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.
