Weishi Liu

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

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.

CENTER MANIFOLDS FOR SMOOTH INVARIANT MANIFOLDS

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9947-00-02443-0. First published in Trans. Amer. Math. Soc. in 2000, published by the American Mathematical Society.

Relaxation Oscillations in a Class of Predator-Prey Systems

Abstract We consider a class of three-dimensional, singularly perturbed predator–prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model.

Geometric Singular Perturbation Approach to Steady-State Poisson--Nernst--Planck Systems

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 ...

Poisson–Nernst–Planck Systems for Narrow Tubular-Like Membrane Channels

We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previously studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous as the radius of the channel tends to zero.

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

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.

Poisson--Nernst--Planck Systems for Ion Flow with a Local Hard-Sphere Potential for Ion Size Effects

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 ...

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

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

Lorenz type attractors from codimension one bifurcation

Q_0

Reversal permanent charge and reversal potential: case studies via classical Poisson–Nernst–Planck models

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