Girsanov reweighting for path ensembles and Markov state models

Bettina Keller
Freie Universität Berlin

Many molecular processes, such as protein-protein interactions or protein-ligand binding events, are not accessible with direct MD simulation - even on a supercomputer. Enhanced sampling techniques, such as metadynamics or umbrella sampling, in which a biasing potential U(x) is added to the unbiased force field V(x) increase the sampling of rare events. However, the distortion of the timescales in the system due to the biasing potential is not uniform. The resulting biased trajectories can hence not be used to estimate models of the molecular dynamics, e.g. Markov state models.

I will present the Girsanov reweighting method with which one can estimate the the expected path ensemble average of an unbiased dynamics for a set of biased paths. The method 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 of molecular dynamics can be formulated as a combined phase-space and path ensemble average, the method can be extended to reweight these models by combining it with a reweighting of the Boltzmann distribution. I will explain 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 I will demonstrate the construction of Markov state models from biased simulations for several test systems. Besides its use in enhanced sampling simulations, the Girsanov reweighting can also be used to test the response of the slow dynamic processes to perturbations of the potential energy surface. It will therefore likely be useful in the development of new force fields and surrogate models.

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