Mathematical Modeling of Biomembranes

Miksis1312935该项目的目标是开发一种有效的零厚度模型，用于流体包埋的脂质双层膜。模型方程的预测被用来解释观察到的膜行为的特征，如尖锐边缘的出现，囊泡破裂，以及电融合过程中膜间薄液膜的稳定性。还考虑了复合囊泡的动力学问题。研究了用于解释毛孔的脂双层模型的推广。从数学上讲，这些都是具有挑战性的自由边界问题，表现出复杂的动力学。连续介质理论被用来模拟脂膜界面和周围液体的运动，例如，描述体液流动的Stokes方程和描述界面演化和膜电荷的耦合偏微分方程组。界面条件与界面上应力和电势的跳跃有关，它们取决于界面的几何形状和几个物理常数，如弯曲刚度和膜电容。提出了一种求解这类复杂的三维自由边界问题的计算方法。利用渐近法和摄动法对极限情况进行了解析研究。最近的实验工作被用作我们选择数学模型的指导。为了充分理解这些模型，并作为开发更现实的细胞模型的指南，我们进行了超出实验情况范围的调查。生物膜的动力学和稳定性是细胞功能的核心。生物膜的机电学在很大程度上是一个未被探索的领域。该项目使用应用数学中的方法和连续统理论中的思想，并将它们应用于生物学环境中。这扩大了这些学科的边界。该项目的结果直接影响使用膜电穿孔(例如，基因转染法)或细胞融合的生物医学技术。电能和机械能的相互转换，例如薄膜的挠曲电性(电场响应的弯曲)，可以被用于新颖的微米和纳米流体设计。该项目是跨学科的，这对与该项目相关的本科生和研究生的教育和发展非常有益。

Miksis1312935 The objective of the project is the development of an effective zero-thickness model for a fluid-embedded lipid bilayer membrane. The predictions of the model equations are applied to explain observed features of membrane behavior such as the appearance of sharp edges, vesicle bursting, and stability of the thin liquid film between membranes undergoing electrofusion. The dynamics of a compound vesicle are also considered. Generalizations of the lipid bilayer model to account for pores are studied. Mathematically, these are challenging free boundary problems exhibiting complex dynamics. Continuum theory is used to model the motion of the lipid membrane interface and the surrounding liquids, e.g., the Stokes equations for the flow in the bulk and a coupled system of partial differential equations describing the interface evolution and membrane charging. The interface conditions are related to the jump in stress and electric potential across the interface and they depend on the geometry of the interface and several physical constants, e.g., the bending rigidity and membrane capacitance. A computational method is developed to solve these complicated transient three-dimensional free-boundary problems. Limiting cases are investigated analytically, using asymptotic and perturbation methods. Recent experimental work is used as a guide in selecting our mathematical models. An investigation beyond the range of the experimental situations is done to fully understand the models and as a guide to the development of more realistic models of cells. The dynamics and stability of biomembranes is at the heart of cellular function. The electromechanics of biomembranes is an area that is largely unexplored. This project uses methods from applied mathematics and ideas from continuum theory, and applies them in a biological setting. This expands the frontiers of these disciplines. The outcomes of this project directly impact biomedical technologies that employ membrane electroporation (e.g. gene transfection) or cell fusion. The interconversion of electric and mechanical energies, one example being the membrane flexoelectricity (bending in response to electric fields), could be exploited for novel micro- and nano-fluidic designs. The project is interdisciplinary and this is very beneficial for the education and development of the undergraduate and graduate students associated with the project.