Pattern formation in systems of partial differential equations modeling genetic networks
Doctor of Philosophy
One of the big problems in developmental biology is understanding how multi-cellular organisms grow and organize from what was originally one cell. In order for essential structures and organs to form in precise locations, cells must acquire spatial information regarding its location relative to the whole. The fruit fly has long been a model organism for studying the spatial organization of the body axes. However, recent work of Houchmandzadeh et al.  and Gregor et al.  indicated robust scaling of protein gradients (1) across natural variations in embryo lengths, and (2) across multiple species with a four-fold difference in length. In particular, the gradient Hunchback was shown to sharply drop at the midpoint of the embryo, despite variations in upstream regulators and environment. While the networks of genes involved in axis specification are among the most studied and understood, this robust scaling was not fully explained by the current knowledge. Therefore, alternative methods for understanding this phenomenon are needed, including an excellent opportunity to use mathematical models to uncover biological function. I will present a general reaction-diffusion system modeling the process and the analysis of specific systems mimicking the biological phenomenon, in that solutions representing the Hunchback gradient are symmetric around the midpoint of space, drop sharply at the midpoint, and are robust to changes in the length of space and initial conditions.
Biology; Genetics; Mathematics