Capturing Brownian dynamics with an on-lattice model of hard-sphere diffusion

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Capturing Brownian dynamics with an on-lattice model of hard-sphere diffusion

Claudia Cianci, Stephen Smith, and Ramon Grima

Phys. Rev. E 95, 052118 – Published 11 May 2017

Abstract

Conventional master equation approaches approximate the diffusion of molecules in continuum space by the process of particles hopping on a spatial lattice. The hopping probability from one voxel (spatial lattice point) to its neighbor is usually considered to be constant throughout space. Such an assumption is only consistent with pointlike molecules and thus neglects volume-exclusion effects due to finite particle size. A few studies have attempted to introduce volume-exclusion effects by choosing the hopping probability from one voxel to a neighboring one to be a linear function of the number density. Here, we formulate an alternative master equation in which the hopping probability is equal to the fraction of available space in the neighboring voxel as estimated using scaled particle theory. This leads to the hopping probability being a nonlinear function of the number density. A mean-field approximation (mfa) leads to a partial differential equation of the advection-diffusion type. We show that the time evolution of the particle number density sampled using the stochastic simulation algorithm associated with the new master equation and the number density obtained by numerical integration of the mfa are in good agreement with each other. They are also distinctly different than the time evolution predicted by the conventional master equation and those with hopping probabilities which are linear functions of the number density. The results from the new lattice description are also shown to be in very good agreement with the lattice-free method of Brownian dynamics, even for highly crowded scenarios.

Received 4 October 2016
Revised 13 February 2017

DOI:https://doi.org/10.1103/PhysRevE.95.052118

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Diffusion Fluctuations & noise Intracellular transport Random walks

Statistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Claudia Cianci, Stephen Smith, and Ramon Grima

School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh EH93JR Scotland, United Kingdom

Article Text

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Issue

Vol. 95, Iss. 5 — May 2017

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Images

Figure 1
Comparison of the SPT modified approaches with Brownian dynamics (BD) for a one species system for various times (a) $t = 0$ , (b) $t = 0.8$ , (c) $t = 1.0$ , and (d) $t = 3.0$ . Both the SSA modified using SPT (SPT-SSA) and the associated mean-field theory (SPT-mfa) reproduce well the BD dynamics, in particular the strongly non-Gaussian behavior at short to intermediate times. Note that position $i$ on the $x$ axis denotes the end of voxel $i$ , not the center of voxel $i$ . This convention is used throughout all the plots. The large differences from pure diffusion (dashed lines) indicate the importance of volume exclusion at intermediate times. See text for further details.
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Figure 2
Quantitative measure of the difference between the SPT-SSA and BD (lower blue line), and the pure diffusion and BD (upper red line) using the Kullback-Leibler (KL) divergence. Parameters as in Fig. 1. Note the accuracy of the SPT-SSA is reflected in a small KL divergence at all times. The error in the pure diffusion estimate is maximum at intermediate times when the non-Gaussian behavior is most evident.
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Figure 3
Comparison of the SPT modified approaches with Brownian dynamics (BD) for a two species system for various times (a) $t = 0$ , (b) $t = 0.1$ , (c) $t = 0.4$ , and (d) $t = 3.0$ . Red (dark gray) denotes the species with the larger radius and blue (light gray) the smaller sized species. Both the SSA modified using SPT (SPT-SSA) and the associated mean-field theory (SPT-mfa) reproduce well the BD dynamics, in particular, the strongly non-Gaussian behavior in the central spatial region where the two types of particles interact strongly. The large differences from pure diffusion (dashed lines) indicate the importance of volume exclusion at all times. See text for further details.
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Figure 4
Accuracy of SPT vs linear theory for the single particle case. The SPT equation (5) excellently agrees with the proportion of available volume computed from Monte Carlo simulations. The latter involves calculating the fraction of time that a randomly allocated particle location does not intersect with any of the particles in a random mixture of particles in a cubic box of unit volume. The radius of all particles is $1 / 20$ . The proportion of occupied volume is increased through the particle numbers. See text for discussion.
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Figure 5
Accuracy of SPT vs linear theory for a two species particle case. The SPT equation (4) excellently agrees with the proportion of available volume computed from Monte Carlo simulations. The latter involves calculating the fraction of time that a randomly allocated position of a particle of species 1 does not intersect with any of the particles in a random mixture of particles in a cubic box of unit volume. The radius of species 1 is 0.02 and of species 2 is 0.05. The proportion of occupied volume is increased through the particle numbers. See text for discussion.
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Figure 6
Accuracy of SPT as a function of $Λ$ , the ratio of the particle radius, and the box length for the single particle case. The occupied volume fraction is fixed to $ϕ = 0.2$ . Monte Carlo simulations show that the accuracy of SPT equation (5) is excellent for $Λ < 0.2$ . See text for discussion.
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