Multiple solutions of steady-state Poisson-Nernst-Planck equations with steric effects

TL;DR

This paper demonstrates the existence of multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric effects, suggesting they can model the two-level current behavior observed in ion channel gating.

Contribution

It proves the existence of multiple solutions to the PNP-steric equations under specific conditions, providing a potential model for spontaneous ion channel gating without additional modifications.

Findings

Two steady state solutions for three ion species with positive permanent charge.

Two steady state solutions for four ion species with no sign restriction.

Derived formulas for excess currents associated with these solutions.

Abstract

Experiments measuring currents through single protein channels show unstable currents. Channels switch between 'open' or 'closed' states in a spontaneous stochastic process called gating. Currents are either (nearly) zero or at a definite level, characteristic of each type of protein, independent of time, once the channel is open. The steady state Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) describe steady current through the open channel quite well, in a wide variety of conditions. Here we study the existence of multiple solutions of steady state PNP-steric equations to see if they themselves, without modification or augmentation, can describe two levels of current. We prove that there are two steady state solutions of PNP-steric equations for (a) three types of ion species (two types of cations and one type of anion) with a positive constant permanent charge, and (b) four types of ion species (two types of cations and their counter-ions) with a constant permanent charge but no sign condition. The excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that PNP-steric equations may become a useful model to study spontaneous gating of ion channels. Spontaneous gating is thought to involve small structural changes in the channel protein that perhaps produce large changes in the profiles of free energy that determine ion flow. Gating is known to be modulated by external structures. Both can be included in future extensions of our present analysis.