...of ligands from umbrella sampling and steered MD
simulations;
...applications to ions, small molecules and toxin peptides.
Contents of this presentation I
Introduction


General principles of free energy
calculations
Illustrations by simple examples


Theoretical Underpinnings


Toy models: ions in small water boxes
Ways to affirm and void the fidelity of
simulation models to reality
Advanced examples

Ligands in gramicidin-A channel, K+ channels
Contents of this presentation II
Introduction

Full analysis of an absolute free energy
of binding calculation


Using toxin peptides (my work)
Summary and notes


Implications of various assumptions
Warning signs to watch out for
...let’s go!
Goal of FE calculations
Introduction

Accuracy
 Chief motivation of conducting calculations is
to predict/replicate experimental values
 We must not lose sight of this

Therefore, it is worthwhile to test and
verify our foundations
 Especially for calculations that last over a
month
Mechanics of FE calculations
Introduction

Umbrella sampling and steered MD are
path-dependent calculations
e.g. the ligand must physically move out of the
receptor

A well-constructed simulation can:
 Take information from known reaction
mechanisms
 Provide information on how those
mechanisms occure
Foundation of FE calculations
Introduction
F = ma
All simulations are models
 Classical approximation of QM phenomena

 System proceeds via forces imposed on atoms
 Energy calculations occur via manipulation of the
these forces

Small loss in accuracy must be granted
...motivated?
Steered molecular dynamics
A simple manipulation
F= ½ k ( r – r0 + tv )

SMD is a “moving spring”
 Force is imposed on an atom or set of atoms
 Equilibrium position moves to create a path

Work done by this spring is used to find
the local free energy surface.
A simple model
Steered Molecular Dynamics

A small box, containing:




~400 water atoms,
one neon atom.
Suppose that we create
an artificial gaussian
barrier (of ~5 kT) for the
Ne atom.
How do I measure the
free energy surface of
that barrier, using the Ne
as a probe?
Simulation design
Steered Molecular Dynamics

Constrain position of Ne
in (x,y,z) to a starting
location



Define path along water
box
Measure the total amount
of work done
W = ∫ F(z) dz
W → manipulations →
free energy surface
Work done(?)
Steered Molecular Dynamics


Resulting potential of
mean force
Includes environmental
influences
 i.e. action of water around
the atom
 NB: notice that PMF does
not return to zero readily
Averaging
Steered Molecular Dynamics

A single trajectory does not “sample” the
reaction sufficiently
 i.e. A free energy surface requires comprehensive
knowledge of the sub-states, ala possible trajectories

Trajectories contain influence from the SMD
pulling itself
 Some of the work is directed at displacing the solvent
around Ne
 This needs to be accounted for
Jarzynski’s Equality
Steered Molecular Dynamics

.˙. Take multiple trajectories
 The average of multiple trajectories should even out
random influences on the PMF
< e -W/kT > ≈ Σi=1n e -Wi / kT / n

.˙. Apply Jarzynski’s Equality (given certain
assumptions)
e -ΔG/kT = < e -W/kT >
Average work
Steered Molecular Dynamics

The Boltzmann average
of 10 calculations

v = 10 Å ns-1

A very fast calculation,
converges because Neon
does not interact with
environment
...next: theory
Criteria of SMD-based FE calculations
SMD Assumptions

Reaction coordinate must be well
chosen


This coordinate measures all contributions to
the real ΔG, and only those contributions.
The simulation must be near-equilibrium

Jarzynski's Equality holds when no dissipative
work is done by the pulling
Reaction coordinates
SMD Assumptions
In SMD, the dimension(s) controlled by the
constraining potential is a
reaction coordinate
 The reaction coordinate can be:
 Distance
 Center of mass (collective variable)
 RMSD to target structure

The path traced through the SMD
simulation is a reaction path
Theoretical interpretation
SMD Assumptions

Our systems are canonical ensembles:
Z(n,P,T)
 We wish to measure particular sub-states of
the system
 e.g. ligand-bound and ligand-unbound.

Thus, steered molecular dynamics
(SMD):
 Constrains the system to certain sub-states
according to some coordinate
 Drives the system along this coordinate to
new sub-states by application of forces.
Implications on path design
SMD Assumptions

Both the reaction-path
and the simulations must
sufficiently sample the
required sub-space

May not be complete, but
must be representative

In a ligand-unbinding
process
Phase space
Bulk
Bound
 Translation must be primary
 Rotation may be secondary
Implications on path design
SMD Assumptions

Sampling and stability of
a system suffers where
large barriers must be
crossed

Therefore, choose path of
least resistance
Simply using the distance between the
ligand and the whole receptor may not
be a good choice…
Criteria of SMD-based FE calculations
SMD Assumptions

Reaction coordinate must be well
chosen


This coordinate measures all contributions to
the real ΔG, and only those contributions.
The simulation must be conducted in a
near-equilibrium state

Jarzynski's Equality holds when no dissipative
work is done by the pulling
Theoretical implications
SMD Assumptions
SMD –(Jarzynski's Equality)→ Free energy
 JE comes with certain qualifications:
 Dissipative work can be done on the system
 Pulling velocity
i.e. System must equilibrate around SMD
perturbation, else perturbation will also
be measured. (We will show this later.)
...next: 
Umbrella Sampling
Collecting local information
F= ½ k ( r – ri), i along path

US is a static potential
 Force is also imposed on an atom or set of
atoms
 Multiple overlapping states are constructed to
cover reaction path.

Each state provides information about
local surface
 Link to derive complete surface
...
Second toy model
Umbrella sampling

A box containing two ions
 sodium and chloride

Solution known
 FE surface related to radial
distribution function

This was done ab-initio
yesterday
Second toy model
Umbrella sampling

Reaction coordinate
 Na – Cl separation

Umbrella potential
 1 Å apart, 2.5-9.5 Å
 k = 10 kcal mol-1 Å-2

Derive original by WHAM
analysis
PMF convergence
Umbrella sampling
Phase space
Bulk
Bound
PMF
Umbrella sampling
Phase space
Bulk
Bound
...pretty...
Criteria of US-based FE calculations
US assumptions

Reaction coordinate must be well
chosen


This coordinate measures all contributions to
the real ΔG, and only those contributions.
Sufficient sampling over the entire path:


Convergence of PMF curve means that
environmental variables are well sampled.
Overlap between adjacent windows.
Reaction coordinate
US assumptions

Same arguments as for SMD (underlying
physics identical)

Umbrella sampling is capable of treating
two/three dimensions
 As long as all dimensions are properly
sampled
Criteria of US-based FE calculations
US assumptions

Reaction coordinate must be well
chosen


This coordinate measures all contributions to
the real ΔG, and only those contributions.
Sufficient sampling over the entire path:


Convergence of PMF curve means that
environmental variables are well sampled
Overlap between adjacent windows
Environmental variables
US assumptions

All coordinates not
included in your
reaction coordinate(s)
must be well sampled

This means that
simulation has visited
dimensions
perpendicular to
reaction path that may
contribute to FEbind
Bound
Bulk
Window Overlap
Bulk
US assumptions

Require enough
sampling to accurately
interpolate between
windows
Bound
Window-overlap
US assumptions




Define measure of
overlap between two
distributions
When underlying surface
is flat, harmonic potential
produces gaussian
distributions
Theoretical overlap:
Ω = [ 1- erf(d/8σ)]
Ω = [ 1- erf(d/8σ)]
US assumptions


In practice, minimum
overlap is ~2%
Overlap should agree
with theoretical value
when in bulk
k = 20 kcal/mol/Å-2
d = 0.5 Å
Ω = 15%
k = 40 kcal/mol/Å-2
d = 0.5 Å
Ω = 4%
Summary
Steered MD –vs– Umbrella Sampling

Constructing FE surfaces via SMD
 Straightforward construction
 Relies on JE conditions: Difficult to achieve in
practice

Constructing FE surfaces via US
 Additional checks and balances

Both dependent on sufficient sampling
...take a break?
JE/US comparisons
J Chem. Phys. 128:155104
(2008)

Using several test
cases of ions and
molecules in
channel systems
 Ion transit through membrane
 Ion binding to gramicidin exterior
 Organic-cation binding to gramicidin-A
JE/US comparisons
J Chem. Phys. 128:155104
(2008)

Comparing PMFs
 Tests balanced by
equalising the total
simulation time
SMD setup @ v=5 Å ns-1 ~ US setup
 SMD: Also use different pulling velocities to
test reversibility of JE
Ion transit (nanotube)
J Chem. Phys. 128:155104
(2008)

Results equivalent
between two methods

Energy surface not equal
at both openings
 resulting from system setup

JE valid
36
Ion transit (gA)
J Chem. Phys. 128:155104
(2008)


Umbrella sampling, not
SMD, gives symmetric
surface
Pulling at different
velocities do not seem to
help
 Can potentially use
v < 1 Å ns-1
 But more time consuming
than equivalent US setup
37
JE: Practical problems?
J Chem. Phys. 128:155104
(2008)

Equilibration time sharply
increases for peptide
environments
 Nanotube highly ordered
.˙. Fast dissipation

Reliance on sampling
“negative work” trajectories
 High v: low probability for the
environment to push SMD
particle
K+-binding to gA
J Chem. Phys. 128:155104
(2008)
( next test case: )
 gA has a weak ion binding
site at the entrances
v =2.5 Å ns-1

Smaller potentials
 Perhaps using a smaller k will
reduce the perturbations
39
K+-binding to gA
J Chem. Phys. 128:155104
(2008)

What about using different
force constants?
 No significant help in repairing
JE assumptions
v =2.5 Å ns-1

Using small k may reduce
perturbations on system
 However, binding site shape
lost
 k must be greater than binding
well ‘potential’
40
K+-binding to gA
J Chem. Phys. 128:155104
(2008)
k = 2 kcal mol-1 Å-2

There are hard limits to
varying parameters
k = 20 kcal mol-1 Å-2
41
EA and TEA binding
J Chem. Phys. 128:155104
(2008)

Ethylammonium (EA) and
tetra-ethylammonium (TEA)
bind weakly to gA
 v = 2.5 A /ns
 k = 20 kcal mol-1 Å-2

Reducing barrier height
produces no difference here
CnErg1
Toy comparison
J Chem. Phys. 128:155104
(2008)

If it doesn’t work for small cations,
it won’t work for a peptide
 Test for a purported binding of
CnERG1 toxin to hERG channel
since rate of dissipation to environment
is less than rate of work done…

SMD-PMFs essentially measures
work done to move solvent
43
Mechanisms
J Chem. Phys. 128:155104 (2008)

Input: work done on system
 Carried out by imposed SMD
forces (irreversible)
 Contribution from underlying FE
surface (reversible)

Output: dissipation to heatbath (NPT systems)
 Equilibration occurs by two
means:
 Forces bleed out to atoms far
from SMD location
 Temperature coupling to
thermostat

JE maintained in O < I
conditions (only in nanotube)
44
Mechanisms
J Chem. Phys. 128:155104 (2008)


The time for equilibration is
such that v << 2.5 Ang/ns is
required for JE condition to
hold
This velocity requirement
become more stringent as:
 ligand size increases
 interactions increase

Not as efficient as umbrella
sampling.
...paper finished.
Organic cations-gA
J. Phys. Chem. B 111:11303 (2007)

Block of GA by various
small molecular ligands
 Energetics dependent on
ligand size and partial charges

Comparison between
ligands, and with extant
experimental data
Organic cations
J. Phys. Chem. B 111:11303 (2007)



Use Autodock3 to find
potential binding sites of
molecules
MD, umbrella sampling to
find PMF and free energy of
binding
COM-coordinates of ligands
10 kcal mol-1 Å-2
0.5 Å ns-1
Molecule list
J. Phys. Chem. B 111:11303
(2007)

Use of six different
molecules

Varying sizes and polarity

Determines strength of
binding, and whether
molecules can permeate
through gA
MA and EA
J. Phys. Chem. B 111:11303
(2007)
FMI and GNI
J. Phys. Chem. B 111:11303
(2007)
TMA and TEA
J. Phys. Chem. B 111:11303
(2007)
Comparison of FEbind
J. Phys. Chem. B 111:11303 (2007)
Molecule
z (Å)
r (Å)
FEbind (kT)
K(M-1) (expt.)
MA
10.7 ± 0.8
0.7 ± 0.4
-1.4
4.1 (4.4)
EA
12.5 ± 0.6
1.4 ± 0.6
1.6
0.2 (~0)
FMI
12.6 ± 0.5
1.5 ± 0.7
0.5
0.6 (23)
GNI
12.8 ± 0.3
2.3 ± 0.5
-2.2
8.9
TMA
13.2 ± 0.6
1.4 ± 0.6
0
1
TEA
14.1 ± 0.5
2.2 ± 0.8
0.9
0.4

FMI and GNI binding
 Channel lifetimes increases by many folds
 Binding must influence the center of pocket
 A binding site likely exists in the centre of the
channel, not in simulation
...wait, how did we get FE?
From PMF to free energy
How did we finally derive FEbind?
Assumption: x-y variations in the PMF
are “small” in the region sampled
 Keq = ∫ ∫ ∫ e-W(z)/kT dx dy dz
= π R2 ∫ e-W(z)/kT dz


∆Gb= -kT ln (Keq Co )
Co is standard concentration
 What is R?

Integration Volume
Bulk
How did we finally derive FEbind?

The measured PMF
occurs over a certain
volume
 This represents the size of
the entire binding pocket
 Larger binding pocket =>
larger effective ∆Gb

This is then standardised
to 1 M of ligand
 Equivalent to 1 ligand per
1661 Å3
Bound
Integration Volume
Bulk
How did we finally derive FEbind?

One dimensional PMF
merely hides the fact that
binding sites have
volume
 Assumption essentially
states that the PMF value
at some position is the
average PMF value of the
local slice

R is the radius of these
local slices
 This depends on the actual
area sampled
Bound
Integration Volume
Bulk
How did we finally derive FEbind?

In practice, R does not
change significantly over the
length of the bound area
 Therefore, can pick uniform
average R at a minimal loss of
accuracy
 e-W(z)/kT
means that only the
sampling around binding site
is critical

The rest of the path connects
this to bulk
Bound
Set R to be average area
visited by ligand in site
From PMF to free energy
How did we finally derive FEbind?

∆Gb= -kT ln (Keq Co )
 Other notes:

These derivations assume a simple two
state mechanism
[L] + [B]  [LB]

In cases of e.g. cooperative binding, the
relationship needs a different derivation
From PMF to free energy
How did we finally derive FEbind?

∆Gb= -kT ln (Keq Co )
 Other notes:
We also ignore possibilities of multiple
binding sites
 Secondary binding pockets within the
same site may contribute – this
depends on your sampling and reaction
paths

...next paper...
K+ permeation
Biophys J. (2011)

PMFs can be used to
study permeation
processes
 subject to classical
assumptions

The K+ channel conducts
ions at a near diffusion
rate
 This implies that only small
barriers exist along the path
 How does this occur?
K+ permeation
Biophys J. (2011)

K+ ions occupy
the filter at all
times in-vivo
 S1/S3,
S0/S2/S4
 If less ions are present, the channel closes
 .˙. conduction must involve concerted movement
Setup
Biophys J. (2011)
Kv1.2 (Shaker)
 Reaction
coordinate along
channel-axis


Define various PMFs for the different
conditions that may exist:
 one K+ approaching two ions within filter
 Two ions moving along filter
 Three ions moving along filter
One ion movt.
Biophys J. (2011)

Approach from
left
One ion movt.
Biophys J. (2011)

Approach from
right

Significant
difference
between the two
occupancy states
Barrier-less permeation?
Biophys J. (2011)

The “barrier-less” transport involves
 Movement of the two ions in the filter
 Filling the hole left behind
 Every pair must be separated by 1 water

Adjacent K+ states are unfavourable
 States like S1/S3/S4 should not exist

We further test the cohesive movement
K+ permeation
Biophys J. (2011)

2-ion movement

S2/S4 -> S1/S3
Barrier less

S1/S3 -> S0/S2
Large barrier?
K+ permeation
Biophys J. (2011)

3-ion movement

S1/S3/‘S5’ ->
S0/S2/S4
also large barrier
Barrier-less permeation?
Biophys J. (2011)

The large barrier cannot
physically exist
 But simulation well converged: there must be a
problem with the simulation itself
 Classical forcefields used here are not
polarisable
 Leads to difference between protein behaviour
near solvent (S0) and within filter (S1-S4)?
Barrier-less permeation?
Biophys J. (2011)

The large barrier cannot
physically exist
 Similar problem with
permeation
 All due to polarisation of
molecules in different
media

Stay tuned for next
generation forcefields
...take a break?
Binding of Charybdotoxin to KcsA

Current paper in submission
 Various aspects shown above in illustrations

We will now cover the project in detail
 Highlighting the tests that affirm accuracy
i.e. Sampling sufficiency, coordinate choices,
control…
Criteria of US-based FE calculations
US assumptions

Reaction coordinate must be well
chosen


Sufficient sampling over the entire path:



This coordinate measures all contributions to
the real ΔG, and only those contributions.
Convergence of PMF curve means that
environmental variables are well sampled.
Overlap between adjacent windows.
Complexity from collective variables:

Influence of internal coordinates on PMF.
Ligand Complexity
US assumptions

Reaction coordinates work by contracting
the entire freedom of the ligand




A ligand molecule has 3N degrees of freedom
A fraction of these are important
Maximum of ~3 coordinates is reasonable.
In complex ligands with multiple sites of
interaction, must either:



select all important coordinates as reaction
coordinates, e.g. charge-centers.
Contract them further, e.g. center-of-mass
Then must deal with how well reaction coordinate
corresponds to the interaction sites
Ligand complexity

Small ligands have
dozens of atoms


Relatively few
internal degrees of
freedoms
Choosing COM
does not introduce
complications
Ligand complexity

Proteins contain 100s
of atoms with tertiary
structure



many internal
degrees of freedom
low vibrational
modes.
Choosing COM –
Internal modes may
respond to umbrella
potential
Background
The binding of charybdotoxin

Scorpion toxins
 Selectively targets neuronal ion channels and
blocks conduction.

Specificity can be altered by mutations
 Modified toxins for therapeutic targets
 Potential for:
therapeutics
in-vivo studies of channel distribution
Background
The binding of charybdotoxin

Potassium channel targets
 Kv1 family, calcium-activated channels
 Structure difficult to obtain.

Bacterial K+ channel, KcsA, is easier
 Mutate residues and bind scorpion toxins

KcsA-ChTX complex obtained by NMR +
previous crystallography
 Park et. al. (2005)
MD system setups
The binding of charybdotoxin

Reaction coordinate:

 Toxin backbone center-of-
mass.
 Path extends along
channel-axis.

Harmonic constraints on
residues 3-35 and 17-20
 Prevent unfolding of protein
during sampling.
~60,000 atoms
 Match experimental ionic
concentration (150 mM)

NPT ensemble

KcsA and membrane
lightly constrained
 Prevent unlikely case of
drifting
The center-of-mass coordinate
The binding of charybdotoxin

Represents the translational freedoms of
the ligand.



Sampling must integrate rotational effects and
internal modes
Works for ions and small ligands with
single important site of interaction
May not work alone for peptides with
multiple sites of interaction
Cartoon example
The binding of charybdotoxin

A ligand is pulled from its binding site to the bulk


However, ligand unfolds over the path of reaction
coordinate.
Then the chosen path must include the energy
of unfolding.
Prior Work
The binding of charybdotoxin


We’ve carried out a direct
umbrella sampling
procedure for a scorpion
toxin
COM coordinate
 Results in partial unfolding
of alpha helix
 Trapped in alternate
conformation
Solution
The binding of charybdotoxin

Parts of the protein are
restrained in order to
prevent unfolding

As a check, calculate the
free energy of restraining
and unrestraining
Solution
The binding of charybdotoxin

Contribution of
thermodynamic cycle
needs to be calculated
 (site) 1.43
 (bulk) 1.47

Negligible in this case

Not negligible if restraints
applied to functional
residues
Spoiler
The binding of charybdotoxin

Parts of the protein are
restrained in order to
prevent unfolding

A different “path” is
sampled when toxin is
restrained
Sampling Overlap
The binding of charybdotoxin

Plot overlap between
successive windows
 Gaps likely at transitional
barriers
 These require additional
windows
 A junction can introduce errors
of up to 0.5 kcal mol-1
Sampling Overlap
The binding of charybdotoxin
Spoiler II
The binding of charybdotoxin

WHAM analysis
interpolates data to
connect two adjacent
windows.
 Gaps in this case introduce
~0.5 kcal mol-1 differences
 Important w.r.t. accuracy
Spoiler II - note
The binding of charybdotoxin

Angular peaks within an
otherwise smooth PMF
curve
 Location of barrier
 Potential indication of
insufficient sampling
PMF convergence

(work computer died)
PMF
B&C:
toxin-restrained
PMFs
-----
A:
non-restrained PMFs
path-dependence of PMF
The binding of charybdotoxin
path-dependence of PMF
The binding of charybdotoxin
k-dependence of PMF
The binding of charybdotoxin
k-dependence of PMF
The binding of charybdotoxin

There is no dependence
of the binding free energy
on k
 The storage of elastic
energy in the peptide is
conservative
(error is calculated by
standard deviations of
subsets of sampling data)
Source set
FE of
binding
A20
-17.1 ± 0.9
A40
-16.8 ± 2.3
B20
-8.7 ± 1.7
C40
-7.6 ± 1.1
Experiment
-8.3
Kinetic analyses
The binding of charybdotoxin

Rotational freedom of ChTX
increases in steps
 Significant increase after
contact disassociation

Clearly there should
be free rotation in bulk
Kinetic analyses
The binding of charybdotoxin

Translational freedom only
achieved outside binding
pocket

Effective binding site is
actually rather small

Transition region with
some charge contacts
Kinetic analyses
The binding of charybdotoxin

Although ligand is 100’s of
atoms, sampling is not as
hard as one might imagine
Kinetic analyses
The binding of charybdotoxin

Binding site is small
due to contacts

Multiple charge
interactions “lock” the
toxin to the pore
 Therefore a very
narrow and deep
binding pocket
Kinetic analyses
The binding of charybdotoxin

Significant residues identifiable
 R25, K27, R34, Y36
 Not K11 or other charges

Correlates with existing
mutational data
Take home message
Addendum

Many types of ligand interactions can be
explored

Important caveats exist, but not
extraordinarily challenging

Most useful in explaining why a process
occurs
 Complementary with experimental data