Abstract
A rigorous statistical mechanical formulation of the equilibrium properties of selective ion channels is developed, incorporating the influence of the membrane potential, multiple occupancy, and saturation effects. The theory provides a framework for discussing familiar quantities and concepts in the context of detailed microscopic models. Statistical mechanical expressions for the free energy profile along the channel axis, the cross-sectional area of the pore, and probability of occupancy are given and discussed. In particular, the influence of the membrane voltage, the significance of the electric distance, and traditional assumptions concerning the linearity of the membrane electric field along the channel axis are examined. Important findings are: 1) the equilibrium probabilities of occupancy of multiply occupied channels have the familiar algebraic form of saturation properties which is obtained from kinetic models with discrete states of denumerable ion occupancy (although this does not prove the existence of specific binding sites; 2) the total free energy profile of an ion along the channel axis can be separated into an intrinsic ion-pore free energy potential of mean force, independent of the transmembrane potential, and other contributions that arise from the interfacial polarization; 3) the transmembrane potential calculated numerically for a detailed atomic configuration of the gramicidin A channel embedded in a bilayer membrane with explicit lipid molecules is shown to be closely linear over a distance of 25 A along the channel axis. Therefore, the present analysis provides some support for the constant membrane potential field approximation, a concept that has played a central role in the interpretation of flux data based on traditional models of ion permeation. It is hoped that this formulation will provide a sound physical basis for developing nonequilibrium theories of ion transport in selective biological channels.
Full Text
The Full Text of this article is available as a PDF (1.0 MB).
Selected References
These references are in PubMed. This may not be the complete list of references from this article.
- Becker M. D., Koeppe R. E., 2nd, Andersen O. S. Amino acid substitutions and ion channel function. Model-dependent conclusions. Biophys J. 1992 Apr;62(1):25–27. doi: 10.1016/S0006-3495(92)81767-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Ben-Tal N., Ben-Shaul A., Nicholls A., Honig B. Free-energy determinants of alpha-helix insertion into lipid bilayers. Biophys J. 1996 Apr;70(4):1803–1812. doi: 10.1016/S0006-3495(96)79744-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Breed J., Sankararamakrishnan R., Kerr I. D., Sansom M. S. Molecular dynamics simulations of water within models of ion channels. Biophys J. 1996 Apr;70(4):1643–1661. doi: 10.1016/S0006-3495(96)79727-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Busath D., Szabo G. Permeation characteristics of gramicidin conformers. Biophys J. 1988 May;53(5):697–707. doi: 10.1016/S0006-3495(88)83151-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Chen D., Lear J., Eisenberg B. Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel. Biophys J. 1997 Jan;72(1):97–116. doi: 10.1016/S0006-3495(97)78650-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Chiu S. W., Jakobsson E., Subramaniam S., McCammon J. A. Time-correlation analysis of simulated water motion in flexible and rigid gramicidin channels. Biophys J. 1991 Jul;60(1):273–285. doi: 10.1016/S0006-3495(91)82049-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Cowan S. W., Schirmer T., Rummel G., Steiert M., Ghosh R., Pauptit R. A., Jansonius J. N., Rosenbusch J. P. Crystal structures explain functional properties of two E. coli porins. Nature. 1992 Aug 27;358(6389):727–733. doi: 10.1038/358727a0. [DOI] [PubMed] [Google Scholar]
- Doyle D. A., Morais Cabral J., Pfuetzner R. A., Kuo A., Gulbis J. M., Cohen S. L., Chait B. T., MacKinnon R. The structure of the potassium channel: molecular basis of K+ conduction and selectivity. Science. 1998 Apr 3;280(5360):69–77. doi: 10.1126/science.280.5360.69. [DOI] [PubMed] [Google Scholar]
- Gilson M. K., Given J. A., Bush B. L., McCammon J. A. The statistical-thermodynamic basis for computation of binding affinities: a critical review. Biophys J. 1997 Mar;72(3):1047–1069. doi: 10.1016/S0006-3495(97)78756-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Hinton J. F., Fernandez J. Q., Shungu D. C., Whaley W. L., Koeppe R. E., 2nd, Millett F. S. TI-205 nuclear magnetic resonance determination of the thermodynamic parameters for the binding of monovalent cations to gramicidins A and C. Biophys J. 1988 Sep;54(3):527–533. doi: 10.1016/S0006-3495(88)82985-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Honig B., Nicholls A. Classical electrostatics in biology and chemistry. Science. 1995 May 26;268(5214):1144–1149. doi: 10.1126/science.7761829. [DOI] [PubMed] [Google Scholar]
- Ichiye T., Karplus M. Anisotropy and anharmonicity of atomic fluctuations in proteins: analysis of a molecular dynamics simulation. Proteins. 1987;2(3):236–259. doi: 10.1002/prot.340020308. [DOI] [PubMed] [Google Scholar]
- Jing N., Prasad K. U., Urry D. W. The determination of binding constants of micellar-packaged gramicidin A by 13C-and 23Na-NMR. Biochim Biophys Acta. 1995 Aug 23;1238(1):1–11. doi: 10.1016/0005-2736(95)00095-k. [DOI] [PubMed] [Google Scholar]
- Jordan P. C., Bacquet R. J., McCammon J. A., Tran P. How electrolyte shielding influences the electrical potential in transmembrane ion channels. Biophys J. 1989 Jun;55(6):1041–1052. doi: 10.1016/S0006-3495(89)82903-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Karplus M., Petsko G. A. Molecular dynamics simulations in biology. Nature. 1990 Oct 18;347(6294):631–639. doi: 10.1038/347631a0. [DOI] [PubMed] [Google Scholar]
- Ketchem R., Roux B., Cross T. High-resolution polypeptide structure in a lamellar phase lipid environment from solid state NMR derived orientational constraints. Structure. 1997 Dec 15;5(12):1655–1669. doi: 10.1016/s0969-2126(97)00312-2. [DOI] [PubMed] [Google Scholar]
- Klapper I., Hagstrom R., Fine R., Sharp K., Honig B. Focusing of electric fields in the active site of Cu-Zn superoxide dismutase: effects of ionic strength and amino-acid modification. Proteins. 1986 Sep;1(1):47–59. doi: 10.1002/prot.340010109. [DOI] [PubMed] [Google Scholar]
- Kurnikova M. G., Coalson R. D., Graf P., Nitzan A. A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel. Biophys J. 1999 Feb;76(2):642–656. doi: 10.1016/S0006-3495(99)77232-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Levitt D. G. Interpretation of biological ion channel flux data--reaction-rate versus continuum theory. Annu Rev Biophys Biophys Chem. 1986;15:29–57. doi: 10.1146/annurev.bb.15.060186.000333. [DOI] [PubMed] [Google Scholar]
- Läuger P. Ion transport through pores: a rate-theory analysis. Biochim Biophys Acta. 1973 Jul 6;311(3):423–441. doi: 10.1016/0005-2736(73)90323-4. [DOI] [PubMed] [Google Scholar]
- Mackay D. H., Berens P. H., Wilson K. R., Hagler A. T. Structure and dynamics of ion transport through gramicidin A. Biophys J. 1984 Aug;46(2):229–248. doi: 10.1016/S0006-3495(84)84016-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
- McGill P., Schumaker M. F. Boundary conditions for- single-ion diffusion. Biophys J. 1996 Oct;71(4):1723–1742. doi: 10.1016/S0006-3495(96)79374-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Neumcke B., Läuger P. Nonlinear electrical effects in lipid bilayer membranes. II. Integration of the generalized Nernst-Planck equations. Biophys J. 1969 Sep;9(9):1160–1170. doi: 10.1016/S0006-3495(69)86443-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Olah G. A., Huang H. W., Liu W. H., Wu Y. L. Location of ion-binding sites in the gramicidin channel by X-ray diffraction. J Mol Biol. 1991 Apr 20;218(4):847–858. doi: 10.1016/0022-2836(91)90272-8. [DOI] [PubMed] [Google Scholar]
- Pomès R., Roux B. Free energy profiles for H+ conduction along hydrogen-bonded chains of water molecules. Biophys J. 1998 Jul;75(1):33–40. doi: 10.1016/S0006-3495(98)77492-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Roux B. Influence of the membrane potential on the free energy of an intrinsic protein. Biophys J. 1997 Dec;73(6):2980–2989. doi: 10.1016/S0006-3495(97)78327-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Roux B., Karplus M. Ion transport in a model gramicidin channel. Structure and thermodynamics. Biophys J. 1991 May;59(5):961–981. doi: 10.1016/S0006-3495(91)82311-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Roux B., Prod'hom B., Karplus M. Ion transport in the gramicidin channel: molecular dynamics study of single and double occupancy. Biophys J. 1995 Mar;68(3):876–892. doi: 10.1016/S0006-3495(95)80264-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Roux B. Valence selectivity of the gramicidin channel: a molecular dynamics free energy perturbation study. Biophys J. 1996 Dec;71(6):3177–3185. doi: 10.1016/S0006-3495(96)79511-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Sigworth F. J. Voltage gating of ion channels. Q Rev Biophys. 1994 Feb;27(1):1–40. doi: 10.1017/s0033583500002894. [DOI] [PubMed] [Google Scholar]
- Smart O. S., Goodfellow J. M., Wallace B. A. The pore dimensions of gramicidin A. Biophys J. 1993 Dec;65(6):2455–2460. doi: 10.1016/S0006-3495(93)81293-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Smith R., Thomas D. E., Atkins A. R., Separovic F., Cornell B. A. Solid-state 13C-NMR studies of the effects of sodium ions on the gramicidin A ion channel. Biochim Biophys Acta. 1990 Jul 24;1026(2):161–166. doi: 10.1016/0005-2736(90)90059-w. [DOI] [PubMed] [Google Scholar]
- Tian F., Lee K. C., Hu W., Cross T. A. Monovalent cation transport: lack of structural deformation upon cation binding. Biochemistry. 1996 Sep 17;35(37):11959–11966. doi: 10.1021/bi961170k. [DOI] [PubMed] [Google Scholar]
- Tieleman D. P., Berendsen H. J. A molecular dynamics study of the pores formed by Escherichia coli OmpF porin in a fully hydrated palmitoyloleoylphosphatidylcholine bilayer. Biophys J. 1998 Jun;74(6):2786–2801. doi: 10.1016/S0006-3495(98)77986-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Tieleman D. P., Berendsen H. J., Sansom M. S. An alamethicin channel in a lipid bilayer: molecular dynamics simulations. Biophys J. 1999 Apr;76(4):1757–1769. doi: 10.1016/s0006-3495(99)77337-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Warwicker J., Watson H. C. Calculation of the electric potential in the active site cleft due to alpha-helix dipoles. J Mol Biol. 1982 Jun 5;157(4):671–679. doi: 10.1016/0022-2836(82)90505-8. [DOI] [PubMed] [Google Scholar]
- Woolf T. B., Roux B. Molecular dynamics simulation of the gramicidin channel in a phospholipid bilayer. Proc Natl Acad Sci U S A. 1994 Nov 22;91(24):11631–11635. doi: 10.1073/pnas.91.24.11631. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Woolf T. B., Roux B. Structure, energetics, and dynamics of lipid-protein interactions: A molecular dynamics study of the gramicidin A channel in a DMPC bilayer. Proteins. 1996 Jan;24(1):92–114. doi: 10.1002/(SICI)1097-0134(199601)24:1<92::AID-PROT7>3.0.CO;2-Q. [DOI] [PubMed] [Google Scholar]
- Woolf T. B., Roux B. The binding site of sodium in the gramicidin A channel: comparison of molecular dynamics with solid-state NMR data. Biophys J. 1997 May;72(5):1930–1945. doi: 10.1016/S0006-3495(97)78839-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Zhong Q., Moore P. B., Newns D. M., Klein M. L. Molecular dynamics study of the LS3 voltage-gated ion channel. FEBS Lett. 1998 May 8;427(2):267–270. doi: 10.1016/s0014-5793(98)00304-4. [DOI] [PubMed] [Google Scholar]