RiteWeight: Randomized Iterative Trajectory Reweighting for Steady-State Distributions Without Discretization Error

TL;DR

The paper addresses the challenge that molecular dynamics sampling often fails to converge to the correct stationary distribution, limiting accurate thermodynamic and kinetic observables. It introduces RiteWeight, an iterative reweighting algorithm that uses random clustering to estimate stationary distributions from unconverged trajectory data, without relying on the Markov property at the cluster level. The authors provide a fixed-point analysis and demonstrate accuracy on synthetic and atomistic Trp-cage data for both equilibrium and nonequilibrium states, including mean first passage times and net fluxes. RiteWeight outperforms traditional MSM reweighting and single-shot approaches by delivering quasi-continuous distributions and accurate path-based observables even at short lag times, with broad implications for Boltzmann-weighted ensemble generation and reaction-trajectory analyses.

Abstract

A significant challenge in molecular dynamics (MD) simulations is ensuring that sampled configurations converge to the equilibrium or nonequilibrium stationary distribution of interest. Lack of convergence constrains the estimation of free energies, rates, and mechanisms of complex molecular events. Here, we introduce the "Randomized ITErative trajectory reWeighting" (RiteWeight) algorithm to estimate a stationary distribution from unconverged simulation data. This method iteratively reweights trajectory segments in a self-consistent way by solving for the stationary distribution of a Markov state model (MSM), updating segment weights, and employing a new random clustering in each iteration. The iterative random clustering mitigates the phase-space discretization error inherent in existing trajectory reweighting techniques and yields quasi-continuous configuration-space distributions. We present mathematical analysis of the algorithm's fixed points as well as empirical validation using both synthetic MD Trp-cage trajectories, for which the stationary solution is exactly calculable, and standard atomistic MD Trp-cage trajectories extracted from a long reference simulation. In both test systems, we find that RiteWeight corrects flawed distributions and generates accurate observables for equilibrium and nonequilibrium steady states. The results highlight the value of correcting the underlying trajectory distribution rather than using a standard MSM

Figures (12)

• Figure 1: The RiteWeight algorithm. In each iteration, a fixed set of trajectories (red arrows) is organized into clusters (colored regions). Based on the discrete clusters and current weights (circle sizes) of the trajectories, the transition matrix $\boldsymbol{T}$ is computed and solved to yield the stationary measure $\boldsymbol{\pi}$ for the given clusters. Each trajectory is then assigned a new weight (filled circles) so that the total cluster weights match $\boldsymbol{\pi}$ but the relative weights of the trajectories starting within each cluster remain unchanged. In subsequent iterations, the process is repeated with new cluster boundaries, enabling changes in the relative weights of trajectories formerly in the same cluster, e.g., trajectories 1 and 2 in iteration 2. Ultimately all the initial weights are changed relative to one another.
• Figure 2: Estimates of the equilibrium distribution for the SynMD Trp-cage system. (a) RiteWeight recovers the true equilibrium distribution (green) starting from a far away estimate (red dashed), using either 10 (dark, dashed orange) or 1,000 clusters (orange). Here, the lag time is 1 ns. (b) Single-shot reweighting deviates from the true distribution using 10 (dashed light blue) or 1,000 (dark blue) clusters. Note that the state index is ordered according to TIC1.
• Figure 3: Estimates of the equilibrium distribution for the atomistic Trp-cage system. Reference data is from a 208 $\mu$s MD trajectory (green with shaded error bars). Starting from mis-distributed input data (red dashed), RiteWeight recovers the true distribution (dark orange). Also shown are MSM "single shot" estimates based on 100--50,000 clusters (blue dashed lines). Both RiteWeight and the MSM employ a lag time of 10 ns, and RiteWeight uses $n = 10$ clusters.
• Figure 4: Estimates of the nonequilibrium distribution for the atomistic Trp-cage system. The plot shows the probability density for the intermediate region, after excluding the folded and unfolded states. The initial distribution (red) represents all MD samples outside the folded and unfolded states, while the MD reference distribution (green) is derived from trajectory segments more recently in the unfolded than the folded state. RiteWeight (dark orange) closely follows the reference distribution.
• Figure 5: Estimates of the mean first passage time for the atomistic Trp-cage system to fold. Reference MD values (dark green) are based on 20 or fewer events with associated uncertainty (green shaded region), and they are compared to RiteWeight estimates based on $10$ clusters (dark orange) and MSM estimates based on $50$ clusters (blue).

Theorems & Definitions (3)

• Definition 1: Fixed point
• Theorem 1: RiteWeight fixed point
• proof