Dynamics of Biological Interfaces

这个项目是关于生物界面动力学的数值和分析研究。我们的兴趣将指向脂质双分子层，这是细胞膜的基本组成部分。我们的目标是开发解决方法，并研究控制这些生物界面动力学的数学模型的行为。数学上我们将处理一个复杂的自由边界问题。连续介质理论将用于模拟这些界面的运动。围绕界面的液体动力学将由一组偏微分方程控制。这些方程将沿界面与一个偏微分方程组耦合。边界积分数值方法将用于解决这些复杂的三维自由边界问题。最近的实验工作将作为我们选择数学模型的指导。为了充分理解我们的模型，将进行超出当前实验结果参数范围的计算研究。几种极限情况将用摄动方法进行分析研究。生物界面动力学是一个重要且相对未开发的领域。它是细胞活动的中心，特别是在囊泡运输和递送过程中。许多疾病与细胞中囊泡运输的功能障碍有关；因此，这项研究的结果可能会导致新的治疗方法和药物。本项目将使用应用数学的方法和连续统理论的思想，并将其应用于生物环境。该项目是跨学科的，因此将加强与项目相关的学生的教育和发展。

This project is concerned with a numerical and analytical investigation of the dynamics of biological interfaces. Our interest will be directed to lipid bilayers, which are the basic component of cell membranes. Our aim is to develop solution methods and to investigate the behavior of the mathematical models governing the dynamics of these biological interfaces. Mathematically we will be dealing with a complex free boundary problem. Continuum theory will be used to model the motion of these interfaces. The dynamics of the liquids surrounding the interface will be governed by a set of partial differential equations. These equations will be coupled to a system of partial differential equations along the interface. A boundary integral numerical method will be used to solve these complicated three-dimensional free-boundary problems. Recent experimental work will be used as a guide in selecting our mathematical models. A computational investigation beyond the parameter ranges of current experimental results will be done in order to fully understand our models. Several limiting cases will be investigated analytically using perturbation methods.The dynamics of biological interfaces is an important and relatively unexplored field. It is central to cellular activities and especially in vesicle transport and delivery processes. A number of diseases are related to a malfunction of vesicle transport in cells; hence the results of this research could lead to new treatments and drugs. This project will use methods from applied mathematics and ideas from continuum theory and applied them in a biological setting. The project is interdisciplinary and hence will enhance the education and development of the students associated with the project.