Path probability ratios for Langevin dynamics – exact and approximate*

Enhanced sampling techniques generate trajectories at a biased potential, such that rare events occur more frequently. Path reweighing techniques recover the transition rates of the unbiased system from the biased trajectories by calculating the path probability ratio. Path reweighing requires that (a) the trajectory has been generated using an integration scheme for stochastic dynamics, and (b) that the formula for the path probability ratio has been tailored for that specific integration scheme. This makes them technically difficult, because a separate reweighing factor for each stochastic integration scheme is needed.

Most published path probability ratios are derived for overdamped Langevin dynamics. Yet, overdamped Langevin dynamics is rarely used in MD simulations. Instead a variety of integration schemes for Langevin dynamics are used. Simply applying the path probability ratio for overdamped Langevin dynamics to a Langevin trajectory introduces a sizeable error.

Here we derive the path probability ratio for the integration scheme of Langevin dynamics implemented in OpenMM. We demonstrate that it to accurately reweights Langevin trajectories.

By comparing this path probability ratio to the path probability ratio for overdamped Langevin dynamics, we then derive an approximate and general path probability ratio for Langevin dynamics. This approximate path probability ratio depends on the random number sequence used to generate the Langevin trajectory and on the bias potential. Because it is independent of the integration scheme, it removes requirement (b) and can be used as multi-purpose probability ratio for any Langevin trajectory. We show that the approximate path probability ratio yields highly accurate results, and discuss the limits of the approximation.

L Donati, BG Keller, J Chem Phys. 149, 072335 (2018)
S Kieninger, L Donati, BG Keller, Curr Opin Struct Biol, 61, 124-131 (2020)