Modeling non-Markovian data using Markov state and Langevin models

Markov processes provide a popular approach to construct low-dimensional dynamical models of a complex biomolecular system. By partitioning the conformational space into metastable states, protein dynamics can be approximated in terms of memory-less jumps between these states, resulting in a Markov state model (MSM). Alternatively, suitable low-dimensional collective variables may be identified to construct a data-driven Langevin equation (dLE). In both cases, the underlying Markovian approximation requires a propagation time step (or lag time) delta t that is longer than the memory time tau (M) of the system. On the other hand, delta t needs to be chosen short enough to resolve the system timescale tau (S) of interest. If these conditions are in conflict (i.e., tau (M) > tau (S)), one may opt for a short time step delta t = tau (S) and try to account for the residual non-Markovianity of the data by optimizing the transition matrix or the Langevin fields such that the resulting model best reproduces the observables of interest. In this work, rescaling the friction tensor of the dLE based on short-time information in order to obtain the correct long-time behavior of the system is suggested. Adopting various model problems of increasing complexity, including a double-well system, the dissociation of solvated sodium chloride, and the functional dynamics of T4 lysozyme, the virtues and shortcomings of the rescaled dLE are discussed and compared to the corresponding MSMs.