Limiting relaxation times from Markov state models

Markov state models (MSMs) are more and more widely used in the analysis of molecular simulations to incorporate multiple trajectories together and obtain more accurate time scale information of the slowest processes in the system. Typically, however, multiple lagtimes are used and analyzed as input parameters, yet convergence with respect to the choice of lagtime is not always possible. Here, we present a simple method for calculating the slowest relaxation time (RT) of the system in the limit of very long lagtimes. Our approach relies on the fact that the second eigenvector's autocorrelation function of the propagator will be approximately single exponential at long lagtimes. This allows us to obtain a simple equation for the behavior of the MSM's relaxation time as a function of the lagtime with only two free parameters, one of these being the RT of the system. We demonstrate that the second parameter is a useful indicator of how Markovian a selected variable is for building the MSM. Fitting this function to data gives a limiting value for the optimal variational RT. Testing this on analytic and molecular dynamics data for Ala5 and umbrella sampling-biased ion channel simulations shows that the function accurately describes the behavior of the RT and furthermore that this RT can improve noticeably the value calculated at the longest accessible lagtime. We compare our RT limit to the hidden Markov model (HMM) approach that typically finds RTs of comparable values. However, HMMs cannot be used in conjunction with biased simulation data, requiring more complex algorithms to construct than MSMs, and the derived RTs are not variational, leading to ambiguity in the choice of lagtime at which to build the HMM.