Spatiotemporal models of intra and extra cellular processes

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Mathematical models are playing an increasingly important role in the understanding of biological processes. My research focuses on models related to cellular processes. Specifically I use methods from asymptotic analysis and perturbation theory to study systems which form localized structures. Such systems play important roles in many areas of biology. The new models being considered include a model of plaque formation in atherosclerosis, a model of the formation of polarity in the early C. elegans development and a model of cell surface receptor aggregation. As well we will continue to work on models of cell signal transduction and general pattern formation. Some details of each project follow.******Atherosclerosis is a leading cause of death in the world today. Briefly the disease consists of the formation of a fatty plaque on the arterial wall resulting in a blockage and reduction in blood flow. We will model the system with a reaction-diffusion system consisting of the interaction between immune cells, foam cells, and low-density lipoproteins (distinguishing between the oxidized and unoxidized forms) and chemo-attractants. Previous studies have shown that such systems can form localized solutions representing a plaque. We will use the methods for studying nonlinear patterns to determine the condition under which a plaque may form and find the size of the plaque.******A great deal is known about the developmental process of the nematode C. elgans. One of the first steps to occur is the formation of anterior and posterior regions the egg after fertilization. This is accomplished by the segregation of proteins to opposite ends of the cell, creating two biochemically distinct regions. We model the system with a reaction-diffusion equation which incorporates the geometry of the egg. We intend to show that the egg geometry is crucial for the formation of a robustly placed barrier.******In previous work, I have considered a three dimensional model of cell signal transduction. A key element of this model was the localization of enzymes which activate messenger molecules in the signal transduction process. This work focused on the events occurring inside the cell. We now propose a model of the formation of surface receptor aggregation. We will use density dependent diffusion to model the effect of reduced motility of large groups of cell surface molecules. **