Analytical solution of the poisson-boltzmann equation for membrane potential

Abstract

Analytical solution of the one-dimensional Poisson-Boltzmann equation for membrane potential is obtained in an equilibrium state of the Nemst-Planck Poisson system. Approximations, e.g. constant field or Debye-Hückel approximation, need not be used. Two types of solution, arctangenthyperbolic and arccotangenthyperbolic, exist for every value of membrane potential. A new approximate solution is obtained where V and V m are electric potential inside the membrane and membrane potential respectively; R, T and F have their usual thermodynamic meanings, κ is Debye constant, τ the membrane thickness. This approximate solution fits the numerical solution by Runge-Kutta method (maximum error being less than 5 × 10 −4) around the potential being 50 mV. The fitness is considerably more accurate than that of Debye-Huckel approximation (error being c. 15 % around the potential 50 mV.).