A Poisson-nernst-planck Model for Biological Ion Channels — an Asymptotic Analysis in a 3-d Narrow Funnel

We wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel, when some features of channel structure are known. We propose to apply singular perturbation analysis to the coupled Poisson-Nernst-Planck equations, which are the basic continuum model of ionic permeation and semiconductor physics. In semiconductor physics the problem is a singular perturbation, because the ratio of the Debye length to the width of the channel is a very small parameter that multiplies the Laplacian term in the Poisson equation. In contrast to semiconductors, the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter, which, surprisingly, renders the problem a singular perturbation in a different sense. We construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for Fick’s and Ohm’s laws. Our aim is to extend the asymptotic analysis to a class of nonlinear problems hitherto intractable. Analytical and numerical results for the mass flux and the electric current serve as a tool for molecular biophysicists and physiologists to understand, study and control protein channels, thereby aiding clinical and technological applications.