Applicability of Goldman's constant field assumption to biological systems

Abstract

The applicability of Goldman's constant field assumption is examined by constructing an asymptotic solution of the Nernst-Planck-Poisson equations, valid as a certain dimensionless parameter α −2 approaches zero. ( α −2 is proportional to the concentration of the fixed charge within the membrane, the square of the width of the membrane, and inversely proportional to the dielectric constant.) Goldman's solution is the zeroth-order term in this expansion. By examining the first-order terms, we are able to estimate the range of values for α −2 for which the constant field assumption is appropriate. For α −2 as large as two, we find that the first-order term is less than 20% of the zero-order term. For such values of α −2 , the magnitude of the membrane thickness, and/or the fixed charge concentration is so small that it is possible that agreement between the theory and experiment is fortuitous. On the other hand, the agreement may enable us to estimate the thickness of the diffusion barrier and the concentration of the fixed charge.