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

This is the published version, also available here: http://dx.doi.org/10.1137/060657480.Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one‐dimensional steady‐state Poisson‐Nernst‐Planck (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

On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail

Coulomb blockade model of permeation and selectivity in biological ion channels

Biological ion channels are protein nanotubes embedded in, and passing through, the bilipid membranes of cells. Physiologically, they are of crucial importance in that they allow ions to pass into and out of cells, fast and efficiently, though in a highly selective way. Here we show that the conduction and selectivity of calcium/sodium ion channels can be described in terms of ionic Coulomb blockade in a simplified electrostatic and Brownian dynamics model of the channel. The Coulomb blockade phenomenon arises from the discreteness of electrical charge, the strong electrostatic interaction, and an electrostatic exclusion principle. The model predicts a periodic pattern of Ca2+ conduction versus the fixed charge Qf at the selectivity filter (conduction bands) with a period equal to the ionic charge. It thus provides provisional explanations of some observed and modelled conduction and valence selectivity phenomena, including the anomalous mole fraction effect and the calcium conduction bands. Ionic Coulomb blockade and resonant conduction are similar to electronic Coulomb blockade and resonant tunnelling in quantum dots. The same considerations may also be applicable to other kinds of channel, as well as to charged artificial nanopores