Unbiased estimation of equilibrium, rates, and committors from Markov state model analysis

TL;DR

This paper develops theoretically grounded, unbiased estimators for key molecular dynamics observables from Markov state models, addressing biases caused by coarse-graining and finite trajectory lengths.

Contribution

It introduces unbiased estimators for equilibrium populations, mean first-passage times, and committors, extending reweighting schemes and accounting for finite trajectory averaging.

Findings

Unbiased estimators can be derived despite coarse-graining.

Extension of reweighting schemes accelerates convergence to unbiased values.

Proper boundary conditions are crucial for unbiased steady-state property estimation.

Abstract

Markov state models (MSMs) have been broadly adopted for analyzing molecular dynamics trajectories, but the approximate nature of the models that results from coarse-graining into discrete states is a long-known limitation. We show theoretically that, despite the coarse graining, in principle MSM-like analysis can yield unbiased estimation of key observables. We describe unbiased estimators for equilibrium state populations, for the mean first-passage time (MFPT) of an arbitrary process, and for state committors - i.e., splitting probabilities. Generically, the estimators are only asymptotically unbiased but we describe how extension of a recently proposed reweighting scheme can accelerate relaxation to unbiased values. Exactly accounting for 'sliding window' averaging over finite-length trajectories is a key, novel element of our analysis. In general, our analysis indicates that coarse-grained MSMs are asymptotically unbiased for steady-state properties only when appropriate boundary conditions (e.g., source-sink for MFPT estimation) are applied directly to trajectories, prior to calculation of the appropriate transition matrix.