What question did this study set out to answer?

This work aims to provide a robust framework for invariant analysis in molecular simulations using Riemannian geometry.

February 21, 2026

BPS2026 – A Riemannian geometry framework for invariant analysis of molecular simulations

Key Points

This work aims to provide a robust framework for invariant analysis in molecular simulations using Riemannian geometry.
Developed a framework treating collective variable space as a Riemannian manifold.
Derived invariant definitions for potential of mean force and minimum free energy path.
Created a model for diffusion on a manifold to estimate kinetic properties.
Computed metric tensor and transition rates from simulation trajectories.
Integrated Bayesian inference for statistical robustness.
Established a method for calculating invariant free energy landscapes.
Demonstrated feasibility using alanine dipeptide as a benchmark system.
Provided definitions for kinetic properties that are invariant under transformations.

Abstract

A fundamental challenge in molecular dynamics (MD) simulations is that key quantities, such as the potential of mean force (PMF) and transition rates, often show an unphysical dependence on the specific choice of collective variables (CVs). This coordinate-dependence complicates the interpretation and rigor of theoretical models. To resolve this, we introduce a comprehensive framework that treats the CV space as a Riemannian manifold, providing a geometrically sound basis for analyzing biomolecular simulations. Our approach yields definitions for the PMF and the minimum free energy path (MFEP) that are guaranteed to be invariant under any smooth transformation of the CVs. We extend this formalism to kinetics by developing a generalized model for diffusion on the manifold, allowing for the rigorous estimation of kinetic properties. From this model, we derive practical numerical methods to compute the metric tensor, position-dependent diffusion constants, and transition rates directly from simulation trajectories. Furthermore, by integrating this geometric framework with Bayesian inference, we provide a statistically robust pipeline for the analysis. We demonstrate the practical feasibility of our method by calculating the invariant free energy landscape and kinetics for alanine dipeptide, a canonical benchmark system. This work establishes a robust foundation for improving the reliability and theoretical consistency of biomolecular simulations.

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