Generalising Reaction-Diffusion and Turing Patterning Systems

自从1952年艾伦·图灵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。尽管由偏微分方程组组成的典型两种群系统已经被全面研究，但研究正在从这一范式转向更接近生物现实的模型。在这个项目中，我们的目标是探索这个基本模型的一些扩展，并走向更接近生物现实的模型。我们已经确定了一些可能性，包括扩展到三个物种系统。这些都没有得到广泛的研究，几乎没有关于空间非均相反应动力学对这类体系的影响的研究。这样的扩展不仅在数学上更有趣和更具挑战性，在生物学上也更现实，而且还有可能解释现有模型和生物学之间尚未解决的差异，例如形态起源如何具有模型预测的不同扩散系数，同时在相同介质中是类似大小的物体。此外，这类系统可能更有可能出现亚临界分叉，与超临界情况相比，亚临界分叉研究得更少，并且可能在健壮模式的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在一起。另一种新颖的研究途径是在发育生物学的背景下研究空间非局部性。非局部效应更多地应用于生态学，然而在细胞间尺度上，它们可以用来描述在许多情况下对模式形成至关重要的长距离细胞突起。例如，斑马鱼黑素载体的细胞突起与黄色素载体相互作用，控制条纹的形成。另一个例子是脊髓神经元的分化模式，它由神经元本身的突起调节。利用带有非局部项的偏微分方程组对这类系统进行建模和仿真将需要在数学分析和数值分析的研究领域取得新的发展。另一种可能性是研究分层结构域中的反应-扩散，这与许多生物学背景有关。最近的工作是关于由界面边界调节的具有分离的体区和表面区的系统。表面层的存在和界面上的耦合极大地改变了图案的形成行为，包括不稳定性准则。对此有许多可能的扩展。其中一个特别是调查不同运输功能在界面上的影响。这将纳入表面科学的研究领域，以促进对生物学的理解。这个跨学科的项目属于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.