# Girsanov reweighting for path ensembles and Markov state models

## TL;DR

This paper introduces a Girsanov reweighting method to estimate how molecular dynamics path ensembles and Markov state models respond to potential energy perturbations, enabling efficient sensitivity analysis and trajectory reweighting.

## Contribution

The paper develops a Girsanov reweighting approach for path ensembles and MSMs, extending stochastic analysis tools to molecular dynamics simulations for the first time.

## Key findings

- Effective reweighting of path ensembles demonstrated on test systems.
- Method enables on-the-fly reweighting during simulations.
- Applicable to both implicit and explicit solvent 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 phase-space and path ensemble average, the method can be extended toreweight 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.