Computational Study on Hysteresis of Ion Channels: Multiple Solutions to Steady-State Poisson-Nernst-Planck Equations

摘要

The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effectivemodel for the description of ionic transport in ion channels. It is observedthat an ion channel exhibits voltage-dependent switching between open andclosed states. Different conductance states of a channel imply that the ssPNPequations probably have multiple solutions with different level of currents. Wepropose numerical approaches to study multiple solutions to the ssPNP equationswith multiple ionic species. To find complete current-voltage (I-V ) andcurrent-concentration (I-C) curves, we reformulate the ssPNP equations intofour different boundary value problems (BVPs). Numerical continuationapproaches are developed to provide good initial guesses for iterativelysolving algebraic equations resulting from discretization. Numericalcontinuations on V , I, and boundary concentrations result in S-shaped anddouble S-shaped (I-V and I-C) curves for the ssPNP equations with multiplespecies of ions. There are five solutions to the ssPNP equations with fiveionic species, when an applied voltage is given in certain intervals.Remarkably, the current through ion channels responds hysteretically to varyingapplied voltages and boundary concentrations, showing a memory effect. Inaddition, we propose a useful computational approach to locate turning pointsof an I-V curve. With obtained locations, we are able to determine criticalthreshold values for hysteresis to occur and the interval for V in which thessPNP equations have multiple solutions. Our numerical results indicate thatthe developed numerical approaches have a promising potential in studyinghysteretic conductance states of ion channels.