Dynamics of Singularly Perturbed Systems and Ion Channel Problems

奇扰动系统动力学和离子通道问题

• 批准号: 0807327
• 负责人: Weishi Liu
• 金额: $ 16.68万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807327&HistoricalAwards=false
• 关键词: Dynamics; Singularly; Perturbed; Systems; Ion

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.

该提案主要包括关于转折点和分析Poisson-Nernst-Planck（PNP）系统的几何研究。可以通过奇异的扰动系统适当地建模大量的多尺度物理现象。如果问题存在于问题中，则转折点会大大增加全球动力学的复杂性水平。它们允许一个看似简单的系统来支持非常丰富的系统，有时甚至是令人惊讶的行为。研究人员建议继续他对转折点问题的研究，尤其是关于多户家庭转折点的集体影响。一类待调查的问题涉及没有光谱差距的多户家庭转折点。除了这项研究之外，由于几何奇异扰动理论的成功，研究者还将扩展该理论的高阶近似值，这对于应用程序中的定性和定量目的至关重要。该提案的另一个主要组成部分涉及PNP系统。 PNP系统是半导体和通过离子通道中离子传输的基本模型。它们是具有多个时间和空间尺度的非线性电扩散系统。除了其重要的应用值之外，PNP系统还为有关基本适当的问题，稳态解决方案的存在和多样性，其渐近行为的复杂性等各种具有挑战性的数学问题提供了丰富的来源。研究人员及其合作者为PNP系统获得了许多重要的结果。成功在很大程度上取决于发现PNP系统的内在结构。研究人员建议对PNP系统进行系统研究。这项活动将增强对这一重要类型多规模系统的理论理解。拟议的关于Poisson-Nernst-Planck（PNP）系统的研究是由直接应用于细胞的跨膜离子通道的动机。了解离子通道的生物学功能对人类健康至关重要。实际上，相对通道的特定缺陷是许多健康问题的根本原因，并且所有药物的很大一部分直接在离子通道上起作用。 PNP系统用于建模离子通道性能的有效性已仔细检查。 PNP系统在很大程度上进行了数值研究，结果在许多情况下与实验数据表现出了极好的一致性。但是，通过数学分析，人们非常需要更好地了解PNP系统。研究人员及其合作者将专注于PNP系统的生物学相关数学问题；例如，可以提取通道的渗透和选择性的电流关系，对于生物信号传播的自动控制至关重要的“门控”现象，以及突变永久性电荷对通道特性的影响。 PNP系统的研究最终将提高我们对离子通道的了解，提出有效有效的实验室设计，并为产生更好的药物提供基本机制。