Abstract
We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.
Highlights
Ion channels are large proteins embedded in cell membranes with a hole down their middle that provides a controllable path for electrodiffusion of ions through biological membranes, establishing communications among cells and the external environment [1,2,3]
To better understand the importance of the relation of ionic flows and permanent charges, we remark that the role of permanent charges in membrane channels is similar to the role of doping profiles in semiconductor devices
This indicates that our work provides a better understanding of the mechanism of ionic flows through single ion channels, which is necessary and important for future studies of ion channel problems
Summary
Ion channels are large proteins embedded in cell membranes with a hole down their middle that provides a controllable path for electrodiffusion of ions (mainly Na+, K+, Ca++ and Cl−) through biological membranes, establishing communications among cells and the external environment [1,2,3]. The study of ion channels consists of two related major topics: structures of ion channels and ionic flow properties. The physical structure of ion channels is defined by the channel shape and the spacial distribution of permanent and polarization charge. Mathematical analysis plays important and unique roles for generalizing and understanding the principles that allow control of electrodiffusion, explaining mechanics of observed biological phenomena and for discovering new ones, assuming a more or less explicit solution of the associated mathematical model can be obtained. In general, the latter is too much to expect. There have been some successes in mathematical analysis of Poisson–Nernst–Planck (PNP) models for ionic flows through membrane channels [5,6,7,8,9,10,11,12,13,14]
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