title: Quantum Dynamics in Continuum for Proton Transport I: Basic Formulation.
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.
authors.affiliation: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
authors.forename: Duan
authors.initials: D
authors.lastname: Chen
authors.forename: Guo-Wei
authors.initials: GW
authors.lastname: Wei
keywords: Poisson-Boltzmann equation; Proton transport; generalized Kohn-Sham equation; multiscale model; quantum dynamics in continuum; variational principle
references.citation: Nature. 2008 Jan 31;451(7178):591-5
references.pmid: 18235503
references.citation: Biophys J. 1997 Jan;72(1):97-116
references.pmid: 8994596
references.citation: J Comput Phys. 2010 Jun 20;229(12):4431-4460
references.pmid: 20396650
references.citation: Biophys J. 1991 Jul;60(1):101-9
references.pmid: 1715764
references.citation: J Phys Chem B. 2005 Jan 13;109(1):583-92
references.pmid: 16851050
references.citation: Proc Natl Acad Sci U S A. 1978 Jan;75(1):298-302
references.pmid: 272644
references.citation: Biophys J. 2000 Dec;79(6):2840-57
references.pmid: 11106593
references.citation: Biophys J. 1996 Jul;71(1):19-39
references.pmid: 8804586
references.citation: Biophys J. 1996 Feb;70(2):693-702
references.pmid: 8789086
references.citation: Physiol Rev. 2003 Apr;83(2):475-579
references.pmid: 12663866
references.citation: Biophys J. 1997 Dec;73(6):2980-9
references.pmid: 9414213
references.citation: J Chem Phys. 2011 May 21;134(19):194101
references.pmid: 21599038
references.citation: Biophys J. 2002 Sep;83(3):1348-60
references.pmid: 12202360
references.citation: J Comput Chem. 2008 Jan 15;29(1):87-97
references.pmid: 17508411
references.citation: Biochim Biophys Acta. 2000 May 12;1458(1):120-34
references.pmid: 10812028
references.citation: IEEE Trans Nanobioscience. 2005 Mar;4(1):81-93
references.pmid: 15816174
references.citation: J Phys Chem B. 2008 Oct 23;112(42):13401-10
references.pmid: 18826179
references.citation: Bull Math Biol. 2010 Aug;72(6):1562-622
references.pmid: 20169418
references.citation: Biopolymers. 1996 Mar;38(3):305-20
references.pmid: 8906967
references.citation: Nucleic Acids Res. 2004 Jul 1;32(Web Server issue):W665-7
references.pmid: 15215472
references.citation: J Comput Phys. 2010 Nov 1;229(22):8231-8258
references.pmid: 20938489
references.citation: Biophys J. 2003 Jun;84(6):3646-61
references.pmid: 12770873
references.citation: J Chem Phys. 2007 Sep 21;127(11):114106
references.pmid: 17887827
references.citation: Biophys J. 2006 Jul 15;91(2):580-7
references.pmid: 16617081
references.citation: J Phys Chem B. 1998 Apr 30;102(18):3586-616
references.pmid: 24889800
references.citation: J Theor Biol. 1978 May 8;72(1):9-16
references.pmid: 661337
references.citation: J Comput Phys. 2011 Jun;230(13):5239-5262
references.pmid: 21552336
references.citation: Biochim Biophys Acta. 2002 Oct 11;1565(2):267-86
references.pmid: 12409200
references.citation: Biophys J. 1996 Apr;70(4):1643-61
references.pmid: 8785323
references.citation: J Math Biol. 2011 Dec;63(6):1139-200
references.pmid: 21279359
references.citation: Int J Numer Method Biomed Eng. 2012 Jan;28(1):25-51
references.pmid: 22328970
references.citation: Ann N Y Acad Sci. 1980;339:8-20
references.pmid: 6156618
references.citation: Biochim Biophys Acta. 2004 Apr 12;1655(1-3):37-44
references.pmid: 15100014
references.citation: Biophys J. 1997 Nov;73(5):2489-502
references.pmid: 9370442
references.citation: Biophys J. 2002 Feb;82(2):628-45
references.pmid: 11806907
references.citation: Annu Rev Biophys Biophys Chem. 1990;19:301-32
references.pmid: 2194479
references.citation: Biochim Biophys Acta. 2000 May 12;1458(1):6-27
references.pmid: 10812022
references.citation: Biochem Soc Trans. 1976;4(3):399-430
references.pmid: 137147
references.citation: J Comput Chem. 2011 Mar;32(4):756-70
references.pmid: 20845420