Mathematical Sciences: Modeling of Pattern Formation by Cellular Tractions on in vitro Extracellular Matrix

牵引力是形态发生过程中图案形成的重要机制。许多类型的细胞，当播撒在基质上时，仅仅由于牵引力而形成聚集体。牵引力使凝胶发生机械变形，在聚集体之间形成纤维轨迹。纤维轨迹通过接触引导诱导细胞伸长并沿其迁移。模式形成的速度非常快，没有细胞增殖、分泌或基质降解的并发症，这为孤立地通过牵引力研究模式形成提供了独特的机会。研究者和他的同事开发并分析了细胞牵引力的数学模型，以及由此产生的矩阵力学响应，包括对基本材料和响应牵引力而产生的行为各向异性的描述。分析包括线性分岔分析，适当的非线性分析，以及数值分析的重要组成部分。后者需要开发算法来模拟两个空间维度的非线性守恒和进化方程。本研究旨在深入了解细胞外基质的生物牵引力和机制，并最终更好地理解细胞外基质的形态发生和重塑。研究人员开发并分析了一个特定情况下细胞模式形成的数学模型，目的是更好地，更基本地理解(1)细胞与其生长介质（自然或人工）之间的牵引力的机械相互作用，(2)构成人体大部分的细胞外基质的微妙各向异性力学，(3)后期重组，如伤口愈合和肿瘤生长，(4)人类胚胎发育，(5)一般生物模式的形成。数学建模技术允许测试生物学理论的逻辑结构，以及用“数学实验”来测试这些理论的方法。此外，对数学模型的分析可以得出预测，然后可以通过适当的实验室实验来证实或驳斥这些预测。正是通过实验与理论的密切联系，人们才得以理解。该项目与实验学家密切合作，利用数学和计算技术解决基本的生物学问题。

Murray Traction forces are a crucial mechanism of pattern formation in morphogenesis. Cells of many types, when seeded on Matrigel, form aggregates due solely to traction forces. The traction mechanically deforms the gel, creating fiber tracks between aggregates. The fiber tracks induce cells to elongate and migrate along them by contact guidance. The pattern formation occurs sufficiently quickly that there are no complications of cell proliferation or secretion or degradation of the matrix, affording a unique opportunity to study pattern formation by traction forces in isolation. The investigator and his colleague develop and analyze a mathematical model of the cellular tractions and the resultant mechanical response of the Matrigel, incorporating a description of the essential material and behavioral anisotropy which develops in response to traction. Analysis consists of linear bifurcation analysis, nonlinear analysis where appropriate, and a strong component of numerical analysis. The latter entails development of algorithms to simulate the nonlinear conservation and evolution equations in two spatial dimensions. The study aims at a deeper understanding of biological traction and the mechanics of extracellular matrix, and ultimately to a greater understanding of morphogenesis and remodeling in general. The investigators develop and analyze a mathematical model of a particular case of cellular pattern formation with the aim of a better, more fundamental understanding of (1) the mechanical interaction by traction forces between cells and their growth medium (natural or artificial), (2) the subtle anisotropic mechanics of the extracellular matrix that forms a large part of the human body, (3) later restructuring such as wound healing and tumor growth, (4) human embryonic development, and (5) biological pattern formation in general. The technique of mathematical modeling allows a logical construction of biological theories to test, and a method o f testing those theories with "mathematical experiments." Furthermore, the analysis of the mathematical model leads to predictions which may then be confirmed or refuted by appropriate laboratory experiments. It is by a close communication between experiment and theory that understanding is gained. The project addresses fundamental biological questions using mathematical and computational techniques, in close collaboration with experimentalists.