Theoretical restrictions on longest implicit timescales in Markov state models of biomolecular dynamics

TL;DR

This paper provides the first analytical proof supporting the rule that the slowest implicit timescale in Markov state models of biomolecular dynamics is comparable to the total simulation time, with implications for improving simulation efficiency.

Contribution

It derives analytical expressions for the slowest timescale in MSMs, clarifying the factors influencing its magnitude and validating a widely used heuristic.

Findings

The slowest implicit timescale equals the product of total sampling and four key factors.

Most practical cases have these factors close to unity, validating the heuristic.

Analytical results can guide more efficient biomolecular simulations.

Abstract

Markov state models (MSMs) have been widely used to analyze computer simulations of various biomolecular systems. They can capture conformational transitions much slower than an average or maximal length of a single molecular dynamics (MD) trajectory from the set of trajectories used to build the MSM. A rule of thumb claiming that the slowest implicit timescale captured by an MSM should be comparable by the order of magnitude to the aggregate duration of all MD trajectories used to build this MSM has been known in the field. However, this rule have never been formally proved. In this work, we present analytical results for the slowest timescale in several types of MSMs, supporting the above rule. We conclude that the slowest implicit timescale equals the product of the aggregate sampling and four factors that quantify: (1) how much statistics on the conformational transitions corresponding to the longest implicit timescale is available, (2) how good the sampling of the destination Markov state is, (3) the gain in statistics from using a sliding window for counting transitions between Markov states, and (4) a bias in the estimate of the implicit timescale arising from finite sampling of the conformational transitions. We demonstrate that in many practically important cases all these four factors are on the order of unity, and we analyze possible scenarios that could lead to their significant deviation from unity. Overall, we provide for the first time analytical results on the slowest timescales captured by MSMs. These results can guide further practical applications of MSMs to biomolecular dynamics and allow for higher computational efficiency of simulations.