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. 1999 Feb;76(2):642–656. doi: 10.1016/S0006-3495(99)77232-2

A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel.

M G Kurnikova 1, R D Coalson 1, P Graf 1, A Nitzan 1
PMCID: PMC1300070  PMID: 9929470

Abstract

A lattice relaxation algorithm is developed to solve the Poisson-Nernst-Planck (PNP) equations for ion transport through arbitrary three-dimensional volumes. Calculations of systems characterized by simple parallel plate and cylindrical pore geometries are presented in order to calibrate the accuracy of the method. A study of ion transport through gramicidin A dimer is carried out within this PNP framework. Good agreement with experimental measurements is obtained. Strengths and weaknesses of the PNP approach are discussed.

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Selected References

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  1. 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]
  2. Andersen O. S., Koeppe R. E., 2nd Molecular determinants of channel function. Physiol Rev. 1992 Oct;72(4 Suppl):S89–158. doi: 10.1152/physrev.1992.72.suppl_4.S89. [DOI] [PubMed] [Google Scholar]
  3. Bernstein F. C., Koetzle T. F., Williams G. J., Meyer E. F., Jr, Brice M. D., Rodgers J. R., Kennard O., Shimanouchi T., Tasumi M. The Protein Data Bank: a computer-based archival file for macromolecular structures. J Mol Biol. 1977 May 25;112(3):535–542. doi: 10.1016/s0022-2836(77)80200-3. [DOI] [PubMed] [Google Scholar]
  4. Chen D. P., Barcilon V., Eisenberg R. S. Constant fields and constant gradients in open ionic channels. Biophys J. 1992 May;61(5):1372–1393. doi: 10.1016/S0006-3495(92)81944-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Chen D. P., Eisenberg R. S. Flux, coupling, and selectivity in ionic channels of one conformation. Biophys J. 1993 Aug;65(2):727–746. doi: 10.1016/S0006-3495(93)81099-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Chen D., Eisenberg R. Charges, currents, and potentials in ionic channels of one conformation. Biophys J. 1993 May;64(5):1405–1421. doi: 10.1016/S0006-3495(93)81507-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. 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]
  8. Chen D., Xu L., Tripathy A., Meissner G., Eisenberg B. Permeation through the calcium release channel of cardiac muscle. Biophys J. 1997 Sep;73(3):1337–1354. doi: 10.1016/S0006-3495(97)78167-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Chernyak Y. B. A universal steady state I-V relationship for membrane current. IEEE Trans Biomed Eng. 1995 Dec;42(12):1145–1157. doi: 10.1109/10.476121. [DOI] [PubMed] [Google Scholar]
  10. Connolly M. L. Solvent-accessible surfaces of proteins and nucleic acids. Science. 1983 Aug 19;221(4612):709–713. doi: 10.1126/science.6879170. [DOI] [PubMed] [Google Scholar]
  11. Cooper K., Jakobsson E., Wolynes P. The theory of ion transport through membrane channels. Prog Biophys Mol Biol. 1985;46(1):51–96. doi: 10.1016/0079-6107(85)90012-4. [DOI] [PubMed] [Google Scholar]
  12. 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]
  13. 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]
  14. Eisenberg R. S. Computing the field in proteins and channels. J Membr Biol. 1996 Mar;150(1):1–25. doi: 10.1007/s002329900026. [DOI] [PubMed] [Google Scholar]
  15. Elber R., Chen D. P., Rojewska D., Eisenberg R. Sodium in gramicidin: an example of a permion. Biophys J. 1995 Mar;68(3):906–924. doi: 10.1016/S0006-3495(95)80267-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Fishman H. M. Relaxations, fluctuations and ion transfer across membranes. Prog Biophys Mol Biol. 1985;46(2):127–162. doi: 10.1016/0079-6107(85)90007-0. [DOI] [PubMed] [Google Scholar]
  17. 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]
  18. Kreusch A., Schulz G. E. Refined structure of the porin from Rhodopseudomonas blastica. Comparison with the porin from Rhodobacter capsulatus. J Mol Biol. 1994 Nov 11;243(5):891–905. doi: 10.1006/jmbi.1994.1690. [DOI] [PubMed] [Google Scholar]
  19. Levitt D. G. General continuum theory for multiion channel. I. Theory. Biophys J. 1991 Feb;59(2):271–277. doi: 10.1016/S0006-3495(91)82220-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  20. Levitt D. G. General continuum theory for multiion channel. II. Application to acetylcholine channel. Biophys J. 1991 Feb;59(2):278–288. doi: 10.1016/S0006-3495(91)82221-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  21. Mazet J. L., Andersen O. S., Koeppe R. E., 2nd Single-channel studies on linear gramicidins with altered amino acid sequences. A comparison of phenylalanine, tryptophane, and tyrosine substitutions at positions 1 and 11. Biophys J. 1984 Jan;45(1):263–276. doi: 10.1016/S0006-3495(84)84153-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
  22. Novak J. P. Calcium ion current from an extracellular electrolyte toward a channel opening in an insulating membrane: quantitative model with rotational symmetry. IEEE Trans Biomed Eng. 1997 Oct;44(10):940–947. doi: 10.1109/10.634646. [DOI] [PubMed] [Google Scholar]
  23. Oiki S., Koeppe R. E., 2nd, Andersen O. S. Asymmetric gramicidin channels: heterodimeric channels with a single F6Val1 residue. Biophys J. 1994 Jun;66(6):1823–1832. doi: 10.1016/S0006-3495(94)80976-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Ramanan S. V., Mesimeris V., Brink P. R. Ion flow in the bath and flux interactions between channels. Biophys J. 1994 Apr;66(4):989–995. doi: 10.1016/S0006-3495(94)80880-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
  25. Riveros O. J., Croxton T. L., Armstrong W. M. Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations. J Theor Biol. 1989 Sep 22;140(2):221–230. doi: 10.1016/s0022-5193(89)80130-4. [DOI] [PubMed] [Google Scholar]
  26. Separovic F., Gehrmann J., Milne T., Cornell B. A., Lin S. Y., Smith R. Sodium ion binding in the gramicidin A channel. Solid-state NMR studies of the tryptophan residues. Biophys J. 1994 Oct;67(4):1495–1500. doi: 10.1016/S0006-3495(94)80623-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
  27. Sharp K. A., Honig B. Electrostatic interactions in macromolecules: theory and applications. Annu Rev Biophys Biophys Chem. 1990;19:301–332. doi: 10.1146/annurev.bb.19.060190.001505. [DOI] [PubMed] [Google Scholar]
  28. Song L., Hobaugh M. R., Shustak C., Cheley S., Bayley H., Gouaux J. E. Structure of staphylococcal alpha-hemolysin, a heptameric transmembrane pore. Science. 1996 Dec 13;274(5294):1859–1866. doi: 10.1126/science.274.5294.1859. [DOI] [PubMed] [Google Scholar]
  29. Urry D. W., Prasad K. U., Trapane T. L. Location of monovalent cation binding sites in the gramicidin channel. Proc Natl Acad Sci U S A. 1982 Jan;79(2):390–394. doi: 10.1073/pnas.79.2.390. [DOI] [PMC free article] [PubMed] [Google Scholar]
  30. Urry D. W., Walker J. T., Trapane T. L. Ion interactions in (1-13C)D-Val8 and D-Leu14 analogs of gramicidin A, the helix sense of the channel and location of ion binding sites. J Membr Biol. 1982;69(3):225–231. doi: 10.1007/BF01870401. [DOI] [PubMed] [Google Scholar]
  31. Wallace B. A. Gramicidin channels and pores. Annu Rev Biophys Biophys Chem. 1990;19:127–157. doi: 10.1146/annurev.bb.19.060190.001015. [DOI] [PubMed] [Google Scholar]
  32. Warshel A., Russell S. T. Calculations of electrostatic interactions in biological systems and in solutions. Q Rev Biophys. 1984 Aug;17(3):283–422. doi: 10.1017/s0033583500005333. [DOI] [PubMed] [Google Scholar]
  33. Weiss M. S., Schulz G. E. Structure of porin refined at 1.8 A resolution. J Mol Biol. 1992 Sep 20;227(2):493–509. doi: 10.1016/0022-2836(92)90903-w. [DOI] [PubMed] [Google Scholar]
  34. 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]
  35. Woolley G. A., Biggin P. C., Schultz A., Lien L., Jaikaran D. C., Breed J., Crowhurst K., Sansom M. S. Intrinsic rectification of ion flux in alamethicin channels: studies with an alamethicin dimer. Biophys J. 1997 Aug;73(2):770–778. doi: 10.1016/S0006-3495(97)78109-8. [DOI] [PMC free article] [PubMed] [Google Scholar]

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