Computational Study on Hysteresis of Ion Channels: Multiple Solutions to Steady-State Poisson-Nernst-Planck Equations (Communications in Computational Physics, 2018)
The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effectivemodel for the description of ionic transport in ion channels. It is observed that an ionchannel exhibits voltage-dependent switching between open and closed states. Differentconductance states of a channel imply that the ssPNP equations probably havemultiple solutions with different level of currents. We propose numerical approachesto study multiple solutions to the ssPNP equations with multiple ionic species. To findcomplete current-voltage (I-V) and current-concentration (I-C) curves, we reformulatethe ssPNP equations into four different boundary value problems (BVPs). Numericalcontinuation approaches are developed to provide good initial guesses for iterativelysolving algebraic equations resulting from discretization. Numerical continuations onV, I, and boundary concentrations result in S-shaped and double S-shaped (I-V andI-C) curves for the ssPNP equations with multiple species of ions. There are five solutionsto the ssPNP equations with five ionic species, when an applied voltage is givenin certain intervals. Remarkably, the current through ion channels responds hystereticallyto varying applied voltages and boundary concentrations, showing a memory effect.In addition, we propose a useful computational approach to locate turning pointsof an I-V curve. With obtained locations, we are able to determine critical thresholdvalues for hysteresis to occur and the interval for V in which the ssPNP equations havemultiple solutions. Our numerical results indicate that the developed numerical approacheshave a promising potential in studying hysteretic conductance states of ionchannels.