BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

A PDE/compartment-based hybrid method for simulating stochastic reaction-diffusion systems Yates, Christian


Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique for reaction-diffusion systems that has predominated due to its analytical tractability and ease of simulation has been the use of partial di↵erential equations (PDEs). However, due 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.\r\n\r\nThe specific stochastic model with which we shall concern ourselves are known as ‘compartment-based’. These models are characterised by a discretisation of the computational domain into a grid/lattice of discrete voxels between which molecules can jump. Molecules are considered to be well-mixed in each one of these voxels and can react stochastically with other molecules in their voxel with prescribed rates.\r\nIn a wide variety of biological situations, stochasticity due to low copy numbers is relevant only in particular regions of the domain. In other regions, copy numbers are su\0ciently high that mean field models su\0ce to capture the important dynamics. Such conditions necessitate the development of hybrid models in which some areas of the domain are modelled using a continuum representation and others using an individual-based representation.\r\n\r\nIn this talk we develop hybrid algorithms which couple a PDE in one region of the domain to a compartment-based model in the other. Rather than balancing flux at the interface, we use a method which is similar to the ghost-cell method. Characteristic of this method is the individual treatment of particles as they cross the interface. A small region of the PDE domain adjacent to the compartment- based region is allowed to contribute particles to the compartment-based regime. When particles cross over the interface into this pseudo-compartment from the compartment-based regime a step-function with the mass of a single particle is added. We test our hybrid method in a variety of different scenarios and analyse the error introduced in each case.

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