Markov State Models with Weighted Ensemble Simulation: How to Eliminate the Trajectory Merging Bias

The weighted ensemble (WE) algorithm is gaining popularity as a rare event method for studying long timescale processes with molecular dynamics. WE is particularly useful for determining kinetic properties, such as rates of protein (un)folding and ligand (un)binding, where transition rates can be calculated from the flux of trajectories into a target basin of interest. However, this flux depends exponentially on the number of splitting events that a given trajectory experiences before reaching the target state and can vary by orders of magnitude between WE replicates. Markov state models (MSMs) are helpful tools to aggregate information across multiple WE simulations and have previously been shown to provide more accurate transition rates than WE alone. Discrete-time MSMs are models that coarsely describe the evolution of the system from one discrete state to the next using a discrete lag time, τ. When an MSM is built using conventional MD data, longer values of τ typically provide more accurate results. Combining WE simulations with Markov state modeling presents some additional challenges, especially when using a value of τ that exceeds the lag time between resampling steps in the WE algorithm, τ_WE. Here, we identify a source of bias that occurs when τ > τ_WE, which we refer to as "merging bias". We also propose an algorithm to eliminate the merging bias, which results in merging bias-corrected MSMs, or "MBC-MSMs". Using a simple model system, as well as a complex biomolecular example, we show that MBC-MSMs significantly outperform both τ = τ_WE MSMs and uncorrected MSMs at longer lag times.