Numerous processes across both the physical and biological sciences are driven by diffusion, for example transport of proteins within living cells, and some drug delivery mechanisms. Diffusion is an unguided process which is of great importance at small spatial scales. Partial differential equations (PDEs) are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic (probabilistic) models instead to capture the behaviour of a system accurately, especially when the number of diffusing particles is low, such as in gene regulation.
Exploring the underlying mathematics behind these models is an important current area of research. Mathematicians need to understand these models better, so that they can be applied more meaningfully and so that they can be made more efficient while still preserving their accuracy (as computational power and time are often limiting factors). Oxford Mathematicians Paul Taylor and Ruth Baker, working with colleagues Christian Yates of the University of Bath and Matthew Simpson of the Queensland University of Technology, have been seeking to explore stochastic models of diffusion that are 'compartment-based'. In their paper, published in the Journal of Royal Society Interface, the domain under consideration is discretized into compartments, with particles jumping between compartments, possibly with constraints such as that a compartment cannot contain more than a certain number of particles. Previous work by these authors has concentrated on situations where the compartments all have the same size, but these can be unhelpfully restrictive for some applications, where it is important to focus at a high resolution in some parts but impractical to apply this same high resolution across the whole domain. This latest piece of work brings together a number of aspects, including allowing different compartments to have different sizes.
Crucially, this research demonstrates that these new approaches will be of value to researchers working on multi-scale systems, as they can speed up simulations while preserving precision where needed.