A Riemannian Framework for Tackling Large-Scale Conformational Changes of Proteins using All-ATOM Molecular Dynamics Simulations

Abstract

Simulation studies of biomolecular processes often suffer from the curse of dimensionality both in sampling and in analysis. Dimensionality reduction, either using intuitive reaction coordinates or automated algorithms, is thus an inevitable part of advanced biomolecular simulation studies. Most free energy calculation methods and path-finding algorithms, implicitly or explicitly, rely on an effective diffusion model in a low-dimensional space as their underlying theoretical framework. Dimensionality reduction, however, comes with a price and the effective dynamics of the reduced system could be nontrivial. For instance, assuming the diffusivity of the effective dynamics, the diffusion tensor is likely to be position-dependent and anisotropic.We have developed a Riemannian framework for free energy calculation methods and path-finding algorithms in molecular dynamics simulations. We show that a Riemannian diffusion model may describe the dynamics of the reduced system such that a Riemannian metric replaces the position-dependent diffusion tensor. In addition, the Riemannian formulation provides a suitable mathematical framework for generalizing one-dimensional free energy relations to multi-dimensional spaces in a rigorous manner such that the model is invariant under any smooth coordinate transformation.Within the proposed mathematical framework, a novel combination of free energy calculation methods and path-finding algorithms is developed which can be used for accurate calculation of free energies along protein conformational transition pathways using all-atom molecular dynamics simulations. The method is particularly useful for describing large-scale and complex conformational changes of proteins. We note that the simplified Euclidean version of the method was recently employed successfully in reconstructing the transport cycle of a membrane transporter at an atomic level (see M. Moradi, et al, “Atomic-level characterization of transport cycle thermodynamics in the glycerol-3-phosphate:phosphate antiporter.”, Nat. Commun., 6:8393, 2015).