Non-localness of Excess Potentials and Boundary Value Problems of Poisson–Nernst–Planck Systems for Ionic Flow: A Case Study

• Published in: Journal of dynamics and differential equations 2018-06, Vol.30 (2), p.779-797
• Main Authors: Sun, Lili, Liu, Weishi
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Boundary conditions; Boundary value problems; Computer simulation; Differential equations; Ion channels; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Well posed problems
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-017-9578-2

Poisson–Nernst–Planck (PNP) type systems are basic primitive models for ionic flow through ion channels. Important properties of ion channels, such as current-voltage relations, permeation and selectivity, can be extracted from solutions of boundary value problems (BVP) of PNP type models. Many issues of BVP of PNP type systems with local excess potentials (including particularly classical PNP systems that treat ions as point-charges) are extensively examined analytically and numerically. On the other hand, for PNP type systems with nonlocal excess potentials , even the issue of well-posedness of BVP is poorly understood. In fact, the formulation of correct boundary conditions seems to be overlooked, even though complications of ionic behavior near the boundaries (locations of applied electrodes) have been long experienced in experiments and simulations. PNP type systems with nonlocal excess potentials can be viewed as functional differential systems and, for many approximation models of nonlocal excess potentials, as differential equations with both delays and advances. Thus PNP type systems with nonlocal excess potentials have infinite degree of freedoms and BVP with the traditional “two-point-boundary-conditions” would be severely under determined. The mathematical theory for PNP with nonlocal excess potential would be significantly different from that for PNP with local excess potentials. Taking into considerations of experimental designs of ionic flow through ion channels and in a relatively simple setting, we present a form of natural “boundary conditions” so that the corresponding BVP of PNP type systems with nonlocal excess potentials are generally well-posed. This work, at an early stage toward a better understanding of related issues, provides some insights on interpretations of experimental designs of imposing boundary conditions and for correct formulations of numerical simulations, and hopefully, will stimulate further mathematical analysis on this important issue.