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

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.