Abstract
The movement of ions in the aqueous medium as they approach the mouth (radius a) of a conducting membrane channel is analyzed. Starting with the Nernst-Planck and Poisson equations, we derive a nonlinear integrodifferential equation for the electric potential, phi(r), a less than or equal to r less than infinity. The formulation allows deviations from charge neutrality and dependence of phi(r) on ion flux. A numerical solution is obtained by converting the equation to an integral equation that is solved by an iterative method for an assumed mouth potential, combined with a shooting method to adjust the mouth potential until the numerical solution agrees with an asymptotic expansion of the potential at r-a much greater than lambda (lambda = Debye length). Approximate analytic solutions are obtained by assuming charge neutrality (Läuger, 1976) and by linearizing. The linear approximation agrees with the exact solution under most physiological conditions, but the charge-neutrality solution is only valid for r much greater than lambda and thus cannot be used unless a much greater than lambda. Families of curves of ion flux vs. potential drop across the electrolyte, phi(infinity)-phi (a), and of permeant ion density at the channel mouth, n1(a), vs. flux are obtained for different values of a/lambda and S = a d phi/dr(a). If a much greater than lambda and S = O, the maximum flux (which is approached when n1(a)----0) is reduced by 50% compared to the value predicted by the charge-neutrality solution. Access resistance is shown to be a factor a/[2 (a + lambda)] times the published formula (Hille, 1968), which was derived without including deviations from charge neutrality and ion density gradients and hence does not apply when there is no counter-ion current. The results are applied to an idealized diffusion-limited channel with symmetric electrolytes. For S = O, the current/voltage curves saturate at a value dependent on a/lambda; for S greater than O, they increase linearly for large voltage.
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- Andersen O. S. Ion movement through gramicidin A channels. Interfacial polarization effects on single-channel current measurements. Biophys J. 1983 Feb;41(2):135–146. doi: 10.1016/S0006-3495(83)84415-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Andersen O. S. Ion movement through gramicidin A channels. Studies on the diffusion-controlled association step. Biophys J. 1983 Feb;41(2):147–165. doi: 10.1016/S0006-3495(83)84416-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Dani J. A. Ion-channel entrances influence permeation. Net charge, size, shape, and binding considerations. Biophys J. 1986 Mar;49(3):607–618. doi: 10.1016/S0006-3495(86)83688-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Hainsworth A. H., Hladky S. B. Effects of double-layer polarization on ion transport. Biophys J. 1987 Jan;51(1):27–36. doi: 10.1016/S0006-3495(87)83308-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Hall J. E. Access resistance of a small circular pore. J Gen Physiol. 1975 Oct;66(4):531–532. doi: 10.1085/jgp.66.4.531. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Hille B. Pharmacological modifications of the sodium channels of frog nerve. J Gen Physiol. 1968 Feb;51(2):199–219. doi: 10.1085/jgp.51.2.199. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Jordan P. C. How pore mouth charge distributions alter the permeability of transmembrane ionic channels. Biophys J. 1987 Feb;51(2):297–311. doi: 10.1016/S0006-3495(87)83336-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Levitt D. G., Decker E. R. Electrostatic radius of the gramicidin channel determined from voltage dependence of H+ ion conductance. Biophys J. 1988 Jan;53(1):33–38. doi: 10.1016/S0006-3495(88)83063-7. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Levitt D. G. Strong electrolyte continuum theory solution for equilibrium profiles, diffusion limitation, and conductance in charged ion channels. Biophys J. 1985 Jul;48(1):19–31. doi: 10.1016/S0006-3495(85)83757-7. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Läuger P. Diffusion-limited ion flow through pores. Biochim Biophys Acta. 1976 Dec 2;455(2):493–509. doi: 10.1016/0005-2736(76)90320-5. [DOI] [PubMed] [Google Scholar]