported.97 In this study, helices with a mismatch were observed at the dimer
interface, similar to those on an earlier report.100 However, the population
analysis did not completely match the extent of the mismatch, i.e., the
transmembrane helices with the maximum mismatch were not necessarily
those maximally observed at the dimer interface.98 These studies appear to
suggest that the driving forces for GPCR association are much more complex
than just hydrophobic mismatch or cholesterol occupancy. Both direct and
indirect membrane effects contribute toward the association of receptors.
Coarse-grain simulations have become a preferred choice for analyzing
several aspects of membrane proteins. For example, a combined coarsegrain and atomistic simulation study has demonstrated the conserved
interactions of lipids across the aquaporin family.101 These authors were
able to show that sites of phospholipid interactions matched well with the
electron density observed in two dimensional crystals. Interestingly, similar
to GPCRs, the sites of interaction were suggested to be dynamic and exhibit
fast interchange with bulk lipids. Due to the limitations of the coarse-grain
force-field in incorporating electrostatics, detailed electrostatics simulations
with ion channels are limited. A recent study on transient receptor potential
(TRP) channels suggested that channel gating is robust to lipid perturbation.102 Further studies are necessary to analyze the details of these
interactions.
The interaction of surface-bound and lipid-anchored proteins has been
elucidated with coarse-grain simulations. Due to the longer time scales of
the simulations, an improved sampling of the lipid interactions has been
achieved, which allows a close comparison with experiments. In simulations
of caveolin-1, a protein involved in endocytosis, modulation of protein
orientation and association was observed in cholesterol-rich bilayers.103
A direct comparison with experiments regarding the depth of association in
different bilayers was made and a good correlation was observed. Interestingly, a specific interaction between cholesterol and the protein was not
observed, despite previous speculations about cholesterol-binding sites on the
protein. Similarly, despite a clear dependence on charged lipid types, a direct
lipid interaction has not been observed in lipid-anchored proteins such as
LC3.104 Additional studies are required to delineate the role of specific and
non-specific effects.
5.6.4
Enhanced Sampling Methods
With current computational resources, it is difficult to probe phenomena
occurring beyond the ms time scale with adequate sampling. Enhanced
sampling methods help to overcome this limitation by biasing the simulations toward a particular path or state. Such methods include umbrella
sampling, force pulling and accelerated molecular dynamics approaches.
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Although unbiased molecular dynamics simulations can better represent the
equilibrium evolution of the system without any external bias (potential/
force) or reaction coordinate, they are limited by the phase space they
sample. Biased simulations help to improve sampling, and with careful
analysis can be used to estimate the underlying unbiased true energy
landscape. In the case of membrane proteins, umbrella sampling has been
employed to calculate a potential of mean force (PMF) of GPCR association
along a given reaction coordinate (inter-helical distance).105 A 1D PMF was
calculated for only limited dimer interfaces (e.g., a 1/8 helix interface) and
the sampling of the other dimer interfaces was absent. Importantly, no
lipid–protein interactions were discerned and the membrane composition
dependent association is yet to be elucidated.
5.7 Future Perspectives: The Road Ahead
In this article, we have highlighted representative experimental and computational approaches to address contemporary questions in membrane
biophysics related to membrane organization and lipid–protein interactions.
These are exciting times for membrane researchers since we are able to use
complimentary approaches to address membrane problems at a spatiotemporal resolution that was not possible to achieve even a few years back.
For example, recent advances in spatiotemporal resolution106 and computational power107 would enable us to address membrane spatiotemporal
heterogeneity with robust experimental measurements and simulation. We
envision that the knowledge generated using these tools on membrane organization and interactions will provide novel insight in understanding
membrane phenomena in healthy and diseased states.
Acknowledgements
D. S. and A. C. gratefully acknowledge the support of Ramalingaswami
Fellowship from the Department of Biotechnology, and J. C. Bose Fellowship
from the Department of Science and Technology, Govt. of India, respectively.
G. A. K. and X. P. thank the Council of Scientific and Industrial Research
(Govt. of India) and the University Grants Commission (India) for the award
of Senior Research Fellowships. A. C. is an Adjunct Professor of the Tata
Institute of Fundamental Research (Mumbai), RMIT University (Melbourne,
Australia), Indian Institute of Technology (Kanpur), and Indian Institute
of Science Education and Research (Mohali). We thank members of the
Chattopadhyay laboratory for their comments and discussions.
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00161
CHAPTER 6
Computer Simulation of Ion
Channels
BEN CORRY
Research School of Biology, Australian National University, Canberra
ACT 2601, Australia
Email: ben.corry@anu.edu.au
6.1 Introduction to Ion Channels
The movement of charged and polar solutes across the cell membrane is of
critical importance in a wide range of biological phenomena including
electrical signalling, the adsorption of nutrients, regulation of cell volume,
the transduction of sensory and chemical input into electrical signals and
the conversion of electrical signals into cellular responses such as neurotransmitter release and muscle contraction. The nerve impulse, for example,
is generated by the electrical changes created by the influx of sodium and
delayed efflux of potassium ions through selective pathways across the
membrane. Because such solutes do not like to enter the hydrophobic interior of lipid bilayer membranes, protein mediated pathways are required to
allow for the rapid movement of charged and polar molecules across the
membrane. These proteins can be divided into groups that either (i) facilitate the passive movement of solutes down the electrochemical gradient, or
(ii) actively move them against their electrochemical gradient by utilising an
energy source such as the hydrolysis of ATP or the coupled downhill
movement of another solute.
Ion channels are integral membrane proteins that provide the primary
route for passive transport by forming a water filled pore for polar and
RSC Theoretical and Computational Chemistry Series No. 10
Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
Published by the Royal Society of Chemistry, www.rsc.org
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Chapter 6
charged compounds to cross the cell membrane away from the hydrophobic
core of the lipid bilayer. Most channels also contain gates that can open and
close the pore in response to external stimuli allowing ion movement to be
carefully controlled. In contrast to active transport, the motion of ions down
electrochemical gradients can be extremely fast. Even a narrow pore can
move 107 ions across the membrane every second, as ion transport in an
open channel does not require large conformational changes of the protein.
In contrast, active transporters must cycle through distinct protein conformations to move ions yielding a smaller transport rate.
There are many different types of ion channels, as these proteins carry out a
very large range of functions. Channels differ in their localisation within the
organism, when they open and close the types of ions that they will pass and
the rate at which they do so. Channels are commonly classified according to
the kind of stimulus that leads to gating. Ligand-gated channels, for example,
open in response to the binding of a small molecule, such as neurotransmitters or calcium ions. Prominent examples of ligand-gated channels
include the acetylcholine and glutamate receptors found at the neuromuscular
junction and synapses. Voltage-gated channels, on the other hand, open or
close in repose to changes in the electrical potential across the cell membrane.
Noteworthy examples of such channels include the voltage-gated sodium and
potassium channels involved in propagating electrical signals in excitable
cells, and voltage-gated calcium channels that provide a mechanism of converting electrical stimuli into cellular responses. Channels may also open in
response to mechanical forces, such as tension in the bilayer; the pull from
attached proteins, such as those involved in hearing or osmoregulation; or to
changes in pH, such as in acid sensing channels. Channels can also be classified according to the types of ions that can pass through, as many of these
proteins are highly selective allowing just one of the predominant physiological ion species to permeate. For this reason, we often speak of potassium
channels, sodium channels, calcium channels or chloride channels. Some
channels are less able to discriminate between ion species, such as transient
receptor potential (TRP) channels, meaning that it is possible to use the
classification of cation channels, anion channels and non-selective channels.
To help appreciate some of the roles and diversity of ion channels,
Figure 6.1 shows some of those present at the neuromuscular junction. The
initial nerve impulse (action potential) is primarily carried by the influx of
Na1 through voltage-gated sodium channels, which increases the electrostatic potential in the nerve cell and the delayed efflux of K1 through voltagegated potassium channels that brings the potential back to resting values.
The wave of increased potential is used to activate voltage-gated calcium
channels, allowing Ca21 into the cell, which initiates the release of neurotransmitters (in this case acetylcholine) from intracellular vesicles. The
neurotransmitter diffuses across the synapse (the space between the adjacent cells) and, upon binding to ligand-gated cation channels, opens a
pore inducing the influx of sodium and the increase in membrane potential
in the muscle cell. This is the signal to open voltage-gated calcium channels
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Computer Simulation of Ion Channels
Figure 6.1
163
A schematic diagram showing some of the ion channels present at the
neuromuscular junction. Voltage-gated sodium and potassium channels
are responsible for conveying the action potential (nerve impulse) along
the nerve cell. Voltage-gated calcium channels are utilised to initiate the
release of neurotransmitter acetylcholine (Ach). Ligand-gated cation
channels then create a rise in the membrane potential of the muscle
cell, activating voltage dependent calcium channels. These prompt the
opening of calcium channels that allow the release of calcium from
intracellular calcium stores such as the sarcoplasmic reticulum (SR). The
intracellular calcium then stimulates the contraction of the muscle
fibres (MF).
in the muscle cell, located in narrow invaginations of the membrane known
as T-tubules. The influx of calcium and/or the conformational change in the
calcium channel protein is the signal to open ligand-gated calcium channels
in the membrane of the sarcosplasmic reticulum, which contains very high
concentrations of Ca21. The release of this calcium into the cytoplasm of the
muscle cell is the signal for muscle contraction.
Understanding the mechanisms by which channels work at the molecular
level is a fundamental problem in biophysics, because these are essential
building blocks of cellular function. In addition, a large range of diseases
including epilepsy, cystic fibrosis, cardiac arrhythmias, myasthenia and
migraine are associated with channel malfunction.1 For these reasons, ion
channels have been a focus of significant research ever since they were first
proposed in the 1950s.2
The field of ion channel research has entered a rapid phase of development over the past two decades, thanks largely to the determination of the
molecular structure of many channel forming proteins. The first potassium
channel structure was published in 1998 by the group of Rod Mackinnon
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Chapter 6
and, because much of how these channels worked was immediately visible
in the structure,3 this led to the award of a Nobel Prize in 2001. Gaining
structures of these large membrane proteins is not easy, and considerable
patience and expertise is required. The trickle of channel structures that
followed in the late 1990s and early 2000s is slowly increasing to a flood with
representatives from many of the major classes of channels now available
(see, for example, ref. 4–11). However, many of these structures come from
bacteria, so questions remain as to the similarity and differences of bacterial
and eukaryotic channels.
An example structure of a voltage-gated ion channel is shown in
Figure 6.2. When viewed looking down upon the plane of the membrane
(Figure 6.2A), a central pore can be seen created by four pore forming domains, surrounded by four voltage sensing domains. The voltage sensing
domains are comprised of four transmembrane helices (S1–S4), one of which
(S4) contains a number of charged residues that move in response to a
changing membrane potential. The pore forming domain contains two
transmembrane helices (named S5 and S6) connected by a loop that creates
a narrow part of the pore responsible for discriminating between ions,
known as the selectivity filter (Figure 6.2B). Although the basic architecture
is similar across all voltage-gated cation channels, there are many differences between members of the family. Voltage-gated potassium channels
and bacterial voltage-gated sodium channels are composed of four separate
protein subunits, as shown by the different colours in Figure 6.2A.
Eukaryotic sodium and calcium channels, on the other hand, are generated
by one long protein chain containing four homologous domains. As one
might expect, the selectivity filters are also different in these families, as
required to allow the passage of the different ion types.
In potassium channels, the filter is long and narrow, with the backbone
carbonyl atoms of the highly conserved TVGYG amino acid sequence lining
the pore, as seen in Figure 6.2C. Amino acid side chains, including a ring of
four glutamate residues, line the shorter and wider filter of bacterial sodium
channels (Figure 6.2D). Calcium channels are also known to contain a ring
of glutamate residues; however, eukaryotic sodium channels have a different
amino acid sequence in this region.
The emergence of atomic resolution ion channel structures has also
allowed for detailed molecular simulations to be conducted relating the
conformations of these channels to their function. This chapter aims to
introduce how molecular simulation has been used to tackle some of the
major questions in ion channel biophysics. We start by listing some of the
most critical and often studied questions about ion channels that simulations can hope to address. Because the computational approach used to
tackle these questions is dependent on the timescale over which the relevant
process takes place, we examine the timescales involved in each. We then go
through examples of published studies, aimed at addressing one or more
of these issues to highlight the utility of simulation in gaining biological
insight into these essential building blocks of living organisms.
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Computer Simulation of Ion Channels
Figure 6.2
165
Structure of a voltage-gated cation channel. (A) The structure of a
bacterial voltage-gated sodium channel, NavAb (accession code 3RVY4)
is shown looking down onto the plane of the membrane with
each subunit in a different colour. (B) Two of the four pore-forming
domains are shown viewed from within the plane of the membrane.
(C) The selectivity filter of a potassium channel with pore-lining
backbone-carbonyl atoms shown. Positions at which ions bind are
shown by gold spheres. (D) Structure of a bacterial sodium channel
selectivity filter. Pore-lining side chains and backbone atoms are
shown. The likely positions at which sodium ions dwell is indicated
by yellow spheres.
6.2 Questions that can be Addressed and Associated
Timescales
The brief introduction to ion channels given above raises some immediate
questions about channel function that simulations may help to address.
For example, how do ions move through the protein? Do they simply diffuse through the pore, or are ion–protein interactions essential to regulate
ion flow? How do selective channels discriminate between ions and, more
specifically, how can they do so while still passing ions at such large rates?
How does a stimulus such as ligand binding or a voltage change lead to
opening and closing of the channels? What is the nature of the pore gate
itself? How do disease-causing mutations alter the behaviour of the
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166
Figure 6.3
Chapter 6
Timescales of protein motion and biological processes. Motions occurring in a range of proteins are shown at the bottom, while dynamic
phenomena specific to ion channels are shown in the grey region.
channels? How and where do drugs, toxins or accessory proteins interact
with the channel, and how does their presence affect channel function?
Can the properties of biological channels be mimicked in synthetic
materials?
Before trying to address these questions, it is important to consider the
timescales over which the physiological behaviour of channels takes place
and to compare this to the timescales accessible in simulations, as this will
dictate the methods that can be used to answer them. Molecular motions
and biologically relevant phenomena take place over a very wide range of
timescales, as illustrated in Figure 6.3. The fastest motions are the vibrations of covalent bonds, especially those to light atoms, which occur in
the fs to ps timescale. The rotation of amino acid side chains is also quite
fast, typically arising every 1–100 ps, although the ability of side chains to
move in this way is heavily dependent on their environment. Flexible
protein loops oscillate in the 1–100 ns regime. While these fast motions are
not typically associated with functioning of ion channels, they are very
important for both the simulation methodology and for dictating the
overall protein conformation. As has been noted earlier in this book, fast
motions dictate the time step required in molecular simulations; otherwise
the atoms have the chance to move to unrealistic positions during each
iterative step. All-atom molecular dynamics (MD) simulations therefore
require time steps of around 1 fs. Coarse grained approaches that do not
include covalent bond vibrations, such as Brownian dynamics (BD) simulations can get away with longer time steps, usually in the order of ps. Small
scale fluctuations can dictate the magnitude of local interactions, and have
been shown to be important in influencing the steps involved in ion
transport in channels12 and the mechanism of selective ion binding.13
Because local vibrations result in a large number of possible atomic configurations, they are also critical to describing the entropy of the system,14
and this fact can make local motions one of the primary factors dictating
the overall protein structure, as well as the probability of transitions to
nearby conformational states.
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Computer Simulation of Ion Channels
167
The movement of molecules between kinetically distinct states separated
by energy barriers of several kT gives rise to longer timescale motions taking
place over the range of 100 ns or longer. Typically, these involve the collective
motion of groups of atoms between a small number of states, giving rise
to major structural changes responsible for many of the most interesting
biological processes, such as enzyme catalysis, the movement between
functional states of proteins, the working of molecular machines, protein
folding and protein–protein interactions.
Ion channels display dynamics over the whole range of length scales (grey
region in Figure 6.3). Channels typically carry currents of 1–100 pA, meaning
that ions pass through the channels at a rate of 1 per 10 ns–1 ms. The
opening and closing of channels in response to stimuli is slower, as it requires the collective motion of protein domains. Such motion occurs on the
millisecond time scale. Some channels are also known to inactivate, meaning that the pore becomes non-conductive to ions even though the stimulus
prompting them to open remains present. Channel inactivation is about an
order of magnitude slower than gating, allowing the channels to conduct for
a short period before inactivating. Rates for ligands or drugs to associate/
dissociate with the channels are generally in the ms regime, while the trafficking and insertion of the proteins into the membrane can be expected to
occur over seconds to minutes. Finally, many channels rely on interactions
with other proteins to function, and this can occur over a very large range of
timescales, from milliseconds to seconds.
The duration for which MD simulations can be run depends upon the
number of atoms being included. For this reason, early MD simulations
focused on small proteins to reduce the computational load. Simulations of
ion channels, however, typically require large system sizes, as these are large
proteins and need to be placed in a realistic environment of lipids, water
and ions. A typical ion channel simulation system, for example, might include about 10 000 protein atoms, 15 000 lipid atoms (roughly a 70 70 Å
membrane patch), 20 000 water atoms and a small number of ions: a total of
more than 45 000 atoms.15 If a large protein such as a voltage-gated channel
with voltage sensors is being simulated, the total atom count can easily reach
more than 150 000.16
While the focus of this chapter is on all-atom molecular dynamics, there
are other theoretical or simulation approaches that can be used to study
channels, as illustrated in Figure 6.4. These include methods such as coarse
grained MD, BD, rate models, and continuum models such as Poisson–
Boltzmann (PB) and Poisson–Nernst–Planck (PNP) calculations, explained in
detail in previous chapters.
In the continuum models, none of the atoms are represented explicitly.
Instead, the ions are represented as a continuous distribution while the
protein and water are replaced by dielectric media, as depicted in
Figure 6.4A. Specific fixed charges representing charged or polar portions
of the protein can be included. The electrostatic potential of the system
can be determined using the PB equation. The steady state current in a
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Figure 6.4
Chapter 6
Comparisons of continuum (a), BD (b) and all-atom MD (c) methods
applied to the system of an a-hemolysin channel embedded in a lipid
bilayer membrane and surrounded by an electrolyte solution. In the
continuum Poisson–Nernst–Planck (PNP) model, the ions are described
as a continuous density, and the water, protein and membrane by
dielectric media. In the BD model, only ions are represented explicitly,
whereas all atoms are treated explicitly in the all-atom MD method.
Reprinted with permission from C. Maffeo, S. Bhattacharya, J. Yoo,
D. Wells and A. Aksimentiev, Chem. Rev., 2012, 112, 6250–6284.181
Copyright (2012) American Chemical Society.
non-equilibrium system can be derived with the PNP equations, which
couple the determination of the electrostatic potential from Poisson’s
equation with the drift-diffusion of ions in the Nernst–Planck equations.
Both these approaches have provided invaluable information about ion
permeation and selectivity;17–27 however, questions remain as to the applicability of these mean field approaches in confined spaces.28–31
In Brownian dynamics, some of the degrees of freedom are projected out
of the simulation, reducing the number of calculations needed. For ion
channels, this typically means including the ions explicitly in the simulation, representing the solvent, lipid and protein in average terms as depicted in Figure 6.4B (although there are examples in which some protein
atoms are represented explicitly32). Since Brownian dynamics simulations
of ion channels will usually only follow the motion of the ions and not the
protein, lipid or water atoms, they have a relatively low computational
demand. With this approach, it is relatively easy to simulate for many ms
such that a large number of ion permeation events can be witnessed.
By their nature, such simulations cannot readily be used to follow conformational changes of the protein, such as in channel gating, however
there are examples of BD simulations examining the binding of small
ligands to channels.33 There also has been little effort at making use of
parallel computational architecture within BD, so there is scope to reach
much longer timescales, although the popularity of the approach is lessened by the lack of generally accessible codes and the rise of long MD
simulations.
We now review how molecular simulation have been used to address the
mechanisms of ion permeation, ion selectivity, channel gating and the
binding of drugs and toxins to channels.
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6.3 Ion Permeation
Simulating the passage of ions through a channel may sound like one of
the simplest problems that can be tackled, but in practice it is surprisingly
difficult. Because the timescale of ion permeation is in the order of B1 ms,
direct MD simulation of such events has only been possible in the last few
years. Such simulations also require access to open state structures of
channels, something that is only recently available for a few channel types.
Prior to this, simulations tended to focus on simplified channel structures to
understand the steps involved in ion passage or utilised non-atomistic
simulation methodologies.
For example, because the use of Brownian dynamics allows much longer
simulation times to be reached than in MD, ionic currents were determined
for a range of channels in many conditions long before this was possible
with MD (for example see the studies of potassium, calcium and sodium
channels by Chung and colleagues34–36). While many of these studies used
simplified channel models or rigid proteins and simplified electrostatic
interactions, more recent versions of this approach have used prior molecular dynamics simulations to determine the average forces on an ion at
each position, which are utilised in the Brownian dynamics simulation,37–40
or have directly included limited degrees of protein flexibility in the model.32
Continuum diffusion theory can also be used to approximately link the
forces on ions found from detailed simulations to channel currents using an
additional layer of approximation.30,37,41,42
In principle, BD simulations or diffusion models can be used to address
many of the same questions about ion permeation that could be tackled with
MD. While the extra levels of approximation are often seen as a disadvantage, the added simplicity can also help in finding the causes of transport
phenomena without being distracted by large quantities of data. For example, such models have been effective at understanding the reasons for
current saturation with increased concentration,34,35 the steps in knock on
conduction,34 the relationship between the number of permeating ions and
the so called flux ratio exponent,43 the basis of cation/anion44,45 or Na1/Ca21
selectivity,35 which available structure best reproduces the experimental
data,46 or the change in current passing through a channel during partial
blockage by DNA.47 As an example, in Figure 6.5A and B we show how
Brownian dynamics simulations on a simplified potassium channel model
can reproduce (and explain) the current saturation seen with the increasing
ion concentration, and how the width of the open channel gate can modify
the channel current.
The first direct all-atom MD simulation of multiple ion permeation events
involved a simplified model channel composed of rings of fixed atoms under
a 1.1 V driving force and lasted 100 ns.48 (Note previous short simulations
that observed a single permeation event.49) This model allowed the authors
to see the permeation of 12 Na1 ions, a current of the same order of magnitude seen in biological pores. An alternative to using simple structures is
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Figure 6.5
Simulations of ion currents. BD simulations utilising a simplified model
of a potassium channel (A) can reproduce the current concentration
curves seen experimentally (B). Results are shown for a range of channels
with differing widths of the channel gate, indicating how small structural changes can alter the channel currents. Reprinted from S.-H.
Chung et al., Modeling Diverse Range of Potassium Channels with
Brownian Dynamics, Biophys. J., 83, 263–277, Copyright 2002 with
permission from Elsevier.182 (C) Currents in KV1.2/2.1 determined
from direct MD simulations and compared to experimental data. The
different coloured data points refer to a range of different simulation
parameters. r2014 Jensen et al. J. Cell Biol., 141(5), 619–632.58
doi:10.1085/jgp.201210820.
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to make use of high conductance channels which allow permeation to be
seen in a shorter time. Among the first examples of simulations of solutes
through all-atom protein models of channels, it was actually in aquaporin
channels in which natural water conduction occurs at a rate of approximately three molecules per ns.50–52 In this case, simulations of membraneembedded channels lasting 10 to 12 nanoseconds were sufficient to see
multiple permeation events to determine the steps involved in water translocation and to suggest a mechanism of water versus proton selection.52,53
There are now many publications describing the direct MD simulation of
ion permeation events. The first to show that atomistic simulations could
accurately determine channel conductance for ions examined the a-hemolysin
channels.54 This is a toxin produced by Staphylococcus aureus that is secreted
as monomers but assembles into a homoheptomeric membrane channel (see
Figure 6.4) that can destroy solute, ionic and electric gradients across the host
membrane. The pore carries relatively large currents (conductance in the
order of 1 nS),55 allowing for multiple permeation events to be witnessed in
simulations lasting tens of ns, although the protein itself is large, meaning
that a large computational effort was still required to achieve these results.
Low conductance channels have always been more difficult targets, but
the mechanisms of permeation are often more interesting. An early study of
a potassium channel witnessed only three permeation events in 50 ns with a
large driving force (1 V),56 although this was sufficient to reinforce the notion
of knock on ion conduction. In contrast, recent studies that involve many ms
and hundreds of conduction events have been used to determine current–
voltage (I–V) curves under physiological conditions (see Figure 6.5C).57
Some of the difficulties in directly simulating ion currents were highlighted in a recent study by the same authors.58 Making use of extensive
computational resources and more than 1 ms of simulation time, the ionic
current passing through a voltage-gated potassium channel and the simpler
gramicidin A channel was calculated over a range of voltages, allowing for a
direct comparison with one of the most fundamental experimentally measureable properties. As seen in Figure 6.5C, the simulated currents were
about 40 times lower than the equivalent (highly accurate) experimental
measurements on the potassium channel with a range of different simulation parameters. For gramicidin A, the current was 300 times less than the
experimental measurements. This was a disappointing result from the point
of view of being able to accurately reproduce experimental measurements.
The authors suggested that the most likely culprit for this is the accuracy of
the non-polarisable force fields, something that has been in simulators’
minds since the earliest simulations of proteins almost 40 years ago.59
Although this has demonstrated some of the limitations in trying to directly
simulate ion currents, such long MD simulations provide invaluable information on the microscopic behaviour of ions in these proteins, as well as
helping to understand how simulation force fields can be improved. The
ability to directly simulate I–V curves for comparison with experimental data,
as shown in Figure 6.5, is invaluable.
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In order to directly simulate ion currents, a driving force must be applied
to generate a net flux of ions through the channel. However, because of
the use of periodic boundary conditions, typical MD simulations cannot
easily generate a concentration gradient or charge imbalance across the
membrane. Thus, generating a transmembrane potential is not trivial.
The simplest and most commonly used technique to achieve this is to
apply a homogeneous electric field across the entire simulation, perpendicular to the plane of the membrane.48,49,60 The mobile ions in the
system and the periodic boundary generate a realistic electrostatic potential, and the method has been carefully tested.61,62 However, due to the use
of periodic boundaries, this approach cannot yield asymmetric ion concentrations without additional measures. Simple ion concentration gradients, but not transmembrane potentials, can be generated by applying one
way boundaries for specified ion species at the periodic boundary.63
A method that uses a non-periodic energy step at the edge of the simulation box to maintain a chemical potential difference across the aqueous
reservoirs has also been shown to be able to generate ionic gradients and
asymmetric salt concentrations while maintaining the conventional periodic boundary conditions.64 Alternatively, asymmetric ion distributions
can be obtained by including a vacuum layer to prevent the movement
of solutes from one reservoir to another65 or two membranes in the
simulation system.66 While the use of two membranes requires a larger
simulation system and thus a greater computational effort, some of this
cost can be retrieved through the improved statistics gained by studying
two pores simultaneously.
An alternative to using MD to directly simulate currents is to calculate the
energy landscape for ion transport and deduce the key steps in the transport
process, and potentially the channel current, from this. To achieve this, one
can use many short MD simulations rather than one or a few long simulations, such as in the commonly applied technique of umbrella sampling.67
The free energy map for multiple K1 ions passing through the KcsA potassium channel published by Berneche and Roux in 200168 provides an
excellent example of the energy landscape approach for understanding ion
permeation. Figure 6.6 shows the free energy map (or more correctly the
potential of mean force (PMF)) they found as a function of the position of
three ions in the pore. This shows the most likely configurations of the ions
as the free energy minima, as well as the most likely pathways between them.
Conduction was seen to proceed in a ‘knock on’ manner, alternating
between states with 2 and 3 ions in the selectivity filter separated by
water molecules. Notably, the barriers along this pathway are small
(2–3 kcal mol1), meaning the current will be large – in the order of magnitude expected for this channel. Recent potential of mean force and direct
simulation studies on potassium channels have found other possible landscapes and mechanisms of ion permeation, such as the ‘direct knock on’ of
adjacent ions without water molecules between them.42,69,70 A similar
energy landscape approach has been used to elucidate ion permeation in a
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Figure 6.6
173
Example free energy surface constructed to understand ion permeation
in the KcsA potassium channel. Each level of shading corresponds to an
energy of 1 kcal mol1. The axis coordinates represent the position of the
centre of mass of either a single ion (1 subscript) or a pair of ions (two
subscripts). The most likely conduction pathways are shown by dashed
lines and representative snapshots of the ions in the selectivity filter of
the channel corresponding to each minimum are shown.
Reproduced by permission from Macmillan Publishers Ltd: Nature,68
Copyright (2001).
sodium-selective channel.16 In this case, although permeation involves
knock on conduction, it can arise with two or three ions in the channel. The
wider pore also allows for greater independence in the motion of each ion
and the potential for ions to pass each other, yielding the so called ‘loosely
coupled knock on’ mechanism.
It is possible to determine currents from a free energy profile using the
diffusion theory approximation30,37,41,42 but, as yet, there has not been a
clear test of this approach through comparison to directly simulated currents from MD.
Advanced sampling methods continue to be developed that can be used to
study ion conduction and selectivity in channels. For example, automated
approaches now exist for selecting the regions of interest for umbrella
sampling simulations of ion conduction,71 and bias exchange metadynamics72 has been shown to be an efficient method for generating multiion free energy surfaces for understanding ion permeation and selectivity.73
The weighted ensemble method74 has also been shown to provide a very
efficient way to calculate ionic currents and I–V curves.75 All of these
methods provide exciting alternatives to brute force simulations.
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6.4 Ion Selectivity
As described in the introduction to this chapter, the ability to discriminate
between different ion types is essential to the function of many ion channels.
A large range of physical mechanisms can underlie ion selection but these
can be broadly categorised as either arising from the thermodynamic differences of ions binding to the protein, or the kinetic differences in the
passage of ions. The fact that the thermodynamics of ion binding plays a
large contribution to selectivity means that much has been learnt about this
subject from short simulations studying the location and interactions of
ions with the proteins without necessarily running long simulations to
capture permeation events. Uncovering the kinetic aspects of ion selectivity
is more difficult, as it requires simulations to capture the dynamic movement of multiple ions through the channel. Because the literature on ion
selectivity in channels is vast, it is impossible to cover its entirety here. Instead, we focus on a few examples that examine the selection between ions of
differing valence (Na1 and Ca21) or selection between the very similar Na1
and K1 ions in order to emphasise the different simulation approaches that
have been used.
6.4.1
Na1/Ca21 Selection
Na1 and Ca21 perform very different biological roles, thus it is essential that
calcium channels and sodium channels can distinguish between these ion
types. Voltage-gated calcium channels are extremely selective, choosing
calcium over sodium at a ratio of over 1000 : 1,76 yet the picoampere currents
they carry require over 1 million ions to pass every second.77 The degree of
specificity is important for these channels, as Na1 ions are typically orders of
magnitude more numerous than Ca21 in the extracellular space. Monovalent
ions conduct through the voltage-gated calcium channels at much higher
rates than divalent ions,76,78 but are blocked when the calcium concentration reaches only 1 mM.79,80 The ability of some ion species to block
currents carried by others gave the first clues to the origin of ion selectivity in
these channels.81 These properties were first explained with the so-called
‘sticky-pore’ hypothesis, in which ions that are bound with higher affinity
pass through the channel more slowly and so have a lower conductance.82 In
addition, the fact that the current is lower when two permeating ions are
mixed suggested that ion conduction must be a multi-ion process.78,83 The
nature of the sticky pore was elucidated by site-directed mutagenesis which
showed that the four glutamate residues on the P-loop of the channel (often
called the EEEE locus) were responsible for the high affinity of the calcium
binding site.84–88
Because we lack high resolution structures of eukaryotic voltage-gated
calcium channel selectivity filters, studies of Na1/Ca21 selection in this
family of proteins provide a great example of what can be learnt about
channel selectivity using simpler simulation models that do not include all
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protein atoms (note that we do now have a low resolution structure and a
structure of a calcium selective mutant bacterial channel90). Two interesting modelling approaches were used to connect the calcium channel
structure to its selectivity without doing an all-atom MD simulation. The
first of these used Monte Carlo simulations or a Mean Spherical Approximation method to calculate the energetics of placing a different number of
ions in a charged pore. The glutamate residues were represented by eight
partially charged oxygens which could freely diffuse within a limited region
of the channel, as illustrated in Figure 6.7A.91–93 Cation binding is achieved
due to the electrostatic attraction of the negatively charged oxygen atoms.
The selection of Ca21 ions is proposed to be the result of ions competing
to achieve charge neutrality in a selectivity filter having finite space. Ca21
ions are preferred to Na1 in the model as they have the same chargeneutralising effect as two Na1 ions while occupying less of the limited
volume of the filter. The charge versus volume explanation for selectivity
has been extended to explain the selectivity between a large range
of monovalent and divalent ions in a range of channels including the
ryanodine receptor.94,95
An alternative approach is to use Brownian dynamics simulations to follow the trajectories of ions as they pass through a simple model of the
channel and to observe how the ions interact with the protein and each
other. In these studies a rigid model of the channel was derived from analogy to other channels and a variety of experimental data.35,45,96 The channel
shape includes a relatively narrow region in which ions cannot pass each
other, surrounded by four glutamate residues, as shown in Figure 6.7B.
Unlike the previous models, the glutamate charges do not compete for space
within the pore, but the concentration of negative charge does attract cations
within this region. In this model, the electrostatic attraction of the protein is
all that is required to account for ion permeation and selectivity. The charge
of the glutamate residues attracts multiple ions and, in a process akin to that
suggested by earlier rate models,78,83 the repulsion between two resident
Ca21 ions is found to speed their exit. In this model, selectivity arises from
the fact that divalent Ca21 ions are more strongly attracted by the channel,
and thus they can displace Na1 to occupy this region. But once there, the
Ca21 can only be moved by the repulsion from another divalent ion and not
by the lesser repulsion from Na1 (Figure 6.7B).
Both these simplified simulation models are able to provide a plausible
explanation of ion selectivity that may have been difficult to extract from
more detailed all-atom trajectories, but both also contain a degree of
approximation (e.g. completely mobile oxygen atoms, or a rigid protein).
The mechanisms of selectivity are related, and both can explain a range of
experimental data such as I–V curves, current concentration curves, the
anomalous mole fraction effect, and the attenuation of Ca21 currents by
Na1. The ‘electrostatic’ explanation is less specific about the exact size of
the pore than the ‘charge–volume’ explanation, while the ‘charge–volume’
theory differentiates more easily between ions of the same charge but
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Figure 6.7
Simple models of Ca21/Na1 selectivity. (A) A model of a calcium channel
selectivity filter used to demonstrate the charge–volume competition
model of selectivity. Mobile oxygen atoms representing glutamate side
chains compete with permeating ions for space in the filter. Reprinted
from W. Nonner et al., Binding and Selectivity in L-Type Calcium
Channels: A Mean Spherical Approximation, Biophys. J., 79, 1976–1992,
Copyright 2001 with permission from Elsevier.92 (B) Brownian dynamics
model. The shape of the channel is shown, with squares representing
negative charges from glutamate residues and diamonds electrostatic
dipoles. Ca21 permeation involves knock on conduction between 2 ions.
Na1 cannot displace a resident Ca21 meaning that only divalent ions can
permeate once one enters the pore.
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differing size. As we gain more structural information, we will be in a better
position to understand which aspects of these models are correct.
There have been some efforts to use all-atom models of calcium channels
to help understand selectivity, but the lack of structural data has meant that
they either rely on homology models97,98 or simplified pore models.99,100 In
contrast, the recent availability of X-ray structures of bacterial sodium
channels has allowed for the preference of Na1 over Ca21 in these channels
to be addressed by all-atom methods. For example, Corry used multi-ion
potential of mean force calculations to show that the preference for Na1 in
the sodium channel from Arcobacter butzleri (NavAb) arises from the fact that
Na1 can bypass resident Ca21 ions and that Ca21 faces an unfavourable
dehydration penalty to pass through the internal section of the selectivity
filter.101 There is, however, still much to be explained in this field. For example, how does the ring of glutamates create sodium selectivity in bacterial
sodium channels, yet calcium selectivity in calcium channels? How does
Na1/Ca21 selection arise in eukaryotic sodium channels that have a different
amino acid sequence? The determination of the structures of calcium
channels and eukaryotic sodium channels will help to address these
questions.
6.4.2
Na1/K1 Selection
Na1 and K1 are both spherical monovalent ions with similar radii (B0.95
and 1.33 Å, respectively) and so it is not straightforward to understand how
ion channels can distinguish between them. The simple models described
above, such as Brownian dynamics, have difficulty in capturing the subtle
differences between the ions and, while it is possible to see size dependent
effects, there is a greater emphasis in using all-atom models that explicitly
include hydration energy differences for the ions when studying Na1/K1
selectivity. In addition, the availability of structures for potassium channels
since 19983 and sodium channels since 20114 has meant that there are a
great number of all-atom simulations examining the differentiation between
Na1 and K1. We examine some of the different methods used and results
obtained here.
One way to start elucidating how an ion channel discriminates between
ions is to simply run two short simulations, each containing one of the ion
types. By examining where the ion binds to the protein and the specific
interactions involved, it may be possible to spot factors that influence the
thermodynamics of binding. For example, studies appearing soon after the
publication of the first potassium channel structure were able to highlight
likely differences in ion hydration for Na1 and K1 in the pore as the origin of
ion selectivity.102,103 The success of this approach requires the simulations
to be able to predict the most likely binding locations (or for these to be
known) and to be able to sample all the protein conformations consistent
with this ion binding position in order to work out realistic structures and
interaction energies. Furthermore, it is desirable to be able to predict
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selectivity ratios from the simulations in order to compare the results with
experimental measurements. As this cannot be done from short equilibrium
simulations, more complex approaches have generally been employed.
The most common approaches to examine the free energy difference for
the binding of different ion types in a channel (or any protein) are the
methods of alchemical free energy perturbation (FEP)104 and thermodynamic integration.105 In these, one group of atoms slowly disappears
during the simulation while another group slowly appears. In the context of
selectivity, this simply means that Na1 is slowly replaced with K1 within a
binding site in the channel during the simulation or vice versa, and the free
energy change during this simulation is determined. By comparing this free
energy change with that associated with swapping the ions in bulk water, the
total free energy representing the selectivity of a specific binding site can
be estimated.106 Again, simulations have to be run long enough for the
environment of the ions (protein/water/lipid) to sample all available conformations to ensure that the final free energy value converges to a global
minimum. Not only is this difficult to achieve, it is also difficult to demonstrate that adequate sampling has been obtained. The protein may
undergo a slow conformational change as the ions are swapped and it can be
difficult to run the simulation for long enough to allow this to occur.
Although statistical tests can be applied to ensure statistical convergence,
these will not be able to tell you if there is a slow conformational change that
has not been captured within the timescale of the simulations. Further
complicating the issue is the fact that many channels hold multiple ions,
and the presence of additional ions can alter the selectivity of any given site.
Thus, a simulation must capture all realistic ion/protein configurations
during the free energy perturbation calculation to be able to capture all the
mechanistic details.
Early applications of the FEP approach towards understanding ion channel selectivity were conducted using less than 10 ns to perform the ion replacement.67,107,108 Although this may not be sufficient to capture any larger
structural changes occurring in the protein, it was enough to indicate which
positions and ion configurations in the crystallised structure of the pore
would be selective for K1 over Na1. Once locations that are selective for the
ions were identified, the interactions at that site were analysed to determine
what might cause selectivity. For example, it was shown that the greater
desolvation penalty for Na1 and better coordination of K1 in the pore could
generate a preference for K1 binding.108
To ensure better sampling one can simplify the system, an approach described as the use of ‘‘toy models’’. By including fewer atoms and removing
slow motions, accurate convergence can be ensured, and the ability to gain
many results with different parameters enables a range of situations to be
surveyed to see which yield selectivity. One of the first examples of this
method employed a simple model of one of the ion binding sites in the
potassium channel selectivity filter, as shown in Figure 6.8A, to highlight the
critical importance of the interactions between the ion and backbone
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carbonyl atoms in generating selectivity.
It was shown that, even in a
flexible pore, the specific magnitude of the ion–carbonyl electrostatic
interaction can create a preference for K1, as the balance of attractive ion–
carbonyl and repulsive carbonyl–carbonyl interactions differ as the ion size
changes.
The toy model concept was extended by many groups, which has enabled
the selectivity to be calculated as a function of a range of parameters such as
the magnitude of charge on the coordinating protein atoms, the number of
coordinating atoms in the site, and the presence or absence of water molecules as shown in Figure 6.8B.110–114 Such toy models also allow for
quantum mechanical or density functional theory calculations to be conducted110,115,116 to include a greater level of detail and avoid some of the
difficulties in ion parameterisation that could influence the classical studies.
The toy model approach has been a great success in mapping out the factors
that can create selective binding and for generating ways to rapidly predict
the selectivity of a given binding site. However, the removal of the more
distant parts of the protein means that this focuses on local interactions and
it is difficult to take into account the strain in the protein, conformational
changes, kinetic and multi-ion effects.
One way to take account of the entirety of the protein, different binding
positions, and multi-ion effects is to calculate the free energy landscape for
different ions in the pore in a similar way to that described for elucidating
ion permeation. By recreating maps such as those shown in Figure 6.6 for
multiple ion types, the location and size of energy minima and barriers for
each can be ascertained. Although this can be done for single ions in the
pore (as in the early examples of Allen et al.107), it is most informative in
multi-ion channels when multiple ions are included, although this does
require considerable computational effort. For example, Egwolf and Roux
constructed multi-ion free energy landscapes with either K1 in a potassium
channel pore or a mixture of K1 and Na1.117 This was a computationally
intensive task at the time, as the different coordinates of the three ions in the
pore meant that almost 4000 unique configurations of ion positions had to
be simulated, each for 0.2 ns, to generate each free energy surface. The
outcome, however, shows the location of each free energy minimum, the
selectivity of each one, and the height of the barriers between them. In this
case, it was found that one ion binding site is more selective than the others,
and that the barriers between the minima are greater for Na1 than K1.
A combined approach of FEP and free energy landscapes has also been recently employed to highlight the importance of kinetic barriers between
multi-ion states in generating ion selection in potassium channels.118
More recently, free energy landscapes have been used to highlight the
location at which ion selectivity arises in a voltage-gated sodium channel,16,119
which is known to display about a 20-fold preference for Na1 over K1. These
studies, similar to that of Egwolf and Roux, created free energy landscapes
with either two Na1 in the pore, two K1 or a mixture of Na1 and K1. As shown
in Figure 6.9, with pure Na1, sodium ions can move between the energy
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Figure 6.8
Toy models of selectivity. (A) The toy model used by Noskov and Roux96
to mimic a binding site in the KcsA potassium channel. The ion (sphere)
is shown and sections from the four surrounding protein chains are
included in the model. Reproduced by permission from Macmillan
Publishers Ltd: Nature,109 Copyright (2004). (B) A free energy ‘map’
indicating how the selectivity of a binding site for Na1 or K1 is influenced by the number of coordinating ligands in the binding site and the
charge on these ligands. Regions on the plot are selective for Na1 and
those selective for K1 are indicated by two different shades. The degree
of selectivity is quantified by the difference in free energy to place each
ion in the site from bulk water as indicated by the contour lines.
Reproduced from M. Thomas et al., Mapping the Importance of Four
Factors in Creating Monovalent Ion Selectivity in Biological Molecules,
Biophys. J., 100, 60–69, Copyright 2011 with permission from Elsevier.111
minima with only small barriers of height B2 kcal mol1 (roughly the
accuracy of the simulations themselves). However, when K1 is present, the
height of the largest barrier to conduction increases dramatically, indicating
that discrimination between the ions is likely to arise at the location of this
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Figure 6.9
181
Example free energy surface used to understand ion permeation and
selectivity in a voltage-gated sodium channel. The PMF is plotted as a
function of the axial position of each of two ions assuming (A) both ions
are Na1, and (B) the inner ion is K1 and the outer ion is Na1. Energy
values are shown in the scale bar in units of kcal mol1 and the contour
interval is 1 kcal mol1. The steps involved in conduction and their
positions on the 2-ion PMF are shown on the left.
Reprinted with permission from B. Corry and M. Thomas, J. Am. Chem.
Soc., 2012, 134, 1840–1846.16 Copyright (2012) American Chemical
Society.
energy barrier. Further work then focused on this location to show how
differing hydration geometries created by the narrow charged pore could
generate the differing energy barrier for Na1 and K1.16
6.5 Channel Gating
As the opening and closing of ion channels takes place on the millisecond
time scale, it is difficult to simulate explicitly. Until recently, MD simulations
addressing gating had to extrapolate from local fluctuations in structure,
add additional forces or use another advanced sampling technique to bias
the channel towards a different functional state.
Mechanosensitive channels in the membranes of bacteria have been a
common target for studying channel gating via the application of addition
forces to promote conformational changes. We have atomic resolution
starting structures,5,120 and the physiological role of these proteins is to
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directly respond to mechanical forces. The best studied of these channels is
the family of mechanosensitive channels of large conductance from bacteria,
MscL, which open in response to membrane tension to rescue cells from
osmotic shock.121 After the first MscL structure was published in 1998, a
number of simulations were conducted in which either radial forces or
membrane pressure were applied to the protein to mimic the influence of
membrane tension.122–125 However, all these simulations were of relatively
short duration (r10 ns). Therefore, in order to observe conformational
changes within a feasible time frame, the magnitude of the added forces
were significantly greater than that found under physiological conditions.
An alternative approach has been to bias the structure using experimentally
determined restraints to force a conformation change.126,127 While this can
provide information about alternative conformations, in this case gaining a
possible open channel structure from the closed state crystal structure, it
cannot describe the pathway by which the protein moves from one conformation to another.
The construction of free energy surfaces, similar to those described previously for ion permeation and selectivity, can also be used to examine the
conformational changes involved in channel gating as a function of one or
more structural parameters of the protein. One of the challenges in this
approach is that it is difficult to select the most relevant structural parameters to restrain without prior knowledge. In addition, this requires a
large number of simulations restraining the protein around different values
of the structural parameter(s) and is therefore not computationally trivial.
However, the pathway between conformational states can, in principle, be
determined from this approach as well as the energetics of the process.
While there are not as yet examples in which the energy landscape for the
full structural change involved in gating has been elucidated, a recent study
by Fowler et al. took a step in this direction by finding the energy landscape
for opening the activation gate of a potassium channel.128 The free energy as
a function of the position of the pore-lining S6 helices indicates that, in the
absence of the surrounding portion of the protein responsible for sensing
membrane potential, the channel prefers to reside in an open conformation.
Presumably, work must be done on the pore-lining helices by the voltage
sensors to close the channel, otherwise the channel will spring open. This
study had a system containing 72 000 atoms simulated for 0.7 ms, but the full
conformational change in the presence of the voltage sensors was neither
mapped out, nor was the result assessed in the presence or absence of an
activating membrane potential, as this would require significantly more
atoms and more simulation time.
There are now a number of cases in which we have experimentally determined atomic resolution structures of a channel, or closely related
channels, in multiple functional states. Thus, it should be possible in
principle to use simulations to understand the pathway by which the protein
moves between these conformational states and to elucidate the conditions
that favour one state over another. This approach requires a clear knowledge
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of what state each structure represents, which is not always trivial to determine. For example, the family of ligand-gated ion channels are believed to
prevent ion conduction in the closed state without physically occluding the
pore using a hydrophobic gating mechanism.129–132 Thus, one cannot simply
infer that a continuous pore represents an open state of the channel. MD or
BD simulations that can predict the currents passing through experimentally
determined structures or the hydration states of the pores in such structures
can help to assign the likely functional state.133,134
A conceptually simple way to study the transition from one conformational state to another is to use targeted molecular dynamics simulations. In
these, subsets of atoms are guided toward their target positions via the application of a bias force on the root mean square deviation (RMSD) of the
atomic positions from the target values. For example, a simulation may start
from a closed structure, but progress toward an open state by forcing some
of the atoms toward their positions in an open state target structure. In
general, this is not a good way to see the pathway between the starting and
target states, as large changes will be biased to occur before small changes,
even if this is not what happens in reality. However, it can help to see how a
change in one part of the protein can alter the conformation of another. For
example, by forcing the ligand binding domain of a ligand-gated ion channel
from a resting to an activated state, it is possible to see how this might influence the structure of the pore forming domain.135
Alternatively, the crystal structures obtained in different conformational
states can be used to help choose suitable structural parameters for more
subtle biasing approaches, either to create free energy surfaces, or to speed
the conformational change using an approach such as metadynamics.136
To directly simulate the gating of a channel, one needs to start from a
known conformational state and provide a stimulus for the channel to
change state. Two studies used this approach to examine the first steps in
the gating of ligand-gated ion channels. One studied the bacterial pentameric ligand-gated ion channel from Gloeobacter violaceus, known as GLIC,
which is known to be sensitive to the pH. An open channel structure was
obtained in acidic conditions and then subjected to MD simulations after
instantaneously changing the protonation state of the ionisable residues to
that expected in neutral conditions.137 Rapid closure of the pore was seen
during the 1 ms simulations, indicative of the early steps in channel gating.
Another simulation study focussed on the eukaryotic glutamate-gated
chloride channel from Caenorhabditis elegans (GluCl), whose open channel
structure was determined complexed with the positive allosteric modulator
ivermectin.138 In these simulations, ivermectin was removed and partial
channel closure was seen during the following 150 ns. Although only one
simulation was conducted in each case, making it hard to be certain about
the reproducibility and significance of the conformational changes seen, the
twisting motions of the helical elements moved the channels closer to the
related structures of the closed state channels, yielding some confidence
that these motions represent real gating motions.
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Due to the impressive computational advances made by the D.E. Shaw
Research Institute, very long simulations have now been conducted that
capture the transition of a channel between open and closed states.139 To do
this, the crystallographically determined open structure of the Kv1.2/Kv2.1
potassium channel chimera was subjected to hyperpolarising voltages that
can be expected to induce the channel to close. As can be seen in Figure 6.10,
the initial structure contained a continuous pore through which water molecules could pass and which remained open under a depolarising membrane
potential. However, within 20 ms of applying the hyperpolarising potential, the
size of the pore rapidly shrunk and water molecules became excluded from
the narrow gate, indicative of the channel progressing to a closed state. Upon
reversing the membrane potential, the channel was seen to open again.
Analysis of the voltage sensors showed that the charged residues moved in
response to an electric field during simulations lasting 260 ms, and the
magnitude of this movement was in line with experimentally measured
‘gating charges’ (Figure 6.10B). Despite the magnitude of the membrane
potentials being very large, the uncertainty about the applicability of the
current force fields for such long simulations58,140–142 and the potential for
errors in the algorithms to propagate over time, these direct simulations of
channel gating allow for the possible molecular motion of the protein during
gating to be studied in a way not feasible with any other approach.
6.6 Interactions of Channels with Drugs and Toxins
Because ion channels are critical to so many aspects of human biology, small
molecules or peptides that alter the channel function can have significant
physiological effects. Indeed, a large number of commonly used drugs work
modulating the function of one or more channels and so there is a great deal
of interest in using simulations to better understand the mechanism of
action of these compounds. Channel modulators can work in a range of
ways, such as by blocking the ion conducting pore, locking the pore in an
open state, or altering the likelihood of activation or inactivation of the
channel. There is a vast literature examining protein–drug interactions and
so, here, we focus on just a couple of examples involving voltage-gated cation
channels to highlight the ways in which MD simulations have been used to
elucidate this topic. We refer the reader to some recent reviews for further
information about simulations of ligand–protein interactions.143–145
6.6.1
Toxin–Channel Interactions
A vast number of channel modifying peptides have evolved in nature in the
form of toxins in the venoms of poisonous creatures such as scorpions, cone
snails, sea anemones, spiders and snakes. Toxins affecting voltage-gated ion
channels can be broadly categorised into those that bind to the pore forming
domain of the channel, either directly blocking it (e.g. the sodium channel
blocker tetrodotoxin146) or holding it open (e.g. Batrachotoxin147), or those
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Figure 6.10
Example of MD simulations used to directly monitor channel gating. (A)
The number of water molecules occupying the narrow portion of a
Kv1.2/Kv2.1 potassium channel chimera is plotted versus time. Initially,
a depolarising voltage is applied to the system (blue region of graph),
which is then switched to a hyperpolarising voltage (red region). Rapid
dewetting of the channel occurs within 20 ms of switching the voltage,
indicative of pore closure. Intracellular views of activated (þV, conducting) and resting (V, non-conducting) states are also shown. (B)
Displacement of charges in the voltage-sensing region as a function of
time after switching to a hyperpolarising potential. The total gating
charge, 13.3 0.4 e, was estimated as the difference between the final
displacements at hyperpolarising and depolarising potentials.
From ref. 139. Reprinted with permission from AAAS.
that interact with the voltage sensor stabilising either the activated
(e.g. Scorpion b-toxins148,149) or resting conformation (e.g. hanatoxin150).
Because of the great interest in using these peptides themselves, or peptide
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Chapter 6
mimics for therapeutic use or as insecticides or pesticides, simulations have
been used to better characterise the peptide–protein interactions.
The same timescale limitations that make it difficult to directly simulate
channel gating also make it difficult to directly simulate the binding of a
peptide to the channel. To overcome this, most studies either use experimentally determined constraints to help define the binding site (such as
residues determined from site-directed mutagenesis) or resort to more
approximate docking approaches. Docking does not try to reproduce the dynamics of the protein or ligand, but rather aims to predict docked poses
representing complexes between the ligand and receptor. Because the goal is
to rapidly find and rank docked poses, the calculations use simplified interatom interactions or scoring schemes. Thus, there is often a considerable
degree of approximation and pragmatism employed, yielding an ongoing
discussion as to the utility of the predictions. Traditional docking methods use
a rigid protein and ligand; however, the inherent flexibility of peptides (and
the channel protein) means that approaches to include protein flexibility are
often employed.143 Despite the limitations, docking has been used on many
occasions to predict toxin–channel binding modes. Poses that do not match
known mutagenesis data can be excluded, or new mutagenesis experiments
are designed to differentiate between alternate poses. Binding poses can be
further refined by relaxing the system in MD simulations, as was done for the
potassium channel blockers MTx and ShK.151–153 Alternatively, docking can be
circumvented altogether by applying harmonic restraints directly in MD
simulations to maintain the distances between certain toxin–channel residue
pairs within certain ranges inferred from mutagenesis experiments.154,155
In order to compare the affinity of predicted binding modes to experimental measurements, it is essential to determine the free energy of binding
from the simulations. In addition, the binding free energy is the best computational approach to contrast the efficacy of binding by different ligands to
a channel protein, or to examine the selectivity of the ligand to different
channels. For example, Chen and Chung calculated the free energy of
binding of MTx to each of the voltage-gated potassium channels Kv1.1, Kv1.2
and Kv1.3. The magnitude of the free energy of binding to Kv1.2 is almost
twice that seen for binding to the other channels, consistent with the experimentally observed selectivity. The stronger binding was seen to arise
from the formation of a greater number of salt bridges with Kv1.2.
However, accurately predicting the binding affinity using simulations is
very difficult, especially for large peptide toxins for which long simulation
times are required to adequately sample all the binding modes and protein–
ligand conformations. Determining the binding free energy requires a
comparison of the energetics of the separated protein and ligand and the
bound ligand–protein complex. In general, this can be done in two ways: by
calculating the potential of mean force to bring the ligand from bulk to the
binding site, or by using an alchemical transformation in which the ligand
slowly appears or disappears from the binding site.41,156–160 The approach of
determining the potential of mean force has been most commonly applied
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to studies of ion–channel toxin interactions, allowing dissociation constants
to be determined for comparison with the experimental data.33,161–165
Because the energy values determined depend upon sampling all conformations of the peptide, protein and water along the entire length of the
pathway, such simulations are extremely computationally expensive, and
convergence to the true PMF over practical computational timescales is often
uncertain. Dissociation constants within an order of magnitude of experimental values have been determined with this method; however, the results
can be dependent upon the specific parameters used, such as restraining
potentials and simulation duration.
6.6.2
Channel Blockage by Small Molecules
Four recent studies examining the blockage of bacterial voltage-gated
sodium channels highlight approaches that can be used to study the interaction of small molecules with ion channels. Local anaesthetic compounds,
as well as many anti-arrythmic and anti-epileptic molecules, are known
to act by binding in the interior of these channels where they block ion
currents and/or stabilise the inactivated state of the channel. As sodium
channels are used to initiate action potentials, inhibitors will diminish the
nerve impulses to generate an anaesthetic effect. While we know from
mutagenesis studies that local anaesthetics bind in the interior of the
pore,166–168 the exact mode of binding and the path by which they reach this
site have remained elusive. Given that there is a strong desire to develop
compounds that can selectively target the different types of voltage-gated
sodium channels found in the body, there has been great interest in using
simulations to better understand the mechanisms of local anaesthetics since
the first bacterial sodium channel structures were published.4,169–173
Simulating the binding of small molecules to channels is slightly easier
than for toxins due to a smaller number of internal degrees of freedom in
these compounds. However, the fact that binding takes place over timescales
much longer than most simulations and accurate simulations still need to
sample many conformations of the protein makes it difficult to accurately
predict binding modes and affinities. Unless you have a large amount of
computer power and are lucky enough to see a rapid binding event, direct
simulation of physiological concentrations of drugs around channels is unlikely to yield significant results. However, given that local anaesthetics are
known to bind in the confined interior of the pore, results are much more
likely to be achieved by limiting the drugs to the small volume of the channel
interior. Indeed, simulations lasting 125 ns starting with the local anaesthetic
compounds benzocaine and phenytoin inside the pore were sufficient for the
drugs to find high affinity binding sites.15,174 Although it is possible that such
simulations may not capture all the important binding sites, they can be
repeated multiple times to ensure reproducibility. The affinity of the binding
sites found in this way was determined using free energy perturbation for
comparison with experimental data.
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Figure 6.11
Simulations of small molecule interactions with a voltage-gated sodium
channel. (A) Example of a system used for flooding simulations reprinted
with permission from Boiteux et al.175 Two of the four protein (Nav Ab)
subunits are shown as ribbons, surrounded by a lipid bilayer (grey lines)
and water (red/white sticks). Multiple copies of the drug phenytoin are
included in the simulation (dark grey sticks). (B) Potential of mean force
calculation for benzocaine (line with red region of error) and phenytoin
(line with blue region of error) to enter the interior of the pore via the lateral
fenestrations. The snapshots below show the position of the drug (cyan)
within the protein (grey surface) at points of interest along the profile.
Adapted with permission from ref. 15.
Rather than confining the search space to a localised region, an alternative
approach known as ‘flooding’ uses a high concentration of ligand to speed
up the binding process as shown in Figure 6.11A. Two recent studies
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employed this method to seek binding sites of anaesthetics on sodium
channels starting the simulations with the drugs in the aqueous phase.175,176
In both cases, drugs were found to be able to make their own way into the
interior of the pore and potential binding sites could thus be identified.
Long (ms) simulations were required to capture this process, and such
simulations can also be complicated by the aggregation of compounds. In
principle, a potential of mean force can be derived from these unbiased
simulations provided the drugs adequately sample all the available space.
Such free energy surfaces have helped to approximately determine the
affinity of each site;175 however, it is difficult to ensure convergence of the
free energy values or to witness very slow events.
Biasing forces can also be used to help sample drug positions in and
around the channel. For example, metadynamics has been used to ensure
that the drugs sampled all available positions in the interior of the
pore.15,32,174 Not only does this ensure that the drugs try out all positions in
the pore, the results can be used to create a free energy map from which the
comparative affinity of the sites can be determined. Drug positions in bulk
were not determined in these studies, meaning that absolute binding free
energies were not available. To help understand how the drugs find their
binding site in the interior of the pore, umbrella sampling was also used to
create a potential of mean force for the drugs to move from the bulk through
the lipid filled lateral fenestrations of the protein, as shown in
Figure 6.11B.15 The results indicated that it is feasible for benzocaine and
phenytoin to enter the pore by this route, something also seen in the
flooding simulations. In addition, the energy minima in the interior of the
pore were consistent with the binding sites found previously by equilibrium,
flooding and metadynamics methods.
6.7 Conclusions
This chapter has described a number of examples in which computer
simulations have been used to understand the properties of ion channels.
While many different approaches have been used, MD is becoming more
popular with time, as software and hardware improvements have allowed
for longer simulations to be run, improved conformational sampling of the
proteins, and larger data sets to be reproduced. As a consequence, the
amount of insight that can be gained from these simulations has increased
accordingly. While it was not computationally feasible to directly simulate
the opening of an ion channel a few years ago, simulation timescales have
now reached the point where this can be done in some situations. However,
caution should continue to be applied since, while the very long simulations that have recently been published have led to insight into some
important biological processes, they have also highlighted some shortcomings that continue to dog the simulation method. The inability to
precisely reproduce measured currents, for example, serves as a warning
that inaccuracies remain.
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This chapter has introduced some of the different simulation methods
that have been used to study ion channels. In some cases, brute force MD
has led to significant insight. In others, the use of biased simulations for
speeding up events, improving conformational sampling, or quantifying free
energies has been essential. Such advanced simulation methods are becoming more common and probably account for the majority of simulations
currently being conducted. Calculating free energies from simulations often
allows for a more direct comparison with experimentally measured properties, while improved conformational sampling can allow for alternative
conformational states of a protein to be found without having to conduct
individual long simulations. A number of recent reviews on these topics are
available for interested readers.177–180
Molecular dynamics simulations provide a high level of temporal and
spatial resolution that is not available with experimental techniques. Because of this, they can provide detailed insight into the molecular basis of a
number of ion channel phenomena, such as the way ions pass through these
channels at large rates, how the channels can distinguish between different
ion types, how channels open and close and how they are modulated by
drugs and toxins. While the technique has limitations, improvements are
continuously being made and, with the availability of free easy-to-use software, molecular dynamics has become an essential technique for studying
the function of ion channels.
Acknowledgements
The author would like to acknowledge financial support from the Australian
Research Council (Future Fellowship FT-130100781).
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00197
CHAPTER 7
Computational
Characterization of Molecular
Mechanisms of Membrane
Transporter Function
NOAH TREBESCH, JOSH V. VERMAAS
AND EMAD TAJKHORSHID*
Center for Biophysics and Quantitative Biology, Department
of Biochemistry, and Beckman Institute for Advanced Science
and Technology, University of Illinois at Urbana-Champaign,
405 N. Mathews Ave., Urbana, IL 61801, USA
*Email: emad@life.illinois.edu
7.1 Membrane Transport – A Fundamental
Biological Process
All living cells depend on the continuous exchange of diverse molecular
species, e.g., nutrients, precursors, and reaction products, across the cellular
membrane for their proper function,1 and membrane transporters are one of
the major classes of proteins that perform this vital task. Transporters are
molecular machines that couple the use of various sources of cellular energy
to structural transitions that mediate the transport of their specific substrates from one side of the membrane to the other in a highly efficient
manner. As gate-keepers that control the traffic of metabolically important
substances in to and out of the cell, they are vital to a wide variety of
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Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
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Chapter 7
biological and physiological processes. This fundamental role has rendered
transporters important drug targets, stimulating widespread interest in
mechanistic studies of these proteins at a molecular level.2
From a mechanistic perspective, membrane transporters are thought to
act through the alternating access mechanism, in which transporter-bound
substrates are accessible from only one side of the membrane at a time
(Figure 7.1).3 This mechanism is often cited as the feature that distinguishes
membrane transporters from membrane channels, which commonly form
conduits for passive diffusion of their substrates across the membrane.4
Within the context of the alternating access mechanism, a transporter is said
to be in an inward-facing (IF) state when the substrate is accessible from the
intracellular side of the membrane. When the substrate is accessible from
the opposite side of the membrane, the transporter is considered to be in an
outward-facing (OF) state. Additionally, when a substrate is bound to a
transporter, the transporter is said to be in a bound state. Otherwise, it is
said to be in an apo state.
Transporters are primarily classified according to the source of cellular
energy they utilize during transport. Primary active transporters utilize
metabolic energy, most commonly ATP hydrolysis, to transport substrates
across the membrane (usually up their electrochemical gradients), while
secondary active transporters utilize the energy of pre-established electrochemical gradients (of ions, most commonly) to accomplish this task. Any
substance transported by a secondary active transporter that is not a substrate, including but not limited to gradient constituents, is called a cofactor.
If the downhill direction of the gradient utilized in secondary active
Figure 7.1
Schematic representation of the alternating access mechanism and the
major intermediate states in the transport cycle of a secondary transporter. IFa represents the inward-facing apo state, OFa represents the
outward-facing apo state, IFb represents the inward-facing bound state,
and OFb represents the outward-facing bound state.
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Computational Characterization of Membrane Transporter Function
Figure 7.2
199
Schematic comparison of (A) antiporters, (B) symporters, and (C)
uniporters.
transport is in the same direction as substrate transport, cofactors and
substrates bind from the same side of the membrane, and the transporter is
called a symporter (Figure 7.2B). Conversely, if the downhill direction of the
gradient is in the opposite direction of transport, cofactors and substrates
bind on opposite sides of the membrane, and the transporter is called an
antiporter (Figure 7.2A). Finally, passive transporters utilize the electrochemical gradient of the substrates themselves for transport (Figure 7.2C).
These transporters are also called uniporters, and the transport they enable
is called facilitated diffusion.
In the alternating access mechanism, a transporter undergoes structural
transitions between the OF and IF states, temporarily residing in a variety of
intermediate states along the way. High-resolution crystal structures for
several of these proposed states across various families of transporters have
provided strong support for this mechanism.5–11 However, structures for
multiple functional/conformational states of the same protein or even
within the same family are not abundant.12–24 Even when multiple conformational states have been characterized for the same transporter,
knowledge about the transitions between these states and their coupling to
an energy source cannot be accurately gleaned from static X-ray crystal
structures. Experimentally, single-molecule measurement techniques such
as Förster resonance energy transfer (FRET) and electron paramagnetic
resonance (EPR) spectroscopy can be used to probe the dynamics involved
in transport mechanisms,25–28 but these techniques are limited in comprehensiveness because they can only investigate the changes in distance
between up to a few pairs of residues at a time. Molecular dynamics (MD)
simulations are able to successfully bridge this experimental gap between
resolution and dynamics by computing the changing positions of all atoms
in the transporter and its environment throughout time.
Along with MD simulations, molecular modeling and other computational
techniques have found wide applicability in the description of the dynamic
and mechanistic aspects of biomolecular function.29 Over the years, the
potential for overlap between computational and experimental observations
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Chapter 7
has expanded considerably due to long time scale simulations enabled by
better computational algorithms and the increased availability of faster and
more powerful computing resources.30,31 Computational methods are now
routinely utilized for predictive or descriptive studies that parallel or complement those involving experimental techniques, and they have enabled
significant discoveries about the way in which biomolecules function in
a biological context.32–68 In our laboratory, we use an array of advanced
MD-based simulation techniques to elucidate and characterize all aspects of
the transport cycles from a diverse set of active transporters at atomistic
resolution, and our studies allow us to make experimental predictions that
can be used to validate and build support for the proposed mechanisms.
In the remainder of this chapter, we discuss a number of examples of our
investigations to illustrate the power and versatility of computational studies
of membrane transporters. We begin with examples in which MD simulations have been able to capture substrate/ion binding/unbinding events,
processes that are essential components of the transport cycle. Given the
centrality of structural changes in the mechanism of membrane transporters, we discuss several studies designed to describe the nature of conformational changes involved in the function of these proteins. First, we
provide examples in which small-scale, localized structural transitions, such
as gating motions, have been captured by MD simulations. We then turn to a
recently developed approach that aims to describe large-scale, global structural transitions in membrane transporters. We conclude the chapter by
presenting examples and discussion on the involvement of water and lipid
molecules, two essential components of the environment for any membrane
protein or process, on the mechanism of membrane transporters.
7.2 Substrate Binding and Unbinding
Substrate binding and unbinding are among the fundamental steps in the
functional cycle of a membrane transporter. For a transporter to be effective
and to ensure its mode of transport (e.g., symport versus antiport), the
transporter must couple its global conformational changes to substrate
binding/unbinding events. This process is particularly difficult to study
experimentally as many substrates are small, and tagging them frequently
blocks the conformational changes essential to function. Thus, while
(apparent) dissociation constants for transporter substrates are commonly
reported, experimental descriptions of the functional details of substrate
binding and unbinding are sorely lacking and come solely from static crystallographic snapshots of the transporter, which are fundamentally limited.
MD simulations are a powerful complementary tool that breathe life into
the static structures crystallography provides, yielding unique insight into
binding and unbinding processes in membrane transporters. For example,
diffraction-grade crystals can often only be obtained when a substrate analog
locks the transporter into a single conformational state, as was the case for
the dopamine transporter dDAT69 and the dicarboxylate transporter
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VcINDY. In these situations, MD simulations can be used to obtain a more
complete, dynamic picture of substrate–transporter interactions. Another
shortcoming of crystallography is its inability to distinguish between
molecules with the same number of electrons, leading to ambiguities of
assignment between species, particularly between water and light ions such
as Na1 or between heavier ions such as K1, Ca21, and Cl.71 MD simulations
can help resolve such ambiguities by determining the relative stability of the
transporter in the presence of different species.
7.2.1
Spontaneous Binding Simulations Revealing a Binding
Mechanism and Site
One obvious application of MD simulation is to observe binding events as
they might happen under physiological conditions. We have applied this
approach to GlpT (glycerol-3-phosphate transporter), a secondary active
antiporter that drives glycerol-3-phosphate (G3P) uptake against its concentration gradient by using the gradient of inorganic phosphate (Pi).72–76
GlpT is a member of the major facilitator superfamily (MFS),72,76–81 which
is the largest known superfamily of secondary active transporters. This
superfamily includes transporters that have been implicated in antibiotic
resistance82 and diseases such as diabetes mellitus.83 GlpT is considered to
be an exemplary structural and functional model for the members of this
superfamily and has been used as a model for other MFS transporters.75,84,85
GlpT was crystallized in an IFa state, although a putative substrate translocation pathway was observed.72 Two highly conserved arginine residues
within the pathway (R45 and R269, Figure 7.3A) were suggested to bind to
the negatively charged substrates of GlpT.72–76 Without solid experimental
evidence, it was left to simulation to test this hypothesis and to elucidate the
details of the substrate binding mechanism in GlpT.44,51
Even in simulations without the substrate, we observed a negatively
charged Cl spontaneously binding to the putative binding site within the
translocation pathway, indicating a strong driving force for binding to that
site. Electrostatic interactions are a primary driving force in this and other
cases,87 as they are by far the strongest force acting at the molecular scale.88
For GlpT, there is a strong positive potential near the binding site
(Figure 7.3B) that draws in negative charges on its substrates. In equilibrium
simulations, we have observed spontaneous binding of both Pi and G3P to
the same binding site, and we have observed that the arginines coordinate
the phosphate groups.44,51 As these events are spontaneous, observables
such as the binding rate, binding residues, and local conformational
changes are available and can be used to inform future experiments.
This approach works more broadly but has an important caveat. If the
initial position of the substrate is too far away from the binding site, ergodicity arguments disfavor sampling of a binding event within reasonable
simulation time scales, as even strong electrostatic attractive forces are only
effective at close range. Thus, in general, some form of biasing is required to
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Figure 7.3
An overview of the structure of the GlpT substrate translocation pathway,
highlighting the residues that contribute to substrate binding (A) and
the electrostatic potential of the simulation system (B).
Reprinted from G. Enkavi et al., Annual Reports in Computational
Chemistry, Elsevier Books, pp. 77–125.86 Copyright 2014 with permission
from Elsevier.
observe substrate binding in simulations. The bias may be implicit, like
placing the substrate near the binding site initially to maximize the binding
probability, or explicit, like using external forces to drive binding.
7.2.2
Proposing Substrate Binding Sites through Molecular
Docking
Rather than only using MD simulations to capture binding, molecular
docking techniques, originally developed for drug design, can be used to
probe putative substrate binding sites in transporters for further investigation by MD simulations.89 The basics of docking and MD are rather
similar when it comes to characterization of binding sites, as an empirical
scoring or potential energy function based on molecular geometries is used
to guide the substrate to a binding site in both cases.90 However, in contrast
to MD, where dynamics are dutifully maintained by simulating time-evolving
atomic coordinates, docking uses a Monte Carlo style search, where many
different substrate conformations are attempted and only the most promising binding candidates are kept without consideration of the time scale
required to obtain them. Additionally, MD can provide information on the
binding and unbinding pathways and mechanisms and the involvement of
gating elements and residues in the process. The main benefit of docking is
that it takes substantially less computation time than conventional MD
simulations. However, conventional docking is usually used with static
structures (typically the crystollographically solved structure), which cannot
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Computational Characterization of Membrane Transporter Function
203
account for the conformational flexibility of the binding site that is available
in MD simulations.90
To combine the benefits of traditional docking and MD, a technique
called ensemble docking91 was developed and is becoming increasingly
common. In this technique, an ensemble of structures are obtained from
MD simulations, and docking is performed for the set of conformations that
best represent the conformational flexibility of the protein and its binding
pocket. Once an initial set of bound conformations is obtained, it can be
used for additional simulations in which the substrate and the binding site
can undergo further conformational relaxation, resulting in a more optimal
set of interactions. In our lab, we have performed ensemble docking
on P-glycoprotein92 (Pgp), an exporter of amphiphillic substrates, including
a large number of drugs.93,94 Starting from a set of conformations taken
from an equilibrium simulation (Figure 7.4A), we have determined a number
of potential binding sites for the drug of interest (Figure 7.4B). With this
result, we are now poised to use further equilibrium and nonequilibrium
steered simulations to probe the behavior of the substrate as it transitions
between different points along the translocation pathway, thus providing
a description of a putative binding/unbinding pathway. By combining
docking with MD, the computational effort can be focused on the topic of
Figure 7.4
(A) Multiple overlaid Pgp structures taken from an MD trajectory are
used in ensemble docking to take into account the natural motions of
the protein which might result in formation of higher-affinity and/or
additional binding sites for the docked substrate than the crystal structure. (B) In this case, the docked ensemble resulted in five potential
binding sites for the drug being docked to Pgp.
Reprinted from J. V. Vermaas, N. Trebesch, C. G. Mayne, S. Thangapandian,
M. Shekhar, P. Mahinthichaichan, J. L. Baylon, T. Jiang, Y. Wang, M. P.
Muller, E. Shinn, Z. Zhao, P.-C. Wen and E. Tajkhorshid, Microscopic
Characterization of Membrane Transporter Function by In Silico Modeling
and Simulation, Methods Enzymol., 578, 373–428, Copyright 2016, with
permission from Elsevier.
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interest, i.e., the bound state and how the bound substrate moves during the
transition, rather than the binding process, which may not be the central
focus of study.
7.2.3
Unraveling Substrate Release Pathways
Similar to binding, molecular simulation also offers unique insight into
unbinding processes, mechanisms, and pathways. Given a structure with a
bound substrate, simple equilibrium MD simulations may be able to capture
full or partial unbinding events. However, for substrates with strong interactions to their binding site, biased simulations may be required to drive
substrate unbinding, as the same electrostatic forces that draw in the substrate prevent it from being easily dislodged. This is particularly pronounced
for cations and anions, which interact strongly with other charged species.
In cases where tight binding prevents the substrate from unbinding, advanced reaction coordinates that reduce the contact number between the
ion/substrate and its coordinating atoms (binding sites) can be employed to
drive unbinding. This technique is particularly useful when the unbinding
pathway/direction is not clear from the examination of the bound state
structure because the contact number reaction coordinate does not favor any
unbinding pathway over another.
Nevertheless, there have been clear cases where spontaneous unbinding
events have been successfully captured by equilibrium MD simulations. One
such example is the case of the bacterial Na1-coupled galactose transporter
vSGLT.46,54 vSGLT was the first solute sodium symporter (SSS) resolved at high
resolution48,95 and is part of a large family of exchangers structurally homologous with LeuT (leucine transporter) known as the LeuT-fold transporters.5,96 The vSGLT crystal structure was the first IFb structure of a transporter
with the LeuT-fold.95 In the structure, Na1 was also modeled at a particular
binding site, even though the density around this region was not conclusive.
One of the most basic questions about crystallographic structures of
membrane transporters that can be asked is: Which particular functional
state does the structure represent? In the case of vSGLT, the answer was
unclear. Was the structure an ion-bound state (as the crystallographers
claimed95), or was it a state in which the ion binding site was open to the
solution? Through a set of equilibrium MD simulations, we showed that the
Na1 modeled in the structure consistently unbound incredibly rapidly (in
under 10 ns),46 which suggested that the crystallographic structure likely
represents an ion-free state. While this result has been observed in other
related transporters,15 it is not typical in general as, again, electrostatic
interactions are the strongest molecular interactions at molecular length
scales.88 The observation of ion release also provided information on the
sequence of unbinding events for the substrate and the ion, as it suggested
that the release of the ion precedes that of the substrate.46
In a subsequent study on vSGLT, we observed the spontaneous release of
the substrate (galactose) from the binding site, allowing us to elucidate a
previously unproposed binding pathway and mechanism.54 As shown in
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Figure 7.5
205
Galactose binding site of vSGLT with hydrogen bonds stabilizing the
binding site drawn as dashed lines and with the residues of the
transporter in direct hydrogen bonding contact labeled.
Figure derived from work published by Li and Tajkhorshid.54
Figure 7.5, several hydrogen bonds stabilize the substrate in its binding
pocket, such that each hydroxyl is a hydrogen bond donor or acceptor to
either protein side chains or water molecules within the binding pocket.
This example demonstrates the unique perspective the computational
microscope brings to bear in capturing events on the molecular scale. Based
on visual inspection of the crystallographic structure, it was suggested that a
specific tyrosine residue (Y263) played a key role in keeping galactose confined to the binding site, as the bulk of the side chain was seen to block the
passage of substrate out of the binding pocket. Interestingly, the unbinding
event captured during our equilibrium simulation revealed that the substrate followed a curved path around Y263, and Y263 was not displaced or
rotated during the substrate release event.54
7.3 Capturing Localized Transporter Motions with
Equilibrium Molecular Dynamics
All transporters undergo mechanistically important local conformational
rearrangements during their transport cycles. These local structural changes
often occur in response to substrate/cofactor binding/unbinding and can be
associated with the opening/closing of structural elements called the inner
and outer gates. Gates close prior to transport after substrates/cofactors have
bound to the transporter, thus making them solvent inaccessible, and open
after transport to allow the substrates/cofactors to become solvent accessible
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once again. Local conformational changes often involve the rearrangement
of side chains or the movement of loops and sometimes even helices within
the transporter, and they often precipitate the global conformational
changes principally responsible for active transport. In contrast to global
conformational changes, these local changes often occur on time scales that
are accessible to equilibrium MD simulations. In this section, we will
describe our investigations into the local conformational changes involved
in the transport mechanisms of two structurally and mechanistically distinct
transporters.
7.3.1
Substrate-induced Structural Changes of an Antiporter
We use the case of GlpT (introduced in the preceding section) as our
first example in which localized protein motions were captured using
equilibrium MD simulations. Structurally, GlpT is organized into two
transmembrane helix bundles, which exhibit a structural pseudo-symmetry
with respect to the membrane.72,73,75 The transporter is thought to operate
through the ‘‘rocker-switch’’ mechanism, in which the two domains of the
transporter reorient with respect to one another such that the accessibility of
the substrate binding site changes from one side of the membrane to the
other.72,75,76
In equilibrium MD simulations of the IFa state of GlpT, we have observed
spontaneous Pi binding events that resulted in two important effects on the
antiporter.51 The first main effect was a conformational change in which two
of the transmembrane helices straightened, slightly closing the cytoplasmic
opening of GlpT (Figure 7.6). During simulations of apo GlpT, these helices
exhibited significant flexibility, a property which is also affected (reduced) by
Pi binding. The substrate-induced cytoplasmic closure of GlpT is consistent
with the rocker-switch mechanism of transport, as it likely represents the
early conformational changes of the transporter. The second main effect of
Pi binding was on key salt bridges in GlpT. Because salt bridges form strong
but breakable bonds between residues, they can play an important role
in stabilizing particular states that are present in transport cycles. During
our MD simulations, we have observed that Pi binding results in a local
conformational change of a basic residue (K46) that is able to form two salt
bridges. This conformational change destabilizes the salt bridge favored in
the IF state, and we believe this destabilization may lower the activation
energy required to induce the transition from the IF to the OF state.
7.3.2
Gating Elements in a Neurotransmitter Transporter
Communication between nerve cells is accomplished primarily by neurotransmitters. These chemicals are released by presynaptic neurons in
response to electrical activities, then detected by postsynaptic ones. In order
to prepare a new signal, the neurotransmitters must be rapidly removed
from the synapse.98 The glutamate transporter (GlT) is a neurotransmitter
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Computational Characterization of Membrane Transporter Function
Figure 7.6
207
Substrate-induced structural changes in GlpT. (A) The crystal structure of
GlpT with the two bent helices H5 (green) and H11 (pink) highlighted.
The region where the substrate-induced closure takes place is highlighted using a yellow bar. (B) The radius profile of the lumen calculated
using HOLE.97 Substrate-binding simulations show a decrease in radius
around the highlighted region. (C) Substrate-induced straightening of
H5 and H11 toward the lumen. Black, blue, and red helices represent the
structures at t ¼ 0 ns, t ¼ 25 ns, and t ¼ 50 ns taken from the substratebinding simulation. (D) Distances between the Ca atoms of residue pairs
on H5 and H11 as a measure of the distance between the two helices. The
residue pairs lie approximately parallel to the plane of the membrane.
Reprinted with permission from G. Enkavi and E. Tajkhorshid,
Simulation of Spontaneous Substrate Binding Revealing the Binding
Pathway and Mechanism and Initial Conformational Response of GlpT,
Biochemistry, 2010, 49, 1105–1114.51 Copyright 2010 American Chemical
Society.
uptake pump found in neurons and astrocytes that is responsible for terminating glutamate-mediated excitatory signaling.99,100 Mammalian GlTs
are secondary active symporters that couple the uphill import of glutamate
to the downhill transport of positively charged monovalent ions (Na1, K1,
and/or H1).101,102 A homologous transporter from the archaeon Pyrococcus
horikoshii called GltPh has been crystallized in multiple OF and IF states with
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20–22
bound and unbound substrates/cofactors.
GltPh cotransports three Na1
along with its substrate (aspartate) in each transport cycle.
The crystal structures of GltPh provided preliminary insight into the
chemical and structural principles that control the transition of the transporter between the OF and IF states. They revealed that GltPh is a homotrimer and that each protomer is composed of eight transmembrane helices
(TM1–TM8) and two highly conserved helical hairpins (HP1 and HP2) that
form the binding sites for the substrate and three Na1, though GltPh has only
been crystallized with Na1 in two distinct locations. Starting with these
structures, we employed equilibrium MD simulations of a membraneembedded model of the transporter in its apo and a variety of its different
bound states to investigate molecular events that might be controlled by
binding and unbinding of ions and the substrate.50
In the OF state of GltPh, the substrate binding site is occluded by the
loops of HP1 and HP2 (Figure 7.7A). Crystal structures with and without the
substrate bound have shown HP2 in two very different conformations,
suggesting that HP2 is an important part of the OF gate and that it must be
structurally flexible.21 Furthermore, inhibition studies in a mutant homolog
Figure 7.7
Dynamics of the extracellular gate in GltPh. (A) HP2 loop motions
responsible for extracellular gating. Superimposed snapshots of HP2
(gray) show opening motions after the removal of the substrate. Yellow
and pink snapshots show the closed conformation of HP2 and its open
conformation in one of the last simulation frames, respectively. (B) Time
evolution of the root-mean-square displacements (RMSDs) of the helical
hairpins HP1 and HP2 in the presence and absence of the substrate.
Figure adapted from work published by Shaikh et al.103
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of a mammalian GlT suggest that HP2 serves as the extracellular gate of the
transporter and that substrate induces distinct conformations of HP2.104
Comparisons of the dynamics of the substrate-bound and apo simulations
also suggest that HP2 is the extracellular gate.43 Invariably, in all of the
simulations performed in the presence of the substrate, HP2 displays a
stable conformation, while removing the substrate results in a large opening
motion and complete exposure of the substrate binding site to the extracellular solution (Figure 7.7).43 Importantly, these results suggest that the
opening and closure of HP2 are controlled directly by substrate binding.43
An independent MD study has also provided support for this role of HP2.38
Interestingly, despite being pseudo-symmetrically related to HP2, HP1 was
found to exhibit a high degree of conformational stability regardless of the
presence of the substrate (Figure 7.7).43 This result, which might be attributed either to the shorter length of HP1 or to its much closer contact with
TM2 in the OF state, suggests that, at least during the extracellular half of the
transport cycle, HP1 does not play a direct role in gating, and its involvement
might be limited to stabilization of the structure of HP2 upon substrate
binding. The possibility of a gating role for HP1 during the cytoplasmic
half of the transport cycle and its coupling to substrate binding/unbinding
could be determined by performing equivalent simulations on the IF GltPh
structure.43
Through our simulations, we have also evaluated the role the substrate
(aspartate) and cofactors (Na1) play in stabilizing the binding site, which
might suggest the order in which these species bind. In addition to controlling the conformation of HP2, a second major consequence of substrate
binding revealed by our simulations is the formation of a Na1 binding site
(called the Na2 site)43 at a position between two half-helical structures (HP2a
and TM7a). In the apo state, the dipole moments of these half-helices are
misaligned but, upon substrate binding, the dipole moments converge on
the same region, resulting in the formation of the Na2 site. These results
strongly suggest that Na2 binding can only take place after binding of the
substrate.43 Binding to Na2 further stabilizes HP2, resulting in a completely
occluded form, in which water molecules and Na1 can no longer access the
binding sites from the extracellular side of the membrane and suggesting
that binding to Na2 is the last step that occurs before the transition to the IF
state. These simulation results are supported by various experimental
studies on GlT, which have shown that substrate binding enables the
binding of one of the co-transported Na1.21,105
7.4 Computational Description of Global Structural
Transitions in Membrane Transporters
The alternating access mechanism of transport requires most transporters to
undergo coordinated global structural rearrangements when transitioning
from the IF to OF state and vice versa. One of the remarkable and unique
functional aspects of membrane transporters is that these structural
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rearrangements can and often do involve large-scale conformational changes
engaging a large portion of the protein, particularly within the transmembrane region.15,64,106 Regardless of their scale, these global changes naturally
occur on time scales on the order of milliseconds to seconds,107–109 which are
inaccessible to standard atomistic equilibrium MD simulations. For context,
the longest time scales currently accessible to atomistic MD simulations for
transporter-sized biomolecular systems are on the order of tens to hundreds
of microseconds, and such simulations require the use of specialized hardware (i.e., Anton supercomputers).110–112 Because of the time scale limitation,
nonequilibrium methods are required to induce and describe large-scale
structural changes in membrane transporters113 and, more generally, in
enzymes.114 Here, we describe one such nonequilibrium method.
7.4.1
Nonequilibrium Simulation of Structural Changes
In our lab, we have recently developed a nonequilibrium computational
approach to structurally elucidate and thermodynamically characterize the
transport cycles of membrane transporters.113,115,116 We specifically developed our approach to describe the large-scale structural transition between two well-defined conformational states.115–117 Thus, in order to use
our approach, we must first identify a pair of structures to serve as the
endpoints of the transition we want to investigate (Step 1, Figure 7.8). These
structures can be X-ray crystal structures, or they can be created computationally using techniques such as homology modeling.52 Once these endpoint structures are defined, we must perform a detailed analysis of the
structural differences between them. This analysis, together with any existing (low resolution) information on the nature of the structural changes
involved in the process, allows us to define transporter-specific collective
variables (i.e., reaction coordinates) that can be used to simulate the global
conformational changes involved in the transition between the endpoints
(Step 2, Figure 7.8). Using nonequilibrium MD,118–120 we then induce
changes along our collective variables to transition the transporter between
the conformational endpoints. The collective variables used to induce the
transition and the way in which the collective variable values are changed are
collectively referred to as a biasing protocol. To widely sample many possible
transition pathways, we test dozens to hundreds of different biasing protocols (Step 3, Figure 7.8). To compare the sampled pathways, we calculate the
nonequilibrium work required to induce the transitions, which can be used
as a coarse quantitative metric to evaluate how likely a transition is to take
place naturally.115–117
For the next step of our approach to be computationally feasible, we must
find a transition pathway that can be induced using less than B75 kcal mol1
of nonequilibrium work. Once we have found such a pathway, we apply a
computationally expensive MD-based path-refining technique called the
string method with swarms of trajectories (SMwST) to relax it (Step 4,
Figure 7.8).121 In SMwST, hundreds to thousands of copies of the molecular
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Computational Characterization of Membrane Transporter Function
1. Identify Endpoints
2. Define Collective Variables
θ
z
θ
Inside
3. Test Biasing Protocols
Inside
4. Relax Pathway (SMwST)
Trial 2
z (Å)
5. Enhance Sampling (BEUS)
OF
IF
Relaxed Pathway
Free Energy
Trial 200
Original Pathway
θ (°)
θ (°)
IF
Free Energy
OF
Trial 1
z (Å)
6. Calculate Thermodynamics
IF
z (Å)
Figure 7.8
Free Energy
Free Energy
OF
θ (°)
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z
Trial 100
211
z (Å)
Overview of our computational approach to characterize large-scale,
complex transitions in membrane transporters.
systems are simulated simultaneously (or serially or in stages, depending on
the version of the algorithm122) using equilibrium MD starting from different
points along the transition pathway, and their mean diffusion in collective
variable space is used to relax the transition pathway to the nearest minimum
free energy pathway. We then apply bias-exchange umbrella sampling (BEUS,
also called Hamiltonian replica-exchange123,124 or window exchange125 umbrella sampling) to increase sampling along the pathway determined with
SMwST. In BEUS, dozens of copies of the transporter are simulated while
being restrained to dozens of different points along the pathway, and these
biases are periodically exchanged to enhance sampling along the pathway
(Step 5, Figure 7.8).
Next, we reweight this nonequilibrium sampling to obtain equilibrium
thermodynamic information (e.g., free energy) along the transition pathway
(Step 6, Figure 7.8). The weighted histogram analysis method (WHAM)126 is
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considered standard for this task, but alternative and more general methods
also exist.127,128 Finally, we computationally assess the quality of our final
pathway and derive experimentally validatable predictions from it. For
example, we can identify the location of inhibitory cross-link sites, or we can
measure the changes in distance between pairs of residues, which can be
tested using experimental techniques like FRET or EPR spectroscopy. The
results of these computational analyses and tested experimental predictions
allow us to refine the steps we took earlier in our approach, which allows us
to iteratively apply our methodology to improve our transition pathway. By
applying our entire methodology to the transitions between multiple pairs of
conformational states within a transport cycle, we are able to structurally
and thermodynamically characterize entire transport cycles.
7.4.2
Application to an ABC Transporter
The methodology described above was initially developed and employed to
study the OFa-IFa transition of MsbA,116,117 a primary active transporter
responsible for exporting a variety of substances in bacteria. MsbA belongs
to the largest superfamily of primary active transporters called ATP-binding
cassette (ABC) transporters, which are identifiable by their unique architecture.129,130 ABC transporters are composed of two nucleotide binding
domains (NBDs) responsible for ATP hydrolysis and two transmembrane
domains (TMDs) responsible for substrate transport.131,132 In these transporters, dimerization/dissociation of the NBDs and the associated ATP
binding/hydrolysis/release induces the large-scale conformational changes
in the TMDs associated with substrate transport.27,131,133,134 In MsbA, we
found that the global conformational changes associated with substrate
transport were quite complex.113
MsbA has been crystallized in three distinct states,135 and the first step in
our methodology required us to determine collective variables that capture the
large-scale conformational differences between these states. Ultimately, we
selected four collective variables and labeled them a, b, g, and d (Figure 7.9).
The collective variables a and b represent the relative orientation of the TMDs,
g represents the relative orientation of the NBDs, and d represents the distance
between the NBDs. To determine the optimal biasing protocol for these collective variables, we simulated more than 200 different pathways and classified
them into fourteen groups based on the order in which the changes in the
collective variables were induced.116,117 We used the nonequilibrium work
required to induce transitions for each group (Figure 7.10) to identify the best
transition pathway. This pathway involves inducing changes along d followed
by changes along b, then g, then finally a. Structurally, this pathway corresponds to NBD dissociation, followed by closure of the OF gate, then twisting of
the NBDs, and finally the opening of the cytoplasmic gates (Figure 7.11).
After identifying this pathway, we used BEUS to increase the sampling
along the pathway. The pathway was relaxed enough that SMwST was not
needed in this study. The changes in free energy along a particular collective
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A
β
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γ
OF
IF (Closed)
α
IF (Open)
IF (Closed)
B
d
γ
OF
Figure 7.9
IF (Closed)
The three crystal structures of MsbA (A) and an alternative view of the
NBDs (B) with the collective variables that capture the large-scale
conformational differences between the conformations.
Figure adapted from work published by Moradi and Tajkhorshid.113
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Figure 7.10
Nonequilibrium work profiles for fourteen groups of simulations
formulated on the basis of the order in which we applied biases to
different collective variables. The bias orders are shown to the right of
the plot, and the fourteen groups are further classified into three
supersets and colored accordingly. The group requiring the lowest
total work on average is colored black.
Figure adapted from work published by Moradi and Tajkhorshid.113
Figure 7.11
The principal stages of the apo transition of MsbA obtained using our
nonequilibrium methodology.
Figure adapted from work published by Moradi and Tajkhorshid.113
variable is called the potential of mean force (PMF), and the PMF along a is
plotted in Figure 7.12. By analyzing the PMF along this and other collective
variables, we determined that the pathway we found was thermodynamically
reasonable, and it did not require further refinement. While we cannot rule
out the possibility that a different transition pathway is used in nature, the
pathway we have proposed provides insight into the global structural
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PMF (kcal/mol)
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5
4
3
2
1
0
10
Figure 7.12
20
30
α (°)
40
50
PMF governing the opening and closing of the NBDs (measured by a)
in MsbA. The values of a associated with the closed IF (circle) and open
IF (triangle) crystal structures are marked on the horizontal axis. This
PMF profile reveals great conformational flexibility in the resting state
of apo MsbA.
Figure adapted from work published by Moradi and Tajkhorshid.113
dynamics and underlying thermodynamics that govern the transport mechanism of MsbA at an unprecedented level of detail. Since developing this
methodology and applying it to MsbA, we have also applied it successfully to
the transporter GlpT,115 and we are currently using it to investigate the global
conformational changes of a variety of other transporters in our lab. Our
methodology provides the framework necessary to investigate and characterize the large-scale conformational changes vital to the function of many active
transporters and, more generally, to many other biomolecules as well.
7.5 Water within Transporters
Up to this point, we have primarily discussed transporters in terms of their
functional motions related to transporting substrates from one side of the
membrane to the other. However, this focus neglects the role water can play
in transporter function and how to use computational techniques to
investigate this role.136 For proton-coupled transporters, water itself can
have a more integral role in the transport cycle, e.g., by guiding protons to
their respective binding sites or by forming transient proton-conductive
states that can pump protons unidirectionally.137–140 Even in the absence of
major protein conformational changes, water itself has been found to form
transient water wires141–143 or to leak within membrane transporters144
without known functional significance. Water interaction with membrane
transporters is incredibly challenging to describe without animating a
static structure, as hydrogen atoms and mobile water molecules cannot be
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captured crystallographically (with the exception of ultrahigh resolution
structures145). In this context, MD offers a powerful tool to capture transporter
conformational fluctuations and to directly observe water interactions on an
atomic scale unattainable by experimental methods.110,146
7.5.1
Water Leaks in Transporters
Membrane transporters rely on highly coordinated structural transitions between major conformational states for their function and, by the alternating
access mechanism, simultaneous access of the substrate binding site to both
sides of the membrane is considered prohibited. Although such mechanical
precision successfully accounts for the efficient exchange of the primary substrate across the membrane, accruing evidence on significant water transport
and even uncoupled ion transport mediated by transporters has challenged
the concept of perfect mechanical coupling and coordination of the gating
mechanism in transporters which might be expected from the alternating
access mechanism. Using a large set of extended equilibrium MD simulations
performed on several membrane transporters from different families and in
different conformational states,147 we have demonstrated that water leaks are
likely a universal phenomenon in all classes of membrane transporters.
These simulations not only allowed us to test for the presence of ‘‘leaky’’
states in diverse transporter architectures and conformational states, but
they also allowed us to observe these states in microscopic detail, thus
making it possible to investigate the underlying molecular mechanism of
water transport. In our simulations, we have observed spontaneous formation of transient water-conducting (channel-like) states allowing passive
water diffusion through the transporters (see an example with vSGLT in
Figure 7.13). In all simulated transporters, the water transport pathways
were found to coincide with the putative pathway for substrate transport and
involve the substrate binding site directly.147 We found that these channellike states are permeable to water but occluded to substrate, which means
that the alternating access mechanism remains applicable to the substrate.
The rise of such water-conducting states during the large-scale structural
transitions of the transporter protein is indicative of imperfections in the
coordinated closing and opening motions of the cytoplasmic and extracellular gates. Based on the results of these MD simulations, we proposed
that the observed water-conducting states likely represent a universal phenomenon in membrane transporters, which is consistent with their reliance
on large-scale motion for function.
7.5.2
Water in Proton Pathways
The conduction of protons across membranes is a fundamental feature of
biology.148 Indeed, metabolism in eukaryotes revolves around generating a
proton gradient across a bilayer and then exploiting that gradient to generate
ATP for use throughout the cell.149,150 Water plays a key role in this process
by contributing to a pathway (formed by water and protein side chains) for
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Figure 7.13
217
Spontaneous water channel formation in vSGLT, comparing a ‘‘leaky’’
state (A) and a closed state (B). vSGLT is shown in a white cartoon
representation, with the galactose substrate in peach, and water is
shown both atomistically and as a surface (red). The three hydrophobic
gating residues that control the water flow, Y87, F424, and Q428, are
shown explicitly.
Figure adapted from work published by Li et al.147
rapid proton translocation via a Grotthuss mechanism, in which a hydrogen
bonded chain of hydroxyl groups can accept a proton at one end and,
through a series of proton transfer events, rearrange itself to release a proton
at the other.151,152 In membrane proteins, water used in this way complements proton wires formed by side chains153 and spatially links together
distant segments of the proton conduction pathway, as in the electron
transport chain138,154–156 or in photosynthesis.148,157 Tracking individual
water molecules and the structures they dynamically form is impossible
using experimental techniques but are straight-forward tasks to accomplish
via computational simulations with explicit water models. One example of
this comes from our recent study142 on the H1–Cl antiporter ClC-ec1.
The chloride channel (CLC) family158,159 includes both passive Cl channels
and secondary active H1-coupled Cl transporters.160–165 The latter, also
known as H1/Cl exchangers, drive the uphill movement of H1 by coupling
the process to the downhill movement of Cl, or vice versa, thereby exchanging
the two types of ions across the membrane with a fixed stoichiometry.166
ClC-ec1, a CLC from Escherichia coli, has served as the archetypal CLC for
biophysical studies because of its known crystal structures,167,168 its experimental tractability, and its structural and mechanistic similarities to mammalian CLC transporters.160–165,169–174 Prior to our study of ClC-ec1, a putative
pathway for Cl had been established, but the pathway of H1 translocation
remained obscure. Experimental studies had provided some information on
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1
166,170,171,175–178
the involvement of specific residues in H transport.
Specifically, extensive site-directed mutagenesis studies had zeroed in on two glutamate residues essential for H1 transport: E148 (Gluex), which acts as the
main extracellular H1 binding site,166,168,178 and E203 (Gluin), which plays a
similar role on the cytoplasmic side of the membrane.175–177 However, the
discovery of these H1 binding sites raised a mechanistic puzzle:160,179 How do
protons translocate between the two sites which are separated by a B15 Å-long,
largely hydrophobic region within the protein?
Using a 400 ns equilibrium MD simulation of membrane-embedded
dimeric ClC-ec1, we observed transient but frequent hydration of the central
hydrophobic region by water molecules from the intracellular bulk phase via
the interface between the two subunits (Figure 7.14).142 To validate the idea
that these water wires are central to the H1-transport mechanism, we
identified I109 as the residue that exhibits the greatest conformational
coupling to water-wire formation and experimentally tested the effects of
mutating this residue. As predicted, mutations at this position specifically
Figure 7.14
Example water wire observed during a simulation of CIC-ec1 within a
lipid bilayer, with an enlarged view (bounded by an orange box) to
highlight the wire itself. The protein is shown in a transparent
representation, with the exception of E148 and E203, which constitute
the terminal ends of the water wire formed only when Cl (green
sphere) is present. Water molecules within the wire are shown with
each atom represented by a red (oxygen) or white (hydrogen) sphere.
The atomic coordinates for this model were provided by Tao Jiang,
based on simulation results from work published by Han et al.142
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1
disrupt the H -transport function of ClC-ec1. Our simulations also allowed
us to identify a portal region which acts as the main gateway for the hydration of the hydrophobic region between Gluex and Gluin. This portal region is lined by three specific residues (E202, E203, and A404) whose size
and amino acid identity were expected to impact the formation of water
wires and, thus, the efficiency of the coupled H1 transport. Site-specific
mutagenesis experiments of the proposed portal region showed that, as
expected, ClC-ec1 ion transport rates decrease as the size of the portal
residue at position 404 is increased.
Finally, in this study, we also found that the water wires in ClC-ec1 require
the presence of an anion in the central binding site (green sphere in
Figure 7.14) to form.142 This finding explains the previously mystifying experimental observation that Cl occupancy correlates with the ability to
transport protons. Overall, this study represented a highly concerted computational and experimental approach and highlights how easily molecular
simulations can inform experimental studies by providing mechanistic details critical to deconvoluting the relationship between steps in the transport
cycle, particularly of small, highly mobile molecules such as water whose
interactions cannot otherwise be probed.
7.6 The Lipid Frontier
Besides water, one other vital player in transporter function is the membrane
itself. Various functional aspects of transporters can be targeted and
modulated by the change in lipid molecules of the membrane,180–183 going
so far as to actually be actively transported by some ABC transporters.184
The molecular underpinnings of these phenomena can be quite difficult to
probe experimentally, and much of what we know about the lipid dependence of protein function derives from functional assays in different lipid
environments. Using MD to probe these interactions directly has only
recently become routine and represents an emerging field in computational
biophysics.
7.6.1
Why Now? Initial Barriers to Simulating Lipid–Protein
Interactions
One of the drawbacks of conventional atomistic MD is that of time scale. We
have previously established that the time scales attainable with conventional
supercomputers are unable to capture the large-scale conformational
changes involved in transporters’ function, but they are also unable to
effectively capture important lipid dynamics. Due to the slow lateral diffusion of lipids, measured to be B8 108 cm2 s1,185,186 individual lipids
exchange with their neighbors infrequently, approximately once every 100 ns
when in the vicinity of proteins.187 This fact renders a membrane effectively
static in a conventional atomistic MD simulation.
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Chapter 7
As such, many older bilayer simulations used only a single lipid species in
the membrane and were unable to investigate specific lipid–protein interactions arising from the lipid heterogeneity of real biological membranes. We
note that a large fraction of experimental studies were also performed in artificial membranes composed of only a fraction of lipid types present in
biological membranes. As the importance of heterogeneous lipid compositions of the membrane on protein function became more apparent, tools
to easily generate mixed bilayers such as CHARMM-GUI188,189 were introduced. Later enhancements to CHARMM-GUI190 and competing approaches191,192 have been a boon to the field in simplifying the generation of
mixed lipid systems. More importantly, access to faster computers and
the ability to run much longer simulations have allowed us to begin to sample
lipid–protein interactions more effectively. These tools and techniques allow
us to address new questions, such as how microclusters might form in mixed
bilayers193 and how those clusters might affect protein function.
7.6.2
Computational Probes of Lipid–Protein Interactions
Buoyed by fascinating new results indicating important physiological roles
for lipids in regulating the transport cycle of membrane transporters,183,194–197 there is a growing sense that lipid–protein interactions cannot
be ignored. Lipid tail intercalation into the protein and other lipid–protein
interactions have been noted in several systems and are thought to have
functional relevance,26,197–199 although, at this early stage, no definitive
mechanism has been identified. Additionally, in many systems where the
original structure of the membrane protein was detergent-solubilized,
computational studies have demonstrated substantial changes in transporter structure due to removal of the detergent.200–202
For example, by combining molecular simulation and activity measurements, the structure and function of the mitochondrial anion carrier UCP2
was shown to be significantly altered by a zwitterionic detergent.200 Similarly, the occupancy of a secondary substrate-binding site of LeuT203,204 by
detergent in MD simulations201 represents another remarkable example of
the direct functional impact of detergents. Finally, computational studies of
Pgp have revealed that the transporter both has specific lipid interactions
and can be perturbed by detergent. In our own simulations, we have observed that phospholipids may intercalate between helices in Pgp
(Figure 7.15), and simulations of Pgp under its crystallization conditions
(i.e., in the presence of an anionic detergent) have shown that the detergent
binds to the soluble regions of the protein in addition to membraneembedded regions,202 providing a strong warning that crystallographic
artifacts are a real threat in membrane protein simulations.
As with any new field, there are a wealth of unexplored problems to
which analysis of lipid–protein interactions could be applied. In our own
studies, we have unexpectedly observed so many instances of important
lipid–protein interactions that our new favorite motto has become: ‘‘There
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Computational Characterization of Membrane Transporter Function
Figure 7.15
221
Example of a phospholipid intercalating between the helices of Pgp,
with carbon atoms represented by yellow spheres.
Atomic positions derived from studies published by Wen et al.26
are two types of membrane proteins: those that are known to be affected by
lipid composition and those where it has not yet been studied.’’ With the
advancement of computational tools, such as new atomistic membrane
models with accelerated sampling,205 prodigious growth is expected in our
understanding of how the lipid environment around membrane proteins
influence their function. Together with the determination of large-scale
conformational changes during the transport cycle, lipid dependence will
be on the forefront of computational transporter studies for the foreseeable
future.
7.7 Concluding Remarks
Membrane transporters are molecular devices that have evolved to efficiently
harvest various forms of chemical energy in a living cell to drive transport of
materials across the membrane, often against established electrochemical
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Chapter 7
gradients. They use a universal mechanism known as the alternating access
mechanism, in which protein conformational changes of various forms and
scales switch the accessibility of the transported substrate from one side of
the membrane to the other. While the change in substrate accessibility is
largely governed by global and usually large-scale structural transitions of
the transporter, coupling of these changes to small-scale structural and
chemical changes, most prominently binding and unbinding of the substrates
and cofactors, and binding and hydrolysis of ATP (for most primary transporters), is key to the process. Additionally, the selectivity of a transporter (or
lack thereof) for specific substrates is closely affected by the atomistic details
of the interaction between the substrate and the protein. Thus, to address
fundamental mechanistic questions in transporters, methodologies that are
able to provide the high spacial and temporal resolutions needed to describe
the molecular processes underlying the mechanism are necessary.
MD simulations have provided us with such a tool and, when combined
with rationally designed experiments, such simulations offer a powerful
methodology to dissect and reveal highly relevant functional details of
transporter mechanisms and function. We have demonstrated in this
chapter how such an approach can be used to gain insight into such aspects
as substrate binding/unbinding mechanisms, substrate/cofactor-induced
(allosteric) conformational changes, structural transition pathways, water–
protein interactions, and lipid–protein interactions, most of which cannot
be satisfactorily studied using currently available experimental techniques
alone. MD and other computational modeling techniques have become an
indispensable component of modern structural biology and molecular sciences. In the future, we can only expect expansion of the application of these
methods in a variety of biological problems. Empowered by increased
computational power, better and more plentiful structures of biological
macromolecules in multiple functional states, and experimental biophysical
techniques with higher resolutions, we will be witnessing more and more
examples in which molecular simulation technologies will provide the most
detailed and dynamic picture of how biology works.
Acknowledgements
We would like to recognize the efforts of the entire Tajkhorshid
group in generating the results described here. Of special note are
Dr Sundarapandian Thangapandian, who provided Figure 7.4, Dr Jing Li,
who provided the structure for Figure 7.5, Tao Jiang, who provided the
underlying structure for Figure 7.14, and Dr Po-Chao Wen, who provided
the underlying structure for Figure 7.15.
The simulation studies presented in this chapter were supported in part
by the National Institutes of Health (Grants R01-GM101048, R01-GM086749,
U54-GM087519, and P41-GM104601 to E.T.) and computationally through
XSEDE (grant TG-MCA06N060 to E.T.), DOE INCITE, and Blue Waters. N.T.
acknowledges support by the National Science Foundation Graduate Research
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Computational Characterization of Membrane Transporter Function
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Fellowship Program under Grant No. 1144245. J.V.V. acknowledges support
from the Sandia National Laboratories Campus Executive Program, which is
funded by the Laboratory Directed Research and Development (LDRD) Program. Sandia is a multi-program laboratory managed and operated by Sandia
Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for
the US Department of Energy’s National Nuclear Security Administration
under Contract No. DE-AC04-94AL85000.
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CHAPTER 8
Computational Studies
of Receptors
MARIA MUSGAARD* AND PHILIP C. BIGGIN*
Department of Biochemistry, University of Oxford, South Parks Road,
Oxford, OX1 3QU, UK
*Email: maria.musgaard@bioch.ox.ac.uk; philip.biggin@bioch.ox.ac.uk
8.1 Introduction
Cell surface receptors are protein molecules that transmit a chemical signal,
for example the message from a neurotransmitter, from one side of a
membrane to the other. There are many different types of cell surface receptors but some of the most studied are the ligand-gated ion channels
(LGICs) and the G-protein coupled receptors (GPCRs). Indeed, the latter
form the largest single class of targets for which known drugs have been
developed against. Due to their centrality to many neurological functions
and their consequent implication in disease, both of these receptor families
have been intensively studied by computational methods.
The types of computational studies of receptors covered in this book all
require a structural model of the molecule under study. Although the
number of available atomistic resolution structural models for membrane
proteins has increased exponentially in recent years, many computational
studies of receptors still rely on some degree of structural prediction. Often,
the structure for the specific receptor the study is aimed at may not be
available. However, if the structure of a related protein has been determined,
homology modelling can generally be applied. Homology modelling is based
on the expectation that the structure is more conserved than the sequence
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Computational Biophysics of Membrane Proteins
Edited by Carmen Domene
r The Royal Society of Chemistry 2017
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1
for homologous proteins and uses the structure of the homologous protein
as a template to which the target sequence is aligned. The most probable
structure for the target sequence can then be predicted, e.g. based on satisfying spatial restraints obtained from the template structure, preferences
extracted from databases of known protein structures and force-field terms
from molecular mechanics.2
In addition to the potential difficulty of obtaining or constructing a
structural model of the receptor of interest, getting the required functional
state can also be very challenging, if possible at all. Indeed, one of the major
problems once one has a structure is to relate it back to the large amount
of functional data, as has been documented for the pentameric Cys-loop
receptors.3 There is no rulebook concerning the interpretation, and it usually
relies on e.g. which ligands, if any, are bound to the protein, along with
specific geometric properties of the structure, which might be consistent
with the prevailing understanding of function.
Many receptors are also modulated by the presence of additional accessory
molecules, which may be proteins or other ligands such as steroids for
example. Ideally, a structure would be solved for every different complex,
but this, at least currently, is not feasible. Furthermore, as many of these
receptors are found in the central nervous system, and are potential drug
targets, we are often interested to know if a ligand can behave as a partial
agonist as this may provide better long term therapeutic options: for many
neurological diseases, a complete activation (or shutdown) of a signalling
process does not provide a therapeutic response and, indeed, may even exacerbate the condition.4
Central to many of these issues is the structural dynamics of these
receptors. Communication across a membrane mediated by a membrane
receptor is dependent on conformational changes, which allow the flow of
information. For LGICs, this means opening of the transmembrane ion
channel such that ions can diffuse across the membrane and cause a response inside the cell. For GPCRs, these conformational changes allow the
binding of different cytosolic proteins, such as G-proteins or beta arrestin,
which can subsequently trigger further downstream signalling pathways.
In this chapter, we will show how computational methods have been used
in recent years to investigate conformational changes in cell surface receptors. Rather than trying to cover the vast literature associated with this area,
we will use specific examples to illustrate the general approaches that are
typically used. In order to do that, we will focus on studies of GPCRs and
LGICs including ATP-gated P2X receptors, ionotropic glutamate receptors
(iGluRs) and Cys-loop receptors. Despite all being LGICs, their architectures
are remarkably different. P2X receptors are trimeric, iGluRs are tetrameric
and Cys-loop receptors are pentameric complexes (Figure 8.1). Nevertheless,
all LGICs display long-range intra-molecular communication. For all of
them, the ligand-binding site is located 40–50 Å away from the ion channel,
yet binding of a ligand, sometimes as small as glycine, can trigger channel
opening and signal transduction.
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Figure 8.1
239
Examples of cell surface receptors. From left: a P2X receptor in the
resting state10 (PDB:4DW0), an ionotropic glutamate receptor in the
resting state18 (PDB:3KG2), a Cys-loop receptor in the resting state54
(PDB:3JAD) and a GPCR in the active state with G-protein bound75
(PDB:3SN6). For the three ligand-gated ion channels, the different chains
are shown in different colours. For the GPCR complex, the receptor itself
is shown in blue, Gas in red, Gb in orange and Gg in purple. Antagonists
bound to the iGluR and the Cys-loop receptors are shown with yellow
surfaces, along with the agonist bound to the GPCR. One of the three
agonist-binding sites of the P2X receptor is indicated with a yellow star.
Membrane positions are taken from ‘‘Orientations of proteins in membranes’’ (opm.phar.umich.edu76).
8.2 Network Models Can Provide Insight into
Large-scale Conformational Changes
Because of the difficulty in obtaining atomistic resolution structures for
membrane proteins, they are frequently solved in only one functional state.
However, as mentioned above, conformational changes are vital to the
function of receptors, and obtaining large-scale conformational changes
with molecular dynamics simulations is often very expensive, if even
feasible, because of the simulation time required to observe a large conformational change. A coarser, but also much cheaper, alternative method
involves using a single structure as the starting point for normal model
analysis (NMA) or related methods, which can identify inherent lowfrequency motions. The directions of motion are obtained by diagonalising
the Hessian matrix, which contains the mass-weighted second derivatives of
the potential energy matrix. The eigenvectors of the Hessian matrix are the
normal modes, which describe the directions of motion, and the corresponding eigenvalues are the squares of the frequencies with which these
motions occur. As the theory behind NMA is based on the harmonic
approximation being valid around a minimum, NMA requires an extensively
and accurately energy-minimised structure, otherwise negative eigenvalues
are obtained and the analysis becomes meaningless.5 For that and other
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reasons, in recent years, the simpler elastic network modelling (ENM)
method has been preferentially used. In ENM, ‘‘bonds’’, described with
harmonic potentials, connect all particles within a given cut-off and, as no
other types of interactions are included, this gives a much simpler potential
energy function. Furthermore, ENM for proteins is often performed with the
Ca atoms only, which heavily reduces the number of dimensions compared to
using all atoms in the structure (indeed, quite often many sidechain atoms are
missing in low resolution structures). A further advantage of ENM, relative to a
full NMA, is that one usually assumes that the crystal structure corresponds to
a minimum on the potential energy surface, such that no complete force-field
based energy minimisation is required. Thus, most often you have a model of
the Ca atoms of your protein where all Ca atoms separated by less than the
cut-off are connected by ‘‘bonds’’, and for this model the normal modes are
easily calculated. The lowest six modes correspond to the rigid-body translational and rotational degrees of freedom and can therefore be disregarded,
but the remaining modes can provide insight into the low frequency (and
hence largest amplitude) motions that the protein will exhibit as part of what
one might refer to as ‘‘inherent dynamics’’ (Figure 8.2).
ENM has been used in various computational studies and has provided
information on both conformational changes potentially functionally important for receptors, e.g. guiding further experimental work, but also as a
Figure 8.2
Elastic network modes illustrated on a Cys-loop receptor (the GlyR alpha
1 in the resting state54 (PDB:3JAD). (A) shows the elastic network illustrated by bonds between all Ca atoms that are separated by less than 8 Å.
(B) shows the lowest, non-trivial elastic network mode for the receptor,
illustrating a twisting motion. The movement goes from the darker
representation to the lighter one, and shows that the transmembrane
domain is twisting clockwise while the extracellular domain is twisting
anticlockwise. The obtained twisting mode was the lowest energy mode
calculated using elNemo5 with default settings.
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tool to investigate whether different structures of related proteins are able to
interconvert or whether the proteins indeed fold into slightly different
structures.
8.3 Network Models to Examine Gating
P2X receptors, P2XRs, are trimeric receptors gated by extracellular ATP.
Seven types of P2X receptors, P2X1–7, are found in humans. Upon activation
by ATP binding, a non-selective cation channel opens and allows the flow of
sodium, potassium and calcium ions across the membrane.6 P2XRs are
implicated in various processes, e.g. the modulation of synaptic transmission, taste, pain sensation and regulation of immune responses. The first
structure of a P2X receptor was resolved in 2009 and captured the zebrafish
P2X4 receptor in the ATP-free resting state with the ion channel closed.7 The
structure showed that the three subunits are organised with three-fold
symmetry (see Figure 8.1). ATP was suggested to bind to the extracellular
domain at three symmetrical sites (‘‘binding jaws’’, each consisting of an
upper and a lower ‘‘jaw’’), located approximately 45 Å from the ion channel,
one at each subunit interface, and cause conformational changes leading to
channel opening.7 However, the structure did not reveal how ATP binding
would trigger channel opening. Du et al.8 studied the gating mechanism
using both normal mode analysis and molecular dynamics (MD) simulations. They treated the closed-channel zebrafish P2X4 receptor with ENM
and generated the 100 lowest modes, searching for modes that would show
channel opening along with closure of the binding jaw. Two modes with
these characteristics were identified. Opening of the pore was caused by an
outward motion of the N-terminal half of the channel-lining helix.8 The
other helix also moved outward, and the helical motions appeared coupled
to the extracellular domain, for which some of the 14 beta strands underwent rotation that lead to a closure of the ATP binding site. The results from
the ENM analysis were supported by MD simulation of the extracellular
domain, where it was observed that the membrane-proximal ends of b1 and
b14 were pulled apart, consistent with opening of the pore.
Jiang et al.9 generated a homology model of rat P2X2 based on the closed
zebrafish structure. In this model, two histidine residues that are believed to
form a modulatory zinc-binding site were located too far away from each
other. To try to resolve this paradox, the authors investigated conformational
changes used ENM analysis. The 10 lowest-energy non-trivial modes were
included, and some of these modes did indicate pore opening, though it is
unclear whether this would be sufficient for ion flow. Instead, the changes
that happen upon ATP-binding were examined by docking ATP into the
binding site and recalculating the modes. In one of these modes, the distance between the two histidine residues was shortened and it was possible
to model in a zinc ion between the two histidine residues in the perturbed
structure. Thus, this mode could indeed be relevant for the overall dynamics
of the P2X receptors despite not revealing significant pore opening. Based on
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a comparison between the resting state and the structure perturbed along
the identified normal mode, a new zinc site was designed that would not be
present in the closed state but would form upon closure of the binding jaws.
This new site was tested experimentally by first abolishing the native zinc
site followed by inserting new histidine residues and regaining zinc potentiation, supporting the motions observed in the given normal mode.
Thus, closure of the ATP-binding jaw favours pore opening.9
An open state of P2X4 from zebrafish was determined in 201210 along with
a new apo structure with slightly higher resolution than the previous one.
The conformational change from the apo to the ATP-bound open state included expansion of the extracellular domain close to the transmembrane
domain (TMD), as predicted by the ENM analysis,8 pulling the helices open
and allowing closure around ATP. Transmembrane helices rearrange in an
iris-like motion when opening the channel.10 ATP binding promotes closure
of the binding jaw, partly by closure of the top of the binding site, but also by
pulling the lower part of the binding site upwards.
The Cys-loop receptor family includes neuronal acetylcholine (ACh) receptors, GABAA receptors, 5-hydroxytryptamine (5-HT3) receptors and glycine
receptors.11 Structurally, the pentameric Cys-loop receptors are composed of
a large extracellular N-terminal ligand-binding domain, a transmembrane
ion channel and an intracellular domain (Figure 8.1). The agonist binding
site is approximately 40 Å from the membrane.12 The transmembrane domain is constructed from four TM helices from each subunit with the
transmembrane helix M2 lining the channel. The first reasonably highresolution structural information for full-length Cys-loop receptors came
from a 4 Å electron microscopy structure of the AChR from Torpedo (electric
ray) in 2005.12 A few years later, structures of bacterial homologues of
Cys-loop receptors were published; the ELIC and GLIC structures.13–15 The
first, presumably closed, structure of ELIC was determined to a resolution of
3.3 Å,15 then a potentially open state of the proton-gated GLIC at 3.1 Å
resolution13 and another, similar, potentially open GLIC structure at 2.9 Å
resolution.14 It was noted that, if assuming the GLIC structures were open
and the ELIC one closed, a twisting mode with the extracellular domain
twisting in one direction and the transmembrane domain in the other,
similar to the motion associated with the wringing of a wet towel, seemed to
be involved in the transition between the open and the closed states.14 This
type of motion had been suggested previously on the back of normal mode
analysis performed on models of homomeric a7 nicotinic acetylcholine
receptors,16 suggesting that major motions can be captured quite well,
though detailed analysis tends to show that the calculated modes cannot
quite account for all the transitions observed structurally.14
8.4 Network Models to Compare Dynamics
Individual structures of domains from the extracellular part of iGluRs started to appear in 1998, the first one being a monomeric structure of the
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ligand binding domain (LBD) from the GluA2 a-amino-3-hydroxy-5-methyl-4isoxazolepropionic acid (AMPA) receptor from rat in complex with the partial
agonist kainate.17 However, until 2009, computational studies of iGluRs
were limited to studying the extracellular part at the domain level as no
structure encompassing both the extracellular domains and the transmembrane channel domain had been determined. The first full-length
(C-terminal part removed) iGluR crystal structure of a slightly modified rat
GluA2 receptor was published in 2009.18 In 2014, a number of structures
followed, including rat GluA2 crystallised in different functional states,19 rat
GluA2 and rat GluK2 determined by cryo-electron microscopy in different
functional states,20 along with crystal structures of heteromeric N-methyl-Dasparate (NMDA) receptors from Xenopus laevis (African clawed frog)21 and
rat.22 The architecture observed in these structures is overall similar, however, when comparing AMPA- and NMDA-selective iGluRs (AMPARs and
NMDARs, respectively), it is clear that the extracellular part is more compact
for NMDARs21,22 (see Figure 8.3). This leads to the question of whether this
indicates actual architectural differences between AMPARs or NMDARs or
whether the receptors are captured in potentially different functional states.
It was suggested that the higher level of compaction of the extracellular
domains would allow for better allosteric communication from the amino
terminal domains to the rest of the receptor in NMDARs relative to
AMPARs.21,22 To investigate how different the structures are under more
dynamic conditions, Dutta et al.23 used ENM analysis (Ca atoms) to study the
inherent dynamics in the structures of the full-length AMPA and NMDA
receptors. The aim was to investigate whether the difference in packing
between the amino terminal domains and the LBDs would suggest differing
dynamic patterns. It was found that several of the low-frequency modes were
Figure 8.3
The AMPA GluA2 receptor (left, PDB:3KG218) is much more extended and
loosely packed than the NMDA GluN1a/GluN2B receptor (right,
PDB:4PE522). The proteins are rotated so that their LBDs are oriented
in the same way. Membrane positions are taken from ‘‘Orientations of
Proteins in Membranes’’.76
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conserved between the AMPA and the NMDA receptor and, for both structures, strong coupling was found both within and between domains, no
matter whether the domains packed loosely (AMPAR) or more compactly
(NMDAR).23 This supports a more general hypothesis that dynamics is more
dependent on the overall architecture than on the specific domain interactions. Furthermore, a number of modes were identified which could be
linked to the structural changes observed from crystal structures20,21 as e.g.
intra- and inter-domain bending and rotation or contraction and expansion
between the domains.23 Such modes could potentially be used to generate
alternative structures perturbed along some of the mode vectors for further
simulation studies.
Similarities in ENMs derived from different structures can be quantified
by analysing the overlap. The overlap is found by projecting the mode
vector onto the vector describing the conformational change. The overlap
gives the contribution of the individual mode to the conformational
change. Comparing NMDA and AMPA receptor ENMs, Dutta et al. also
found that, by combining a reasonably small number of low-energy modes,
the two receptor structures can in fact interconvert despite an root-meansquare deviation (RMSD) between the two crystal structures of more than
17 Å.23 The first 12 modes of the AMPA receptor combine to describe 80%
of the transition of the NMDA receptor conformation, whereas the reverse
transition requires around 23 modes for the NMDAR receptor. This
difference is probably a reflection of the previously found result that an
open-to-closed conformational change is generally better described by lowenergy normal modes than a closed-to-open change.24 Overall, these results
together suggest that the two structures may in fact just reflect alternative
conformations rather than suggest an inherent difference in the architecture between AMPARs and NMDARs.23
8.5 Network Models to Suggest Novel Mechanisms
for Modulation
Network models have furthermore been used to generate new ideas about
interactions. For example, ENM analysis for iGluRs has also been used to
suggest some surprising major conformational changes during the functional cycle.25 The amino-terminal domain, located furthest away from the
membrane, has been suggested to interact with transmembrane AMPA receptor regulatory proteins (TARPs).26 For the amino-terminal domain to
reach TARPs, which sit mainly in the membrane with a smaller extracellular
domain,27 a major deformation of the protein would be required. However,
ENM analysis for full length structures of an AMPA receptor have revealed
low-energy normal modes with large bending motions, which could allow for
interactions between the amino-terminal in GluA2 and membrane-bound
auxiliary subunits.25 Thus, there may be much larger conformational
changes happening for iGluRs than currently anticipated,19,20 and thereby
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ways in which the amino-terminal domain can interact with the TARPs,
though experimental structural evidence is currently lacking.
ENM approaches have also been applied to investigate the influence of
oligomerisation on the dynamics of GPCRs. Niv et al.28 examined the behavior of Rhodopsin and found that there was a significant perturbation of
the normal modes of the monomer upon oligomerisation. They also found
the highest inter-residue positive correlation at the interfaces between
protomers and were able to suggest experimentally testable hypothesis
concerning putative oligomeric arrangements. The precise role and nature
of GPCR dimerisation is complex and likely to be concentration and lipiddependent among other things.29,30 More recent work has even suggested
that heteromerisation might be important.31 Modulation of the underlying
motions by the formation of higher-order assemblies may become an increasingly important area of investigation. The underlying motions have
been explored extensively with MD (see below) but ENM methods have also
been used. Kolan et al.32 found that, for a series of GPCRs, modes could be
identified that link the contraction of the extracellular ligand-binding
cavity with the expansion of the intracellular G-protein binding cavity,
consistent with hypotheses concerning how the underling dynamics
in these receptors relates to signaling. Interestingly, they found that the
normal modes of rhodopsin do not correlate with the motion of other
GPCR family members, suggesting that rhodopsin may not be a good
generic template.
8.6 Molecular Dynamics to Aid Crystallographic
Interpretation
Whereas the computationally cheap ENM methods can be useful for some
first insight into the dynamics of a receptor, as illustrated above, the much
more costly MD simulations allow for a much broader range of studies and
ultimately can provide free energy landscapes, giving a much more detailed
view. As mentioned in the Introduction, one of the key issues is mapping the
structure obtained back to functional states (as defined predominantly by
electrophysiological experiments). In that sense, MD can provide a good
initial view as to whether a crystal structure is conformationally stable or not,
and indeed MD simulations have been shown to help resolve ambiguity
when multiple structures appear, as shown recently for the NTR1 transporter
protein.33 As we alluded to earlier, MD simulations were performed on the
GLIC structure, which was crystallised with six detergent molecules inside
the pore.14 To check whether the open state would still be stable when the
detergent molecules were removed, a simulation was performed in which
the protein without the detergent molecules was inserted into a lipid bilayer.
The conformation was stable throughout the simulation time of 20 ns and
this was interpreted to mean that this conformation was likely to represent
an open state.14
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As well as providing information on conformational stability, atomistic
MD simulations have been used to characterise receptors in terms of their
most likely functional state – something that is not always obvious from the
geometry alone. For example, in the case of Cys-loop receptors, even when
the crystallisation conditions reflect those expected to yield a desensitised
state, it was not clear whether the narrowest constriction in the pore was
consistent with a closed channel. Beckstein and colleagues explored this first
by exploring the underlying physics of the system. They constructed an
artificial toy system that allowed them to vary the geometry and charge of a
pore that was of similar dimensions to the constriction in nicotinic acetylcholine receptors. They found that, at geometries and charges similar to
those found in the resting state structure of the nicotinic acetylcholine
receptor, water molecules would not pass through, consistent with a closed
channel. They also demonstrated that both small (1–2 Å) increases in
diameter and/or increasing polarity could open the channel, leading to the
hydrophobic gating hypothesis34 (Figure 8.4). They have since applied this to
the actual nicotinic acetylcholine structure and the results are consistent.35
A similar approach has also been applied by Zhu and Hummer,36 who
performed umbrella sampling of a sodium ion though the transmembrane
pore of the ‘‘open’’ GLIC structure and through a homology model of the
transmembrane part of GLIC in the closed state, constructed based on the
structure of ELIC. Based on these simulations, the free energy of a single
sodium ion moving through the channel was calculated36 and it was found
that, for the open state of the channel, a free energy barrier of less than
4 kcal mol1 is located in the middle of the channel. For the model of the
closed state, a barrier of more than 10 kcal mol1 was identified. Thus, the
properties of the crystal structures with GLIC being in an open state and
ELIC in a closed state could be confirmed by free-energy calculations.36 As
more and more structures become available, knowing which functional state
the crystal (or electron microscopy) structure corresponds to is likely to
become increasingly important.37
Relating structural information to functional states has been an issue for
GPCRs as well. Indeed, extensive unbiased simulations of the b2-adrenergic
receptor38 led to the conclusion that agonists work by stabilising a selection
Figure 8.4
Principle of a hydrophobic gate. The black spheres symbolise two rings
of hydrophobic residues, for example isoleucine. If they are in close
enough proximity (which may well still appear physically open) then the
channel will exclude water (and by inference, ions) from that region of
the pore and the channel will effectively close.
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of pre-existing states. A corollary of this is that many of these states may not
be amenable to crystallographic techniques and thus, the use of MD in this
area is likely to increase. More recently, MD has been exploited to provide
insight as to how mutations can provide thermostabilisation in GPCRs.39 It
appears that thermostable mutants are less flexible than wild-type proteins
and that most of the thermostabilising effect comes from improved interhelical packing compared to the wild-type receptors.
8.7 Molecular Dynamics to Move between States
In theory, if one starts say, from a resting state, ‘‘adds’’ (in silico) an agonist
and subsequently runs a sufficiently long MD simulation, likely milliseconds, it should be possible to observe, directly, the precise conformational change that occurs. This has been attempted multiple times, but the
stochastic nature of the process and the timescales involved prevent this
from becoming a mainstream approach (at the moment). Nevertheless,
various efforts have been reported. To study the closing mechanism of GLIC,
which is initiated by an increase in pH, Nury et al. performed extended MD
simulations40 starting from the open-channel crystal structure14 inserted
into a lipid bilayer. After equilibration with amino acid protonation states
corresponding to pH 4.6, where the channel should remain open, a change
in pH was mimicked by instantaneously changing the protonation states of
all amino acids to their standard protonation state at pH 7. A 1 ms long
simulation was then performed at neutral pH to try and observe the channel
returning to a resting conformation. Despite the extended simulation time,
the channel did not completely reach the resting state. However, the simulation did capture some major changes. Within 50 ns, one of the five subunits undergoes a conformational change toward a more closed state, and
after 450 ns its neighbouring subunit undergoes a similar change.40 The
subunit undergoing the first motion shows the channel-lining helix moving
into the pore at around 25 ns, which more or less closes the channel. The
observed transition, despite not capturing the full motion, does reveal an
overall twist for which the extracellular domain (ECD) and the TMD move in
opposite directions,40 as observed when comparing the open GLIC and the
closed ELIC14 and predicted from normal mode approaches. At around
50 ns, all the channel-lining helices twist around the pore axis, and furthermore, at around 400–600 ns, the channel-lining helices of four of the
subunits undergo a tilt in their extracellular end, moving towards the pore,
while the channel-lining helix of the last subunit tilts away from the pore.
The continuous water occupancy of the channel is disturbed quite early on,
suggesting that the channel holds a hydrophobic gate, and in a similar way,
cations are observed to accumulate within the channel, but leave a gating
region completely free of cation occupancy.40
Zhu and Hummer illustrated how the two potential end-points in the
transition, GLIC and a closed-channel homology model of GLIC based on
ELIC, might be connected by a pathway using various computational
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36
simulation tools, including mixed elastic network models and MD simulations. The minimum free energy path between the open and the closed
state was calculated, and the simulations suggested that the channel closure
happens as a two-stage process with an iris-like motion of the pore-lining
helices. The channel-lining M2 helices tilt in a way such that the extracellular ends of the helices move towards the centre of the channel with a
clockwise rotation, whereas the intracellular ends are rotating in the anticlockwise orientation.36 This is in agreement with the rotation suggested
from the crystal structures14 and observed in other simulations.40 The
extracellular ends of the channel-lining helices move faster than the intracellular ends.36 A gate similar to the one observed by Nury et al.,40 consisting
of two rings of isoleucine residues inside the channel, was reported and
observed to be fully hydrated in the open channel state versus completely dry
for the closed channel36 (see Figure 8.4). For the isolated transmembrane
channel domain, the closed state is energetically favoured,36 which was
probably to be expected since channel opening is gated by protons. In
agreement with this, unrestrained MD simulations of the GLIC open state
transmembrane domain relaxed towards the closed state in less than
100 ns.36
For glutamate receptors, MD simulations have been used to study the
process of channel opening for both an AMPA and an NMDA receptor. Based
on the first full-length structure of the GluA2 AMPA receptor,18 Dong and
Zhou studied the transitions from the apo, closed-channel resting state
through the activated, open-channel state to the ligand-bound, closedchannel desensitised state by targeted MD simulations. They ignored the
amino-terminal domain of the receptor to reduce the cost of the simulations,
and thus simulated a system consisting of the tetrameric TMD-LBD complex
embedded in a solvated lipid bilayer.41 As structures for the LBD dimer in
presumably the active and at least part-desensitised state were known at the
time of study, atoms in the LBD could be ‘‘targeted’’ in the simulations,
meaning extra forces were imposed on these atoms, encouraging a conformational change towards the active and the desensitised state, respectively. The assumption was that, upon changes in the LBD, the TMD would
follow along as a response to these changes and obtain a structure relevant
to the given state.41 The LBD was forced to move from the resting conformation to the active one and the response of the TMD was studied over
the course of around 100 ns.41 The pull in the linkers between the LBD and
the TMD appears to promote opening-like motions of the TMD. The three
most restricted positions towards the extracellular side of the membrane
inside the channel seem to open up in response to the conformational
changes enforced on the LBD. However, the channel diameter is still rather
restricted and does not appear wide enough to be consistent with an open
channel. Upon activation, the receptor appears to contract along the z-axis,41
which is in agreement with more recent structural data.19 However, other
more recent data also suggests that the LBD-TMD linkers undergo asymmetric rearrangement,20 which is not accounted for by this model of the
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active state. Targeted MD simulations of the LBD to a model of the
desensitised state, in which the upper part of the dimer interfaces in the LBD
open up, reverted the channel back to a closed state. The receptor overall
expanded along the z-axis again and the extracellular end of the channellining helices moved inwards towards the centre of the pore, closing off the
channel.41
In a similar fashion, channel gating has been studied for a heterometic
NMDA receptor42 prior to structural data of the NMDARs. A full-length
homology model in the resting state was generated based on the first GluA2
structure,18 combined with information from structures of the LBD of
NMDARs43,44 and a model of the LBD tetramer in the active state. Targeted
MD simulations pulling the LBD tetramer from the resting state to the
active state were performed over 20 ns, after which the LBD structure was
restrained to the agonist-bound conformation while the TMD was expected
to adapt to the active, open-channel state.42 The channel did in fact open, at
least partly, as a response to the LDB tetramer changing conformation with
the minimum pore diameter increasing. Again, a reduction in the length of
the receptor along the z-axis was observed, and the overall opening motions
are similar to the ones for the AMPAR receptor, which is probably not that
surprising considering the template and the procedure for the study. An
open-channel NMDA receptor has yet to be determined experimentally.
8.8 Molecular Dynamics to Refine Working Models
In the absence of direct structural information, homology modelling is often
used to provide at least an initial model. MD should be able to refine these
models closer to what one might expect for the native state, but the evidence
that this is definitely the case is somewhat lacking, in part because it is not
clear whether there is a sampling problem or a limitation of current forcefields. Raval et al.45 have argued, through extensive sampling, that the
limitations of the force-fields are the most likely problem. Nevertheless, this
approach may still provide useful insight at least in terms of being able to
generate an experimentally testable hypothesis, especially in the case of
GPCRs where there are many receptor subtypes for which no structure has
been solved.46,47 It may have particular value when combined with additional information48 or when the state of the receptor needs to be considered more carefully.49 Even if one is fortunate enough to have a crystal
structure of the GPCR of interest, it is frequently solved in only one state and
therefore, the working model may necessitate the modelling of other states,
which may be relevant for recognition by different types of ligands. MD is the
only tool that can help in this respect.50 It is also clear that MD can provide
useful detailed analysis on a timescale consistent with drug-discovery
programs.51
The usual test of whether the refinement/modelling process has been
useful is via identification of the critical residues whose effect on function
can be tested via site-directed mutagenesis. For example, a homopentameric
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Chapter 8
homology model of hGlyR a1 was constructed based on the glutamate-gated
chloride channel structure from C. elegans,52 and MD simulations were
performed in both the apo, agonist (glycine) and antagonist (strychnine)
bound states.53 In the glycine bound simulations, it was noted that, early on
in the simulations, a water molecule would enter the pocket and help to
stabilise the glycine within the binding pocket (Figure 8.5A). This binding
mode identified a residue, Ser129 (Figure 8.5A), which had not previously
been implicated in agonist binding. Site-directed mutagenesis and electrophysiological recordings confirmed the prediction made by MD, that this
residue was critical to the efficacy of the ligand.53 Subsequently, several
structures have been solved in different states for the glycine receptor (from
zebrafish54 and human55). When comparing the refined model of the glycine
bound hGlyR with the glycine-bound structure, the agreement is remarkably
good (Figure 8.5B and C).
Figure 8.5
(A) The important water molecule between glycine and Glu157 and the
extended hydrogen-bonding network in the hGlyR model. The glycine
agonist is shown with green carbon atoms. Hydrogen bonds are indicated with black lines. (B and C) Comparison of a glycine-bound homology model53 of hGlyR a1, based on the C. elegans GluCl structure, after
150 ns of unbiased MD simulation (dark colours) and the glycine-bound
zebrafish GlyR structure (brighter colours) (PDB: 3JAE54). The glycine
ligands bound to the homology model are shown in yellow surface
representation. The line in (B) indicates the clipping plane applied in
(C) to ease the visualisation of the ligand-binding region.
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8.9 Molecular Dynamics to Explain the Effects of
Ions and Water
One of the key advantages of MD as a technique is the ability to investigate
the fine detail of systems. It has been known for a long time that changes in
conformation are heavily influenced by the behaviour of water and ions. MD
has been used to investigate the role of sodium 56 within the transmembrane
region of GPCRs.57–60 The role of both cations and anions as modulatory
agents within GPCRs is receiving increasing attention, though it is still unclear what the physiological role might be.61
A number of studies focusing on properties of the LBD of iGluRs have also
been performed. As mentioned, the first structural insight for iGluRs was a
monomeric structure of the LBD construct,17 and many studies have illustrated that studying the LBD construct on the domain level, in the monomeric
or dimeric state (Figure 8.6), is of value to the iGluR field and, of course,
computationally much cheaper than studying the full length construct.
Computational studies have proven useful to understand the regulatory
mechanism of extracellular monovalent ions, especially with regards to
kainate receptors. It has been shown experimentally that the GluK2 kainate
receptor subtype cannot activate in the absence of monovalent extracellular
ions.62 An anion binding site nested at the interface of the LBD dimer was
identified crystallographically, presumably stabilising the dimer interface,63
followed later by the identification of two symmetrical sodium sites, each
Figure 8.6
iGluR LBD monomer (left) and dimer (right). Shown is the glutamatebound GluK2 kainate receptor (PDB 3G3F77). The bound glutamate is
shown in yellow surface representation. Left: the two lobes of the LBD
monomer are shown in orange and red, and the residues used to define
the 2D order parameter in umbrella sampling are shown as a blue (x1)
and a purple (x2) pair, respectively.72 For each ‘‘group’’ the centre of mass
between residues in the upper lobe and the lower lobe are used for the
order parameter. The dimer on the right shows the regulatory sodium
(purple) and chloride (green) ions bound at the apex of the interface. The
position of the engineered disulphide bond mentioned in the text is
shown with grey spheres for the Ca atoms.
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Chapter 8
located 8 Å from the anion, at the apex of the LBD dimer interface, residing
in two electronegative pockets64 (Figure 8.6). When measuring the desensitisation rates, sodium provided the largest stabilisation of the active state,
followed by lithium, potassium, ammonium, rubidium and caesium in the
stated order, and all ions in crystal structures have been observed to bind to
the cation pocket, which experimentally showed a weak preference for sodium.64 MD simulations were employed first to understand the interplay
between the cations and the anions; unbiased simulations suggested that
anion binding was stabilised by occupancy of the cation site,64 and that the
binding of chloride was destabilised by the dissociation of cations from the
cation pocket. Potential of mean force calculations (via umbrella sampling)
for the unbinding of the interface-bound chloride ion in the presence and
absence of bound sodium ions, supported the claim that the presence of
sodium stabilised the binding of chloride.65
A later study calculated the relative binding free energies for monovalent
cations by thermodynamic integration, for which the rank order was in close
agreement with the experimental ordering.65 In general, the cation with
higher affinity, as determined from thermodynamic integration,65 is also the
best stabiliser of the active state,64 the exception being lithium, which is
predicted to bind stronger than sodium, even though sodium experimentally
shows the largest efficacy. It is a possibility that lithium does in fact bind
stronger, but that the stabilising effect is larger when sodium is bound.
The binding of sodium ions to the interfacial binding sites have later been
shown to be a key regulator to the onset of desensitisation,66 supporting the
view that cation site occupancy prevents the onset of desensitisation.
Simulations illustrated how two mutants, previously believed to prevent
desensitisation, showed very different dynamics for the LBD dimer, explaining their different single channel phenotypes.66 For a non-desensitising
lysine mutant,67 sodium ion binding was observed to be destabilised and,
after sodium ion dissociation, the lysine sidechain was observed to reach
across the interface and insert the charged amine group to the cation site in
unbiased MD simulations,66 in agreement with the structural information.68
In this way, the interface was tethered together and the receptor showed a
non-desensitising phenotype. Another interface tether, a double-cysteine
mutant,69 however, showed a very different phenotype; it only showed few
and sporadic channel openings over the course of agonist application, which
was rather surprising considering that the crystal structure suggested that
the interface was locked together with a disulphide bond69 (see position in
Figure 8.6). However, unbiased MD simulations showed how this tethered
structure quickly relaxed to a state for which the interface is more open,
despite the presence of the disulphide bond.66 Furthermore, the modulatory
ions did not obtain stable binding at the interface. Two further interface
mutants were studied; one directly in the cation binding site and one located
further down at the interface. Unbiased MD simulations suggested that, for
the sodium site mutation, which removed one of the acidic ligands from
sodium, exchanging it for glycine, promoted sodium unbinding whereas the
interface remained fairly stable and chloride remained bound.66
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For the other mutation, a leucine to a cysteine, which had no direct relation to the sodium site, it was seen that a hydrophobic core in the middle
of the interface was heavily destabilised, causing water to disturb the
interface packing and push out the sodium ions and later the chloride ion,
leading to an opening motion of the interface. These observations were in
agreement with the functional experiments, for which both mutants showed
to be inactive or to have only very little activity, despite good surface expression.66 Most recently, MD simulations of the GluA2 AMPA receptor have
suggested that the experimentally observed stabilising effect of lithium, but
no other cations, on the sustained activity of GluA2 can be explained by
lithium binding to the corresponding site in GluA2 and hereby stabilising
other cross-dimer interactions.70 Removing these cross-dimer interactions
was then shown experimentally to abolish the stabilising effect of lithium,
underlining the strong predictive power of MD simulations in the quest for a
better understanding of the fine details in the mechanisms of membrane
receptors.70
8.10 Molecular Dynamics to Quantify Free Energy
Requirements
Agonist binding and the accompanying conformational changes have
also been studied in fine detail for iGluRs by examining the free energy
landscapes obtained from umbrella sampling simulations of the solvated
LBD monomer.71–73 A two-dimensional order parameter was used for the
umbrella sampling, more specifically two inter-lobe distances, both related
to the opening of the clamshell-shaped bi-lobed LBD, one at each side of the
ligand-binding cleft (x1 and x2, Figure 8.6). Calculations for an antagonist
(DNQX) show a rather well-defined minimum at a position with a relatively
open cleft, and the free energy landscape suggests that more than
9 kcal mol1 would be required to close the cleft to a level similar to the
agonist-bound state, supporting that the particular antagonist hinders
clamshell closure and thereby channel opening (Figure 8.7).72 For GluA2
AMPAR receptors in the apo state, a broad basin is observed, indicating
high flexibility (Figure 8.7). Water molecules occupy the ligand binding site
in the apo state and likely stabilise the open-cleft apo structure.72 Specific
positions of water molecules in the agonist binding site have been studied
independently and shown that a distinct water molecule appear to be conserved for all three main iGluR families, and that this water molecule binds
to both apo and agonist bound conformations and contributes favourably to
the interaction energy in agonist bound states.74 The glutamate bound
GluA2 LBD monomer has a narrow and deep basin with almost 9 kcal mol1
released relative to the open-cleft glutamate-bound state (Figure 8.7),
meaning this free energy can go towards pulling the ion pore open.72
The free energy has furthermore been split into the contribution from ligand binding to the open-cleft apo state and the contribution from clamshell
cleft closure, the sum of which gives the absolute binding free energy for
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Figure 8.7
Chapter 8
Free energy landscapes for the conformational changes in the AMPA
receptor GluA2 LBD monomer as described by the order parameter
(x1, x2). (A) 2D and (B) 1D potential of mean force (PMF) plots are
shown for the apo, DNQX- and glutamate-bound LBD monomer. In (A),
each colour contour corresponds to 1 kcal mol1. The positions marked
with ‘‘X’’ indicate X-ray crystal structure conformations (only chain A
from each crystal structure is marked; the other chains are positioned
very close to chain A). In (B), the PMF along the reduced coordinate
x12 ¼ (x1 þ x2)/2 is shown. The dashed lines indicate the X-ray crystal
structure conformations.
Reprinted from A. Lau and B. Roux, The Free Energy Landscapes
Governing Conformational Changes in a Glutamate Receptor LigandBinding Domain, Structure, 15, 1203–1214. Copyright 2007 with
permission from Elsevier.
both agonists, partial agonists and antagonists.73 The global free energy
minima were, for all nine tested ligands, in good agreement with the crystal
structures of the GluA2-ligand complexes. For kainate, which is a weak
partial agonist at AMPARs, the free energy landscape suggested that the LBD
resides in a relatively open conformation, with rare transitions to more
closed conformations that would trigger channel opening. Transferring
glutamate from the bulk solvent to the open-cleft LBD was unfavourable;
however, this was fully compensated by a considerable gain in free energy
from clamshell closure and for the nine ligands under study, a very strong
correlation was obtained between the calculated binding affinities and
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73
the experimentally determined ones. Similar calculations have been performed for three different subunits of NMDA receptors, two glycine binding
subunits, GluN1 and GluN3, and a glutamate binding subunit, GluN2, with
free energy landscapes calculated for both the apo state and the agonist
bound state and compared to the corresponding calculations for GluA2.71 The
results show that apo-state NMDAR LBDs can sample a much wider range of
cleft motion than GluA2, and that they can even visit the fully closed cleft
conformation expected to lead to channel activation. This suggests that
NMDARs might bind the agonists using conformational selection rather than
by the induced fit mechanism observed for GluA2.71 In the agonist bound
states, the basins become much narrower, as expected, showing that agonist
binding stabilises the closed cleft conformation, promoting channel opening.
8.11 Conclusions
Computational methods have made a large and extremely valuable contribution to our interpretation of how many receptor proteins work at the
atomistic level. This information is vital not only for our general understanding of what links structure to function, but is crucial if we are to adopt a
rational approach to future drug design. After all, how can we realistically
design new compounds if we are not even clear about which functional state
a structure corresponds to? With increased computing power becoming
evermore available, along with rapid developments in algorithms, it seems
likely that computational approaches such as those outlined in this chapter
will play an increasingly important role.
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00259
Subject Index
a-helical bundles, 7–8
b-barrels, 8–9
ABC. See ATP-binding cassette (ABC)
transporters
ab initio molecular dynamics,
33–34
actin cytoskeleton, and GPCRs,
147–149
adaptive biasing force method,
78–80
additive force fields, 22–24
alchemical transformations, 70–73
AMBER. See Assisted Model Building
with Energy Requirement
(AMBER)
anesthetic ligands, and receptors,
82–84
Assisted Model Building with Energy
Requirement (AMBER), 22
ATP-binding cassette (ABC)
transporters, 212–215
Brownian dynamics, 124–127
methods to include atomic
detail, 129–131
CGenFF. See CHARMM General FF
(CGenFF)
channel gating, 181–184
channels, 10–11
CHARMM. See Chemistry at Harvard
Molecular Mechanics (CHARMM)
CHARMM General FF (CGenFF), 44
Chemistry at Harvard Molecular
Mechanics (CHARMM), 22
classical molecular dynamics
additive force fields, 22–24
applications, 29–30
overview, 20–22
polarisable force fields, 24–25
practical and technical
considerations, 25–29
coarse-grained (CG) molecular
dynamics, 30–33, 153–154
computational description, of global
structural transitions in
ATP-binding cassette (ABC)
transporters, 212–215
nonequilibrium simulation of
structural changes, 210–212
overview, 209–210
computational studies of receptors
molecular dynamics
to aid crystallographic
interpretation, 245–247
to explain effects of ions
and water, 251–253
to move between states,
247–249
to quantify free energy
requirements, 253–255
to refine working models,
249–250
and network models, 239–241
to compare dynamics,
242–244
to examine gating, 241–242
to suggest novel
mechanisms for
modulation, 244–245
overview, 237–239
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260
computer simulation of ion
channels
channel blockage by small
molecules, 187–189
channel gating, 181–184
ion permeation, 169–173
ion selectivity
Na1/Ca12, 174–177
Na1/K1, 177–181
overview, 161–165
questions and timescales,
165–168
toxin–channel interactions,
184–187
conformational transitions, in
receptors, 86–88
continuum distributions, and ion
channels
atomic detail in, 132–133
improvements of electrolytes,
119–122
overview, 111–112
Poisson–Boltzmann (PB)
equation, 112–117
Poisson–Nernst–Planck (PNP)
differential equations,
117–119
crystallographic interpretation,
and molecular dynamics,
245–247
Cys-loop receptors, 238
enhanced sampling
and free energy differences,
34–40, 80–81
and lipid–protein interactions,
154–155
enzymes, 14
ergodicity, 63–65
FCS. See fluorescence correlation
spectroscopy (FCS)
fluorescence correlation
spectroscopy (FCS), 143
fluorescence recovery after
photobleaching (FRAP), 143,
147–149
Subject Index
fluorescence resonance energy
transfer (FRET), 142–147
FRAP. See fluorescence recovery after
photobleaching (FRAP)
free energies, of ions across
channels, 84–86
free energy differences, for
membrane receptors
applications of
binding of anesthetic
ligands to receptors,
82–84
conformational
transitions in
receptors, 86–88
free energies of ions
across channels, 84–86
and enhanced sampling,
34–40, 80–81
ergodicity, 63–65
non-equilibrium properties
Leucine–Serine channel,
93–96
overview, 88–90
theoretical background,
90–93
overview, 59–60
parametric formulation of,
60–63
probability distribution
methods, 73–75
thermodynamic integration
adaptive biasing
force method, 78–80
theoretical background,
75–78
transition coordinate, 63–65
variance reduction strategies,
63–65
free energy perturbation methods
alchemical transformations,
70–73
theoretical background, 65–70
free energy requirements, and
molecular dynamics, 253–255
FRET. See fluorescence resonance
energy transfer (FRET)
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00259
Subject Index
gating, and network models, 241–242
gating elements, in
neurotransmitter transporter,
206–209
GPCRs. See G-protein coupled
receptors (GPCRs)
G-protein coupled receptors
(GPCRs), 237
iGluRs. See ionotropic glutamate
receptors (iGluRs)
ion channels
blockage by small molecules,
187–189
channel gating, 181–184
and continuum distributions
improvements of
electrolytes, 119–122
overview, 111–112
Poisson–Boltzmann (PB)
equation, 112–117
Poisson–Nernst–Planck
(PNP) differential
equations, 117–119
ion permeation, 169–173
particle-based methods of
Brownian dynamics,
124–127
Monte Carlo (MC)
simulations, 127–128
overview, 122–124
ionotropic glutamate receptors
(iGluRs), 238
ion permeation, 169–173
ion selectivity
Na1/Ca12, 174–177
Na1/K1, 177–181
Leucine–Serine channel, 93–96
LGICs. See ligand-gated ion channels
(LGICs)
ligand-gated ion channels (LGICs),
237
lipid polymorphism, 5–6
lipid–protein interactions
of actin cytoskeleton with
GPCRs, 147–149
261
computational approaches to
study membrane
organization, 149–155
atomistic simulations
elucidating, 151–153
coarse-grain methods,
153–154
enhanced sampling
methods, 154–155
simulating single
component and
multi-component
bilayers, 151
lipid frontier
computational probes of,
220–221
simulating, 219–220
of melittin with membrane
cholesterol utilizing FRET,
144–147
membranes and
components of, 138–139
mechanisms for, 140–142
organization and
function, 139–140
overview, 137–138
range of time scales,
142–143
localized transporter motions with
equilibrium MD
gating elements in
neurotransmitter
transporter, 206–209
overview, 205–206
substrate-induced structural
changes of antiporter, 206
membrane proteins
classes of
overview, 7
a-helical bundles, 7–8
b-barrels, 8–9
complexes, 15–17
description, 2–5
functions of, 9–15
channels, 10–11
enzymes, 14
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262
membrane proteins (continued)
receptors, 14–15
transporters, 11–14
lipid polymorphism, 5–6
overview, 1–2
membranes, and lipid–protein
interactions
components of, 138–139
mechanisms for, 140–142
non-specific effects, 142
specific effects, 140–142
organization, 149–155
atomistic simulations
elucidating, 151–153
coarse-grain methods,
153–154
enhanced sampling
methods, 154–155
and function, 139–140
simulating single
component and
multi-component
bilayers, 151
overview, 137–138
range of time scales,
142–143
membrane transport
computational description of
global structural transitions
in
ATP-binding cassette
(ABC) transporters,
212–215
nonequilibrium
simulation of
structural changes,
210–212
overview, 209–210
lipid frontier
computational probes of
lipid–protein
interactions,
220–221
simulating lipid–protein
interactions, 219–220
Subject Index
and localized transporter
motions with equilibrium
MD
gating elements in
neurotransmitter
transporter, 206–209
overview, 205–206
substrate-induced
structural changes of
antiporter, 206
overview, 197–200
substrate binding and
unbinding, 200–205
molecular docking,
202–204
spontaneous binding
simulations, 201–202
substrate release
pathways, 204–205
water within transporters
leakage, 216
in proton pathways,
216–219
MM. See molecular mechanics (MM)
modulation, and network models,
244–245
molecular docking, 202–204
molecular dynamics, and receptors
to aid crystallographic
interpretation, 245–247
to explain effects of ions and
water, 251–253
to move between states,
247–249
to quantify free energy
requirements, 253–255
to refine working models,
249–250
molecular dynamics (MD)
simulations
ab initio, 33–34
classical molecular dynamics
additive force fields, 22–24
applications, 29–30
overview, 20–22
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Published on 30 November 2016 on http://pubs.rsc.org | doi:10.1039/9781782626695-00259
Subject Index
263
polarisable force fields,
24–25
practical and technical
considerations, 25–29
coarse-grained (CG) molecular
dynamics, 30–33
enhanced sampling techniques
and free energy methods,
34–40
overview, 19–20
molecular mechanics (MM), 22
Monte Carlo (MC) simulations,
127–128
non-equilibrium properties, and free
energy differences
Leucine–Serine channel,
93–96
overview, 88–90
theoretical background, 90–93
non-specific membrane effects, 142
normal model analysis (NMA), 239
Na1/Ca12 ion selectivity, 174–177
Na1/K1 ion selectivity, 177–181
network models, and receptors,
239–241
to compare dynamics, 242–244
to examine gating, 241–242
to suggest novel mechanisms
for modulation, 244–245
NMA. See N-methylacetamide (NMA);
normal model analysis (NMA)
N-methylacetamide (NMA), 23
non-atomistic simulations of ion
channels
and continuum distributions
improvements of
electrolytes, 119–122
overview, 111–112
Poisson–Boltzmann (PB)
equation, 112–117
Poisson–Nernst–Planck
(PNP) differential
equations, 117–119
methods to include atomic
detail, 128–133
overview, 107–110
particle-based methods
Brownian dynamics,
124–127
Monte Carlo (MC)
simulations, 127–128
overview, 122–124
particle-based methods, of ion
channels
Brownian dynamics, 124–127
Monte Carlo (MC) simulations,
127–128
overview, 122–124
PB. See Poisson–Boltzmann (PB)
equation
PNP. See Poisson–Nernst–Planck
(PNP) differential equations
Poisson–Boltzmann (PB) equation,
112–117
Poisson–Nernst–Planck (PNP)
differential equations, 117–119
polarisable force fields, 24–25
probability distribution methods,
73–75
OPLS. See Optimized Potentials for
Liquid Simulations (OPLS)
Optimized Potentials for Liquid
Simulations (OPLS), 22
receptors, 14–15
RMSD. See root mean square
deviation (RMSD)
root mean square deviation (RMSD),
244
specific membrane effects,
140–142
spontaneous binding simulations,
201–202
substrate binding and unbinding,
200–205
molecular docking, 202–204
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264
Subject Index
substrate binding and unbinding
(continued)
spontaneous binding
simulations, 201–202
substrate release pathways,
204–205
substrate-induced structural
changes, of antiporter, 206
toxin–channel interactions,
184–187
transition coordinates,
63–65
transporters, 11–14
thermodynamic integration
adaptive biasing force method,
78–80
theoretical background, 75–78
water, and membrane transport
leakage, 216
in proton pathways,
216–219
variance reduction strategies,
63–65