Mathematical modelling of spatial patterning on evolving surfaces

几个世纪以来，模式形成的问题一直吸引着实验家和理论家。理解空间模式是如何产生的是发育生物学中一个核心但仍未解决的问题。很明显，基因在胚胎学中起着至关重要的作用，但单独的遗传学研究无法解释决定细胞命运的复杂的机械和化学时空信号信号是如何在早期胚胎中建立和调节的。这些信号是许多非线性相互作用的结果，数学建模和数值计算在理解和预测这种复杂相互作用的结果方面发挥着重要作用。令人惊讶的是，关于生长如何影响图案形成的研究很少。在过去的15年里，许多研究小组从实验和理论的角度表明，生长对模式选择有深远的影响。在这个拟议的项目中，我们想邀请日本中部大学生物化学系的Sekimura教授访问萨塞克斯大学和牛津大学，就鱼类从早期发育到成年的数学模型进行合作。Sekimura是一位数学生物学家，在发育生物学的模式形成方面具有专业知识，我们与他合作了多年。我们的目的是在鱼类色素沉着模式的范式模型中解决生物模式的形成。由于其实验的可追溯性，该模型可能会揭示一般模式形成的重要见解。在分析结构域生长的影响以及将遗传水平信息与宏观水平模式结果联系起来的多尺度分析方面已经取得了进展。至关重要的是，Sekimura的实验室已经获得了关于两种不同鱼类在生长过程中模式如何变化的详细数据，这使我们能够测试关于这些模式如何产生的各种假设。研究表明，反应扩散（RD）型模型对于描述该系统的总体模式行为似乎是很好的。然而，我们的研究表明，传统模型不足以描述更复杂的细节，人们必须在复杂几何的异质、不断增长的领域中考虑这些模型。这为系统的数值求解、数学分析和建模本身提出了新的问题。这三个关键问题（建模、分析和数值）将是本研究的重点。Sekimura已经获得了大量关于模式在成长过程中如何演变的实验数据，使我们能够验证我们的模型。关键的挑战将是将已知的生物学纳入数学模型，然后解决这些模型。我们预计这些模型将是具有空间变化参数的RD型模型，将在复杂形状的生长域和演化表面上求解。我们打算调查以下内容：扩展（非常少的）可用于空间非同构环境中的RD系统的分析。在复杂的非均匀生长域上对问题进行建模——在这方面的分析和数值工作仍然很少。用实验数据对模型进行验证。将讨论Sekimura已获得实验数据的特定应用。最近，我们首次将RD系统的扩散驱动不稳定性分析从固定增长域扩展到任意增长域。这项研究解决了对图灵机制的主要反对意见之一，即它只能在非常严格的、生物学上不现实的条件下运行。我们将开始一项详细的研究，以发现反应动力学，这些反应动力学可能只在存在域生长的情况下产生模式，而这些模式不一定是标准的短程激活，长抑制形式。

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.