From Alanine Dipeptide to Glycerol-3-phosphate transpor: Invariant PMFs and Rates

Analyses of molecular dynamics (MD) often project high-dimensional motion onto collective variables (CVs), yet thermodynamic and kinetic quantities computed in CV space can spuriously depend on the coordinates themselves. We present a Riemannian framework that treats CV space as a manifold with a natural metric, ensuring invariance of the potential of mean force (PMF), minimum free-energy path (MFEP), and rate constants under smooth reparameterizations. We first diagnose why conventional PMF definitions and path constructions violate invariance, using simple examples to clarify the resulting biases. We then define geometry-aware PMF and MFEP and introduce a generalized diffusion model on the manifold with position-dependent metric and diffusivity. From this model, we derive practical estimators for the along-the-path metric, PMF, and transition rates from unbiased trajectories. Applied to alanine dipeptide, the method recovers a single, invariant free-energy surface and consistent MFEPs and rate estimates across distinct CV choices, showing that apparent discrepancies are coordinate artifacts. We further validate the framework on the glycerol-3-phosphate transporter (GlpT), a major facilitator superfamily protein that alternates between inward- and outward-facing states, obtaining the highest barrier of 7.3 kcal/mol (vs. ~8.4 kcal/mol experimentally) and transition rates of the same order of magnitude as experiment.