Tests of Continuum Theories as Models of Ion Channels. I. Poisson−Boltzmann Theory versus Brownian Dynamics

Abstract

Continuum theories of electrolytes are widely used to describe physical processes in various biological systems. Although these are well-established theories in macroscopic situations, it is not clear from the outset that they should work in small systems whose dimensions are comparable to or smaller than the Debye length. Here, we test the validity of the mean-field approximation in Poisson-Boltzmann theory by comparing its predictions with those of Brownian dynamics simulations. For this purpose we use spherical and cylindrical boundaries and a catenary shape similar to that of the acetylcholine receptor channel. The interior region filled with electrolyte is assumed 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 the two methods show that the shielding effect due to counterions is overestimated 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 force grows rapidly. The implication for membrane channels, whose radii are typically smaller than the Debye length, is that Poisson-Boltzmann theory cannot be used to obtain reliable estimates of the electrostatic potential energy and force on an ion in the channel environment.

Numerical framework models of single proton conduction through Gramicidin

“…Moreover, it does take into account correlations between ions, which mean field theories of ion permeation do not (19,(24)(25).…”

Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

“…It is worth noting here that Moy et al 44 used again the linear approximation of the above Boltzmann distribution in Eq. ͑21͒ ͑this is the approximation used to get the LPBE from the NPBE͒ to derive an analytical solution, q͑r͒ = e ͫ 1 − 1 + r 1 + a exp͑− ͑r − a͒͒ ͬ .…”

Molecular dynamics simulations of membrane proteins

“…Since the structure of a potassium channel became available, there has been an intense effort to study the structure-function relations in this channel using a variety of approaches ranging from the particle-based molecular dynamics (MD) and Brownian dynamics (BD) simulations to the continuum Poisson-Nernst-Planck equations (for reviews, see Kuyucak et al 2001;Tieleman et al 2001;Roux 2005). Because the continuum theories were shown to be invalid in narrow pores (Moy et al 2000;Corry et al 2000), BD (Chung et al , 2002a and MD simulations (Allen et al , 2000Aqvist and Luzhkov 2000;Roux 2001, 2002;Shrivastava and Sansom 2000;Shrivastava et al 2002) have been the two main approaches used in studies of potassium channels.…”