Advisor: Professor Linda Petzold Title: "Combining Biochemical Signaling and Mechanics to Understand Yeast Mating Morphogenesis" Abstract: How biological systems are able to form and maintain such a wide variety of patterns and structures is one of the central questions in science. In this dissertation we focus on one example of pattern formation and morphogenesis found in yeast cells. Specifically, we present our work related to understanding how yeast cells are able to change their physical structure and form projections during mating. This is an interesting example of a problem that deals with both intracellular protein signaling and cell mechanics. One issue that has become increasingly important to understanding the dynamics of proteins inside of single cells is the inherent randomness or stochasticity of biochemical reactions. As mathematical modeling and computational techniques have become essential tools in systems biology over the last half century, we first mention our software framework for the efficient simulation of spatial stochastic reaction-diffusion problems which can leverage high-performance computing and cloud infrastructure. This work serves as the basis for our investigation into yeast mating morphogenesis. The first step of yeast mating projection growth is the localization (or polarization) of proteins on the cell membrane. This is a well-studied, yet not fully understood, example of pattern formation in biology. In this dissertation we discuss several mathematical models of polarization and their various properties. When a yeast cell forms a mating projection the cell shape naturally changes in time. To deal with this from a mathematical modeling standpoint, we have developed a novel algorithm for the simulation of spatial stochastic dynamics on moving domains. These technical advances have led to new insight into the biology of yeast mating morphogenesis. In particular, we have elucidated the effects that complex geometries can have on current models of polarization. While polarization is certainly necessary for yeast mating morphogenesis, it is not the whole story. Yeast cells have a cell wall that is responsible for defining cell shape and providing mechanical integrity. To further explore mating projection growth, we have developed methods to couple models of polarization with physically based models for the mechanics of the cell wall. This coupling of biochemical signaling and mechanics allows for a more systems level understanding of yeast mating morphogenesis. We conclude by summarizing our findings about the coupling of polarization and mechanics, and discussing which biological links between the two are important from a mathematical modeling perspective.