Abstract
We simulate sodium chloride currents through the gramicidin A channel using the spectral element method to solve the three-dimensional Poisson–Nernst–Planck (PNP) equations. Using the spectral element method, we are able to simulate the entire channel, plus large enough portions of the lipid bilayer and baths to ensure that all boundary conditions are realistic. In these simulations, we rely on the 3D charge distribution of the gramicidin molecule plus diffusion coefficients and dielectric coefficients. Our main results, which match the experimental data, are current-voltage (IV) curves for gramicidin at various concentrations of Na+Cl− in the surrounding baths. We give a detailed description of the numerical algorithms used to solve the PNP equations, and we present various sensitivity analyses which we have performed to determine which parameters of the model most affect the IV curves.
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REFERENCES
Abramowitz, M., and Stegun, I. A., (ed.) (1972). Handbook of Mathematical Functions, National Bureau of Standards.
Apostol, T. M., (1967). Calculus, 2nd ed., Wiley.
Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., and van der Vorst, H. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Press.
Bockris, J. O'M., and Reddy, A. K. N. (1998). Modern Electrochemistry, 2nd ed., Plenum.
Chen, D.-P., Lear, J., and Eisenberg, R. S. (1997). Permeation through an open channel. Poisson-Nernst-Planck theory of a synthetic ionic channel. Biophys.J. 72, 97.
Courant, R., and Hilbert, D. (1989). Methods of Mathematical Physics(translation of german original), Wiley.
Ghaddar, N. K., Karniadakis, G. E., and Patera, A. T. (1986). a conservative isoparametric spectral element method for forced convection; application to fully developed flow in periodic geometries. Num.Heat Transfer 9, 227.
Golub, G. H., and Van Loan, C. F. (1989). Matrix Computations, 2nd ed., Johns Hopkins.
Gummel, H. K. (1964). A self-consistent iterative scheme for one-dimensional steady-state transistor calculations. IEEE Trans.Electron Devices ED-11, 445.
Hestenes, M., and Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. J.Res.Nat.Bur.Stand. 49, 409.
Hollerbach, U., Chen, D.-P., Busath, D. D., Eisenberg, and R. S. (2000). Predicting Function from Structure Using the Poisson-Nernst-Planck Equations: Sodium Currents in the Gramicidin A Channels, Langmuir, Vol. 16, p. 5509.
Jackson, J. D. (1975). Classical Electrodynamics, 2nd ed., Wiley.
Jacoboni, C., and Lugli, P. (1989). The Monte Carlo Method for Semiconductor Device Simulation, Springer Verlag.
Jerome, J. W. (1996). Analysis of Charge Transport: A Mathematical Study of Semiconductors, Springer Verlag.
Kurnikova, M. G., Coalson, R. D., Graf, P., and Nitzan A. (1999). A lattice relaxation algorithm for 3D Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel. Biophys.J. 76, 642.
Nicholson, D. R. (1983). Introduction to Plasma Theory, Wiley.
Patera, A. T. (1984). A spectral element method for fluid dynamics: laminar flow in a channel expansion. J.Comput.Phys. 54, 468.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1986). Numerical Recipes, Cambridge University Press.
Robinson, R. A., and Stokes, R. H. (1959). Electrolyte Solutions, Butterworths Scientific Publications.
Roux, B. (1997). Influence of the membrane potential on the free energy of an intrinsic protein. Biophys.J. 73, 2980.
Roux, B. (1999). Statistical mechanical equilibrium theory of selective ion channels. Biophys.J. 77, 139.
Selberherr, S. (1984). Analysis and Simulation of Semiconductor Devices, Springer-Verlag.
Strang, G., and Fix, G. J. (1973). An Analysis of the Finite Element Method, Prentice-Hall.
Zienkiewicz, O. C. (1971). The Finite Element Method in Engineering Science, 2nd ed., McGraw-Hill.
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Hollerbach, U., Chen, DP. & Eisenberg, R.S. Two- and Three-Dimensional Poisson–Nernst–Planck Simulations of Current Flow Through Gramicidin A. Journal of Scientific Computing 16, 373–409 (2001). https://doi.org/10.1023/A:1013203223798
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DOI: https://doi.org/10.1023/A:1013203223798