Mathematical modelling 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 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.Surprisingly, very little research has been carried out on how growth affects pattern formation. In the past 15 years a number of research groups have shown both from experimental and theoretical viewpoints, that growth can have a profound effect on pattern selection. In this proposed project we would like to invite Prof Sekimura, Department of Biological Chemistry, Chubu University, Japan to visit the universities of Sussex and Oxford to develop collaborations on mathematical modelling of fish patterns during development from the early stages to adulthood. Sekimura is a mathematical biologist with expertise in pattern formation in developmental biology with whom we have collaborated for a number of years. Our aim is to address biological pattern formation in the paradigm model of fish pigmentation pattern. Because of its experimental tractability, this model could potentially reveal important insights for pattern formation in general. Progress has already been made in analysing the effects of domain growth and in multiscale analysis linking genetic level information to macroscopic level patterning outcome. Crucially, Sekimura's laboratory has acquired detailed data on how patterns on two different kinds of fish change during growth, allowing us to test various hypotheses on how these patterns could be generated.Studies have shown that reaction-diffusion (RD) type models appear to be excellent for describing gross patterning behaviour in this system. Our studies have shown, however, that the traditional model is inadequate to describe the more complex details and that one has to consider these models on heterogeneous, growing domains, of complex geometry. This raises new problems for numerically solving the system, carrying out mathematical analyses and indeed doing the modelling itself. These three key issues (modelling, analysis and numerics) will be the focus of this research. Sekimura has acquired extensive experimental data on how patterns evolve during growth to enable us to verify our models. The key challenges will be to incorporate known biology into mathematical models and then solving these models. We anticipate that these models will be of RD type with spatially varying parameters to be solved on complex-shaped growing domains and evolving surfaces. We intend to investigate the following:1. Extending the (very little) analysis available for RD systems in spatially non-homogeneous environments.2. Modelling the problem on complex non-uniform growing domains - again very little analytical and numerical work has been done in this context.3. Verifying the model with experimental data. Particular applications will be addressed for which Sekimura has acquired experimental data.Recently we extended for the first time, diffusion-driven instability analysis for RD systems from fixed to arbitrary growing domains.This study addressed one of the main objections to the Turing mechanism, namely that it operates only under very restrictive, biologically unrealistic, conditions. We will initiate a detailed study to discover reaction kinetics which might give rise to patterns only in the presence of domain growth and these need not necessarily be of the standard short-range activation, long-inhibition form.

First and foremost, the proposed research project will contribute to the understanding of the formation of patterns during growth and development in biology with particular applications to fish pigmentation patterning. Secondly, the derivation of the mathematical models in non-homogeneous environments will necessitate significant extension of prevailing mathematical techniques for analysing systems reaction-diffusion equations. Numerical analysis will benefit from the development of innovative numerical methods to solve the model equations on continuously deforming and growing domains and evolving surfaces. In the short term, academic beneficiaries will benefit from interacting directly on a face-to-face basis with Professor Sekimura during the visit. In the long term, the wider academic communities will benefit through peer-reviewed papers, attending conferences, seminars or workshops delivered by the investigators. The results of this research will be available as open-source codes easily accessible from public search engines such as Google. Our group has developed software packages for moving grid finite element methods which can be down-loaded freely from Madzvamuse's website (see: http://www.maths.sussex.ac.uk/~anotida/software.php). Since 2000, these packages have been used widely within the scientific community by several researchers interested in the solution of reaction-diffusion systems on fixed and growing domains. In the same vein, the numerical algorithms developed during this visit will enrich software development for partial differential equations on evolving surfaces. Since the packages can be down-loaded freely, programmers and researchers outside the academic research community interested in solving partial differential equations on continuously changing environments will be able to access the algorithms and adapt them to their specific problems. For example, partial differential equations are widely used in bio-medicine and financial mathematics; their extensions to continuously changing environments will be of interest to medical researchers and financial analysts. 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). For a detailed impact summary, see attachment.