Abstract
We propose a discrete transition-based reweighting analysis method (dTRAM) for analyzing configuration-space-discretized simulation trajectories produced at different thermodynamic states (temperatures, Hamiltonians, etc.) dTRAM provides maximum-likelihood estimates of stationary quantities (probabilities, free energies, expectation values) at any thermodynamic state. In contrast to the weighted histogram analysis method (WHAM), dTRAM does not require data to be sampled from global equilibrium, and can thus produce superior estimates for enhanced sampling data such as parallel/simulated tempering, replica exchange, umbrella sampling, or metadynamics. In addition, dTRAM provides optimal estimates of Markov state models (MSMs) from the discretized state-space trajectories at all thermodynamic states. Under suitable conditions, these MSMs can be used to calculate kinetic quantities (e.g., rates, timescales). In the limit of a single thermodynamic state, dTRAM estimates a maximum likelihood reversible MSM, while in the limit of uncorrelated sampling data, dTRAM is identical to WHAM. dTRAM is thus a generalization to both estimators.
Highlights
The dynamics of complex stochastic systems are often governed by rare events – examples include protein folding, macromolecular association, or phase transitions
In Ref. 18, we have introduced the first transition-based reweighting analysis method (TRAM) estimator that is applicable to practical molecular dynamics data, and could show that it can provide superior estimates of equilibrium probabilities and free energy compared with weighted histogram analysis method (WHAM)
We have shown that discrete transition-based reweighting analysis method (dTRAM) is a proper generalization of both methods, i.e., both the WHAM estimator and the reversible Markov state models (MSMs) estimator can be derived as special cases from the dTRAM equations
Summary
The dynamics of complex stochastic systems are often governed by rare events – examples include protein folding, macromolecular association, or phase transitions. When the probability density of trajectories can be evaluated, MBAR can be applied to trajectories instead of sample configurations, obtaining estimates of dynamical expectations.12–15 Both WHAM and MBAR are statistically optimal under specific assumptions as they can be derived from maximum-likelihood or minimum variance principles.. The transition-based reweighting analysis method (TRAM) aims at combining the advantages of reweighting estimators and MSMs. In Ref. 19, we have defined TRAM as a class of estimators that (1) take the statistical weights of samples at different thermodynamic states into account, in order to reweigh these samples; and (2) exploits transitions observed in the sampled trajectories, without assuming that these trajectories are sampled from equilibrium. A number of applications are shown to demonstrate the usefulness and versatility of dTRAM
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