Spatiotemporal models of intra and extra cellular processes

数学模型在理解生物过程中发挥着越来越重要的作用。我的研究重点是与细胞过程相关的模型。具体地说，我使用渐近分析和微扰理论的方法来研究形成局域结构的系统。这样的系统在生物学的许多领域扮演着重要的角色。正在考虑的新模型包括动脉粥样硬化斑块形成模型、线虫发育早期极性形成模型和细胞表面受体聚集模型。此外，我们还将继续研究细胞信号转导和一般模式形成的模型。每个项目的一些细节如下。动脉粥样硬化是当今世界主要的死亡原因。简而言之，这种疾病包括在动脉壁上形成脂肪斑块，导致动脉阻塞和血流减少。我们将用免疫细胞、泡沫细胞、低密度脂蛋白(区分氧化和非氧化形式)和化学诱导剂之间的相互作用组成的反应-扩散系统来模拟该系统。以前的研究表明，这种系统可以形成代表斑块的局部溶液。我们将使用研究非线性模式的方法来确定斑块形成的条件和斑块的大小。我们对线虫的发育过程有很多了解。受精后卵子的前部和后部的形成是发生的第一步。这是通过将蛋白质分离到细胞的相反两端，创建两个生化上截然不同的区域来实现的。我们用包含鸡蛋几何形状的反应扩散方程对系统进行建模。我们打算证明卵子的几何形状对于坚固放置的屏障的形成是至关重要的。在以前的工作中，我已经考虑了细胞信号转导的三维模型。这个模型的一个关键因素是在信号转导过程中激活信使分子的酶的定位。这项研究的重点是细胞内发生的事件。我们现在提出了一个表面受体聚集形成的模型。我们将使用密度依赖扩散来模拟大量细胞表面分子的运动性降低的影响。**

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. **