Energetic Variational Approaches in Complex Fluids

本研究的重点是各向异性复杂流体的数学理论，如粘弹性材料、液晶、蛋白质和生物溶液中的离子流体。研究将包括两类问题：(1)复杂流体中常见的具有基本变分结构的系统的数学分析。(2)这种总体框架的应用，特别是对细胞和蛋白质中的离子流体的研究。复杂的流体，包括混合物和溶液，在日常生活中大量存在。这些材料所表现出的复杂现象和性质反映了微观相互作用与宏观动力学之间的耦合和竞争。该项目将研究所有这些材料共同的潜在能量变分结构。为了理解这些多尺度多物理场系统，将开发新的数学建模、分析和数值模拟方法。主要的工作将致力于研究蛋白质中的离子溶液。分子生物学表明，了解蛋白质和核酸在生物等离子体（离子溶液）中的行为对于理解和控制许多生物系统至关重要。几乎所有的生物过程都涉及带电离子的传输。离子通道对大量的生物现象起着重要作用，许多疾病（如囊性纤维化）和药物治疗都依赖于离子通道的功能。在这些现象的数学分析中，变分方法显示出很大的希望，因为它们自然地处理具有许多在确定边界之间流动的组件的系统。在这项研究中，我们将应用能量变分方法来研究生物等离子体，钠离子溶液，钾离子溶液和氯离子溶液流过特定蛋白质通道的过程。

The focus of this research is on the mathematical theory of anisotropic complex fluids, such as viscoelastic materials, liquid crystals, and ionic fluids in proteins and bio-solutions. The research will include two categories of problems: (1) Mathematical analysis of the systems with basic variational structures that are common to complex fluids. (2) The application of such a general framework, in particular to the study of ionic fluids in cells and proteins. Complex fluids, including mixtures and solutions, are abundant in daily life. The complicated phenomena and properties exhibited by these materials reflect the coupling and competition between microscopic interactions and macroscopic dynamics. The project will study the underlying energetic variational structures that are common to all these materials. New mathematical approaches in the modeling, analysis and numerical simulations will be developed in order to understand these multiscale-multiphysics systems.Major efforts will be devoted to the study of ionic solutions in proteins. Molecular biology has shown that understanding behavior of proteins and nucleic acids in biological plasma (ionic solutions) is essential for understanding and controlling many biological systems. Nearly all biological processes involve the transport of electrically charged ions. Ion channels are instrumental for a vast number of biological phenomena, with many diseases (such as cystic fibrosis) and drug treatments dependent on channel functioning. In mathematical analysis of these phenomena, variational methods show a great promise because they deal naturally with systems with many components that flow between definite boundaries. In this research we will apply energy variational methods to study biological plasma, ionic solutions of sodium, potassium, and chloride ions as they flow through specific protein channels.