A Coupled 3-D PNP/ECP Model for Ion Transport in Biological Ion Channels

Abstract

Ion channels are proteins embedded in the membrane of all biological cells, folded in a manner that creates nanoscopic pores that control the flow of ions in and out of the cell. All ion channels carry a highly localized distribution of permanent charge and possess specific properties (e.g., selectivity and switching) that are interesting to the device engineering community. The Poisson-Nernst-Planck (PNP), or drift-diffusion theory, can be used to compute macroscopic current in ion channels quickly. However, PNP theory can be problematic when applied to regions of constricted volume, such as the interior of ion channels, because, firstly, PNP theory replaces discrete charges with a continuous distribution function, grossly underestimating the dielectric boundary force. Additionally, traditional PNP theory ignores the finite volume occupied by the ions and water molecules, as well as the nonsingular distribution of charge on the ion. As a result, Coulombic screening can be overestimated, particularly in highly charged regions, leading to unphysically high ion densities. PNP theory also fails to explain the specific selectivity observed in certain ion channels. The entropic effects of finite-sized ions and water molecules, and the nonsingular charge distribution on the ion can be introduced into the PNP formulism by including an additional component to the electrochemical potential. The so-called excess chemical potential (ECP) represents the difference between the electrochemical potential of a "real" ionic solution and that of an idealized solution. Our goal was to incorporate in a 3D PNP solver (realized with the computational platform PROPHET) an available model for ECP correction (Gillespie, Nonner and Eisenberg, 2002). The ECP has a hard-sphere component, arising from the fact that ions are not point particles so there is an upper limit to the ion density in any given confined volume, and an electrostatic component, due to the fact that ions are not point charges. Both components of the ECP are functions of local charge carrier and water density, and are obtained using density functional theory (DFT). The ECP terms are added to the electrostatic potential in the flux equations, yielding a modified set of PNP equations. To solve the 3D system of PNP/ECP equations self-consistently we use a decoupled feedback method similar to Gummel's iteration: the discretized PNP model is solved with Newton's method. The resulting charge carrier densities are used to update the ECP equations. The new values for the ECP are fed back into the PNP equations and the entire process is iterated until convergence.