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、液晶和蛋白质和生物溶液中的离子流体。这项研究将包括两类问题：(1)具有复杂流体常见的基本变分结构的系统的数学分析。(2)这种一般框架的应用，特别是在细胞和蛋白质中离子流体的研究。复杂的流体，包括混合物和溶液，在日常生活中大量存在。这些材料表现出的复杂现象和性质反映了微观相互作用和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存在的潜在能量变分结构。为了理解这些多尺度-多物理系统，将在建模、分析和数值模拟方面开发新的数学方法。将主要致力于蛋白质中离子溶液的研究。分子生物学已经表明，了解蛋白质和核酸在生物血浆(离子溶液)中的行为对于理解和控制许多生物系统是必不可少的。几乎所有的生物过程都涉及带电离子的运输。离子通道是大量生物现象的工具，许多疾病(如囊性纤维化)和药物治疗依赖于通道功能。在对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在确定边界之间流动的具有许多组件的系统。在这项研究中，我们将应用能量变分方法来研究生物血浆，钠、钾和氯离子的离子溶液，当它们流经特定的蛋白质通道时。

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.