An improved hybrid An improved hybrid Monte Carlo method for Monte Carlo method for conformational sampling conformational sampling

of large biomoleculesof large biomolecules

Department of Computer Science Department of Computer Science and Engineeringand Engineering

University of Notre DameUniversity of Notre DameNotre Dame, IN 46556-0309Notre Dame, IN 46556-0309

Scott Hampton and Jesus A. Izaguirre

shampton@cse.nd.edu izaguirr@cse.nd.edu

SummarySummary

What is the problem?What is the problem? Why are we interested?Why are we interested? Why is it challenging?Why is it challenging?

Multiple-minima problemMultiple-minima problem Size of the moleculesSize of the molecules Multiple time scalesMultiple time scales

Our contributionOur contribution

Molecular SimulationMolecular Simulation

Molecular Molecular

DynamicsDynamics Monte Carlo Monte Carlo

methodmethod Sampling:Sampling:

f Ma

HMC AlgorithmHMC Algorithm

Start with some initial configuration Start with some initial configuration (q,p)(q,p)

Perform Perform cyclelengthcyclelength steps of MD, steps of MD, using timestep using timestep t,t, generating generating (q’,p’)(q’,p’)

Compute change in total energyCompute change in total energy H = H = H(q’,p’) - H(q’,p’) - H(q,p)H(q,p)

Accept new state based on Accept new state based on exp(-exp(- H H ))

Hybrid Monte CarloHybrid Monte Carlo

Hybrid Monte Carlo Method (HMC)Hybrid Monte Carlo Method (HMC) Combination of MD and MC methodsCombination of MD and MC methods Poor scalability of sampling rate with Poor scalability of sampling rate with

system size Nsystem size N Improvement with higher order Improvement with higher order

methods (Creutz, et. al.)methods (Creutz, et. al.) Our method scales better than HMCOur method scales better than HMC

Shadow HamiltonianShadow Hamiltonian

Based on work by Skeel and Hardy [1]Based on work by Skeel and Hardy [1] Hamiltonian: Hamiltonian: H=1/2pH=1/2pTTMM-1-1p + U(q)p + U(q) Modified Hamiltonian: Modified Hamiltonian: HHMM = H + O( = H + O(t t pp))

Shadow Hamiltonian: Shadow Hamiltonian: HHSS = H = HMM + O( + O(t t 2p2p)) Arbitrary accuracyArbitrary accuracy Easy to computeEasy to compute Stable energy graphStable energy graph

HH4 4 = H – f( q= H – f( qn-1n-1, q, qn-2n-2, p, pn-1n-1, p, pn-2n-2 ) )

Shadow HMCShadow HMC

Replace total energy Replace total energy HH with shadow with shadow energyenergy HHSS = = HHSS (q’,p’) - (q’,p’) - HHSS (q,p) (q,p)

Nearly linear scalability of sampling Nearly linear scalability of sampling raterate

Extra storageExtra storage Small overheadSmall overhead

Acceptance RatesAcceptance Rates

More Acceptance RatesMore Acceptance Rates

Sampling rateSampling rate

ConclusionsConclusions

SHMC has a much higher acceptance SHMC has a much higher acceptance rate, particularly as system size and rate, particularly as system size and timestep increasetimestep increase

SHMC discovers new conformations SHMC discovers new conformations more quicklymore quickly

SHMC requires extra storage and SHMC requires extra storage and moderate overhead.moderate overhead.

SHMC works best at relatively large SHMC works best at relatively large timestepstimesteps

Future WorkFuture Work

Are results valid?Are results valid? Theoretically validTheoretically valid BiasBias

What’s next?What’s next? Multiple Time Stepping (MTS)Multiple Time Stepping (MTS) Combining SHMC with other methodsCombining SHMC with other methods

AcknowledgementsAcknowledgements

This work was supported by NSF This work was supported by NSF Grant BIOCOMPLEXITY-IBN-Grant BIOCOMPLEXITY-IBN-0083653 and NSF CAREER award 0083653 and NSF CAREER award ACI-0135195ACI-0135195

SH was also supported by an Arthur SH was also supported by an Arthur J. Schmitt fellowship from the J. Schmitt fellowship from the University of Notre DameUniversity of Notre Dame

ReferencesReferences

1.1. R. D. Skeel and D. J. Hardy. R. D. Skeel and D. J. Hardy. Practical construction of modified Practical construction of modified Hamiltonians. Hamiltonians. SIAM J. on Sci. SIAM J. on Sci. Computing,Computing, 23(4):1172-1188, Nov. 23(4):1172-1188, Nov. 2001.2001.

2.2. GaSh00GaSh00

3.3. Sampling method paperSampling method paper

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