Test of Poisson-Nernst-Planck Theory in Ion Channels

The recent Perspectives on Ion Permeation have brought the debate about the applicability of the Poisson-Nernst-Planck (PNP) theory in ion channels to a sharp focus. Despite the differences in opinion, all sides of the debate agree that the mean field approximation in PNP theory needs to be checked by comparison with a more accurate theory; e.g., Brownian dynamics (BD). Clearly, for such a test to be meaningful it has to be carried out in a three-dimensional (3-D) channel. We have been performing 3-D BD simulations in ion channels for the last few years (Li et al. and have been aware of the differences between the two theories. Therefore, we would like to contribute to the debate by providing a simple test of the PNP theory in a cylindrical channel, which appears to be the most common geometry used in applications of PNP. Due to space limitations, we will not elaborate on either theory, but refer to Chung et al. (1998, 1999) for details of 3-D BD simulations, and Kurnikova et al. (1999) for 3-D–PNP calculations. Reviews of the 1-D BD and PNP can be found, respectively, in Cooper et al. (1985) and Eisenberg (1996). We have written a code similar to the one in Kurnikova et al. (1999) for solving the PNP equations in 3-D. As a control study, the PNP and BD calculations are compared in bulk conditions, and are found to yield the same results for flux and concentration within the computational errors. A cross section of the channel shape used is shown in Fig. 1 (top). The rounding of corners is required due to the difficulty of solving Poisson's equation with sharp corners. A reservoir with radius 30 Å and variable length is added on both ends of the channel. The length is adjusted so as to keep the concentration fixed at 300 mM when the channel radius is varied. In BD simulations, this concentration is represented by 12 Na ϩ and 12 Cl Ϫ ions in each reservoir. The reason for using a larger value than the physiological range (150 mM) is entirely statistical. Otherwise, almost identical results are obtained for conductance and concentration at 150 mM, once they are normalized to 300 mM. The applied potential in BD is represented with a uniform electric field of E ϭ 10 7 V/m. The potential difference between the top and bottom boundaries is determined from the potential …