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.
