The goal of this project is to investigate the dynamical behavior of ion flow through membrane channels in living cells. The membrane channels are three-dimensional tubular-like structures whose radii are much smaller than their lengths. The ion flow is modeled by the Poisson-Nernst-Planck (PNP) system, which consists of at least two nonlinear parabolic equations for the sodium and calcium ion concentrations, coupled with an elliptic equation for the electric potential. The PNP system is singularly perturbed by the presence of small physical parameters multiplying the highest-order derivatives. This project focuses on the effects of the singular parameters on the global dynamics of the PNP model. It combines approaches from the theory of singular perturbations, dynamical systems, and partial differential equations. The PI will (i) identify the one-dimensional limiting system of the original three-dimensional PNP system when the radius of the membrane channel approaches zero; (ii) justify the limiting system by examining the relationship between the dynamics of the limiting system and the perturbed system; (iii) study the existence and stability of the steady state solutions of the systems; and (iv) explore the dynamics of the perturbed three-dimensional system based on the dynamical information of the one-dimensional limiting system.
Ion channels exist in the membranes of all cells and control a variety of biological functions. Each channel has some specific physiological function; for instance, the movement of sodium and calcium ions through the channels plays a crucial role in conducting electrical signals down the nervous system and activating muscle contraction. Any defects in this physical chemistry process may cause human muscle disease. Hence, studying the mechanism of ion flow through membrane channels is of importance for the practice of medicine. The objective of this project is to establish a practical mathematical model which can be justified theoretically as well as experimentally. The new system will be effective to capture the essential physiological information of charged ions and can be used by biologists to study the complicated behavior of ion flows by numerical as well as experimental approaches. The results of this research are expected to provide new treatments for diseases and hence help life. It is interesting to note that the transport process of charged particles in semiconductor devices and plasma physics is governed by the same model as ion flow through membrane channels. This suggests that the results of this project will also have potential applications in semiconductor technology and plasmas. In particular, the results can be used to design transistors and integrated circuits and control the physical behavior of these devices. Another objective of this project is to provide opportunity for training students in the related research areas.
