Modelling, analysis and simulation 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学家和理论家。了解生长发育过程中的空间格局是如何产生的，是发育生物学中一个核心但尚未解决的问题。很明显，基因在胚胎学中起着至关重要的作用，但单靠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线索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生长发育过程中这种复杂相互作用的结果方面发挥着重要作用，一些研究表明，反应扩散型模型似乎是描述发育生物学中总模式行为的极好模型。自从图灵在1952年的开创性工作表明，反应和扩散化学形态发生剂的系统可以从最初均匀的空间分布演变为空间变化的浓度分布-空间模式-以来，已经提出了许多模型，利用Meinhardt阐明的短程激活、长程抑制的一般模式原理，图灵模型是其中的一个例子，事实上，这是基于不同生物学假设的许多模式范式所共有的。图灵的假设是，一种或多种形态发生素起着信号化学物质的作用，因此细胞的命运由形态发生素浓度的水平决定。虽然无效的固定域，我们最近的研究结果证明，在域的增长，短程抑制，远程激活以及激活剂-激活剂机制的存在下，有可能引起的模式只在生长发育的有机体的形成。这些结果为我们提供了一个独特的机会来建模，分析和模拟新的非标准机制，图案形成的不断变化的表面，在很大程度上是未知的研究领域。此外，实验生物化学家现在能够设计涉及非标准机制的新实验来验证我们的理论预测。本研究旨在解决图灵机制的一个主要问题，即它只在非常严格和生物学上不现实的条件下运行，因此，我们建议推导数学模型，进行理论稳定性分析和计算数值解的现实，几何精确的复杂不断变化的表面，以及进行在发育生物学和细胞运动的应用。更具体地说，我们想（a）推导出在演化表面上形成图案的模型，（B）推导出仅在表面演化期间能够产生图案的非标准机制，（c）推导出在演化表面上的扩散驱动的不稳定条件，（d）推导出分叉理论以研究演化域和表面上的偏微分方程，（e）数值计算模型的解，以及（f）使用生物学，化学和生物医学数据来验证我们的理论预测。所获得的结果将在发育生物学、细胞运动性、生物医学、纺织品、生态学、半导体物理学、材料科学、流体力学、天体物理学、化学、气象学、经济学、癌症生物学、数学、数值分析以及其他非传统领域（如语言）中产生更广泛的影响。例如，人们可以研究（作为竞争模型）由于居民环境不断变化的迁移而导致的语言的生存或消亡。

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.