Tests of continuum theories as models of ion channels. I. Poisson-Boltzmann theory versus Brownian dynamics

Continuum theories of electrolytes are widely used to describe physical pro cesses in various biological systems. Although these are well-established t heories in macroscopic situations, it is not clear from the outset that the y should work in small systems whose dimensions are comparable to or smalle r than the Debye length. Here, we test the validity of the mean-field appro ximation in Poisson-Boltzmann theory by comparing its predictions with thos e of Brownian dynamics simulations. For this purpose we use spherical and c ylindrical boundaries and a catenary shape similar to that of the acetylcho line receptor channel. The interior region filled with electrolyte is assum ed to have a high dielectric constant, and the exterior region representing protein a low one. Comparisons of the force on a test ion obtained with th e two methods show that the shielding effect due to counterions is overesti mated in Poisson-Boltzmann theory when the ion is within a Debye length of the boundary. As the ion gets closer to the boundary, the discrepancy in fo rce grows rapidly. The implication for membrane channels, whose radii are t ypically smaller than the Debye length, is that Poisson-Boltzmann theory ca nnot be used to obtain reliable estimates of the electrostatic potential en ergy and force on an ion in the channel environment.