Abstract
A modern approach to studying the detailed dynamics of biomolecules is to simulate them on computers. Framework models have been developed to incorporate information from these simulations in order to calculate properties of the biomolecules on much longer time scales than can be achieved by the simulations. They also provide a simple way to think about the simulated dynamics. This article develops a method for the solution of framework models, which generalizes the King–Altman method of enzyme kinetics. The generalized method is used to construct solutions of two framework models which have been introduced previously, the single-particle and Grotthuss (proton conduction) models. The solution of the Grotthuss model is greatly simplified in comparison with direct integration. In addition, a new framework model is introduced, generalizing the shaking stack model of ion conduction through the potassium channel.
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References
Agmon, N., 1995. The Grotthuss mechanism. Chem. Phys. Lett. 244, 456–462.
Agmon, N., Hopfield, J.J., 1983. Transient kinetics of chemical reactions with bounded diffusion perpendicular to the reaction coordinate: Intramolecular processes with slow conformational changes. J. Chem. Phys. 78(11), 6947–6959.
Allen, T.W., Chung, S.H., 2001. Brownian dynamics study of an open-state KcsA potassium channel. Biochim. Biophys. Acta 1515, 83–91.
Apaydin, M.S., Brutlag, D.L., Guestrin, C., Hsu, D., Latombe, J.-C., 2003. Stochastic roadmap simulation: An efficient representation and algorithm for analyzing molecular motion. J. Comput. Biol. 10, 257–281.
Arseniev, A.S., Barsukov, I.L., Bystrov, V.F., Lomize, A.L., Ovchinnikov, Y.A., 1985. 1H-NMR study of gramicidin A transmembrane ion channel. Head-to-head right-handed, single-stranded helices. FEBS Lett. 186, 168–174.
Berne, B.J., Pecora, R., 1976. Dynamic Light Scattering. Wiley, New York.
Bernèche, S., Roux, B., 2001. Energetics of ion conduction through the K+ channel. Nature 414, 73–76.
Bernèche, S., Roux, B., 2003. A microscopic view of conduction through the streptomyces lividans K+ channel. Proc. Natl. Acad. Sci. U.S.A. 100, 8644–8648.
Chen, D., Lear, J., Eisenberg, B., 1997. Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel. Biophys. J. 72, 97–116.
Chiu, S.W., Novotny, J.A., Jakobsson, E., 1993. The nature of ion and water barrier crossings in a simulated ion channel. Biophys. J. 64, 98–109.
Cohen, J., Schulten, K., 2004. Mechanism of anionic conduction across ClC. Biophys. J. 86, 836–845.
Crouzy, S., Woolf, T.B., Roux, B., 1994. A molecular dynamics study of gating in dioxolane-linked gramicidin a channels. Biophys. J. 67, 1370–1386.
Eisenman, G., Enos, B., Hagglund, J., Sandbloom, J., 1980. Gramicidin as an example of a single-filing ionic channel. Ann. N.Y. Acad. Sci. 339, 8–20.
Finkelstein, A., Andersen, O.S., 1981. The gramicidin A channel: A review of its permeability characteristics with special reference to the single-file aspect of transport. J. Membr. Biol. 59, 155–171.
Gowen, J.A., Markham, J.C., Morrison, S.E., Cross, T.A., Busath, D.D., Mapes, E.J., Schumaker, M.F., 2002. The role of Trp side chains in tuning single proton conduction through gramicidin channels. Biophys. J. 83, 880–898.
Grote, R.F., Hynes, J.T., 1980. The stable states picture of chemical reactions II. Rate constants for condensed and gas phase reaction models. J. Chem. Phys. 73, 2715–2732.
Heckmann, K., Vollmerhaus, W., 1970. Zur theorie der ``single-file'' diffusion. Z. Physik 71, 320–328.
Hill, T.L., 1977. Free Energy Transduction in Biology. Academic, New York.
Hille, B., 1992. Ionic Channels of Excitable Membranes. Sinauer, Sunderland, MA.
Ketcham, R.R., Roux, B.B., Cross, T.A., 1997. High-resolution polypeptide structure in a lamellar phase lipid environment from solid state NMR derived orientational constraints. Structure 5, 1655–1669.
King, E.L., Altman, C., 1956. A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J. Phys. Chem. 60, 1375–1378.
Kramers, H.A., 1940. Brownian motion in a field of force. Physica 7, 284–304.
Läuger, P., 1973. Ion transport through pores, a rate-theory analysis. Biochim. Biophys. Acta 311, 423–441.
Levitt, D.G., 1986. Interpretation of biological ion channel flux data: Reaction rate versus continuum theory. Ann. Rev. Biophys. Biophys. Chem. 15, 29–57.
Mapes, E., Schumaker, M.F., 2001. Mean first passage times across a potential barrier in the lumped state approximation. J. Chem. Phys. 114, 76–83.
Mashl, R.J., Tang, Y., Schnitzer, J., Jakobsson, E., 2001. Hierarchical approach to predicting permeation in ion channels. Biophys. J. 81, 2473–2483.
McGill, P., Schumaker, M.F., 1996. Boundary conditions for single-ion diffusion. Biophys. J. 71, 1723–1742.
Morais-Cabral, J.H., Zhou, Y., MacKinnon, R., 2001. Energetic optimization of ion conduction rate by the K+ selectivity filter. Nature 414, 37–42.
Mori, H., 1965. Transport, collective motion and Brownian motion. Prog. Theor. Phys. 33, 423–455.
Nadler, B., Naeh, T., Schuss, Z., 2001. The stationary arrival process of independent diffusers from a continuum to an absorbing boundary is poissonian. SIAM J. Appl. Math. 62(2), 433–447.
Nelson, P.H., 2002. A permeation theory for single-file ion channels: Corresponding occupancy states produce Michaelis– Menten behavior. J. Chem. Phys. 117(24), 11396–11403.
Nelson, P.H., 2003. Modeling the concentration-dependent permeation modes of the KcsA potassium ion channel. Phys. Rev. E 68(061908).
Pomès, R., Roux, B., 1996. Structure and dynamics of a proton wire: A theoretical study of H+ translocation along the single-file water chain in the gramicidin a channel. Biophys. J. 71, 19–39.
Pomès, R., Roux, B., 2002. Molecular mechanism of H+ conduction in the single-file water chain of the gramicidin channel. Biophys. J. 82, 2304–2316.
Roux, B., 2002. Theoretical and computational models of ion channels. Curr. Opin. Struct. Biol. 12, 182–189.
Roux, B., Allen, T., Bernèche, S., Im, W., 2004. Theoretical and computational models of biological ion channels. Q. Rev. Biophys. 37, 15–103.
Roux, B., Karplus, M., 1993. Ion transport in the gramicidin channel: Free energy of the solvated right-hand dimer in a model membrane. J. Am. Chem. Soc. 115, 3250–3262.
Schumaker, M.F., 1992. Shaking stack model of ion conduction through the Ca2-activated K+ channel. Biophys. J. 63, 1032–1044.
Schumaker, M.F., 2002. Boundary conditions and trajectories of diffusion processes. J. Chem. Phys. 116(6), 2469–2473.
Schumaker, M.F., 2003. Numerical framework models of single-proton conduction through gramicidin. Front. Biosci. 8, s982–s991.
Schumaker, M.F., MacKinnon, R., 1990. A simple model for multi-ion permeation. Biophys. J. 58, 975–984.
Schumaker, M.F., Pomès, R., Roux, B., 2000. A combined molecular dynamics and diffusion model of single proton conduction through gramicidin. Biophys. J. 79, 2840–2857.
Schumaker, M.F., Pomès, R., Roux, B., 2001. A framework model for single proton conductance through gramicidin. Biophys. J. 80, 12–30.
Schumaker, M.F., Watkins, D.S., 2004. A framework model based on the Smoluchowski equation in two reaction coordinates. J. Chem. Phys. 121, 6134–6144.
Tolokh, I.S., White, G.W.N., Goldman, S., Gray, C.G., 2002. Prediction of ion channel transport from Grote-Hynes and Kramers theories. Mol. Phys. 100, 2351–2359.
Tripathi, S., Hladky, S.B., 1998. Streaming potentials in gramicidin channels measured with ion-selective microelectrodes. Biophys. J. 74, 2912–2917.
Tuckerman, M.E., Berne, B.J., 1991. Stochastic molecular dynamics in systems with multiple timescales and memory friction. J. Chem. Phys. 95, 4389–4396.
Yin, H.-M., 2004. On a class of parabolic equations with nonlocal boundary conditions. J. Math. Anal. Appl. 294, 712–728.
Yu, C.-H., Cukierman, S., Pomès, R., 2003. Theoretical study of the structure and dynamic fluctuations of dioxolane-linked gramicidin channels. Biophys. J. 84, 816–831.
Zhou, M., MacKinnon, R., 2004. A mutant KcsA K+ channel with altered conduction properties and selectivity filter ion distribution. J. Mol. Biol. 338, 839–846.
Zwanzig, R., 2001. Nonequilibrium Statistical Mechanics. Oxford University Press, New York.
Zwanzig, R.W., 1961. Statistical mechanics of irreversibility. In: Brittin, W.E., Downs, B.W., Downs, J. (Eds.), Lectures in Theoretical Physics, vol. 3. Interscience, New York, pp. 106–141.
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Mapes, E.J., Schumaker, M.F. Framework Models of Ion Permeation Through Membrane Channels and the Generalized King–Altman Method. Bull. Math. Biol. 68, 1429–1460 (2006). https://doi.org/10.1007/s11538-005-9016-1
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DOI: https://doi.org/10.1007/s11538-005-9016-1