Dynamics of Singularly Perturbed Systems and Ion Channel Problems

本文主要研究带拐点的奇摄动问题的几何性质，并对PNP系统进行分析。大量的多尺度物理现象可以用奇异摄动系统恰当地模拟。转折点，如果在问题中存在，会显著增加全球动态的复杂性。它们允许一个看似简单的系统支持非常丰富，有时甚至令人惊讶的行为。研究者建议继续研究转折点问题，特别是多家庭转折点的集体效应。要研究的一类问题涉及无谱隙的多族拐点。随着这项研究，由零阶近似的几何奇异摄动理论的成功，研究者将扩展理论的高阶近似，这是至关重要的定性和定量的目的在应用中。该建议的另一个主要组成部分涉及PNP系统。PNP系统是半导体和离子通道中离子传输的基本模型。它们是具有多重时间和空间尺度的非线性电扩散系统。除了具有重要的应用价值外，PNP系统还为各种具有挑战性的数学问题提供了丰富的资源，这些问题涉及基本的适定性问题、稳态解的存在性和多重性、其渐近行为的复杂性等。研究者和他的合作者已经获得了PNP系统的一些重要结果。成功与否很大程度上取决于PNP系统内在结构的发现。研究者建议对PNP系统进行系统研究。这项活动将加强对这类重要的多尺度系统的理论认识。提出的泊松-能-普朗克（PNP）系统的研究动机是直接应用于细胞的跨膜离子通道。了解离子通道的生物学功能对人类健康至关重要。事实上，相对通道的特定缺陷是许多健康问题的潜在原因，而且大部分药物直接作用于离子通道。PNP系统对离子通道特性建模的有效性已经得到了仔细的检验。对PNP系统进行了大量的数值研究，结果在许多情况下与实验数据非常吻合。然而，迫切需要通过数学分析来更好地理解PNP系统。研究者和他的合作者将专注于生物相关的PNP系统的数学问题；例如，可以从中提取通道的渗透和选择性的电流-电压关系，对生物信号传播的自动控制至关重要的“门控”现象，以及突变的永久电荷对通道特性的影响。PNP系统的研究将最终提高我们对离子通道的认识，提出有效和高效的实验室设计，并为生产更好的药物提供基本机制。

This proposal consists mainly of geometric studies of singularly perturbed problems with turning points and analyzing Poisson-Nernst-Planck (PNP) systems. A great deal of multi-scale physical phenomena can be suitably modeled by singularly perturbed systems. Turning points, if present in the problem, increase dramatically the level of complexity of the global dynamics. They allow a seemingly simple system to support very rich and, sometime, surprising behaviors. The investigator proposes to continue his research on turning point problems, particularly on the collective effects of multi-family turning points. A class of problems to be investigated concerns multi-family turning points without spectral gaps. Along with this investigation, motivated by the success of geometric singular perturbation theory for zero-order approximations, the investigator will extend the theory for higher-order approximations which are crucial for both qualitative and quantitative purposes in applications. Another major component of this proposal concerns PNP systems. PNP systems serve as fundamental models for ion transport in semi-conductors and through ion channels. They are nonlinear electro-diffusion systems that possess multiple time and space scales. In addition to their significant application values, PNP systems provide a rich source for a variety of challenging mathematical questions concerning the basic well-posedness problem, the existence and multiplicity of steady-state solutions, the complexity of their asymptotic behavior, et cetera. The investigator and his collaborators have obtained a number of important results for PNP systems. The success depends heavily on the discovery of the intrinsic structure of PNP systems. The investigator proposes to conduct a systematic study of PNP systems. This activity will enhance the theoretical understanding of this important class of multi-scale systems.The proposed study on Poisson-Nernst-Planck (PNP) systems is motivated by direct applications to transmembrane ion channels of cells. Understanding the biological function of ion channels is critical to human health. In fact, specific defects of relative channels are the underlying causes of many health problems, and a large fraction of all drugs work directly on ion channels. The validity of PNP systems for modeling ion channel properties has been carefully examined. PNP systems have been studied numerically to a great extent, and the results have demonstrated excellent agreement with experimental data in many cases. There is, however, a serious need for a better understanding of PNP systems through mathematical analysis. The investigator and his collaborators will focus on biologically relevant mathematical problems of PNP systems; for example, the current-voltage relations from which the permeation and selectivity of channels can be extracted, the "gating" phenomena that are critical for auto-controlling of biological signal propagation, and the effect of mutating permanent charges on channel properties. The study of PNP systems will ultimately advance our knowledge of ion channels, suggest effective and efficient lab designs, and provide fundamental mechanisms for producing better drugs.