Shadow hybrid Monte Carlo: an efficient propagator in phase space of macromolecules

Shadow hybrid Monte Carlo (SHMC) is a new method for sampling the phase space of large molecules, particularly biological molecules. It improves sampling of hybrid Monte Carlo (HMC) by allowing larger time steps and system sizes in the molecular dynamics (MD) step. The acceptance rate of HMC decreases exponentially with increasing system size N or time step δt. This is due to discretization errors introduced by the numerical integrator. SHMC achieves an asymptotic O(N^1/4) speedup over HMC by sampling from all of phase space using high order approximations to a shadow or modified Hamiltonian exactly integrated by a symplectic MD integrator. SHMC satisfies microscopic reversibility and is a rigorous sampling method. SHMC requires extra storage, modest computational overhead, and a reweighting step to obtain averages from the canonical ensemble. This is validated by numerical experiments that compute observables for different molecules, ranging from a small n-alkane butane with four united atoms to a larger solvated protein with 14,281 atoms. In these experiments, SHMC achieves an order magnitude speedup in sampling efficiency for medium sized proteins. Sampling efficiency is measured by monitoring the rate at which different conformations of the molecules' dihedral angles are visited, and by computing ergodic measures of some observables.