Effects of Large Permanent Charges on Ionic Flows via Poisson--Nernst--Planck Models

Permanent charge is the major structural quantity of an ion channel. It defines the ion channel and its interaction with boundary conditions plays the predominate role for ionic flow properties or functions of the ion channel. In this work, we investigate effects of large magnitude permanent charges of a simple form on the ionic flow of a 1:1 solution (an ionic mixture with one positive charged ion species and one negatively charged ion species). The analysis is based on a quasi-one-dimensional classical steady-state Poisson--Nernst--Planck model. Our findings include (i) large permanent charges produce flux and current saturations at large transmembrane electric potentials; (ii) large permanent charges inhibit the flux of co-ions (ions with the same charge sign) but could either enhance or reduce the flux of counter-ions (ions with opposite charge signs), depending on boundary conditions and the channel geometry; (iii) the magnitude of the co-ion flux is decreasing with an increase in magnitude of the large permanent charge but the counter-ion flux could either decease or increase with an increase in magnitude of the large permanent charge, depending on boundary conditions and the channel geometry, and quite significantly; (iv) large permanent charges are responsible for the counterintuitive declining phenomenon---an increase in the electrochemical potential of counter-ion species in a particular manner leads to a reduction of the counter-ion flux. We would like to stress that the model in this paper includes only the ideal components of the electrochemical potentials so the model cannot be used to study some critical properties of ion channels such as selectivity and gating. Our work should be viewed as the first step of future analyses/numerics with more structural detail and more correlations between ions included. The basic findings in this work should provide guidance for further investigation.