Development of high-order numerical methods for the dynamics of suspended lipid membranes with internal structure

细胞膜的形状和刚性是关系到细胞功能的动态特征。包含这些特征的细胞膜的理想模型是脂质双分子层囊泡，由两种或两种以上能够相分离的化学成分组成，并具有不同的固有刚度。例子包括几种脂质-胆固醇系统。通常，组分通量和弹性模量可以由一个共同的能量势导出，从而形成一个紧密耦合的系统。这些囊泡的特征厚度远小于特征直径，表明它们被处理为协维1表面。本建议将三个研究课题联系起来，以开发流体介质中此类囊泡的连续体模型。首先是高阶接口跟踪。嵌入边界方法将被开发来耦合膜的运动悬浮，并包含粘性流体。其次是膜的动力学——化学和力学——随时间变化的形状。解决这一问题的一种方法是采用笛卡尔网格有限体积法和相场法来计算环空中的动力学（化学中的Cahn-Hilliard和运动中的固体力学），并在零厚度极限下获得所需的行为。前沿跟踪和界面动力学问题必须紧密耦合，例如，通过预测校正和松弛策略。第三个问题涉及本构建模——在实验相图、弹性测量和框架不变性的对称要求的约束下，发展能量势，将化学效应与形状耦合起来。这一工作将促进对时间相关域上混合双曲-椭圆型数值偏微分方程的理解。为了更好地理解非均质囊泡（天然的和仿生的）的实验，并最终帮助理解细胞膜化学、形状和功能之间的联系，将开发一个囊泡动力学的数值模型，将形状和化学动力学联系起来。

The shape and rigidity of cell membranes are dynamic features related to cell function. An idealized model of the cell membrane that includes these features is a lipid bilayer vesicle, comprised of two or more chemical components capable of phase separation and endowed with different inherent rigidities. Examples include several lipid-cholesterol systems. In general, compositional fluxes and elastic moduli may be derived from a common energy potential, resulting in a tightly coupled system. These vesicles have characteristic thicknesses that are much less than characteristic diameters, suggesting their treatment as co-dimension 1 surfaces. This proposal links three research topics to develop a continuum model of such vesicles in fluid media. First is the high-order interface tracking. Embedded boundary methods will be developed to couple the motion of a membrane suspended in, and containing, viscous fluid. Second is the dynamics -- chemical and mechanical -- of a membrane with time dependent shape. An approach to this problem is to employ cartesian grid finite volume methods with a phase field approach to compute dynamics (Cahn-Hilliard for chemistry, and solid mechanics for motion) in an annulus, with the desired behavior obtained in the limit of zero thickness. The problems of front tracking and interface dynamics must be tightly coupled, e.g., through predictor-corrector and relaxation strategies. The third problem concerns constitutive modeling - the development of energy potentials, constrained by experimental phase diagrams, elasticity measurements, and the symmetry requirements of frame invariance, to couple the effects of chemistry with shape.This proposed work will advance the understanding of numerical partial differential equations of mixed hyperbolic-elliptic type on time-dependent domains. A numerical model of vesicle dynamics, linking the dynamics of shape and chemistry, will be developed for the purpose of better understanding experiments on heterogeneous vesicles, both natural and biomimetic, and ultimately to aid in understanding the connections between cell membrane chemistry, shape, and function.