Complex Amphiphilic Structures and the Functionality of Biomaterials

复杂的两亲性结构与相变、材料科学和数学生物学密切相关。本项目主要研究纳米颗粒、蛋白质和疏水表面对两亲结构形态的影响。这项研究极大地促进了对纳米颗粒、生物激发材料和生物膜功能的理解。它帮助我们了解蛋白质运输、药物封装和药物递送等基本过程。它也为聚合物材料中其他复杂结构的性能建模和分析提供了新的方法和技术。教育活动包括研究生研究的指导和培训。教育目标是培养研究生的数学建模能力，以及偏微分方程的分析和数值模拟能力。本研究项目集中于应用数学与科学、技术和工程相结合的跨学科研究。PI是一个应用数学家，而co-PI是一个化学实验家。研究人员在功能化的Cahn-Hilliard （FCH）能量框架下，加上适当的强弱锚定条件，模拟了纳米颗粒在双层脂质体和聚合物胶束中的包封、两亲共聚物对生物材料的包封以及蛋白质引起的形态变化。这种新的数学框架将材料系统转化为具有适当边界条件的非线性偏微分方程。它在其他物理、材料和生物系统的建模中有着广泛的应用。非线性偏微分方程应用现有的和新的数学工具来研究非线性偏微分方程。这些工具包括三个方面：渐近分析、变分方法和数值模拟。它们在研究与非线性偏微分方程有关的其他应用数学问题时非常有用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Complex amphiphilic structures are closely related to phase transitions, materials science, and mathematical biology. This project concentrates on the influence of nanoparticles, proteins and hydrophobic surfaces on the morphology of amphiphilic structures. This research greatly advances the understanding of the functionality of nanoparticles, bio-inspired materials, and biological membranes. It helps us understand essential processes like protein transportation, drug encapsulation, and drug delivery. It also provides new methods and techniques for the modeling and analysis of properties of other complex structures in polymeric materials. The educational activities include the supervision and training of graduate research. The educational goal is for graduate students to be trained in mathematical modeling, and in the analysis and numerical simulation of partial differential equations.This research project concentrates on interdisciplinary research that combines applied mathematics and science, technology, and engineering. The PI is an applied mathematician, and the co-PI is a chemical experimentalist. The investigators model the encapsulation of nanoparticles in bilayer liposomes and polymeric micelles, the coating of biomaterials by amphiphilic copolymers, and the change of morphology caused by proteins, in the framework of the functionalized Cahn-Hilliard (FCH) energy, coupled with proper strong and weak anchoring conditions. This new mathematical framework translates the materials system into nonlinear PDEs coupled with proper boundary conditions. It has wide applications in the modeling of other physical, materials and biological systems. The PIs apply existing and new mathematical tools for the study of nonlinear PDEs. These tools lie in three areas: asymptotic analysis, variational methods, and numerical simulations. They are very useful in the study of other problems in applied mathematics related to nonlinear PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.