Coarse-graining molecular dynamics: stochastic models with non-Gaussian force distributions

Radek Erban

doi:10.1007/s00285-019-01433-5

Abstract

Incorporating atomistic and molecular information into models of cellular behaviour is challenging because of a vast separation of spatial and temporal scales between processes happening at the atomic and cellular levels. Multiscale or multi-resolution methodologies address this difficulty by using molecular dynamics (MD) and coarse-grained models in different parts of the cell. Their applicability depends on the accuracy and properties of the coarse-grained model which approximates the detailed MD description. A family of stochastic coarse-grained (SCG) models, written as relatively low-dimensional systems of nonlinear stochastic differential equations, is presented. The nonlinear SCG model incorporates the non-Gaussian force distribution which is observed in MD simulations and which cannot be described by linear models. It is shown that the nonlinearities can be chosen in such a way that they do not complicate parametrization of the SCG description by detailed MD simulations. The solution of the SCG model is found in terms of gamma functions.

Highlights

With increased experimental information on atomic or near-atomic structure of biomolecules and intracellular components, there has been a growing need to incorporate such microscopic data
Erban (2016) use an acceptance-rejection algorithm to fit the parameters of linear stochastic coarse-grained (SCG) model (6)–(9) for N = 3 to match the velocity autocorrelation functions of ions estimated from all-atom molecular dynamics (MD) simulations of Na+ and K+ in the SPC/E water
We have presented and analyzed a family of SCG models given by Eqs. (2)–(5), which can be parametrized to fit properties of detailed all-atom MD models

Summary

Introduction

With increased experimental information on atomic or near-atomic structure of biomolecules and intracellular components, there has been a growing need to incorporate such microscopic data They use fictitious particles with harmonic interactions with coarsegrained degrees of freedom (i.e. they add quadratic terms to the potential function of the system and linear terms to equations of motions) and each fictitious particle is subject to a friction force and noise An application of such an approach to ions leads to systems of linear stochastic differential equations (SDEs) and can be used, after some transformation, to obtain a simplified version of the SCG model (2)–(5), where functions g j and h j are given as identities, i.e. g j (y) = h j (y) = y for y ∈ R and j = 1, 2, . In order to find the values of four parameters η j suitable for modelling ions, Erban (2016) estimates the diffusion constants D and three second moments V 2 , U 2 and Z 2 from all-atom MD simulations of ions (K+, Na+, Ca2+ and Cl−) in aqueous solutions. In order to do this, we have to consider the SCG model (6)–(9) for larger values of N as it is done

General linear SCG model and autocorrelation functions

One-parameter nonlinear function

Two-parameter nonlinear function

Application to MD simulations

Nolinear SCG model for general values of N

Discussion and conclusions