Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations

Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study

We consider a quasi-one-dimensional Poisson-Nernst-Planck model with two cations having the same valances and one anion. Bikerman's local hard-sphere potential is included to account for ion size effects. Under some further restrictions on the boundary conditions of the two cations, we obtain approximations of the I-V (current-voltage) relations by treating the ion sizes as small parameters. Critical potentials are identified, which play critical roles in characterizing finite ion size effects on ionic flows. Nonlinear interplays between system parameters, such as boundary concentrations and diffusion coefficients, are analyzed. To provide more intuitive illustrations of our analytical results and better understanding of the dynamics of ionic flows through membrane channels, numerical simulations are performed.

Mathematical Analysis on Current-Voltage Relations via Classical Poisson-Nernst-Planck Systems with Nonzero Permanent Charges under Relaxed Electroneutrality Boundary Conditions

We focus on a quasi-one-dimensional Poisson-Nernst-Planck model with small permanent charges for ionic flows of two oppositely charged ion species through an ion channel. Of particular interest is to examine the dynamics of ionic flows in terms of I-V (current-voltage) relations with boundary layers due to the relaxation of neutral conditions on boundary concentrations. This is achieved by employing the regular perturbation analysis on the solutions established through geometric singular perturbation analysis. Rich dynamics are observed, particularly, the nonlinear interplays among different physical parameters are characterized. Critical potentials are identified, which play critical roles in the study of ionic flows and can be estimated experimentally. Numerical simulations are performed to further illustrate and provide more intuitive understandings of our analytical results.

Competition between Cations via Classical Poisson-Nernst-Planck Models with Nonzero but Small Permanent Charges

We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.