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

细胞膜的形状和刚度是与细胞功能相关的动态特征。 包括这些特征的细胞膜的理想化模型是脂质双层囊泡，其由两种或更多种能够相分离并具有不同固有刚性的化学组分组成。实例包括几种脂质-胆固醇系统。 一般来说，组成通量和弹性模量可以从一个共同的能量势，导致在一个紧密耦合的系统。 这些囊泡具有远小于特征直径的特征厚度，这表明它们作为共维1表面处理。 该建议将三个研究主题联系起来，以开发流体介质中此类囊泡的连续模型。 首先是高阶界面跟踪。 嵌入式边界方法将被开发来耦合的运动的膜悬浮在，并包含，粘性流体。 第二个是动态-化学和机械-与时间相关的形状的膜。 解决这个问题的一种方法是采用Carnival网格有限体积法和相场方法来计算环中的动力学（化学的Cahn-Hilliard，运动的固体力学），在零厚度的限制下获得所需的行为。 前沿跟踪和界面动力学问题必须紧密耦合，例如，通过预测-校正和放松策略。 第三个问题涉及本构建模-发展的能量潜力，实验相图，弹性测量，和框架不变性的对称性要求的约束下，耦合的化学与形状的影响，这项工作将推进理解的混合双曲椭圆型的数值偏微分方程的时间依赖域。 囊泡动力学的数值模型，连接形状和化学的动力学，将开发的目的，更好地理解实验的异质囊泡，自然和仿生，并最终帮助理解细胞膜化学，形状和功能之间的联系。

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.