From molecular dynamics to Brownian dynamics

Radek Erban

doi:10.1098/rspa.2014.0036

Abstract

Three coarse-grained molecular dynamics (MD) models are investigated with the aim of developing and analysing multi-scale methods which use MD simulations in parts of the computational domain and (less detailed) Brownian dynamics (BD) simulations in the remainder of the domain. The first MD model is formulated in one spatial dimension. It is based on elastic collisions of heavy molecules (e.g. proteins) with light point particles (e.g. water molecules). Two three-dimensional MD models are then investigated. The obtained results are applied to a simplified model of protein binding to receptors on the cellular membrane. It is shown that modern BD simulators of intracellular processes can be used in the bulk and accurately coupled with a (more detailed) MD model of protein binding which is used close to the membrane.

Highlights

Brownian dynamics (BD) simulations have been used for the modelling of a number of spatio-temporal processes in cellular and molecular biology in recent years, including models of intracellular calcium dynamics [1], the MAPK pathway [2] and signal trasduction in Escherichia coli chemotaxis [3]
This reduces the dimensionality of the problem, making BD less computationally intensive than the corresponding molecular dynamics (MD) simulations
To make one- and three-dimensional models comparable, we keep R fixed in the three-dimensional model and we study the behaviour of all MD models in the limit M/m → ∞

Summary

Introduction

Brownian dynamics (BD) simulations have been used for the modelling of a number of spatio-temporal processes in cellular and molecular biology in recent years, including models of intracellular calcium dynamics [1], the MAPK pathway [2] and signal trasduction in Escherichia coli chemotaxis [3]. As the goal of this paper is to study the behaviour of computational algorithms, we formulate the MD model [A] in a finite domain [−L, L], i.e. we consider a finite number n ≡ n(t) of heat bath particles which are at positions xi ∈ [−L, L] with velocities vi ∈ (−∞, ∞), i = 1, 2, . The distribution of its position at time t = 1, computed using 105 realizations of the algorithm [M1]–[M8], is plotted in figure 3a It is compared with the distribution obtained by the limiting BD model (2.6), which is, for t γ −1, given by [19]. Let us consider the MD model [B] where the positions and velocities of heat bath particles are distributed according to (3.3) and (3.4). The distribution of X1 positions of the heavy particle at time t = 1, computed using 105 realizations of the multi-scale algorithm, is plotted in figure 4a. The mean squared displacement obtained for the BD model (1.1) is plotted as the dot-dashed line

Application to protein binding to receptors

Discussion