Generalising Reaction-Diffusion and Turing Patterning Systems

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Since Alan Turing's 1952 work on morphogenesis, mathematicians have studied reaction-diffusion systems in an attempt to explain and quantify the emergence of organised structures in biological organisms. Although the canonical two species system, comprising a coupled system of partial differential equations, has been comprehensively studied, research is moving away from this paradigm towards more biologically realistic models. In this project, we aim to explore a number of extensions to this base model and move towards more biologically realistic models. We have identified a number of possibilities, including the extension to three species systems. These have not been extensively studied, and little to no research exists on the effects of spatially heterogeneous reaction kinetics on such systems. Such an extension is not only more mathematically interesting and challenging, and biologically realistic, but also has the potential to explain unsolved discrepancies between existing models and the biology, such as how morphogens can have the different diffusivities predicted by the models whilst being similar sized objects in the same medium. In addition, such systems may be more likely to exhibit subcritical bifurcations, which are less studied than the supercritical case, and could play an important role in robust pattern formation. This links the research areas of non-linear systems and mathematical biology. Another route of novel investigation is to study spatial non-locality in the context of developmental biology. Non-local effects are much more commonly applied to ecology, however at an intercellular scale, they can be used to describe long-ranged cell protrusions which are essential for pattern formation in a number of situations. For example, cell protrusions of the melanophores of zebrafish interact with xanthophores to govern the formation of stripes. A further example is the differentiation pattern of spinal neurons which is regulated by protrusions from the neurons themselves. Modelling and simulating such systems using partial differential equations with non-local terms will require new developments within the research area of mathematical analysis and numerical analysis. A further possibility would be to study reaction-diffusion in layered domains, which are relevant to a number of biological contexts. Recent work has been done on systems with separate bulk and surface regions, mediated by an interfacial boundary. The presence of the surface layer and coupling at the interface drastically change the pattern formation behaviour, including the instability criteria. There are many possible extensions to this. One in particular is to investigate the effects of different transport functions at the interface. This would incorporate the research area of surface science in order to contribute to understanding within biology. This cross-disciplinary project falls within the EPSRC's broad mathematical sciences theme. As highlighted above, the research areas it covers include mathematical biology, non-linear systems, numerical analysis, mathematical analysis, and surface science.