The efficient calculation of rare-event kinetics in complex dynamical systems, such as the rate and pathways of ligand dissociation from a protein, is a generally unsolved problem. Markov state models can systematically integrate ensembles of short s
imulations and thus effectively parallelize the computational effort, but the rare events of interest still need to be spontaneously sampled in the data. Enhanced sampling approaches, such as parallel tempering or umbrella sampling, can accelerate the computation of equilibrium expectations massively - but sacrifice the ability to compute dynamical expectations. In this work we establish a principle to combine knowledge of the equilibrium distribution with kinetics from fast downhill relaxation trajectories using reversible Markov models. This approach is general as it does not invoke any specific dynamical model, and can provide accurate estimates of the rare event kinetics. Large gains in sampling efficiency can be achieved whenever one direction of the process occurs more rapid than its reverse, making the approach especially attractive for downhill processes such as folding and binding in biomolecules.
Direct simulation of biomolecular dynamics in thermal equilibrium is challenging due to the metastable nature of conformation dynamics and the computational cost of molecular dynamics. Biased or enhanced sampling methods may improve the convergence o
f expectation values of equilibrium probabilities and expectation values of stationary quantities significantly. Unfortunately the convergence of dynamic observables such as correlation functions or timescales of conformational transitions relies on direct equilibrium simulations. Markov state models are well suited to describe both, stationary properties and properties of slow dynamical processes of a molecular system, in terms of a transition matrix for a jump process on a suitable discretiza- tion of continuous conformation space. Here, we introduce statistical estimation methods that allow a priori knowledge of equilibrium probabilities to be incorporated into the estimation of dynamical observables. Both, maximum likelihood methods and an improved Monte Carlo sampling method for reversible transition ma- trices with fixed stationary distribution are given. The sampling approach is applied to a toy example as well as to simulations of the MR121-GSGS-W peptide, and is demonstrated to converge much more rapidly than a previous approach in [F. Noe, J. Chem. Phys. 128, 244103 (2008)]
