Accelerated Estimation of Long-timescale Kinetics by Combining Weighted Ensemble Simulation with Markov Model “Microstates” using Non-Markovian Theory

Abstract

The weighted ensemble (WE) simulation strategy can provide unbiased sampling of nonequilibrium processes, such as molecular folding or binding. Unbiased kinetic rates can be extracted from any discrete clustering of the configuration space based on a historyaugmented Markov state model (haMSM) at any lag time, in the steady-state. However, the convergence of WE to steady-state may require unaffordably long simulations in complex systems. Here we show that by clustering molecular configurations into many (thousands of) microbins using methods developed in the Markov State Modeling (MSM) community, unbiased kinetics can be obtained from WE data using history-augmented Markov State Models (haMSMs) before SS convergence of the WE simulation itself. Because arbitrarily small lag times can be used within the exact haMSM formulation, accurate kinetics can be obtained with significantly less trajectory data than traditional MSMs, while bypassing the often prohibitive convergence requirements of the non-equilibrium weighted ensemble. We validate the method in a simple diffusive process on a 2D random energy landscape, and apply the method to atomistic protein folding simulations using WE molecular dynamics.We report significant progress towards the unbiased estimation of protein folding times and pathways, though key challenges remain.

A Suite of Tutorials for the WESTPA Rare-Events Sampling Software [Article v1.0]

“…If WE was performed with a "recycling" condition where trajectories reaching B are fed back to A, then the rate constant for the process can be estimated from the probability flux arriving to state B if the simulation achieves steady state and hence constant flux [20,21]. If a WE simulation does not achieve steady state, it is still possible in principle to estimate rate constants using a non-Markovian analysis, also called a history-augmented Markov State Model [11,22,23].…”

Computational estimation of ms-sec atomistic folding times

“…[66][67] We also employed haMSM analysis, which is unbiased for steady-state flux estimation at arbitrary lag times, and small lag times allow fuller use of the extensive WE data. 56,64,68 The approach is of particular interest for Protein G because, in principle, a haMSM can estimate steady-state behavior using trajectories generated in the transient period -i.e., in the approach to steady state. As noted, the flux profile for Protein G indicates those WE simulations clearly remained in the transient regime.…”

“…The novelty of the results is their relatively high precision and unbiased nature due to the theoretical foundations of the WE and haMSM methods. 35,56,64 Hence, although comparison to experimental folding times are shown in Table 1, readers are cautioned that the present study should be considered a first step in assessment of molecular models and initial ensembles. Given these caveats, the rough agreement with experimental values is encouraging but also points to the need for further investigation of solvent modeling and initial ensembles as discussed below.…”

“…[96][97] The continuous trajectories generated by WE allow application of the historyaugmented MSMs at arbitrary lag times, which are unbiased for estimation of steady-state fluxes. 56,64 Using short lag-times for haMSMs in turn allows use of all data generated. Figure 1: The WE procedure and comparison to regular MD simulation.…”

“…Importantly, the continuous WE trajectories spanning from unfolded to folded macrostates enable folding rate estimation using the history-augmented Markov state model (haMSM) formalism which is unbiased at arbitrary lag times. 56,64…”