Simulating biomolecules
Steven O. Nielsen
Department of Chemistry
University of Texas at Dallas
Outline
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Concept of the energy landscape funnel for protein folding
Importance of low frequency normal modes of proteins
Hydrophobic effect and cold unfolding
Free energy
Stress the link between thermodynamics
and biomolecular structure
Energy landscape theory:
the protein folding funnel
Protein folding should be complex. The
classical view of protein folding is of a
nearly sequential series of discrete
intermediates.
In contrast, the energy landscape theory of
folding considers folding as the progressive
organization of an ensemble of partially folded
structures through which the protein passes on
its way to the natively folded structure. As a
result of evolution, proteins have a rugged
funnel-like landscape biased toward the native
structure.
Jose Onuchic and Peter Wolynes (UCSD)
Energy landscape theory:
the protein folding funnel
This organization (the funnel)
is not characteristic of all
polymers with any sequence of
amino acids, but is a result of
evolution.
Evolution achieves robustness
by selecting for sequences in
which the interactions present
in the functionally useful
structure are not in conflict, as
in a random heteropolymer,
but instead are mutually
supportive and cooperatively
lead to a low-energy structure.
Jose Onuchic and Peter Wolynes (UCSD)
Trade-off between entropy and
enthalpy almost balance: why?
Energy landscape theory:
the protein folding funnel
THEORY / SIMULATION
Perfect funnel or Go models include only
interactions that stabilize the native
structure.
Very simple force field: only native
contacts are favorable.
This theoretical construct has yielded
enormous insight into protein folding,
even though it is highly simplified.
Nobuhiro Go
Normal mode analysis: low-frequency
modes important for proteins
Normal mode analysis (NMA) is a powerful
tool for predicting the possible movements
of a given macromolecule. It has been shown
recently that half of the known protein
movements can be modelled by using at
most two low-frequency normal modes.
Applications of NMA cover wide areas of
structural biology, such as the study of
protein conformational changes upon ligand
binding, membrane channel opening and
closure, potential movements of the
ribosome,and viral capsid maturation.
apo (left) and holo (right) forms
of lactoferrin
High frequency modes are usually localized
– a bond stretch for example, and are not
important. Low frequency modes are usually
delocalized (eg. breathing modes)
http://www.igs.cnrs-mrs.fr/elnemo/examples.html
Hydrophobic effect
D. Chandler, Nature, 417, 491 (2002).
The separation of oil and water in ambient
conditions is not due to repulsion between water
and oil molecules, but to particularly favourable
hydrogen bonding between water molecules.
Strong mutual attractions between water molecules
induce segregation of oil from water and result in an
effective oil–oil attraction called the hydrophobic
interaction: primary source of protein stability.
But: depends on length scale
Hydrophobic effect
Water–water interactions persist even in the presence of small oily species,
(< 10 carbon alkane). In the close vicinity of the oily molecules, the possible
configurations of hydrogen bonding may be restricted, but the overall
amount of hydrogen bonding remains unchanged. Thus, the cost of
hydrating a small, hydrophobic solute has more to do with the number of
ways in which hydrogen bonds can form than with their strength. That
is, the solvation free energy of the system is largely entropic and not
enthalpic.
However, this geometric picture breaks down for an extended oily region,
because not all hydrogen bonds can persist near to its surface. The nature of
hydrophobicity therefore changes when the size of oily surfaces depletes the
number of hydrogen bonds. This energetic effect — the loss of hydrogen
bonding — drives the segregation of oil from water.
(VOLUME vs SURFACE AREA: crossover on the nanometer scale)
Hydrophobic effect
D. Chandler, Nature, 417, 491 (2002).
The assembly of hydrophobic structures requires the
removal of water from regions between these groups; this
is the same as vaporization. Hence, the closer water is to
the liquid–vapour phase transition, the stronger is the
tendency for hydrophobic assembly. I [Chandler] believe
this explains why proteins are denatured by cold.
Cold unfolding
These (experimental) results demonstrate the potential of cold
denaturation as a means to dissect the cooperative
substructures of proteins and to provide a rigorous framework
for testing statistical thermodynamic treatments of protein
stability, dynamics, and function. -- Babu, Hilser, Wand, Nature Struct.
Mol. Biol. 11 352 (2002)
Usually have to go below the freezing
temperature of water to see cold-unfolding,
but there are experimental tricks to get as
far down as -35oC and still have a liquid.
Cold unfolding : thermodynamics
temperature and pressure phase diagram
Cold unfolding : thermodynamics
Hawley theory: starts from the assumption
that there are only two distinct states of the
protein (native and denatured) and the
transition between them is a two state
process. The Gibbs free energy difference
between these states is defined as:
Upon integration of this equation from an arbitrarily chosen
reference point T0,p0 to T,p we get:
where D means the change of the
corresponding parameter during
denaturation (i.e. the value in the
denatured state minus that in the
native state).
Cold unfolding : thermodynamics
310K
V

ARG compressib ility
p T
CP  T
S
T

P
H
T
278K
ARG

V
T

P
S
 thermal expansivit y
p T
 heat capacity
P
The transition line, where the protein
denatures (or refolds depending on the
direction of the crossing), is defined by DG=0.
VAL
VAL
apo-myoglobin at 310K
(left) and 278K (right).
Water oxygens yellow.
Free energy ( G = H – TS )
Free energy is the fundamental measure of
macromolecular stability. For example, the relative free
energy (Gibbs, G, constant temperature and pressure
conditions) between the native and unfolded states of a
protein is what is measured experimentally.
Also might want to know the free energy barrier
for a reaction (or in general for an event). This is
one of the most important quantities to know in
chemistry.
Will give an example for events in biological membranes
Biological membranes:
packaging
In its fight for resources, bacterium Staphylococcus aureus
secretes alpha-hemolysin monomers that bind to the outer
membrane of susceptible cells. Upon binding, the monomers
oligomerize to form a water-filled transmembrane channel that
facilitates uncontrolled permeation of water, ions, and small
organic molecules. Rapid discharge of vital molecules, such as
ATP, dissipation of the membrane potential and ionic gradients,
and irreversible osmotic swelling leading to the cell wall rupture
(lysis), can cause death of the host cell. This pore-forming
property has been identified as a major mechanism by which
protein toxins can damage cells.
Study energetics (free energy) of water going through the pore:
big entropy change and also big change in hydrogen bonding
pattern, so both enthalpy and entropy must be considered
Experimentally, a free energy
difference is determined either
from the relative probability of
finding the system in a given
state, or from the reversible
work required to transform the
system from one state to
another.
F  kBT ln Q
If an equilibrium simulation does not sample enough of the high
free energy regions, a more powerful method is needed. One such
method is the method of constraints, also called thermodynamic
integration.
F
U
DF  
d  


0
0
1
1
Average force of constraint
d

Free Energy Profile of Membrane Meniscus
against Hydrophobic Mismatch
1
DF (u0 )  k (u0  u0opt ) 2
2
Lipids do influence protein
function—the hydrophobic matching
hypothesis revisited (Biochimica et
Biophysica Acta 2004, 1666, 205)
Xibing He, S.O. Nielsen, et. al.,
manuscript in preparation:
first time this effect has been
quantified by computer
simulation
k = 650 kJ/mol
h0 1/ 2 1/ 4 3 / 4
( )  K L
2
Synthetic antimicrobials
aryl amide oligomers
a
b
Angew. Chem. Int. Ed., 43, 1158 (2004)
d
Shai-Matsuzaki-Huang