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

Abstract

Abstract. We investigate higher order matched asymptotic expansions of a steady-state Poisson-Nernst-Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theorem 3.4), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. 1. 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. The PNP system describes the current flow at low resolution; that is, it is an approximate description of the transport process [3] appropriate when channel selectivity (between different chemical species of ions) is not of importance, as, for example, in the numerous classical studies of the gramicidin channel [51] most recently reviewed in [1, 2] (see also [16,36,37,11,25,51,17,20]). Derivation from a Langevin model of ionic diffusion [42] shows how correlations are approximated in PNP systems and suggests extensions of the PNP approach to deal with selectivity arising from excess chemical potentials [13,14,6]. The bio-

Effects of (Small) Permanent Charge and Channel Geometry on Ionic Flows via Classical Poisson--Nernst--Planck Models

“…Numerical studies have shown that classical PNP models provide good qualitative agreement with experimental data for I-V relations [4,5]. Dynamics of classical PNP models has also been analyzed by using asymptotic expansion methods [1,6,32,42,49,50,52,55] and geometric singular perturbation approaches [14,15,35,36,39].…”

One-dimensional steady-state Poisson–Nernst–Planck systems for ion channels with multiple ion species

“…. ,m. In the first stage over [x 0 , x 1 ] = [0, x 1 ], it will start near the point (ν 0 , u [0] , L, J * , x 0 ), follow the singular layer Γ [0,r] toward the slow manifold Z 1 , move along the regular layer Λ 1 , and leave the vicinity of Z 1 along the singular layer Γ [1,l] toward the point (φ [1,l] , u [1] , C [1,l] , J * , x 1 ). It then continues the evolution over each subsequent sub-intervals in a similar fashion until it reaches the vicinity of the…”

“…The electric field is strong throughout these systems and only a few charges (ions) are needed to create significant electrical potentials, compared to the enormous number of ions (10 23 , Avogadro's number) needed to create chemical potentials (and diffusion). That is why the Debye number is so large (see system (1)). The flow through the atomic scale channel is affected by other variables besides the applied boundary potentials in the reservoirs, namely, by the shape of the pore in the channel (through which permanently charged ions flow) and the distribution of permanent and induced (i.e., polarization) charge on the wall of the channel as well as the mobility of ions [13,21,27,28,34,40].…”

“…In particular, the singular boundary value problem for one-dimensional steadystate PNP systems (5) was studied to understand the I-V relations, multiplicity of solutions and many other important properties of channels. Two main streams of methods are the matched asymptotic expansions [1,4,5,24,51,53,54,57,[59][60][61][62] and the geometric theory for singularly perturbed problems [22,42]. Among others, I-V relations are approximated and multiple solutions are found to exist under some conditions.…”

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.…”