Dynamics of Singularly Perturbed Systems and Ion Channel Problems

这一建议主要包括对具有转折点的奇异摄动问题的几何研究和分析Poisson-Nernst-Planck(PNP)系统。许多多尺度的物理现象可以用奇异摄动系统来适当地模拟。转折点如果出现在问题中，将极大地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的、有时令人惊讶的行为。这位研究人员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。一类需要研究的问题涉及没有光谱间隙的多族转折点。在这项研究的同时，由于几何奇异摄动理论对零阶近似的成功，研究者将把这一理论扩展到高阶近似，这对应用中的定性和定量目的都是至关重要的。这项提案的另一个主要组成部分涉及即插即用系统。PNP系统是离子在半导体中和离子通道中传输的基本模型。它们是具有多时间和空间尺度的非线性电扩散系统。除了其重要的应用价值外，PNP系统还为各种具有挑战性的数学问题提供了丰富的来源，这些问题涉及基本适定性问题、稳态解的存在性和多重性、其渐近行为的复杂性等。这位研究人员和他的合作者已经获得了许多关于PNP系统的重要结果。这一成功在很大程度上取决于对PNP系统内在结构的发现。调查人员建议对PNP系统进行系统研究。这一活动将加强对这类重要的多尺度系统的理论理解。泊松-能斯特-普朗克(Poisson-Nernst-Planck，PNP)系统的研究受到直接应用于细胞跨膜离子通道的推动。了解离子通道的生物学功能对人类健康至关重要。事实上，相关通道的特定缺陷是许多健康问题的根本原因，而且很大一部分药物直接作用于离子通道。对PNP系统模拟离子通道性质的有效性进行了仔细的检验。对PNP系统进行了很大程度的数值研究，计算结果在许多情况下与实验数据符合得很好。然而，迫切需要通过数学分析更好地理解即插即用系统。研究人员和他的合作者将专注于PNP系统中与生物相关的数学问题；例如，可以从其中提取通道的渗透和选择性的电流-电压关系，对于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。对PNP系统的研究最终将促进我们对离子通道的了解，建议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.