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.

该提案主要包括具有转折点的奇异摄动问题的几何研究以及分析泊松-能斯特-普朗克（PNP）系统。大量多尺度物理现象可以通过奇异扰动系统进行适当的建模。如果问题中存在转折点，就会极大地增加全球动态的复杂性。它们允许看似简单的系统支持非常丰富且有时令人惊讶的行为。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影响。要研究的一类问题涉及没有光谱间隙的多家庭转折点。随着这项研究的开展，受零阶近似几何奇异摄动理论成功的推动，研究人员将扩展高阶近似理论，这对于应用中的定性和定量目的都至关重要。该提案的另一个主要组成部分涉及 PNP 系统。 PNP 系统是半导体中和通过离子通道的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。除了其重要的应用价值之外，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、渐近行为的复杂性等等。研究人员和他的合作者已经获得了 PNP 系统的许多重要结果。成功很大程度上取决于 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。这项活动将增强对这一类重要的多尺度系统的理论理解。拟议的泊松-能斯特-普朗克（PNP）系统研究的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，相关通道的特定缺陷是许多健康问题的根本原因，所有药物中很大一部分直接作用于离子通道。 PNP 系统用于模拟离子通道特性的有效性已被仔细检验。 PNP 系统已经在很大程度上进行了数值研究，并且在许多情况下结果与实验数据非常吻合。然而，迫切需要通过数学分析更好地理解 PNP 系统。研究者和他的合作者将专注于 PNP 系统的生物学相关数学问题；例如，可以提取通道的渗透性和选择性的电流-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，以及突变永久电荷对通道特性的影响。 PNP 系统的研究最终将增进我们对离子通道的了解，提出有效且高效的实验室设计，并为生产更好的药物提供基本机制。