Deformation, Phase Segregation and Adhesion of Lipid-Bilayer Vesicles

研究者试图研究描述脂质双层膜的形状、相分布和粘附的平衡和进化的偏微分方程。形状、成分分布和压力之间复杂的相互作用使这些膜能够经历各种成分和顺序相关的相变，从而形成复杂的微观结构。这些反过来影响膜的功能。双层膜的力学、相分离和粘附的实验研究是一个非常活跃和有吸引力的领域。一些考虑不同方面和不同近似的理论模型和研究开始出现。这为系统的数学分析提供了一个阶段，该阶段检查了不同模型之间的关系，了解了各种近似的状态，并给出了不同参数和行为机制的映射。这个数学分析是这个项目的目的。研究者不仅寻求对现有的实验观察进行分类和调和，而且希望通过指出潜在的有趣区域来激发新的实验和发现新的现象。该项目建立并扩展了涉及扩散、相变和模式形成的数学问题类别，这些问题已经研究了膨胀空间，受到材料科学问题的启发，到一个涉及可变形流形上这些现象的新问题类别。研究者从单相膜的能量公式开始，并使用正式（匹配渐近）和严格（伽玛收敛）方法推导出极限理论。然后，他考虑了多组分膜，并确定了存在强相分离的情况，从稳定性考虑研究成核，并推导出尖锐的界面模型。进化问题被视为适当能量的梯度流动。最后，他研究了这种分析对细胞粘附的影响。脂质双分子层膜普遍存在于生物体细胞壁、线粒体、高尔基体和许多其他重要细胞器中。它们在生物学中起着至关重要的作用，因为它们通过形成各种复杂的形状和结构来保护、调节血流，并承载许多代谢功能。该项目旨在促进我们对控制这些膜的形状和组成分布的因素的理解。它是通过建立基于现有观察的数学模型，并使用数学分析来探索这些模型的全部含义来实现的。该项目为这一关键的新兴跨学科领域的博士生提供培训。学生不仅要接受现代数学分析方法的训练，还要接触生物学实验和力学与材料科学理论。该项目还培养了三名暑期本科生。

BhattacharyaDMS-0606667 The investigator seeks to study partial differentialequations that describe the equilibrium and evolution of theshape, phase distribution, and adhesion of lipid bilayermembranes. An intricate interaction between the shape, thecomposition distribution, and pressure enable these membranes toundergo a variety of composition- and ordering-related phasetransformations resulting in complex microstructure. These inturn affect the function of the membrane. The experimental studyof the mechanics, phase segregation, and adhesion of bilayermembranes is an extremely active and fascinating field. A numberof theoretical models and studies that consider different aspectsand different approximations are beginning to emerge. This hasset the stage for a systematic mathematical analysis thatexamines the relationship between the different models,understands the status of the various approximations, and yieldsa mapping of the different parameter and behavioral regimes. This mathematical analysis is the aim of this project. Theinvestigator seeks not only to catalog and reconcile availableexperimental observations but hopes to motivate new experimentsand the discovery of new phenomena by pointing out potentiallyinteresting regions. The project builds on and extends the classof mathematical problems involving diffusion, phasetransformation, and pattern formation that has been studied inflat spaces, motivated by issues in materials science, to a newclass of problems that involve these phenomena on deformablemanifolds. The investigator starts with an energetic formulationof the single phase membrane and derives limiting theories usingboth formal (matched asymptotic) and rigorous (Gamma convergence)methods. He then considers multi-species membranes andidentifies circumstances when there is strong phase separation,studies nucleation from stability considerations, and derivessharp interface models. Evolutionary problems are treated asgradient flows of appropriate energy. Finally, he studies theimplications of this analysis for cell adhesion. Lipid bilayer membranes are ubiquitous in living organismsas cell walls, mitochondria, golgi apparatus, and numerous otherimportant organelles. They play a critical role in biology asthey protect, regulate flow, and host many metabolic functions byforming a variety of complicated shapes and structures. Thisproject seeks to advance our understanding of the factors thatgovern the shape and composition distribution of these membranes. It does so by building mathematical models based on existingobservations and using mathematical analysis to explore the fullimplications of these models. The project provides for thetraining of a doctoral student in this critical and emerginginterdisciplinary area. The student is trained in and usescontemporary methods of mathematical analysis, but also isexposed to experiments in biology and theories in mechanics andmaterials science. The project also trains three summerundergraduate research students.