Stochastic lattice-based models of diffusion in biological systems - Taylor, P (2016)
Diffusion is a universal phenomenon, throughout both biological and physical sciences, and a range of deterministic and stochastic models are available to interrogate diffusion-driven processes. Stochastic models are more computationally intensive to simulate, but may be necessary in situations where deterministic models lead to qualitatively different results. Stochastic lattice-based position-jump (LBPJ) models are a popular framework for representing diffusion. Particles reside within discrete compartments, and may jump to other compartments or react with other particles sharing the same compartment. When the number of particles to be simulated becomes large, however, the computational costs may grow infeasibly large. In this thesis, we propose two modified LBPJ models, both of which are significantly less computationally intensive to simulate than the standard framework. The first model uses non-local jumping, allowing particles to move with a distribution of longer but less frequent jumps, rather than jumping exclusively to nearest neighbour compartments. It is seen that boundary conditions must be formulated carefully to maintain agreement with equivalent partial differential equation models. The second model focuses on LBPJ models incorporating volume exclusion. Two common approaches are 'fully-excluding' models, where at most one particle can occupy each compartment, and 'partially-excluding' models, where larger compartments can contain a finite number of particles. We reconcile these two frameworks, showing that they make similar predictions for the mean and variance of particle numbers. Later chapters extend this work to non-uniform lattices, and consider how reactions between particles can be incorporated into partially-excluding models. Throughout the thesis, we present mathematical derivations, and support these with the results of computational simulation, validating the agreement of our models' results with results from established modelling frameworks, and demonstrating the reduced computational cost of our approaches in comparison to standard LBPJ models.