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Developing PDE-compartment hybrid frameworks for modeling stochastic reaction-diffusion processes

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SDBW04 - Spatially distributed stochastic dynamical systems in biology

Co-author: Mark Flegg (University of Monash)

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biology. The modelling technique most commonly adopted is systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, the simulation of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems.

The specific stochastic models with which we shall be concerned in this talk are referred to as `compartment-based' or `on-lattice'. These models are characterised by a discretisation 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.

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 re-defining the continuous PDE concentration as a probability distribution. Whilst this first algorithm shows very strong convergence to analytic 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 to analytic solutions of PDEs for mean concentrations.

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