Reducing Weighted Ensemble Variance With Optimal Trajectory Management

Weighted ensemble (WE) is an enhanced path-sampling method that is conceptually simple, widely applicable, and statistically exact. In a WE simulation, an ensemble of trajectories is periodically pruned or replicated to enhance sampling of rare transitions and improve estimation of mean first passage times (MFPTs). However, poor choices of the parameters governing pruning and replication can lead to high-variance MFPT estimates. Our previous work [J. Chem. Phys. 158, 014108 (2023)] presented an optimal WE parameterization strategy and applied it in low-dimensional example systems. The strategy harnesses estimated local MFPTs from different initial configurations to a single target state. In the present work, we apply the optimal parameterization strategy to more challenging, high-dimensional molecular models, namely, synthetic molecular dynamics (MD) models of Trp-cage folding and unfolding, as well as atomistic MD models of NTL9 folding in high-friction and low-friction continuum solvents. In each system we use WE to estimate the MFPT for folding or unfolding events. We show that the optimal parameterization reduces the variance of MFPT estimates in three of four systems, with dramatic improvement in the most challenging atomistic system. Overall, the parameterization strategy improves the accuracy and reliability of WE estimates for the kinetics of biophysical processes.

💡 Research Summary

Weighted Ensemble (WE) is a path‑sampling technique that runs many independent trajectories in parallel and periodically resamples them—replicating trajectories in under‑populated regions and pruning those in over‑populated regions—while preserving statistical weights. This “pruning‑and‑replication” scheme dramatically improves sampling of rare events such as protein folding, ligand binding, or membrane permeation. However, the variance of the estimated mean first‑passage time (MFPT) across independent WE runs can be very large, and this run‑to‑run variance is highly sensitive to the choice of two hyper‑parameters: (i) the binning of phase space (i.e., how the collective variables are discretized) and (ii) the target number of trajectories allocated to each bin. Poor choices lead to unreliable kinetic estimates even though the WE estimator remains unbiased in the mean.

In a previous study (J. Chem. Phys. 158, 014108, 2023) the authors introduced a systematic strategy to minimize MFPT variance by exploiting local MFPT information obtained from a short “training” WE simulation. The present paper extends that strategy to realistic, high‑dimensional molecular models: synthetic MD models of the Trp‑cage mini‑protein (both folding and unfolding) and all‑atom MD models of the N‑terminal domain of ribosomal protein L9 (NTL9) simulated in high‑friction and low‑friction continuum solvents.

The core of the variance‑minimization method rests on two scalar fields defined over configuration space:

1. Discrepancy, h(x) – the normalized difference between the global MFPT (starting from the steady‑state distribution π) and the local MFPT obtained when all trajectories start from a single point x. Formally, \