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 ...

The one-dimensional Poisson-Nernst-Planck (PNP) system is a basic model for ion flow through membrane channels. If the Debye length is much smaller than the characteristic radius of the channel, the PNP system can be treated as a singularly perturbed system. We provide a geometric framework for the study of the steady-state PNP system involving multiple types of ion species with multiple regions of piecewise constant permanent charge. Special structures of this particular problem are revealed, which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations. Near each singular orbit, an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of a solution of the singular boundary value problem. A new phenomenon on multiplicity and spatial behavior of steady-states involving three or more types of ion species is discovered in an example. (The phenomenon cannot occur when only two types of ion species are involved.)

Abstract. In this work, we examine effects of permanent charges on ionic flows through ion channels via a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. Two ion species, one positively charged and one negatively charged, are considered with a simple profile of permanent charges: zeros at the two end regions and a constant Q 0 over the middle region. The classical PNP model can be viewed as a boundary value problem (BVP) of a singularly perturbed system. The singular orbit of the BVP depends on Q 0 in a regular way. Assuming |Q 0 | is small, a regular perturbation analysis is carried out for the singular orbit. Our analysis indicates that effects of permanent charges depend on a rich interplay between boundary conditions and the channel geometry. Furthermore, interesting common features are revealed: for Q 0 = 0, only an average quantity of the channel geometry plays a role; however, for Q 0 = 0, details of the channel geometry matter; in particular, to optimize effects of a permanent charge, the channel should have a short and narrow neck within which the permanent charge is confined. The latter is consistent with structures of typical ion channels.Key words. ionic flow, permanent charge, channel geometry AMS subject classifications. 34A26, 34B16, 34D15, 37D10, 92C35 DOI. 10.1137/140992527 Introduction.In this work, we analyze effects of permanent charges on ionic flows through ion channels, based on a quasi-one-dimensional classical PoissonNernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. We start with a brief discussion of the biological background of ion channel problems, a quasi-one-dimensional PNP model, and the main concern of our work in this paper.

Abstract. Boundary value problems of a one-dimensional steady-state Poisson-Nernst-Planck (PNP) system for ion flow through a narrow membrane channel are studied. By assuming the ratio of the Debye length to a characteristic length to be small, the PNP system can be viewed as a singularly perturbed problem with multiple time scales and is analyzed using the newly developed geometric singular perturbation theory. Within the framework of dynamical systems, the global behavior is first studied in terms of limiting fast and slow systems. It is rather surprising that a complete set of integrals is discovered for the (nonlinear) limiting fast system. This allows a detailed description of the boundary layers for the problem. The slow system itself turns out to be a singularly perturbed one, too, which indicates that the singularly perturbed PNP system has three different time scales. A singular orbit (zeroth order approximation) of the boundary value problem is identified based on the dynamics of limiting fast and slow systems. An application of the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the boundary value problem.

In this work, we analyze a one-dimensional steady-state Poisson-Nernst-Planck-type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality (zero net charge) boundary conditions, each of these two critical potentials separates the potential into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary concentrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneutrality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations, and critical potentials agree with those with a nonlocal hard-sphere potential examined by Ji and Liu [J. Dynam. Differential Equations, 24 (2012), pp. 955-983].