An improved hybrid Monte Carlo method for conformational sampling of large biomolecules Department of Computer Science and Engineering University of Notre Dame Notre Dame, IN Scott Hampton and Jesus A. Izaguirre

Summary 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 problem Multiple-minima problem Size of the molecules Size of the molecules Multiple time scales Multiple time scales Our contribution Our contribution

Molecular Simulation Molecular Dynamics Molecular Dynamics Monte Carlo method Monte Carlo method Sampling: Sampling:

HMC Algorithm Start with some initial configuration (q,p) Start with some initial configuration (q,p) Perform cyclelength steps of MD, using timestep  t, generating (q’,p’) Perform cyclelength steps of MD, using timestep  t, generating (q’,p’) Compute change in total energy Compute change in total energy  H =  H(q’,p’) -  H(q,p)  H =  H(q’,p’) -  H(q,p) Accept new state based on exp(-   H ) Accept new state based on exp(-   H )

Hybrid Monte Carlo Hybrid Monte Carlo Method (HMC) Hybrid Monte Carlo Method (HMC) Combination of MD and MC methods Combination of MD and MC methods Poor scalability of sampling rate with system size N Poor scalability of sampling rate with system size N Improvement with higher order methods (Creutz, et. al.) Improvement with higher order methods (Creutz, et. al.) Our method scales better than HMC Our method scales better than HMC

Shadow Hamiltonian Based on work by Skeel and Hardy [1] Hamiltonian: H=1/2p T M -1 p + U(q) Hamiltonian: H=1/2p T M -1 p + U(q) Modified Hamiltonian: H M = H + O(  t p ) Modified Hamiltonian: H M = H + O(  t p ) Shadow Hamiltonian: H S = H M + O(  t 2p ) Shadow Hamiltonian: H S = H M + O(  t 2p ) Arbitrary accuracy Arbitrary accuracy Easy to compute Easy to compute Stable energy graph Stable energy graph H 4 = H – f( q n-1, q n-2, p n-1, p n-2 ) H 4 = H – f( q n-1, q n-2, p n-1, p n-2 )

Shadow HMC Replace total energy H with shadow energy Replace total energy H with shadow energy  H S =  H S (q’,p’) -  H S (q,p)  H S =  H S (q’,p’) -  H S (q,p) Nearly linear scalability of sampling rate Nearly linear scalability of sampling rate Extra storage Extra storage Small overhead Small overhead

Acceptance Rates

More Acceptance Rates

Sampling rate

Conclusions SHMC has a much higher acceptance rate, particularly as system size and timestep increase SHMC has a much higher acceptance rate, particularly as system size and timestep increase SHMC discovers new conformations more quickly SHMC discovers new conformations more quickly SHMC requires extra storage and moderate overhead. SHMC requires extra storage and moderate overhead. SHMC works best at relatively large timesteps SHMC works best at relatively large timesteps

Future Work Are results valid? Are results valid? Theoretically valid Theoretically valid Bias Bias What’s next? What’s next? Multiple Time Stepping (MTS) Multiple Time Stepping (MTS) Combining SHMC with other methods Combining SHMC with other methods

Acknowledgements This work was supported by NSF Grant BIOCOMPLEXITY-IBN and NSF CAREER award ACI This work was supported by NSF Grant BIOCOMPLEXITY-IBN and NSF CAREER award ACI SH was also supported by an Arthur J. Schmitt fellowship from the University of Notre Dame SH was also supported by an Arthur J. Schmitt fellowship from the University of Notre Dame

References 1. R. D. Skeel and D. J. Hardy. Practical construction of modified Hamiltonians. SIAM J. on Sci. Computing, 23(4): , Nov GaSh00 3. Sampling method paper

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