Collaborative Research: Mathematical Studies and Refinements of a Reduced Ion Channel Model and a Nonlocal Dielectric Model

离子通道对于生物功能是至关重要的，并且也代表了独立感兴趣的显著电子器件。它们促进和调节带电离子进出细胞的流动，并且它们以尚未完全理解的精确行为来实现这一点。更精确的模型和更有效的数值算法将极大地推动离子通道和其他重要蛋白质的研究。在这个项目中，PI计划研究两个离子通道（钠和钙通道）的模型，以了解介电模型在预测离子通道机制中的作用。特别是，PI将建立在一个现有的模型，使用一个非常简单，几乎天真，但非常有效的，模型的成本效益。然后，他们将对这个模型进行数学分析，以进一步揭示其特性和适用范围。此外，他们将研究一种更复杂的模型，通常称为非本地模型。此外，该项目将为研究生和本科生提供多学科培训机会。 快速计算离子通道模型（包括非局部连续溶剂模型）的数值解对于定量理解关键生理过程和细胞代谢和能量学至关重要，这对于提高我们对健康和疾病的理解至关重要。新的离子通道模型，非局部连续溶剂模型，并在这个项目中开发的有效的数值算法将是一个相当大的贡献，数学生物学，计算生理学和计算数学。它们将在研究各种离子通道、计算溶剂-溶质体系的连续静电场、模拟大规模蛋白质体系等方面发挥重要作用。

Ion channels are critical for biological function and also represent remarkable electronic devices of independent interest. They facilitate and regulate the flow of charged ions in and out of cells, and they do so with precise behaviors that are not fully understood. The more accurate models and more efficient numerical algorithms will significantly advance the research on ion channels and other important proteins. In this project, the PIs plan to study models for two ion channels (sodium and calcium channels) to understand the role of dielectric models in predicting the mechanism of ion channels. In particular, the PIs will build upon an existing model that uses a very simple, almost naive, yet remarkably effective, model of dielectrics. They then will analyze this model mathematically to further reveal its properties and range of applicability. Moreover, they will study a more sophisticated model of dielectrics, commonly known as a non-local model. In addition, this project will provide multi-disciplinary training opportunities to both graduate and undergraduate students. Fast computation of numerical solutions of ion channel models including nonlocal continuum solvent models is essential for quantitative understanding of critical physiological processes and cellular metabolism and energetics, which is central to improving our understanding of health and diseases. The new ion channel models, nonlocal continuum solvent models, and efficient numerical algorithms developed in this project will be a considerable contribution to mathematical biology, computational physiology, and computational mathematics. They will play important roles in studying various ion channels, computing continuum electrostatics for solvent-solute systems, and simulating large scale protein systems.