Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Abstract

Abstract. Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-NernstPlanck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a singular parameter, the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Application of the exchange lemma from geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the solution of the singular boundary value problem near each singular orbit. A set of simultaneous equations appears in the construction of singular orbits. Multiple solutions of such equations in this or similar problems might explain a variety of multiple valued phenomena seen in biological channels, for example, some forms of gating, and might be involved in other more complex behaviors, for example, some kinds of active transport. Key words. singular perturbation, boundary layers, internal layers AMS subject classifications. 34A26, 34B16, 34D15, 37D10, 92C35DOI. 10.1137/060657480 1. Introduction. Electrodiffusion, the diffusion and migration of electric charge, plays a central role in a wide range of our technology and science [53,11,54,14,15,67,41]: semiconductor technology controls the migration and diffusion of quasi-particles of charge in transistors and integrated circuits [75,62,71], chemical sciences deal with charged molecules in water [11,19,8,26,9,10], all of biology occurs in plasmas of ions and charged organic molecules in water [3,16,33,72]. It is no coincidence that the physics of electrodiffusion is of such general importance: systems of moving charge have a richness of behavior that can be sometimes easily controlled by boundary conditions [67,71], and the goal of technology (and much of physical science) is to control systems to allow useful behavior.Control is important to the medical and biological sciences as well. Medicine seeks to control disease and help life. Evolution controls life by selecting those organisms that successfully reproduce. Organisms control their internal environment and external behavior to make reproduction possible, often using electrodiffusion for the mechanism of control [72,33]. Whatever the reason, it is a fact that nearly all biology occurs in ultrafiltrates of blood called plasmas, in which ions move much as they move in gaseous plasmas, or ...

Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

“…The asymptotic matching principle, with a suitable hypothesis, can be also rigorously justified. It turns out that, for the problem handled in this paper, the so-called outer manifold is normally hyperbolic [10,27], and Van Dyke's principle of asymptotic matching is justified (see, for example, [33]). We will thus use the asymptotic matching principle for our matching purpose.…”

“…We remark that, in general, the I-V relation is NOT unique (see [30,38,39,45,46,10] for Q = 0 and see [28] even for Q = 0). Understanding the nonuniqueness issue is critical for ion channels because nonuniqueness might explain the gating behavior of single channels, which switch suddenly and stochastically from one current level to another [16,40,31,15].…”

“…In, e.g., [5,12,27], the zeroth order I-V relation is explicitly obtained for two types of ion species and Q = 0 (see [10,28] for a treatment of general situations and [44,43] for a treatment using a three-dimensional PNP model). The zeroth order I-V relation, in this case, is linear.…”

“…Introduction. The Poisson-Nernst-Planck (PNP) systems are basic electro-diffusion equations modeling, for example, ion flow through membrane channels and transport of holes and electrons in semiconductors (see, for example, [3,4,5,10,21,32,39,18,19,34,35]). In the context of ion flow through a membrane channel, the flow of ions is driven by their concentration gradients and by the electric field modeled together by the Nernst-Planck continuity equations, and the electric field is in turn determined by the concentrations through the Poisson equation.…”

Bias-dependent model of the electrical impedance of ionic polymer-metal composites

“…We use a modified PNP formulation to account for the presence of steric effects [52][53][54], which are shown to play a prominent role in shaping the charge distribution in the composite layers at moderately large voltages, see also the comprehensive review in [55]. Following [37,40,[52][53][54][56][57][58][59][60], we use the method of matched asymptotic expansions, see for example [61], to derive a semi-analytical solution for the electric potential and the charge concentration in the IPMC along with a new circuit model for IPMCs in presence of a DC voltage bias. The equivalent circuit model consists of the series connection of two impedances of the form presented in [24] to account for differences in charge build up at the anode and cathode regions for large voltages.…”