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Singular perturbation analysis of the steady-state Poisson–Nernst–Planck system: Applications to ion channels

Published online by Cambridge University Press:  01 October 2008

A. SINGER ,
D. GILLESPIE ,
J. NORBURY  and
R. S. EISENBERG
A. SINGER
Affiliation:
Program in Applied Mathematics, Department of Mathematics, Yale University, 10 Hillhouse Ave., PO Box 208283, New Haven, CT 06520-8283, USA
D. GILLESPIE
Affiliation:
Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 West Harrison Street, Chicago, IL 60612, USA
J. NORBURY
Affiliation:
OCIAM, Mathematical Institute, Oxford University, 27–29 St Giles', Oxford OX1 3LB, UK
R. S. EISENBERG
Affiliation:
Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 West Harrison Street, Chicago, IL 60612, USA
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Abstract

Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst–Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current–voltage (I–V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I–V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 5 , October 2008 , pp. 541 - 560
Copyright
Copyright © Cambridge University Press 2008

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