# Thermodynamic and Kinetic Analysis of Molecular Conformational Dynamics in a Riemannian Framework

## TL;DR

This paper introduces a new geometric framework using Riemannian geometry to improve the accuracy of molecular simulations by addressing issues with coordinate transformations.

## Contribution

A novel Riemannian framework is introduced to ensure invariance of key thermodynamic and kinetic quantities under coordinate transformations.

## Key findings

- The framework resolves noninvariance issues in conventional PMF definitions through Riemannian geometry.
- A generalized Riemannian diffusion model enables rigorous estimation of kinetic properties in CV spaces.
- Bayesian integration with the framework improves the reliability of free energy and transition rate calculations.

## Abstract

We have formulated
a Riemannian framework for describing the geometry
of collective variable (CV) spaces in molecular simulations and demonstrate
its applicability through both toy model potentials and a biomolecular
example. This formalism addresses significant theoretical challenges
arising from the inherent nonlinearity of CV transformations, ensuring
critical quantities such as the potential of mean force (PMF), minimum
free energy path (MFEP), and rate constant remain invariant under
coordinate transformations. Our framework identifies and addresses
the noninvariance issues of conventional PMF definitions, clearly
illustrating their limitations through simple illustrative examples.
To overcome these issues, we introduce invariant definitions of PMF
and MFEP using Riemannian geometry. Moreover, we propose a generalized
Riemannian diffusion model applicable to diffusive dynamics within
CV spaces, allowing rigorous estimation of kinetic properties. Through
this model, we derive practical numerical methods for determining
the PMF, diffusion constant, metric tensor, and transition rates from
unbiased simulations conducted along identified transition pathways.
By integrating Bayesian approaches with the Riemannian framework,
our method provides a statistically robust technique for accurately
calculating free energy landscapes and transition kinetics, thereby
enhancing the reliability and interpretability of biomolecular simulations.