Girsanov reweighting for path ensembles and Markov state models

Abstract

The sensitivity of molecular dynamics on changes in the potential energy function plays an important role in understanding the dynamics and function of complex molecules. We present a method to obtain path ensemble averages of a perturbed dynamics from a set of paths generated by a reference dynamics. It is based on the concept of path probability measure and the Girsanov theorem, a result from stochastic analysis to estimate a change of measure of a path ensemble. Since Markov state models (MSM) of the molecular dynamics can be formulated as a combined phasespace and path ensemble average, the method can be extended to reweight MSMs by combining it with a reweighting of the Boltzmann distribution. We demonstrate how to efficiently implement the Girsanov reweighting in a molecular dynamics simulation program by calculating parts of the reweighting factor "on the fly" during the simulation, and we benchmark the method on test systems ranging from a two-dimensional diffusion process to an artificial many-body system and alanine dipeptide and valine dipeptide in implicit and explicit water. The method can be used to study the sensitivity of molecular dynamics on external perturbations as well as to reweight trajectories generated by enhanced sampling schemes to the original dynamics.

Dynamical reweighting methods for Markov models

“…This is likely the reason that demonstrations of this reweighting approach have been limited to diffusion in lowdimensional energy landscapes [53,54,55,56] or short trajectories of alanine dipeptide [57], in which writing the trajectory to a disc at every integration time step is still a viable option. The problem can be solved by calculating the reweighting factor on-the-fly during the simulation [58]. This requires a modification of the MD program such that ∇U (x) is added to an internal variable at each integration time, and the variable is written to a file at the same frequency as the positions.…”

“…Ref. [58] reports an implementation for calculating the path reweighting on-the-fly in OpenMM [59]. It has been tested on reweighting MSM transition probabilities [58], as well as mean first hitting times [60] in alanine dipeptide.…”

An Efficient Strategy to Estimate Thermodynamics and Kinetics of G Protein-Coupled Receptor Activation Using Metadynamics and Maximum Caliber

“…Another recently proposed strategy based on Girsanov reweighting of the transition matrix 42 is promising, but has yet to be validated for large biological systems. Here, we test the efficiency and accuracy of the MaxCal principle in extracting kinetic rates from MetaD.…”

Backward Simulation of Stochastic Process Using a Time Reverse Monte Carlo Method

“…3 and 4 and its improved version based on SMC in Sect. 6. There have been several studies using "path reweighting" in computational chemistry.…”