Advanced Sampling Techniques

When molecular dynamics is just not enough.

Why do we need enhanced sampling?

MDNN

Trying to fold a peptide from two different initial conditions, gives two different free energy surfaces !

Rates depend on barriers and T

What can we do to speed things up?

• It’s basically a problem of rates, or kinetics– Rates are exponentially dependent on temperature – higher T = higher KE = ability to cross higher barriers

• Can we just simulate at a higher T?

• Yes! Folding is faster at higher T.

Why is convergence so slow?• We want the average of observable properties- so we

need to include in that weighted average all of the important values that it might have

• Structural averages can be slow to converge- you need to wait for the structure to change, and do it many, many times.

• Sampling an event one time doesn’t tell you the correct probability (or weight) for that event

– What is the proper weight? Boltzmann’s equation

What can we do to speed things up?

• It’s basically a problem of rates, or kinetics– Rates are exponentially dependent on temperature – higher T = higher KE = ability to cross higher barriers

• Can we just simulate at a higher T?

• Yes! Folding is faster at higher T.

but…we mess up the thermodynamics. – Equilibrium constants (weights) also depend on T

Populations depend on ΔE and T

Free energies along a defined reaction coordinate via Umbrella Sampling

Umbrella SamplingHow to force barrier crossings without compromising thermodynamic properties?

Very slow transitions

Umbrella Sampling

trapped, bad ΔG

good for ΔG

One could just run dynamics and wait until all space has been sampled.

Then, if one extracts P(xk) from the trajectory, the PMF can be written as:

However, it takes forever to properly sample all conformations, and to jump over the barrier. The solution is to bias the system towards whatever value of the coordinate we want.

This is called unbiased sampling

AMBER WS 2012CECAM Lausanne

Umbrella Sampling

True PMF

Ideal BiasingPotential

No barrier,perfect sampling

We could BIAS the simulation, but we do not really know how to do it exactly.

AMBER WS 2012CECAM Lausanne

Umbrella Sampling

True PMF

Windows: 1 2 3 4 5 choose i, k and xi system-dependent

Introduce biasing potentials along the reaction coordinate

AMBER WS 2012CECAM Lausanne

Adding a quadratic biasing potential

Check for sufficient overlap

Histograms from neighboring windows should overlap strongly, all points on the RC must be sampled suffciently.

AMBER WS 2012CECAM Lausanne

Umbrella Sampling

Simulation Window Histogram Part of PMF

Final computed PMF from many windows

Solved iteratively using e.g. the WHAM program by Alan Grossfield

Constructing the PMF

AMBER WS 2012CECAM Lausanne

Umbrella Sampling

Histograms from neighboring windows should overlap strongly,

all points on the RC must be sampled suffciently.

Solved iteratively using e.g. the WHAM program by Alan Grossfield (http://membrane.urmc.rochester.edu/content/wham)

Check for sufficient overlap between sampled regions

Histograms & free energy profiles

• Umbrella run needs many simulations

• Do NOT need to sample full range in 1 simulation

G= -RTlnP/P0

Comparing 2 conformations

Song, Hornak, de los Santos, Grollman and Simmerling, Biochemistry 2006

It will take much too long to get precise populations for these 2 minima just by running MD.

8OG binding mode in complex: dihedral umbrella sampling

synanti

Song, Hornak, de los Santos, Grollman and Simmerling, Biochemistry 2006

Simulations reveal how the energy profile changes if a mutation is made

syn

anti

Song, Hornak, de los Santos, Grollman and Simmerling, Biochemistry 2006

Effect of mutations

4 Conformational space has many local minima 4 Barriers hinder rapid escape from high-energy minima

4 How can we search and score all conformations?

Replica Exchange

Coordinate

Ene

rgy

Early tries:

Run MD at Tlow, once in a while heat up to Thigh, run some more.Change to Tlow

Repeat…

The problem is that the statistical weights of the first and second Tlow are unknown and unknowable.

Metropolis Monte Carlo, one of the most important algorithms of the 20th century.

MANIAC computer.Los Alamos national Lab1943-1944

80 Kbytes memory11Khz Clock

When Bugs were real (Mark II computer, 1945)

Coordinate

Ene

rgy

We want to sample all conformations in a way that when we are done, the probability of finding a particular conformation is

Moreover, in equilibrium, we require that the rates from minimum A to B be exactly equal to the rate from minimum B to A. This is called Microscopic Reversibility.

Coordinate

Ene

rgy

Temperature Replica Exchange(Parallel tempering)

375K

350K

325K

300K

REMD

Hansmann, U., CPL 1997

Sugita & Okamoto, CPL 1999

[ ] [ ]min(1,exp{( )( ( ) ( )})i jm n E q E q

vnew vold

Tnew

Told

4 Impose desired weighting (limiting distributionon exchange calculation

375K

350K

325K

300K

4 Impose reversibility/detailed balance

Rewrite using only potential energies, by rescaling kinetic energy for both replicas.

Exchange?

Temperature Replica Exchange4 MD runs over range of T

4 Periodically swap structures

4 Faster convergence than regular MD

4 Populations as a function of T

4 Drawback: ΔE must be small small ΔT for large systems

[ ] [ ]min(1,exp{( )( ( ) ( )})i jm n E q E q

Number of replicas ~ N1/2

Does it really converge faster than regular MD?

Yes !

But not really as much as one would think (or hope, or dream)

5kBT / bG k Tk e Δ

For a barrier of ~5kBTlow, and a Thigh=2Tlow , the barrier crossing rate is x12

Better methods use replicas in Hamiltonian space. Different biasing for each replica

Do you have an MD code? Build yourself a REMD code in minutes ! Just write a wrapper !

START REMD

Spawn replicas and run MD at different temperatures

Stop MDsDecide to exchange or not

Advanced methods can help

MD

Replica exchange

NN

Roe, Hornak and Simmerling, J. Mol. Biol. 2006

What if you don’t know the pathway for the change?

in silico structure interconversions

?

“endpoint method”: • link many simulations • timescale independent

Bergonzo, Campbell, Walker & Simmerling, Intl J Quantum Chem 2009

NEB: use many simulations to study slow dynamics

R

P

Co

ord

inat

e 2

F||

F┴

Coordinate 1

Final path

Standard forces

Spring forces

Reaction Path Calculations: NEB

kcal/mol

•NEB requires coupling of multiple simulations using “springs”

A B

How can we get from A to B?

Other techniques

Accelerated MD

Hamiltonian Replica Exchange

Multi Dimensional Replica Exchange

Constant pH ( + REMD)

Steered Molecular Dynamics

Accelerated Molecular Dynamics

de Oliveira C.A.F., Hamelberg, D., McCammon, J.A., On the Application of Accelerated Molecular Dynamics to Liquid Water Simulations. J. Phys. Chem. B 2006.Hamelberg, D., de Oliveira C.A.F., McCammon J.A. , Sampling of slow diffusive conformational transitions with accelerated molecular dynamics. The Journal of chemical physics, 2007. Grant, B.J., Gorfe, A.A., and McCammon, J.A. , Ras conformational switching: simulating nucleotide-dependent conformational transitions with accelerated molecular dynamics. PLoS computational biology, 2009. de Oliveira, C.A.F., et al., Large-Scale Conformational Changes of Trypanosoma cruzi Proline Racemase Predicted by Accelerated Molecular Dynamics Simulation. PLoS computational biology, 2011.

AMD effect on the Potential

Principal Component Analysis• Build PC space based on the 500ns aMD simulation• Project X-ray, Shaw structures, 500ns cMD, and 1ms cMD

Cost: several million dollars Energy consumption ~116.5KW Code mostly hardwired to the hardware

Cost: $2000 Energy consumption ~0.9KW AMBER 12

Routine Access to Millisecond Time Scale Events with Accelerated Molecular Dynamics. Levi Pierce, Romelia Salomon-Ferrer, Cesar Augusto de Oliveira, J. Andrew McCammon and Ross C. Walker. J. Chem. Theory Comput., 2012, 8 (9), pp 2997–3002 DOI: 10.1021/ct300284c