For many centuries, the problem of pattern formation has fascinated experimentalists and theoreticians alike. Understanding how spatial pattern arises during growth development is a central but still unresolved issue in developmental biology. It is clear that genes play a crucial role in embryology but the study of genetics alone cannot explain how the complex mechanical and chemical spatio-temporal signalling cues which determine cell fate are set up and regulated in the early embryo. These signals are a consequence of many nonlinear interactions and mathematical modelling and numerical computation have an important role to play in understanding and predicting the outcome of such complex interactions during growth development.
Several studies have shown that reaction-diffusion type models appear to be excellent for describing gross patterning behaviour in developmental biology. Since the seminal work of Turing in 1952, which showed that a system of reacting and diffusing chemical morphogens could evolve from an initially uniform spatial distribution to concentration profiles that vary spatially - a spatial pattern - many models have been proposed exploiting the generalised patterning principle of short-range activation, long-range inhibition elucidated by Meinhardt of which the Turing model is an example, and which in fact is common to many patterning paradigms based on different biological hypotheses. Turing's hypothesis was that one or more of the morphogens played the role of a signaling chemical, such that cell fate is determined by levels of morphogen concentration. Although invalid on stationary domains, our recent results prove that in the presence of domain growth, short-range inhibition, long-range activation as well as activator-activator mechanisms have the potential of giving rise to the formation of patterns only during growth development of the organism. These results offer us a unique opportunity to model, analyse and simulate new non-standard mechanisms for pattern formation on evolving surfaces, a largely unchartered research area. Furthermore, experimental biochemists are now able to design new experiments involving non-standard mechanisms to validate our theoretical predictions. This study offers to address one of the main objections to the Turing mechanism, namely that it operates only under very restrictive and biologically unrealistic conditions.
Hence, we propose to derive mathematical models, carry-out theoretical stability analysis and compute numerical solutions on realistic, geometrically accurate complex evolving surfaces as well as carrying-out applications in developmental biology and cell motility. More specifically we want to (a) derive models for pattern formation on evolving surfaces, (b) derive non-standard mechanisms capable of generating patterns only during surface evolution, (c) derive diffusion-driven instability conditions on evolving surfaces, (d) derive bifurcation theory to study partial differential equations on evolving domains and surfaces, (e) numerically compute solutions of the models and (f) to use biological, chemical and biomedical data to validate our theoretical predictions. The results obtained will have wider implications in the areas of developmental biology, cell motility, biomedicine, textiles, ecology, semiconductor physics, material science, hydrodynamics, astrophysics, chemistry, meteorology, economics, cancer biology, mathematics, numerical analysis as well as other non-traditional fields such as languages where such mechanisms are readily applicable. For examples, one could study (as competition models)the survival or extinction of languages due to migration where the inhabitants' environment continuously changes.
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