Download Weighted Ensemble Simulations of Protein Dynamics

Weighted Ensemble
Simulations of Protein
Dynamics
Robert C. Sterner and Matthew C. Zwier
Department of Chemistry
Drake University
Why simulate?
•  With simulation, we have access to atomic length scales and
femtosecond timescales, sufficient to watch many chemical
processes in exquisite detail.
•  We can help explain experimental results and suggest new
experiments.
•  E.g. we can correlate NMR relaxation data of proteins with
motions, giving us a better understanding of protein function.
What are Molecular Dynamics Simulations?
• Molecular dynamics (MD) is a computer simulation
technique, which allows the prediction of the time
evolution of interacting particles.
• Atoms are treated as classical point particles.
• Evolution of the system in time can be followed through
Newton’s Second Law:
• At the smallest scales, atoms follow the laws of quantum
mechanics, but at the scale of hundreds of atoms, a
classical approximation is remarkably effective.
Requirements
• MD has relatively simple requirements
including:
§ Initial positions of atoms
§ Potential energy function describing atomic
interactions
• Typically a simple potential function is used
which employs modeling through balls on
springs.
MD integrates Newton’s equations
Bond Angles Modeled
by Harmonic Springs
Bonds Modeled by
Harmonic Springs
Modeling of
Steric Effects
This accounts for uncharged balls on springs.
MD integrates Newton’s equations
Electrostatic Interactions
+
−
Governed by
Coulomb’s Law
Dispersion Interactions
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Typically modeled by Lennard-Jones potential
MD integrates Newton’s equations
Ø Force F on each atom is
calculated as gradient of
potential energy U
Ø Force is allowed to be
exerted for a short time
1-2fs (10-15)
Ø Results in a change in
velocity
Ø Movement!
Ø Repeated (simulation loop)
The problem: MD takes fs steps, but needs ms reach
Loop motions
Surface
side chain
rotation
Bond vibration
10-15
(fs)
MD timestep
Buried
side chain
rotation
Intermolecular
diffusion
Hinge bending
10-12
(ps)
Protein folding
10-9
(ns)
Allosteric transitions
10-6
(µs)
10-3
(ms)
Current reach of MD simulations
Zwier & Chong, Curr Opin Pharmacol 2010, 10, 745–752.
100
(s)
The solution: enhanced sampling
•  Biologically relevant processes are usually rare — fast but
infrequent.
•  (e.g.) allosteric transitions take ms, but are composed of processes
that take ps – ns.
•  Random nature of molecular motion results in a two-stepsforward-one-step-back kind of progress.
•  If we can focus our computing power on forward steps, we can
improve the efficacy of calculation by orders of magnitude.
The weighted ensemble approach
• Enhanced sampling approach employing independent
simulations.
• Each simulation (“walker”) carries a statistical weight.
• Statistical weight is a factor assigned to a walker to make
its effect on the simulation as a whole reflect its relative
physical importance.
• Statistical weight P and free energy are linked by a
Boltzmann factor:
Weighted Ensemble Continued
• Weight depends on the number of simulations
running and the free energy of the region
• If more simulations are run, each counts as a lower
weight
• Walkers in regions disfavored by entropy or
enthalpy also have low statistical weights
• Focus on regions with high free energy with out
biasing the simulation
Requirements for the weighted ensemble technique
• Function which describes rare event
• Selection of the space of interested
• The following defined initial and final states, an order
parameter, and the partition of space of along the order
parameter into bins
• Bins should be divided in order minimize computer time
spent to achieve a desired degree of precision
Weighted Ensemble Continued
• Flexible
§ Only loose dependence on order parameter
² Loose dependence meaning that a
perfect idea of what defines progress is
not needed (just need to be close)
• Generates ensemble of pathways and rigorous
rate constants
• Straightforward to parallelize
Propagate dynamics
Simulation Bin
Begins
Bin
Weight =
Weight
Lower
Higher
Free Energy
ü Space divided
into bins
ü Simulation
begins
Time = 0–50 ps (0–1τ)
1.0
Progress Coordinate
Split/merge walkers
Weight
Lower
Higher
Free Energy
v After set time
interval, the
bin which
each walker
is in is
determined
v When one bin
is filled the
next bin is
more likely to
be filled
(Ratcheting
Effect)
Time = 50 ps (1τ)
Weight =
0.5
0.5
Progress Coordinate
Propagate dynamics
Weight
Lower
Higher
Free Energy
v Walkers
arriving in
new bins
are split
into a
parent and
daughter
walker
ü Ensures
equal
sampling
Time = 50–100 ps (1–2τ)
Weight =
0.5
0.5
Progress Coordinate
The process is repeated…
• As unoccupied bins become populated more
simulations are run to investigate that region of
space.
• Bins are progressively filling as the rare event
approaches, but rarely (if ever) empty out,
meaning that net progress occurs very quickly.
• In addition to splitting, elimination also occurs,
ensuring one region of space is not
oversampled.
Split/merge walkers
Time = 200 ps (4τ)
v walkers crossing
back into the
preceding bin
typically leads
to eliminations
Weight
Higher
Free Energy
ü  Prevents
oversampling
Lower
Weight =
0.5
0.25
0.125
0.125
Progress Coordinate
Propagate dynamics
Weight
Lower
Higher
Free Energy
v Simulations
continue and
eventually the
walkers reach
their the
target state
Time = 200–250 ps (4–5τ)
Weight =
0.5
0.25
0.125
0.125
Progress Coordinate
Flux = 0.0625 /τ
Does it work? WE reproduces binding pathways (more efficiently).
Na+/Cl-
Methane/methane
7.5x
K+/18-Crown-6
16x
20x
> 1100x
Transition event
duration (ps)
Transition event
duration (ps)
Transition event
duration (ps)
efficiency
gain
Transition event
duration (ps)
Methane/benzene
Brute force
Weighted ensemble
Zwier, Kaus, & Chong, J Chem Theory Comput 2011, 7, 1189–1197.
Does it work? WE reproduces association rate constants
(to higher quality).
Speedup
Zwier, Kaus, & Chong, J Chem Theory Comput 2011, 7, 1189–1197.
Our objectives
1.  Determine WE parameters
necessary for efficient
sampling of protein motion.
2.  Simulate the domain
motions of calmodulin.
3.  Compare to NMR relaxation
data to understand how
detailed protein motion and
NMR data relate to each
other.
Calmodulin bound to ryanodine receptor.
How does motion relate to Ca2+ regulation in muscles?
Acknowlegements
•  Drake University Provost’s Office and Department of Chemistry
•  NSF XSEDE
•  Prof. Lillian Chong (University of Pittsburgh)