Modelling, analysis and simulation of spatial patterning on evolving surfaces

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.

The proposed research will impact four critical research areas: (i) modelling and analysis, (ii) numerical analysis, (iii) computations and (iv) applications to spatial pattern formation on evolving surfaces during growth development in developmental biology and to cell movement and deformation in cell motility. The derivation of the mathematical models in non-homogeneous environments on evolving surfaces will necessitate significant extension of prevailing mathematical techniques for analysing partial differential equations on evolving surfaces. Numerical analysis will benefit from the development of innovative numerical methods to solve the model equations on evolving surfaces. The surface finite element method is a natural candidate for solving partial differential equations on complex real-world evolving biological surfaces. On the other hand, particle methods coupled with generalised level set methods are perfect for treating splitting and reconnecting evolving surfaces; fundamental to understanding models for cell movement and deformation in cell motility.

The main impact of this proposal lies in its applications to biomedical, chemical and biological systems. For example, in cell motility, the modelling and simulations of cell movement will give rise to the possibility of experimentalists being able to introduce mutation within part of the cell population to study how it affects cell behaviour. For the first time, biochemists, bio-engineers and neurologists will be able to study how mutated cells behave in virtual experiments without the complications of in vivo experiments. On the other hand, it will be possible to predict cell population behaviour in the presence of obstacles on a substrate, as well as competition for food (prey).

In developmental biology, new insights on non-standard reaction kinetic models will offer experimentalists avenues to test in experiments the evolution of patterns only during growth development. Madzvamuse and his group have established collaborations with several experimentalists in developmental biology. Profs. Sekimura and Kondo in Japan will be able to test such hypotheses to study the emergence of patterns on evolving fish surfaces. Prof. Cheong's group in California is interested in models that could describe zebra stripe formation during growth development. Prof. Marle at IMPAS in Manaus Brazil has accumulated a huge amount of data on characterising chemical molecules that could be associated with Turing morphogens which could give rise to evolving patterns on the Brazilian Amazon fish known locally as Tambaqui.

The results of this research will be disseminated through: (i) Publications in peer-reviewed journals across the numerical analysis, mathematical and biological communities. (ii) Software packages will be uploaded to Madzvamuse's website which is accessible to the academic research community, either in the public sector, commercial private sector, third sector or the wider public in general. (iii) Delivery of lectures at conferences, seminars and workshops. (iv) Delivery of special seminars/presentations at public sector events such as the Brighton Science Festival or meetings between universities (public) and the private sector (industries).

The University of Sussex and the Medical Research Council, through the School of Mathematics and Physical Sciences has committed and recruited a 3.5-year DTA student to start work on atherosclerosis where reaction-diffusion equations describe the formation of carotid plaque during growth development. The PI (main advisor), the surgeon (Dr Jibawi) and the cardiologist (Dr Cheal), both at Sussex Hospitals, will co-supervise the student. This project has direct impact on modelling blood flow in arteries coupled with the formation of plaque as free moving boundaries. Therefore the PI will explore the possibilities of collaborating with medical companies via the Knowledge Transfer Partnership programme.