Mathematical Sciences: Studies in Pattern Formation and CellMovement

该项目旨在对模式形成和细胞运动进行理论和计算研究，目的是：(1)阐明在脊椎动物四肢发育过程中，能够产生所观察到的导致肱骨、桡骨、尺骨和指骨的冷凝序列的机制类型；（ii）建立非相互作用细胞的微观模型，这些细胞的动力学依赖于外部浓度场，并对这些场表现出不同的行为反应；（iii）为这类细胞的种群水平行为建立连续的描述；（iv）建立密集聚集体中细胞运动的模型。首席研究员计划分析一个模式形成模型，该模型结合了随机运动和趋化性的细胞运动，趋化物种的产生和扩散，以及两种可相互转换的细胞。一个主要的目标是了解这些过程是如何相互作用产生空间格局的。分析和数值计算将用于确定在初始均匀组织中可以发展的模式的参数值，并且将使用数值延拓技术来确定稳定模式如何依赖于参数。在这种情况下，其他过程的影响和肢体特殊区域的作用将被研究。

This project is for theoretical and computational studies of pattern formation and cell movement with the aims of (i) elucidating the types of mechanisms that can produce the observed sequence of condensations leading to the humerus, the radius and ulna, and the digits in developing vertebrate limbs, (ii) developing microscopic models for non-interacting cells whose dynamics depend on external concentration fields and which exhibit various behavioral responses to these fields, (iii) developing continuum descriptions for the population-level behavior of such cells, and (iv) developing models for the motion of cells in dense aggregates. The principal investigator plans to analyze a model for pattern formation that incorporates cell movement by random motion and chemotaxis, the production and diffusion of the chemotatic species, and two types of cells that are interconvertible. A major objective is to understand how these processes interact to produce spatial patterns. Analysis and numerical computations will serve to delimit the parameter values for which pattern can develop in an initially-homogeneous tissue, and numerical continuation techniques will be used to determine how the stable patterns depend on parameters. The effects of other processes and the role of specialized regions of the limb will be investigated in this context.