From molecular mechanics to Brownian dynamics and multi-resolution methods

Molecular dynamics (MD) approaches, based on the rules of classical mechanics, are commonly used to study the behaviour of complex biomolecules in biological applications. They are given as systems of ordinary or stochastic differential equations for the time evolution of positions and velocities of particles, representing either individual atoms or groups of atoms, describing parts of a biomolecule. One of the main limitations of all-atom MD simulations is that their direct application to the modelling of intracellular behaviour is restricted to modelling processes in relatively small domains over relatively short time intervals. In particular, intracellular processes which include transport of molecules between different parts of a cell, are usually only modelled by a much coarser modelling approach, including Brownian dynamics (BD) and other stochastic reaction-diffusion models.

In my talk, I will discuss connections between MD and BD, focusing on limiting theorems guaranteeing convergence of an MD model to a coarser stochastic description, given by the (generalized) Langevin dynamics written in terms of a relatively low-dimensional stochastic dynamical system. I will show how this limiting process can help us to develop and analyze multi-resolution methods for spatio-temporal modelling of intracellular processes. These methods use detailed MD simulations in localized regions of particular interest (in which accuracy and microscopic details are important) and a coarser (less-detailed) model in other regions where accuracy may be traded for simulation efficiency. Three types of multi-resolution methodologies will be considered in detail:

(a) describing the whole biomolecule (biological structure) of interest by the detailed modelling approach which is coupled with a coarse model for the solvent molecules which are far away from the biomolecule;

(b) describing different parts of a biomolecule by using models with different level of resolution; and

(c) considering the region with the most detailed model as a fixed part of the physical space and allowing the biomolecule of interest to pass between this region and its surroundings, where a coarse-grained modelling approach is used.

references:

[1] R. Erban. From molecular dynamics to Brownian dynamics. Proceedings of the Royal Society A 470: 20140036 (2014)

[2] R. Erban. Coupling all-atom molecular dynamics simulations of ions in water with Brownian dynamics. Proceedings of the Royal Society A 472: 20150556 (2016)

[3] R. Gunaratne, D. Wilson, M. Flegg and R. Erban. Multi-resolution dimer models in heat baths with short-range and long-range interactions, to appear in Interface Focus (2019)