Importance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Yates, Christian A.; Baker, Ruth E.

doi:10.1103/physreve.88.054701

Cited by 7 publications

(11 citation statements)

References 8 publications

(8 reference statements)

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“…The approximation is the same with hat functions as basis and test functions in FEM and in a FVM with a voxel boundary between 𝒱 i −i and 𝒱 i at ( x i −1 + x i )/2 as in [50]. …”

Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

Simulation of stochastic diffusion via first exit times

Lötstedt

Meinecke

2015

Journal of Computational Physics

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In molecular biology it is of interest to simulate diffusion stochastically. In the mesoscopic model we partition a biological cell into unstructured subvolumes. In each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion. The jump rates can be computed by discretizing the diffusion equation on that unstructured mesh. If the mesh is of poor quality, due to a complicated cell geometry, standard discretization methods can generate negative jump coefficients, which no longer allows the interpretation as the probability to jump between the subvolumes. We propose a method based on the mean first exit time of a molecule from a subvolume, which guarantees positive jump coefficients. Two approaches to exit times, a global and a local one, are presented and tested in simulations on meshes of different quality in two and three dimensions.

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“…The approximation is the same with hat functions as basis and test functions in FEM and in a FVM with a voxel boundary between 𝒱 i −i and 𝒱 i at ( x i −1 + x i )/2 as in [50]. …”

Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

Simulation of stochastic diffusion via first exit times

Lötstedt

Meinecke

2015

Journal of Computational Physics

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“…For unstructured meshes, the jumping propensity α i,j between two lattice points can be calculated by considering a finite element discretisation of the appropriate PDE over the mesh or, in special cases by considering first passage time problems. For details on the derivation of transition rates on unstructured meshes the reader is referred to (Engblom et al, 2009;Yates and Baker, 2013;Yates et al, 2012). Such a discretisation justifies the notation α i,j (that is, a dependence on the destination for the jump).…”

Section: Compartment-based Modellingmentioning

confidence: 99%

“…Coarse-graining the continuous domain into a lattice of compartments, between which particles undergo a random walk, is a simplification which may be appropriate if the lattice spacing is not too large as to impair good spatial resolution. Both off-lattice (Andrews and Bray, 2004;Erban and Chapman, 2009;van Zon and ten Wolde, 2005) and on-lattice (Baker et al, 2010;Drawert et al, 2010;Engblom et al, 2009;Lampoudi et al, 2009;Yates and Baker, 2013;Yates et al, 2012) individual-based stochastic simulations have remained popular paradigms for stochastic diffusion simulations.…”

Section: Introductionmentioning

confidence: 99%

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion

Yates

Flegg

2015

J. R. Soc. Interface.

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Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as 'compartment-based' or 'on-lattice'. These models are characterized by a discretization of the computational domain into a grid/lattice of 'compartments'. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: 'how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations.

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“…Using the formula stated in (7) to relate the diffusion constant D to the transition rates, in the case where Q = 2, we find that D/2 describes the rate of jumping to a neighbouring lattice site, and D/8 the rate of jumping two lattice sites. In this section, and for the rest of this paper, we discuss implementing boundary conditions at the left-hand boundary without loss of generality.…”

Section: Extending To the Q = 2 Casementioning

confidence: 99%

Deriving appropriate boundary conditions, and accelerating position-jump simulations, of diffusion using non-local jumping

Taylor¹,

Baker²,

Yates³

2014

Phys. Biol.

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In this paper we explore lattice-based position-jump models of diffusion, and the implications of introducing non-local jumping; particles can jump to a range of nearby boxes rather than only to their nearest neighbours. We begin by deriving conditions for equivalence with traditional local jumping models in the continuum limit. We then generalize a previously postulated implementation of the Robin boundary condition for a non-local process of arbitrary maximum jump length, and present a novel implementation of flux boundary conditions, again generalized for a non-local process of arbitrary maximum jump length. In both these cases we validate our results using stochastic simulation. We then proceed to consider two variations on the basic diffusion model: a hybrid local/non-local scheme suitable for models involving sharp concentration gradients, and the implementation of biased jumping. In all cases we show that non-local jumping can deliver substantial time savings for stochastic simulations.

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Importance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Cited by 7 publications

References 8 publications

Simulation of stochastic diffusion via first exit times

Simulation of stochastic diffusion via first exit times

The pseudo-compartment method for coupling partial differential equation and compartment-based models of diffusion

Deriving appropriate boundary conditions, and accelerating position-jump simulations, of diffusion using non-local jumping

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