Reactions, diffusion, and volume exclusion in a conserved system of interacting particles

Daniel B. Wilson, Helen Byrne, and Maria Bruna

Phys. Rev. E 97, 062137 – Published 21 June 2018

Abstract

Complex biological and physical transport processes are often described through systems of interacting particles. The effect of excluded volume on these transport processes has been well studied; however, the interplay between volume exclusion and reactions between heterogenous particles is less well studied. In this paper we develop a framework for modeling reaction-diffusion processes which directly incorporates volume exclusion. We consider simple reactions (unimolecular and bimolecular) that conserve the total number of particles. From an off-lattice microscopic individual-based model we use the Fokker-Planck equation and the method of matched asymptotic expansions to derive a low-dimensional macroscopic system of nonlinear partial differential equations describing the evolution of the particles. A biologically motivated, hybrid model of chemotaxis with volume exclusion is explored, where reactions occur at rates dependent upon the chemotactic environment. Further, we show that for reactions that require particle contact the appropriate reaction term in the macroscopic model is of lower order in the asymptotic expansion than the nonlinear diffusion term. However, we find that the next reaction term in the expansion is needed to ensure good agreement with simulations of the microscopic model. Our macroscopic model allows for more direct parametrization to experimental data than existing models.

Received 30 March 2018

DOI:https://doi.org/10.1103/PhysRevE.97.062137

Physics Subject Headings (PhySH)

Excluded volume interaction

Reaction diffusion systems

Brownian dynamics Coarse graining Fokker–Planck equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Daniel B. Wilson^*, Helen Byrne^†, and Maria Bruna^‡

Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom

^*daniel.wilson@maths.ox.ac.uk
^†helen.byrne@maths.ox.ac.uk
^‡bruna@maths.ox.ac.uk

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Vol. 97, Iss. 6 — June 2018

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Images

Figure 1
Population-level equations (21a) for both blue ([I–IV]-b) and red ([I–IV]-r) particles at time $t = 0.05$ , with data being initially Gaussian with zero mean and standard deviation 0.09. (I) Solutions to Eqs. (21a) for point particles, $ε = 0$ . (II) Histograms for point particles, $ε = 0$ . (III) Solutions to Eqs. (21a) for finite-size particles, $ε = 0.01$ . (IV) Histograms for finite-size particles, $ε = 0.01$ . The data were calculated from $10^{4}$ stochastic simulations with $Δ t = 10^{- 5}$ . Parameter values are $D_{b} = 0.5$ , $D_{r} = 1$ , $k_{b} = k_{r} = 10$ , and $N = 400$ . Initial conditions were $N_{b} (0) = 350$ and $N_{r} (0) = 50$ .
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Figure 2
(a) Total density $b + r$ along $y = 0$ at $t = 0$ (dot-dash line), $t = 0.05$ and $ε = 0$ (dash line), and $t = 0.05$ and $ε = 0.01$ (solid line). (b) Slice of the solutions in Fig. 1 at $y = 0$ , showing the blue particles' density (blue [light grey]) and red particles' density (red [dark grey]). Solid (dashed) lines correspond to the PDE model for finite-size (point) particles, and circles (crosses) to the histograms for finite-size (point) particles. (c) Evolution of the particles numbers $N_{b} (t)$ (blue [light grey]) at $N_{r} (t)$ (red [dark grey]) for $0 \leq t \leq 0.05$ for $ε = 0.01$ . Solution of (22) (solid lines) and particle numbers from the stochastic simulations (circles). Parameter values are as in Fig. 1.
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Figure 3
Comparison between the stochastic hybrid chemotaxis model (23) (crosses for $ε = 0$ and circles for $ε = 0.01$ ) and continuum PDE model (24) (dashed lines for $ε = 0$ and solid lines for $ε = 0.01$ ) for a system with $N = 800$ cells with $N_{b} (0) = N_{r} (0) = 400$ and final time $T_{f} = 0.05$ . (a) Evolution of the blue cells density $b (x, t)$ , with uniform initial density. (b) Evolution of the red cells density $r (x, t)$ , with initial density uniform in $[- 3 / 62, 3 / 62] \times [- 1 / 2, 1 / 2]$ . (c) Evolution of the chemoattractant concentration $c (x, t)$ with initial concentration $c_{0} (x) = 1$ . (d) Evolution of the cell numbers $N_{b} (t)$ (blue [light grey]) and $N_{r} (t)$ (red [dark grey]) and the total concentration $N_{c} = \int_{Ω} c (x, t) d x$ (black). Panels (a), (b), (c) show the solutions averaged over the vertical direction at times $t = 0, 0.01, 0.02, 0.05$ . Other parameters used are $D_{b} = D_{r} = 0.1$ , $D_{c} = 1$ , $χ = 1$ , $k_{b} = 10$ , $k_{r} = 1$ , $γ = 0.01$ , $μ = 0.5$ . Histograms were computed from 5000 realizations and 31 bins in the horizontal coordinate. The PDE model was solved using 500 grid points.
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Figure 4
(a) Evolution of the particle numbers $N_{b} (t)$ (blue [light grey]) and $N_{r} (t)$ (red [dark grey]) for $ε = 0.02$ . Solutions to Eq. (38) are up to $O (ε^{2})$ (dashed lines) and up to $O (ε^{3})$ (solid lines). Circles represent data from stochastic simulations. (b) Evolution of the blues population number, $N_{b} (t)$ for particle sizes $ε = 0, 0.01, 0.02, \dots, 0.05$ . The initial conditions are $N_{b} (0) = 650, N_{r} (0) = 350$ , with blue particles uniformly distributed and red particles normally distributed with zero mean and standard deviation 0.09 in the $(x, y)$ plane and uniformly distributed in the $z$ coordinate in $Ω = {[- 1 / 2, 1 / 2]}^{3}$ . Other parameter values are $D_{b} = 0.5$ , $D_{r} = 1$ , $λ = 10$ , $f_{b} = f_{r} = 0$ , and $N = 1000$ .
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