Dynamics of Poisson-Nernst-Planck systems and applications to ionic channels

Dynamics of Poisson-Nernst-Planck systems and its applications to ion channels are studied in this dissertation. The Poisson-Nernst-Planck systems serve as basic electro-diffusion equations modeling, for example, ion flow through membrane channels and transport of holes and electrons in semiconductors. The model can be derived from the more fundamental models of the Langevin-Poisson system and the Maxwell-Boltzmann equations, and from the energy variational analysis EnVarA. A brief description of the model is given in Chapter 2 including the physical meaning of each equation involved. Ion channels are cylindrical, hollow proteins that regulate the movement of ions ( mainly Na+, K+, Ca++ and Cl-;) through nearly all the membrane channels. When an initial potential is applied at one end of the channel, it will drive the ions through the channel, and the movement of these ions will produce the current which can be measured. Different initial potentials will result in different currents, and the collection of all those data will provide a relation, the so-called I-V (current-voltage) relation, which is an important characterization of two most relevant properties of a channel: permeation and selectivity. In Chapter 3, a classical Poisson-Nernst-Planck system is studied both analytically and numerically to investigate the cubic-like feature of the I-V relation. For the case of zero permanent charge, under electroneutrality boundary conditions at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion speciesis that the third order correction is cubic in the potential V , and furthermore, up to the third order, the cubic I-V relation has three distinct real roots (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Numerical simulations are performed and and they are consistent with our analytical results. In Chapter 4, we consider a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through membrane channels including ionic interaction modeled by a nonlocal hard-sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. Ion size effect on the I-V relations is investigated numerically. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I-V curves. This is accomplished through a numerical implement of the analytical strategy introduced in [46]. The second task is to numerically detect two critical potential values Vc and Vc. Our numerical detections are based on the defining properties of Vc and V c and are designed to use the numerical I-V curves directly. For the setting in the above mentioned reference, our numerical results agree well with the analytical predictions. In Chapter 5, a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through a membrane channel is analyzed, which includes a local hard-sphere potential that depends pointwise on ion concentrations to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singu- larly perturbed differential system. Based on the geometric singular perturbation theory, especially, 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 relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality (zero netcharge) 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 con- centrations. 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 in [46]