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Correction factors for boundary diffusion in reaction-diffusion master equations
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Figures

Image of FIG. 1.

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FIG. 1.

(a) PDE solution at steady state with absorbing boundary condition (Software: COMSOL). (b) “Coarse-grained” solution of the PDE; data obtained from subdomain integration, where mesh refinements were performed until no significant differences were observed.

Image of FIG. 2.

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FIG. 2.

First moment dynamics of the uncorrected RDME over 40 time units for different values of K, k s = 100, D A = 0.1. As K increases, the curves seemingly converge to a solution with a steady-state identical to the domain integral of the PDE solution at steady state (reference solution).

Image of FIG. 3.

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FIG. 3.

(a) β depending on K for different absorption scenarios (Robin boundary conditions), where the value ρ = ∞ corresponds to the fully absorbing boundary. (b) β plotted against the absorption constant ρ for different discretizations K (D = .1, k s = 100).

Image of FIG. 4.

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FIG. 4.

(a) Numerically calculated mean of the RDME at steady state with corrected boundary diffusion (D = .1, k s = 100). (b) Absolute error of the RDME solution as compared to the corresponding PDE solution (i. e., absolute difference between Figs. 1(b) and 4(a)).

Image of FIG. 5.

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FIG. 5.

Comparison between R (dotted lines) and β (solid lines) at different discretizations K for four different values of ρ: 0.1 (red), 0.5 (green), 1 (blue), and 5 (black).

Image of FIG. 6.

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FIG. 6.

(a) Solution of the PDE (2) with mixed boundary conditions: zero-flux (left B.C.), zero-Dirichlet (right B.C.), and Robin (ρ = .1, up; ρ = 5, lower B.C.). b Solution of the first moment of the RDME with corrected boundary diffusion at steady state for K = 50. The summed mean (over all subvolumes) approximates well the solution of the domain integration of the PDE. The discretization-dependent correction factors are chosen according to the boundary condition.

Image of FIG. 7.

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FIG. 7.

(a) Comparison of mean dynamics obtained from Smoldyn (gray) with the solution of the moment equations (red) and the average over 1000 spatial SSA simulations (black) (summed over all subvolumes). The correction factor β2 = 0.3466 was derived from moment equations for the complete system while the reference solution was obtained from 1000 Smoldyn simulations. Upper trajectories show A, lower trajectories show B dynamics. Parameter values are: K = 20, (synthesis of A), (unary reaction), . The squared domain has a side length of . For the reduced system with one diffusing species, a correction factor was derived while its reference solution was obtained from 400 simulations of the corresponding scenario in Smoldyn (trajectories not shown). (b) Coefficient of variation for both simulation types, Smoldyn (red) and corrected spatial SSA (black) for both species A (solid) and B (dotted).

Image of FIG. 8.

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FIG. 8.

Stationary distributions for K = 20 (a) without and (b) with correction and in comparison to the CME. Parameter values: , , , B 0 = 1, β = 0.7933; the domain area has size .

Tables

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Table I.

Discretization criteria for “uncorrected” three-dimensional RDMEs. Variables in each case refer to: (a) d: particle diameter, n v : number of molecules per unit volume; (b) τ S : average lifetime of molecules of species S with diffusion constant D S ; (c) τ C : time between reaction events, τ D : time between diffusion events, d = 2, 3 (2D, 3D), and D a “typical diffusion coefficient”; (d) r max : the maximum reaction radius of all species.

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/content/aip/journal/jcp/135/13/10.1063/1.3634003
2011-10-03
2014-04-23

Abstract

The reaction-diffusion master equation (RDME) has been widely used to model stochastic chemical kinetics in space and time. In recent years, RDME-based trajectorial approaches have become increasingly popular. They have been shown to capture spatial detail at moderate computational costs, as compared to fully resolved particle-based methods. However, finding an appropriate choice for the discretization length scale is essential for building a reasonable RDME model. Moreover, it has been recently shown [R. Erban and S. J. Chapman, Phys. Biol.4, 16 (2007)10.1088/1478-3975/4/1/003; R. Erban and S. J. Chapman, Phys. Biol.6, 46001 (2009)10.1088/1478-3975/6/4/046001; D. Fange, O. G. Berg, P. Sjöberg, and J. Elf, Proc. Natl. Acad. Sci. U.S.A.107, 46 (2010)] that the reaction rates commonly used in RDMEs have to be carefully reassessed when considering reactive boundary conditions or binary reactions, in order to avoid inaccurate – and possibly unphysical – results. In this paper, we present an alternative approach for deriving correction factors in RDME models with reactive or semi-permeable boundaries. Such a correction factor is obtained by solving a closed set of equations based on the moments at steady state, as opposed to modifying probabilities for absorption or reflection. Lastly, we briefly discuss existing correction mechanisms for bimolecular reaction rates both in the limit of fast and slow diffusion, and argue why our method could also be applied for such purpose.

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Scitation: Correction factors for boundary diffusion in reaction-diffusion master equations
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/13/10.1063/1.3634003
10.1063/1.3634003
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