Modeling diffusive systems using flat-bottom restraints

Proteins and other large biomolecules (such as nucleic acids, sugars, phospolipids and their complexes) are known to undergo slow diffusive motions, which motivates a mathematical description of their evolution using overdamped (e.g. 1D Smoluchowski) diffusion theory.  In the present approach, we assume that a progress -- often collective -- variable, or a reaction coordinate (RC) describing a reaction or process of interest in a biomolecular system has been identified, and follows overdamped dynamics on an associated free energy (FE) landscape with a position-dependent diffusion coefficient D.  We describe a method in which flat-bottom restraints applied during molecular simulations are used to determine the position-dependent FE and D, which effectively "fit" the dynamics to the overdamped diffusion model.  Notable features of the method are simplicity, ease of implementation in most software, accuracy, and, at least in the case of diagonal diffusion tensors, trivial generalization to multiple dimensions.  We recently applied the method to the opening transition in the SARS2-Covid19 Spike protein.

References:

V. Ovchinnikov, K. Nam, and M. Karplus. A Simple and Accurate Method to Calculate Free Energy Profiles and Reaction Rates from Restrained Molecular Simulations of Diffusive Processes. J. Phys. Chem. B, 120:8457–8472, 2016.

V. Ovchinnikov and M. Karplus. Free energy simulations of receptor-binding domain opening in the SARS-CoV-2 spike indicate a barrierless transition with slow conformational motions. J. Phys. Chem. B, 127(40):8565–8575, doi=10.1021/acs.jpcb.3c05236, 2023.