Title: Markov State Models from short non-Equilibrium Simulations - Analysis and Correction of Estimation Bias,
Abstract: Many state of the art methods for the thermodynamic and kinetic
characterization of large and complex biomolecular systems by simulation rely
on ensemble approaches, where data from large numbers of relatively short
trajectories are integrated. In this context, Markov state models (MSMs) are
extremely popular because they can be used to compute stationary quantities and
long-time kinetics from ensembles of short simulations, provided that these
short simulations are in "local equilibrium" within the MSM states. However, in
the last over 15 years since the inception of MSMs, it has been controversially
discussed and not yet been answered how deviations from local equilibrium can
be detected, whether these deviations induce a practical bias in MSM
estimation, and how to correct for them. In this paper, we address these
issues: We systematically analyze the estimation of Markov state models (MSMs)
from short non-equilibrium simulations, and we provide an expression for the
error between unbiased transition probabilities and the expected estimate from
many short simulations. We show that the unbiased MSM estimate can be obtained
even from relatively short non-equilibrium simulations in the limit of long lag
times and good discretization. Further, we exploit observable operator model
(OOM) theory to derive an unbiased estimator for the MSM transition matrix that
corrects for the effect of starting out of equilibrium, even when short lag
times are used. Finally, we show how the OOM framework can be used to estimate
the exact eigenvalues or relaxation timescales of the system without estimating
an MSM transition matrix, which allows us to practically assess the
discretization quality of the MSM. Applications to model systems and molecular
dynamics simulation data of alanine dipeptide are included for illustration.
The improved MSM estimator is implemented in PyEMMA as of version 2.3.