Generalising Reaction-Diffusion and Turing Patterning Systems

自从阿兰·图灵1952年关于形态发生的研究以来，数学家们一直在研究反应扩散系统，试图解释和量化生物有机体中有组织结构的出现。尽管由偏微分方程耦合系统组成的典型两种系统已经得到了全面的研究，但研究正在从这种范式转向更现实的生物学模型。在这个项目中，我们的目标是探索这个基础模型的一些扩展，并朝着更现实的生物学模型迈进。我们已经确定了一些可能性，包括扩展到三个物种系统。这些还没有得到广泛的研究，并且很少甚至没有研究存在空间非均相反应动力学对这些系统的影响。这样的扩展不仅在数学上更有趣，更具挑战性，在生物学上更现实，而且有可能解释现有模型和生物学之间未解决的差异，例如形态原如何在模型预测的不同扩散率下在相同介质中是相似大小的物体。此外，这样的系统可能更有可能表现出亚临界分岔，这比超临界情况下的研究少，并且可能在鲁棒模式形成中发挥重要作用。这将非线性系统和数学生物学的研究领域联系起来。另一个新的研究途径是在发育生物学的背景下研究空间非定域性。非局部效应更常应用于生态学，但在细胞间尺度上，它们可用于描述在许多情况下对模式形成至关重要的长期细胞突出。例如，斑马鱼的黑色素细胞突起与黄素细胞相互作用，控制条纹的形成。另一个例子是脊髓神经元的分化模式，这是由神经元本身的突起调节的。利用非局域项的偏微分方程对此类系统进行建模和模拟，需要在数学分析和数值分析研究领域取得新的发展。进一步的可能性是研究与许多生物环境相关的层状域的反应扩散。最近的工作已经完成了系统的分离体和表面区域，调解的界面边界。表面层的存在和界面上的耦合极大地改变了图案形成行为，包括不稳定准则。有许多可能的扩展。其中一个重点是研究界面上不同输运函数的影响。这将纳入表面科学的研究领域，以促进生物学内部的理解。这个跨学科项目属于EPSRC广泛的数学科学主题。如上所述，它涵盖的研究领域包括数学生物学、非线性系统、数值分析、数学分析和表面科学。

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.