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Markov state models (MSMs) have been widely used to analyze computersimulations of various biomolecular systems. They can capture conformationaltransitions much slower than an average or maximal length of a single moleculardynamics (MD) trajectory from the set of trajectories used to build the MSM. Arule of thumb claiming that the slowest implicit timescale captured by an MSMshould be comparable by the order of magnitude to the aggregate duration of allMD 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 analyticalresults for the slowest timescale in several types of MSMs, supporting theabove rule. We conclude that the slowest implicit timescale equals the productof the aggregate sampling and four factors that quantify: (1) how muchstatistics on the conformational transitions corresponding to the longestimplicit timescale is available, (2) how good the sampling of the destinationMarkov state is, (3) the gain in statistics from using a sliding window forcounting transitions between Markov states, and (4) a bias in the estimate ofthe implicit timescale arising from finite sampling of the conformationaltransitions. We demonstrate that in many practically important cases all thesefour factors are on the order of unity, and we analyze possible scenarios thatcould lead to their significant deviation from unity. Overall, we provide forthe first time analytical results on the slowest timescales captured by MSMs.These results can guide further practical applications of MSMs to biomoleculardynamics and allow for higher computational efficiency of simulations.

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