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Many state of the art methods for the thermodynamic and kineticcharacterization of large and complex biomolecular systems by simulation relyon ensemble approaches, where data from large numbers of relatively shorttrajectories are integrated. In this context, Markov state models (MSMs) areextremely popular because they can be used to compute stationary quantities andlong-time kinetics from ensembles of short simulations, provided that theseshort simulations are in "local equilibrium" within the MSM states. However, inthe last over 15 years since the inception of MSMs, it has been controversiallydiscussed and not yet been answered how deviations from local equilibrium canbe detected, whether these deviations induce a practical bias in MSMestimation, and how to correct for them. In this paper, we address theseissues: We systematically analyze the estimation of Markov state models (MSMs)from short non-equilibrium simulations, and we provide an expression for theerror between unbiased transition probabilities and the expected estimate frommany short simulations. We show that the unbiased MSM estimate can be obtainedeven from relatively short non-equilibrium simulations in the limit of long lagtimes and good discretization. Further, we exploit observable operator model(OOM) theory to derive an unbiased estimator for the MSM transition matrix thatcorrects for the effect of starting out of equilibrium, even when short lagtimes are used. Finally, we show how the OOM framework can be used to estimatethe exact eigenvalues or relaxation timescales of the system without estimatingan MSM transition matrix, which allows us to practically assess thediscretization quality of the MSM. Applications to model systems and moleculardynamics simulation data of alanine dipeptide are included for illustration.The improved MSM estimator is implemented in PyEMMA as of version 2.3.

Markov State Models from short non-Equilibrium Simulations - Analysis and Correction of Estimation Bias