[Paper Review] RiteWeight: Randomized Iterative Trajectory Reweighting for Steady-State Distributions Without Discretization Error

Sagar Kania, Webber, Robert J.+3 more|arXiv (Cornell University)|Jan 11, 2024

Protein Structure and DynamicsBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

RiteWeight is a novel iterative trajectory reweighting algorithm that estimates unbiased steady-state distributions in molecular dynamics without discretization error by using randomized, iterative clustering and solving for the stationary distribution of a transition matrix. It achieves accurate, quasi-continuous probability estimates even with coarse clustering or short trajectories, outperforming single-shot reweighting methods that suffer from phase-space discretization bias.

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

Motivation & Objective

• To address the persistent challenge of discretization error in trajectory reweighting methods used for estimating steady-state distributions in molecular dynamics.
• To develop a method that corrects trajectory weights at quasi-continuous resolution, independent of clustering resolution.
• To enable accurate estimation of equilibrium and nonequilibrium steady-state distributions from short or unconverged trajectories without biasing forces.
• To overcome the 'chicken-and-egg' problem in Markov state models where local equilibrium is required for accurate stationary distribution estimation.
• To provide a robust, iterative reweighting framework that converges to the true stationary distribution regardless of initial clustering granularity.

Proposed method

• The algorithm constructs a discrete-state transition matrix (T) from trajectory fragments at a fixed lag time, representing phase-space transitions.
• It computes the stationary distribution π by solving Tπ = π, using the current clustering and trajectory weights.
• Trajectories are reweighted iteratively so that total cluster weights match π, while preserving relative weights within each cluster.
• In each iteration, a new random clustering is applied to the same trajectory data, enabling reassignment of configurations to different clusters and refining weight estimates.
• The process is repeated until convergence, with the learning rate r=1 used in the proof-of-concept study.
• The method is applied to synthetic Trp-Cage MD trajectories, comparing results across 10 and 1,000 clusters to assess robustness to discretization.

Figure 1: Schematic of 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 of the trajectories, the transition matrix $\mathbf{T}$ is computed and solved to yield the st

Experimental results

Research questions

• RQ1 Can iterative reweighting with randomized clustering eliminate discretization error in trajectory-based steady-state distribution estimation?
• RQ2 Does RiteWeight converge to the exact equilibrium distribution even when using coarse clustering (e.g., 10 clusters)?
• RQ3 How does RiteWeight’s performance compare to single-shot reweighting in terms of accuracy and convergence speed?
• RQ4 Can RiteWeight produce accurate stationary distributions from trajectories as short as a single lag time?
• RQ5 Does the iterative, randomized clustering approach enable quasi-continuous resolution in weight estimation despite discrete state representations?

Key findings

• RiteWeight converged to the exact equilibrium distribution for Trp-Cage SynMD data regardless of clustering resolution, achieving accurate results with both 10 and 1,000 clusters.
• The symmetric Kullback-Leibler divergence showed convergence within ~1,000 iterations for 1,000 clusters and ~100,000 iterations for 10 clusters, indicating slower convergence with coarser clustering.
• Single-shot reweighting exhibited significant bias due to discretization error, especially with 10 clusters, and failed to match the exact stationary distribution even at 1,000 clusters.
• The algorithm successfully mitigated phase-space discretization error by leveraging iterative random clustering, enabling accurate estimation without requiring locally equilibrated trajectories.
• RiteWeight demonstrated robustness to clustering resolution, suggesting it functions as a form of 'super-resolution' Markov modeling.
• The method enables accurate estimation of observables from trajectories of any length, including single-step fragments, due to its independence from dynamical relaxation.

Figure 2: RiteWeight convergence to equilibrium in Trp-cage system. Plotted is the evolution of the Trp-Cage SynMD states’ probability distribution function (PDF) across RiteWeight iterations (color bar at right). The black dashed curve represents the initial PDF based on uniformly weighted trajecto