Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T22:03:41.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum Dynamics in Continuum for Proton Transport I: Basic Formulation

Published online by Cambridge University Press:  03 June 2015

Duan Chen  and
Guo-Wei Wei
Duan Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Guo-Wei Wei*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA
*
*Corresponding author.Email:wei@math.msu.edu
Get access

Abstract

Proton transport is one of the most important and interesting phenomena in living cells. The present work proposes a multiscale/multiphysics model for the understanding of the molecular mechanism of proton transport in transmembrane proteins. We describe proton dynamics quantum mechanically via a density functional approach while implicitly model other solvent ions as a dielectric continuum to reduce the number of degrees of freedom. The densities of all other ions in the solvent are assumed to obey the Boltzmann distribution. The impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic level. We formulate a total free energy functional to put proton kinetic and potential energies as well as electrostatic energy of all ions on an equal footing. The variational principle is employed to derive nonlinear governing equations for the proton transport system. Generalized Poisson-Boltzmann equation and Kohn-Sham equation are obtained from the variational framework. Theoretical formulations for the proton density and proton conductance are constructed based on fundamental principles. The molecular surface of the channel protein is utilized to split the discrete protein domain and the continuum solvent domain, and facilitate the multiscale discrete/continuum/quantum descriptions. A number of mathematical algorithms, including the Dirichlet to Neumann mapping, matched interface and boundary method, Gummel iteration, and Krylov space techniques are utilized to implement the proposed model in a computationally efficient manner. The Gramicidin A (GA) channel is used to demonstrate the performance of the proposed proton transport model and validate the efficiency of proposed mathematical algorithms. The electrostatic characteristics of the GA channel is analyzed with a wide range of model parameters. The proton conductances are studied over a number of applied voltages and reference concentrations. A comparison with experimental data verifies the present model predictions and validates the proposed model.

Keywords

87.10.Ed87.15.hj87.16.VyProton transportquantum dynamics in continuummultiscale modelPoisson-Boltzmann equationgeneralized Kohn-Sham equationvariational principle
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 1 , January 2013 , pp. 285 - 324
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akeson, M. and Deamer, D.Proton conductance by the gramicidin water wire. Model for proton conductance in the F0F1ATPases? Biophys J., 60: 101–109, 1991.Google Scholar
[2]Bernal, J. and Fowler, R.A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J. Chem. Phys., 1: 515–548, 1933.Google Scholar
[3]Bothma, J., Gilmore, J., and McKenzie, R.The role of quantum effects in proton transfer reactions in enzymes: Quantum tunneling in a noisy environment? New J. Phys., 12(055002), 2010.Google Scholar
[4]Brandsburg-zabary, S., Fried, O., Marantz, Y., Nachliel, E., and Gutman, M.Biophysical aspects of intra-protein proton transfer. Biochim Biophys Acta, 1458: 120–134, 2000.Google Scholar
[5]Braun-Sand, S., Burykin, A., Chu, Z. T., and Warshel, A.Realistic simulations of proton transport along the gramicidin channel: Demonstrating the importance of solvation effects. J. Phys. Chem. B, 109: 583–592, 2005.Google Scholar
[6]Breed, J., Sankararamakrishnan, R., ID, K., and Sansom, M.Molecular dynamics simulations of water within models of ion channels. Biophys. J., 70: 1643–1661, 1996.Google Scholar
[7]Chen, D., Chen, Z., Chen, C. J., Geng, W. H., and Wei, G. W.MIBPB: A software package for electrostatic analysis. J. Comput. Chem., 32: 657–670, 2011.CrossRefGoogle ScholarPubMed
[8]Chen, D., Chen, Z., and Wei, G. W.Quantum dynamics in continuum for proton transport II: Variational solvent-solute intersurface. Int. J. Numer. Methods Biomed. Engr., 28: 25–51, 2012.Google Scholar
[9]Chen, D., Lear, J., and Eisenberg, B.Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel. Biophys. J., 72(1): 97–116, 1997.Google Scholar
[10]Chen, D. and Wei, G. W.Modeling and simulation of electronic structure, material interface and random doping in nano-electronic devices. J. Comput. Phys., 229: 4431–4460, 2010.Google Scholar
[11]Chen, Z., Baker, N. A., and Wei, G. W.Differential geometry based solvation models I: Eulerian formulation. J. Comput. Phys., 229: 8231–8258, 2010.CrossRefGoogle ScholarPubMed
[12]Chen, Z., Baker, N. A., and Wei, G. W.Differential geometry based solvation models II: Lagrangian formulation. J. Math. Biol., 63: 1139–1200, 2011.Google Scholar
[13]Chernyshev, A. and Cukierman, S.Proton transfer in gramicidin water wires in phosphlipid bilayers: Attenuation by phosphoethanolamine. Biophys. J., 91: 580–587, 1997.Google Scholar
[14]Chung, S.-H., Allen, T., and Kuyucak, S.Conducting-state properties of the KcsA potassium channel from molecular and Brownian dynamics simulations. Biophys. J., 82: 628–645, 2002.Google Scholar
[15]Chung, S.-H. and Kuyucak, S.Recent advances in ion channel research. Biochimica et Bio-physica Acta, 1565: 267–286, 2002.CrossRefGoogle ScholarPubMed
[16]Coalson, R. D. and Kurnikova, M. G.Poisson-Nernst-Planck theory approach to the calculation of current through biological ion channels. IEEE Trans Nanobioscience, 4(1): 81–93, 2005.Google Scholar
[17]Cukier, R.Theory and simulation of proton-coupled electron transfer, hydrogen-atom transfer, and proton translocation in proteins. Biochimica et Biophysical Acta-Bioenergetics, 1655: 37–44, 2004.Google Scholar
[18]Cukierman, S., Quigley, E. P., and Crumrine, D. S.Proton conduction in Gramicidin A and in its dioxolane-linked dimer in different lipid bilayers. Biophys. J., 73: 2489–2502, 1997.Google Scholar
[19]de Falco, C., Jerome, J. W., and Sacco, R.A self-consitent iterative scheme for the onedimensional steady-state transistor calculations. IEEE Trans. Ele. Dev., 11: 455–465, 1964.Google Scholar
[20]Decoursey, T.Voltage-gated proton channels and other proton transfer pathways. Physiol. Rev., 83: 475–579, 2003.Google Scholar
[21]Dolinsky, T. J., Nielsen, J. E., McCammon, J. A., and Baker, N. A.PDB2PQR: An automated pipeline for the setup, execution, and analysis of Poisson-Boltzmann electrostatics calculations. Nucleic Acids Research, 32: W665–W667, 2004.Google Scholar
[22]Dunker, A. and Marvin, D.A model for membrane transport through a-helical protein pores. J. Theor. Biol., 72: 9–16, 1978.Google Scholar
[23]Deamer, D. W.Proton permeability in biological and model membranes. In: Intracellular pH: Its Measurement, Regulation, and Utilization in Cellular Functions. New York: Liss, 1982.Google Scholar
[24]Edwards, S., Corry, B., Kuyucak, S., and Chung, S.-H.Continuum electrostatics fails to describe ion permeation in the gramicidin channel. Biophys. J., 83: 1348–1360, September 2002.Google Scholar
[25]Eisenman, G., Enos, B., Hagglund, J., and Sandbloom, J.Gramicidin as an example of a singlefiling ion channel. Ann. N.Y. Acad. Sci., 339: 8–20, 1980.Google Scholar
[26]Geng, W. H., Yu, S. N., and Wei, G. W.Treatment of charge singularities in implicit solvent models. J. Chem. Phys., 127: 114106, 2007.Google Scholar
[27]Gilson, M. K., Davis, M. E., Luty, B. A., and McCammon, J. A.Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation. J. Phys. Chem., 97(14): 3591–3600, 1993.Google Scholar
[28]Krishtalik, L.The mechanism of the proton transfer: An outline. Biochimica et Biophysical Acta-Bioenergetics, 1458: 6–27, 2000.Google Scholar
[29]Kuyucak, S., Andersen, O. S., and Chung, S.-H.Models of permeation in ion channels. Rep. Prog. Phys., 64: 1427–1472, 2001.CrossRefGoogle Scholar
[30]MacKerell, J., Bashford, A. D. D., Bellot, M., Dunbrack, J., Evanseck, R. L. J. D., Field, M. J., Fischer, S., Gao, J., Guo, H., Ha, S., Joseph-McCarthy, D., Kuchnir, L., Kuczera, K., Lau, F. T. K., Mattos, C., Michnick, S., Ngo, T., Nguyen, D. T., Prodhom, B., Reiher, I., Roux, W. E. B., Schlenkrich, M., Smith, J. C., Stote, R., Straub, J., Watanabe, M., Wiorkiewicz-Kuczera, J., Yin, D., and Karplus, M.All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B, 102(18): 3586–3616, 1998.Google Scholar
[31]Mamonov, A. B., Coalson, R. D., Nitzan, A., and Kurnikova, M. G.The role of the dielectric barrier in narrow biological channels: A novel composite approach to modeling singlechannel currents. Biophys. J., 84: 3646–3661, June 2003.Google Scholar
[32]Mitchell, P.Vectorial chemistry and the molecular mechanics of chemiosmotic coupling: power transmission by proticity. Biochem. Soc. Trans., 4: 399–430, 1976.Google Scholar
[33]Nagle, J. and Morowitz, H.Molecular mechanisms for proton transport in membranes. Proc. Natl. Acad. Sci. U.S.A, 1458(72): 298–302, 1978.Google Scholar
[34]Perlman, D., Case, D., Caldwell, J., Ross, W., Cheatham, T., Debolt, S., Ferguson, D., Seibel, G., and Kollman, P.Amber, a package of computer programs for applying molecular mechanics, normal mode analysis, molecular dynamics and free energy calculations to simulate the structural and energetic properties of molecules. Comp. Phys. Commun., 91: 1–41, 1995.Google Scholar
[35]Pomes, R. and Roux, B.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, 2002.Google Scholar
[36]Roux, B.Influence of the membrane potential on the free energy of an intrinsic protein. Biophys. J., 73: 2980–2989, December 1997.Google Scholar
[37]Sanner, M. F., Olson, A. J., and Spehner, J. C.Reduced surface: An efficient way to compute molecular surfaces. Biopolymers, 38: 305–320, 1996.3.0.CO;2-Y>CrossRefGoogle ScholarPubMed
[38]Sansom, M., Kerr, I., Breed, J., and Sankararamakrishnan, R.Water in channel-like cavities: Structure and dynamics. Biophys. J., 70: 693–702, 1996.Google Scholar
[39]Schnell, J. R. and Chou, J. J.Structure and mechanism of the M2 proton channel of influenza A virus. Nature, 451: 591–596, January 2008.Google Scholar
[40]Schumaker, M. F., Pomes, R., and Roux, B.A combined molecular dynamics and diffusion model of single proton conduction through gramicidin. Biophys. J., 79: 2840–2857, December 2000.Google Scholar
[41]Sharp, K. A. and Honig, B.Electrostatic interactions in macromolecules – Theory and applications. Annual Review of Biophysics and Biophysical Chemistry, 19: 301–332, 1990.Google Scholar
[42]Till, M. S., Essigke, T., Becker, T., and Ullmann, G. M.Simulating the proton transfer in Gramicidin A by a sequential dynamical Monte Carlo method. J. Phys. Chem., 112: 13401–13410, 2008.CrossRefGoogle ScholarPubMed
[43]Wei, G. W.Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72: 1562–1622, 2010.CrossRefGoogle ScholarPubMed
[44]Yu, S. N. and Wei, G. W.Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities. J. Comput. Phys., 227: 602–632, 2007.Google Scholar
[45]Yu, S. N., Zhou, Y. C., and Wei, G. W.Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys., 224(2): 729–756, 2007.Google Scholar
[46]Zheng, Q., Chen, D., and Wei, G. W.Second-order Poisson-Nernst-Planck solver for ion transport. J. Comput. Phys., 230: 5239 – 5262, 2011.Google Scholar
[47]Zheng, Q. and Wei, G. W.Poisson-Boltzmann-Nernst-Planck model. J. Chem. Phys., 134: 194101, 2011.Google Scholar
[48]Zhou, Y. C., Feig, M., and Wei, G. W.Highly accurate biomolecular electrostatics in continuum dielectric environments. J. Comput. Chem., 29: 87–97, 2008.CrossRefGoogle ScholarPubMed
[49]Zhou, Y. C. and Wei, G. W.On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J. Comput. Phys., 219(1): 228–246, 2006.Google Scholar
[50]Zhou, Y. C., Zhao, S., Feig, M., and Wei, G. W.High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys., 213(1): 1–30, 2006.CrossRefGoogle Scholar