August  2016, 21(6): 1775-1802. doi: 10.3934/dcdsb.2016022

Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects

1. 

Department of Mathematics, Qingdao Binhai University, Qingdao, Shandong 266555, China

2. 

Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, United States

Received  August 2015 Revised  November 2015 Published  June 2016

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows through membrane channels with fixed boundary ion concentrations and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. Bikerman's local hard-sphere potential is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive approximations of individual fluxes and I-V (current-voltage) relations, from which qualitative properties of ionic flows related to ion sizes are studied. A detailed characterization of complicated interactions among multiple and physically crucial parameters for ionic flows, such as boundary concentrations and potentials, diffusion coefficients and ion sizes, is provided.
Citation: Yusheng Jia, Weishi Liu, Mingji Zhang. Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022
References:
[1]

N. Abaid, R. S. Eisenberg and W. Liu, Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1507.  doi: 10.1137/070691322.  Google Scholar

[2]

V. Barcilon, Ion flow through narrow membrane channels: Part I,, SIAM J. Appl. Math., 52 (1992), 1391.  doi: 10.1137/0152080.  Google Scholar

[3]

V. Barcilon, D.-P. Chen and R. S. Eisenberg, Ion flow through narrow membrane channels: Part II,, SIAM J. Appl. Math., 52 (1992), 1405.  doi: 10.1137/0152081.  Google Scholar

[4]

V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Perturbation and simulation study,, SIAM J. Appl. Math., 57 (1997), 631.  doi: 10.1137/S0036139995312149.  Google Scholar

[5]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM J. Appl. Math., 67 (2007), 960.  doi: 10.1137/060664689.  Google Scholar

[6]

J. J. Bikerman, Structure and capacity of the electrical double layer,, Philos. Mag., 33 (1942), 384.  doi: 10.1080/14786444208520813.  Google Scholar

[7]

P. W. Bates, W. Liu, H. Lu and M. Zhang, Ion size and valence effaces on ionic flows via Poisson-Nernst-Planck systems,, Commun. Math. Sci., ().   Google Scholar

[8]

A. E. Cardenas, R. D. Coalson and M. G. Kurnikova, Three-dimensional poisson-nernst-planck theory studies: influence of membrane electrostatics on gramicidin a channel conductance,, Biophys. J., 79 (2000), 80.  doi: 10.1016/S0006-3495(00)76275-8.  Google Scholar

[9]

D. P. Chen and R. S. Eisenberg, Charges, currents and potentials in ionic channels of one conformation,, Biophys. J., 64 (1993), 1405.  doi: 10.1016/S0006-3495(93)81507-8.  Google Scholar

[10]

R. D. Coalson, Discrete-state model of coupled ion permeation and fast gating in ClC chloride channels,, J. Phys. A, 41 (2009).  doi: 10.1088/1751-8113/41/11/115001.  Google Scholar

[11]

R. Coalson and M. Kurnikova, Poisson-Nernst-Planck theory approach to the calculation of current through biological ion channels,, IEEE Transaction on NanoBioscience, 4 (2005), 81.  doi: 10.1109/TNB.2004.842495.  Google Scholar

[12]

B. Deng, Homoclinic bifurcations with nonhyperbolic equilibria,, SIAM J. Math. Anal., 21 (1990), 693.  doi: 10.1137/0521037.  Google Scholar

[13]

B. Eisenberg, Y. Hyon and C. Liu, Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids,, J. Chem. Phys., 133 (2010).  doi: 10.1063/1.3476262.  Google Scholar

[14]

B. Eisenberg, Y. Hyon and C. Liu, Energy variational analysis EnVarA of ions in calcium and sodium channels, Field theory for primitive models of complex ionic fluids,, Biophys. J., 98 (2010).   Google Scholar

[15]

B. Eisenberg, Ion channels as devices,, J. Comp. Electro., 2 (2003), 245.  doi: 10.1023/B:JCEL.0000011432.03832.22.  Google Scholar

[16]

B. Eisenberg, Proteins, channels, and crowded ions,, Biophys. Chem, 100 (2002), 507.  doi: 10.1016/S0301-4622(02)00302-2.  Google Scholar

[17]

R. S. Eisenberg, Channels as enzymes,, J. Memb. Biol., 115 (1990), 1.  doi: 10.1007/BF01869101.  Google Scholar

[18]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels,, In New Developments and Theoretical Studies of Proteins, 7 (1996), 269.  doi: 10.1142/9789814261418_0005.  Google Scholar

[19]

R. S. Eisenberg, From structure to function in open ionic channels,, J. Memb. Biol., 171 (1999), 1.  doi: 10.1007/s002329900554.  Google Scholar

[20]

B. Eisenberg and W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges,, SIAM J. Math. Anal., 38 (2007), 1932.  doi: 10.1137/060657480.  Google Scholar

[21]

B. Eisenberg, W. Liu and H. Xu, Reversal permanent charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models,, Nonlinearity, 28 (2015), 103.  doi: 10.1088/0951-7715/28/1/103.  Google Scholar

[22]

A. Ern, R. Joubaud and T. Leliévre, Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes,, Nonlinearity, 25 (2012), 1635.  doi: 10.1088/0951-7715/25/6/1635.  Google Scholar

[23]

J. Fischer and U. Heinbuch, Relationship between free energy density functional, Born-Green-Yvon, and potential distribution approaches for inhomogeneous fluids,, J. Chem. Phys., 88 (1988), 1909.  doi: 10.1063/1.454114.  Google Scholar

[24]

D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels,, Ph.D Dissertation, (1999).   Google Scholar

[25]

D. Gillespie, L. Xu, Y. Wang and G. Meissner, (De)constructing the ryanodine receptor: Modeling ion permeation and selectivity of the calcium release channel,, J. Phys. Chem. B, 109 (2005), 15598.  doi: 10.1021/jp052471j.  Google Scholar

[26]

D. Gillespie and R. S. Eisenberg, Physical descriptions of experimental selectivity measurements in ion channels,, European Biophys. J., 31 (2002), 454.  doi: 10.1007/s00249-002-0239-x.  Google Scholar

[27]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, J. Phys.: Condens. Matter, 14 (2002), 12129.  doi: 10.1088/0953-8984/14/46/317.  Google Scholar

[28]

D. Gillespie, W. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.031503.  Google Scholar

[29]

D. Gillespie, W. Nonner and R. S. Eisenberg, Crowded charge in biological ion channels,, Nanotech, 3 (2003), 435.   Google Scholar

[30]

P. Graf, M. G. Kurnikova, R. D. Coalson and A. Nitzan, Comparison of dynamic lattice monte-carlo simulations and dielectric self energy poisson-nernst-planck continuum theory for model ion channels,, J. Phys. Chem. B, 108 (2004), 2006.  doi: 10.1021/jp0355307.  Google Scholar

[31]

L. J. Henderson, The Fitness of the Environment: An Inquiry into the Biological Significance of the Properties of Matter,, Macmillan, (1927).   Google Scholar

[32]

U. Hollerbach, D.-P. Chen and R. S. Eisenberg, Two and Three-Dimensional Poisson-Nernst-Planck simulations of current flow through gramicidin-A,, J. Comp. Science, 16 (2002), 373.   Google Scholar

[33]

U. Hollerbach, D. Chen, W. Nonner and B. Eisenberg, Three-dimensional Poisson-Nernst-Planck theory of open channels,, Biophys. J., 76 (1999).   Google Scholar

[34]

Y. Hyon, B. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions,, Commun. Math. Sci., 9 (2010), 459.  doi: 10.4310/CMS.2011.v9.n2.a5.  Google Scholar

[35]

Y. Hyon, J. Fonseca, B. Eisenberg and C. Liu, A new Poisson-Nernst-Planck equation (PNP-FS-IF) for charge inversion near walls,, Biophys. J., 100 (2011).  doi: 10.1016/j.bpj.2010.12.3342.  Google Scholar

[36]

Y. Hyon, J. Fonseca, B. Eisenberg and C. Liu, Energy variational approach to study charge inversion (layering) near charged walls,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2725.  doi: 10.3934/dcdsb.2012.17.2725.  Google Scholar

[37]

Y. Hyon, C. Liu and B. Eisenberg, PNP equations with steric effects: A model of ion flow through channels,, J. Phys. Chem. B, 116 (2012), 11422.   Google Scholar

[38]

W. Im, D. Beglov and B. Roux, Continuum solvation model: Electrostatic forces from numerical solutions to the Poisson-Boltzmann equation,, Comp. Phys. Comm., 111 (1998), 59.  doi: 10.1016/S0010-4655(98)00016-2.  Google Scholar

[39]

W. Im and B. Roux, Ion permeation and selectivity of OmpF porin: A theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory,, J. Mol. Biol., 322 (2002), 851.  doi: 10.1016/S0022-2836(02)00778-7.  Google Scholar

[40]

J. W. Jerome, Mathematical Theory and Approximation of Semiconductor Models,, Springer-Verlag, (1995).   Google Scholar

[41]

J. W. Jerome and T. Kerkhoven, A finite element approximation theory for the drift-diffusion semiconductor model,, SIAM J. Numer. Anal., 28 (1991), 403.  doi: 10.1137/0728023.  Google Scholar

[42]

S. Ji and W. Liu, Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part I: Analysis,, J. Dyn. Diff. Equat., 24 (2012), 955.  doi: 10.1007/s10884-012-9277-y.  Google Scholar

[43]

S. Ji, W. Liu and M. Zhang, Effects of (small) permanent charges and channel geometry on ionic flows via classical Poisson-Nernst-Planck models,, SIAM J. on Appl. Math., 75 (2015), 114.  doi: 10.1137/140992527.  Google Scholar

[44]

C. Jones, Geometric singular perturbation theory,, Dynamical systems (Montecatini Terme, 1609 (1994), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[45]

C. Jones, T. Kaper and N. Kopell, Tracking invariant manifolds up tp exponentially small errors,, SIAM J. Math. Anal., 27 (1996), 558.  doi: 10.1137/S003614109325966X.  Google Scholar

[46]

C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems,, J. Differential Equations, 108 (1994), 64.  doi: 10.1006/jdeq.1994.1025.  Google Scholar

[47]

M. S. Kilic, M. Z. Bazant and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations,, Phys. Rev. E, 75 (2007).  doi: 10.1103/PhysRevE.75.021503.  Google Scholar

[48]

M. G. Kurnikova, R. D. Coalson, P. Graf and A. Nitzan, A Lattice Relaxation Algorithm for 3D Poisson-Nernst-Planck Theory with Application to Ion Transport Through the Gramicidin A Channel,, Biophys. J., 76 (1999), 642.   Google Scholar

[49]

B. Li, Minimizations of electrostatic free energy and the Poisson-Boltzmann equation for molecular solvation with implicit solvent,, SIAM J. Math. Anal., 40 (2009), 2536.  doi: 10.1137/080712350.  Google Scholar

[50]

B. Li, Continuum electrostatics for ionic solutions with non-uniform ionic sizes,, Nonlinearity, 22 (2009), 811.  doi: 10.1088/0951-7715/22/4/007.  Google Scholar

[51]

G. Lin, W. Liu, Y. Yi and M. Zhang, Poisson-Nernst-Planck systems for ion flow with density functional theory for local hard-sphere potential,, SIAM J. on Appl. Dyn. Syst., 12 (2013), 1613.  doi: 10.1137/120904056.  Google Scholar

[52]

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points,, J. Differential Equations, 167 (2000), 134.  doi: 10.1006/jdeq.2000.3778.  Google Scholar

[53]

W. Liu, Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems,, SIAM J. Appl. Math., 65 (2005), 754.  doi: 10.1137/S0036139903420931.  Google Scholar

[54]

W. Liu, Geometric singular perturbations for multiple turning points: Invariant manifolds and exchange lemmas,, J. Dynam. Differential Equations, 18 (2006), 667.  doi: 10.1007/s10884-006-9020-7.  Google Scholar

[55]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species,, J. Differential Equations, 246 (2009), 428.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[56]

W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow,, J. Differential Equations, 258 (2015), 1192.  doi: 10.1016/j.jde.2014.10.015.  Google Scholar

[57]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels,, J. Dynam. Differential Equations, 22 (2010), 413.  doi: 10.1007/s10884-010-9186-x.  Google Scholar

[58]

W. Liu, X. Tu and M. Zhang, Poisson-Nernst-Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I-V relations and Critical Potentials. Part II: Numerics,, J. Dynam. Differential Equations, 24 (2012), 985.  doi: 10.1007/s10884-012-9278-x.  Google Scholar

[59]

M. S. Mock, An example of nonuniqueness of stationary solutions in device models,, COMPEL, 1 (1982), 165.  doi: 10.1108/eb009970.  Google Scholar

[60]

B. Nadler, Z. Schuss, A. Singer and B. Eisenberg, Diffusion through protein channels: From molecular description to continuum equations,, Nanotech., 3 (2003), 439.   Google Scholar

[61]

W. Nonner and R. S. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type Calcium channels,, Biophys. J., 75 (1998), 1287.  doi: 10.1016/S0006-3495(98)74048-2.  Google Scholar

[62]

S. Y. Noskov, S. Berneche and B. Roux, Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands,, Nature, 431 (2004), 830.  doi: 10.1038/nature02943.  Google Scholar

[63]

S. Y. Noskov, W. Im and B. Roux, Ion Permeation through the $z_1$-Hemolysin Channel: Theoretical Studies Based on Brownian Dynamics and Poisson-Nernst-Planck Electrodiffusion Theory,, Biophys. J., 87 (2004), 2299.   Google Scholar

[64]

S. Y. Noskov and B. Roux, Ion selectivity in potassium channels,, Biophys. Chem., 124 (2006), 279.  doi: 10.1016/j.bpc.2006.05.033.  Google Scholar

[65]

J.-K. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study,, SIAM J. Appl. Math., 57 (1997), 609.  doi: 10.1137/S0036139995279809.  Google Scholar

[66]

J. K. Percus, Equilibrium state of a classical fluid of hard rods in an external field,, J. Stat. Phys., 15 (1976), 505.  doi: 10.1007/BF01020803.  Google Scholar

[67]

J. K. Percus, Model grand potential for a nonuniform classical fluid,, J. Chem. Phys., 75 (1981), 1316.  doi: 10.1063/1.442136.  Google Scholar

[68]

A. Robledo and C. Varea, On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids,, J. Stat. Phys., 26 (1981), 513.  doi: 10.1007/BF01011432.  Google Scholar

[69]

Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing,, Phys. Rev. Lett., 63 (1989), 980.  doi: 10.1103/PhysRevLett.63.980.  Google Scholar

[70]

Y. Rosenfeld, Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas,, J. Chem. Phys., 98 (1993), 8126.  doi: 10.1063/1.464569.  Google Scholar

[71]

R. Roth, Fundamental measure theory for hard-sphere mixtures: A review,, J. Phys.: Condens. Matter, 22 (2010).  doi: 10.1088/0953-8984/22/6/063102.  Google Scholar

[72]

B. Roux, T. W. Allen, S. Berneche and W. Im, Theoretical and computational models of biological ion channel,, Quat. Rev. Biophys., 37 (2004), 15.  doi: 10.1017/S0033583504003968.  Google Scholar

[73]

B. Roux, Theory of transport in ion channels: From molecular dynamics simulations to experiments,, in Comp. Simul. In Molecular Biology, (1995), 133.   Google Scholar

[74]

B. Roux and S. Crouzy, Theoretical studies of activated processes in biological ion channels,, in Classical and quantum dynamics in condensed phase simulations, (1997), 445.  doi: 10.1142/9789812839664_0019.  Google Scholar

[75]

I. Rubinstein, Multiple steady states in one-dimensional electrodiffusion with local electroneutrality,, SIAM J. Appl. Math., 47 (1987), 1076.  doi: 10.1137/0147070.  Google Scholar

[76]

I. Rubinstein, Electro-Diffusion of Ions,, SIAM Studies in Applied Mathematics, (1990).  doi: 10.1137/1.9781611970814.  Google Scholar

[77]

M. Saraniti, S. Aboud and R. Eisenberg, The simulation of ionic charge transport in biological ion channels: an introduction to numerical methods,, Rev. Comp. Chem., 22 (2005), 229.  doi: 10.1002/0471780367.ch4.  Google Scholar

[78]

S. Schecter, Exchange Lemmas 1: Deng's lemma,, J. Differential Equations, 245 (2008), 392.  doi: 10.1016/j.jde.2007.08.011.  Google Scholar

[79]

S. Schecter, Exchange lemmas. II. General exchange lemma,, J. Differential Equations, 245 (2008), 411.  doi: 10.1016/j.jde.2007.10.021.  Google Scholar

[80]

Z. Schuss, B. Nadler and R. S. Eisenberg, Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.036116.  Google Scholar

[81]

A. Singer and J. Norbury, A Poisson-Nernst-Planck model for biological ion channels-an asymptotic analysis in a three-dimensional narrow funnel,, SIAM J. Appl. Math., 70 (2009), 949.  doi: 10.1137/070687037.  Google Scholar

[82]

A. Singer, D. Gillespie, J. Norbury and R. S. Eisenberg, Singular perturbation analysis of the steady-state Poisson-Nernst-Planck system: applications to ion channels,, European J. Appl. Math., 19 (2008), 541.  doi: 10.1017/S0956792508007596.  Google Scholar

[83]

H. Steinrück, Asymptotic analysis of the current-voltage curve of a $pnpn$ semiconductor device,, IMA J. Appl. Math., 43 (1989), 243.  doi: 10.1093/imamat/43.3.243.  Google Scholar

[84]

H. Steinrück, A bifurcation analysis of the one-dimensional steady-state semiconductor device equations,, SIAM J. Appl. Math., 49 (1989), 1102.  doi: 10.1137/0149066.  Google Scholar

[85]

S.-K. Tin, N. Kopell and C. Jones, Invariant manifolds and singularly perturbed boundary value problems,, SIAM J. Numer. Anal., 31 (1994), 1558.  doi: 10.1137/0731081.  Google Scholar

[86]

X.-S. Wang, D. He, J. Wylie and H. Huang, Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems,, Phys. Rev. E, 89 (2014).  doi: 10.1103/PhysRevE.89.022722.  Google Scholar

[87]

M. Zhang, Asymptotic expansions and numerical simulations of I-V relations via a steady-state Poisson-Nernst-Planck system,, Rocky Mountain J. Math., 45 (2015), 1681.  doi: 10.1216/RMJ-2015-45-5-1681.  Google Scholar

[88]

Q. Zheng and G. W. Wei, Poisson-Boltzmann-Nernst-Planck model,, J. Chem. Phys., 134 (2011).  doi: 10.1063/1.3581031.  Google Scholar

[89]

S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with nonuniform ionic sizes: A numerical approach,, Phy. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.021901.  Google Scholar

show all references

References:
[1]

N. Abaid, R. S. Eisenberg and W. Liu, Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1507.  doi: 10.1137/070691322.  Google Scholar

[2]

V. Barcilon, Ion flow through narrow membrane channels: Part I,, SIAM J. Appl. Math., 52 (1992), 1391.  doi: 10.1137/0152080.  Google Scholar

[3]

V. Barcilon, D.-P. Chen and R. S. Eisenberg, Ion flow through narrow membrane channels: Part II,, SIAM J. Appl. Math., 52 (1992), 1405.  doi: 10.1137/0152081.  Google Scholar

[4]

V. Barcilon, D.-P. Chen, R. S. Eisenberg and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Perturbation and simulation study,, SIAM J. Appl. Math., 57 (1997), 631.  doi: 10.1137/S0036139995312149.  Google Scholar

[5]

M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity,, SIAM J. Appl. Math., 67 (2007), 960.  doi: 10.1137/060664689.  Google Scholar

[6]

J. J. Bikerman, Structure and capacity of the electrical double layer,, Philos. Mag., 33 (1942), 384.  doi: 10.1080/14786444208520813.  Google Scholar

[7]

P. W. Bates, W. Liu, H. Lu and M. Zhang, Ion size and valence effaces on ionic flows via Poisson-Nernst-Planck systems,, Commun. Math. Sci., ().   Google Scholar

[8]

A. E. Cardenas, R. D. Coalson and M. G. Kurnikova, Three-dimensional poisson-nernst-planck theory studies: influence of membrane electrostatics on gramicidin a channel conductance,, Biophys. J., 79 (2000), 80.  doi: 10.1016/S0006-3495(00)76275-8.  Google Scholar

[9]

D. P. Chen and R. S. Eisenberg, Charges, currents and potentials in ionic channels of one conformation,, Biophys. J., 64 (1993), 1405.  doi: 10.1016/S0006-3495(93)81507-8.  Google Scholar

[10]

R. D. Coalson, Discrete-state model of coupled ion permeation and fast gating in ClC chloride channels,, J. Phys. A, 41 (2009).  doi: 10.1088/1751-8113/41/11/115001.  Google Scholar

[11]

R. Coalson and M. Kurnikova, Poisson-Nernst-Planck theory approach to the calculation of current through biological ion channels,, IEEE Transaction on NanoBioscience, 4 (2005), 81.  doi: 10.1109/TNB.2004.842495.  Google Scholar

[12]

B. Deng, Homoclinic bifurcations with nonhyperbolic equilibria,, SIAM J. Math. Anal., 21 (1990), 693.  doi: 10.1137/0521037.  Google Scholar

[13]

B. Eisenberg, Y. Hyon and C. Liu, Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids,, J. Chem. Phys., 133 (2010).  doi: 10.1063/1.3476262.  Google Scholar

[14]

B. Eisenberg, Y. Hyon and C. Liu, Energy variational analysis EnVarA of ions in calcium and sodium channels, Field theory for primitive models of complex ionic fluids,, Biophys. J., 98 (2010).   Google Scholar

[15]

B. Eisenberg, Ion channels as devices,, J. Comp. Electro., 2 (2003), 245.  doi: 10.1023/B:JCEL.0000011432.03832.22.  Google Scholar

[16]

B. Eisenberg, Proteins, channels, and crowded ions,, Biophys. Chem, 100 (2002), 507.  doi: 10.1016/S0301-4622(02)00302-2.  Google Scholar

[17]

R. S. Eisenberg, Channels as enzymes,, J. Memb. Biol., 115 (1990), 1.  doi: 10.1007/BF01869101.  Google Scholar

[18]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels,, In New Developments and Theoretical Studies of Proteins, 7 (1996), 269.  doi: 10.1142/9789814261418_0005.  Google Scholar

[19]

R. S. Eisenberg, From structure to function in open ionic channels,, J. Memb. Biol., 171 (1999), 1.  doi: 10.1007/s002329900554.  Google Scholar

[20]

B. Eisenberg and W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges,, SIAM J. Math. Anal., 38 (2007), 1932.  doi: 10.1137/060657480.  Google Scholar

[21]

B. Eisenberg, W. Liu and H. Xu, Reversal permanent charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models,, Nonlinearity, 28 (2015), 103.  doi: 10.1088/0951-7715/28/1/103.  Google Scholar

[22]

A. Ern, R. Joubaud and T. Leliévre, Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes,, Nonlinearity, 25 (2012), 1635.  doi: 10.1088/0951-7715/25/6/1635.  Google Scholar

[23]

J. Fischer and U. Heinbuch, Relationship between free energy density functional, Born-Green-Yvon, and potential distribution approaches for inhomogeneous fluids,, J. Chem. Phys., 88 (1988), 1909.  doi: 10.1063/1.454114.  Google Scholar

[24]

D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels,, Ph.D Dissertation, (1999).   Google Scholar

[25]

D. Gillespie, L. Xu, Y. Wang and G. Meissner, (De)constructing the ryanodine receptor: Modeling ion permeation and selectivity of the calcium release channel,, J. Phys. Chem. B, 109 (2005), 15598.  doi: 10.1021/jp052471j.  Google Scholar

[26]

D. Gillespie and R. S. Eisenberg, Physical descriptions of experimental selectivity measurements in ion channels,, European Biophys. J., 31 (2002), 454.  doi: 10.1007/s00249-002-0239-x.  Google Scholar

[27]

D. Gillespie, W. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux,, J. Phys.: Condens. Matter, 14 (2002), 12129.  doi: 10.1088/0953-8984/14/46/317.  Google Scholar

[28]

D. Gillespie, W. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.031503.  Google Scholar

[29]

D. Gillespie, W. Nonner and R. S. Eisenberg, Crowded charge in biological ion channels,, Nanotech, 3 (2003), 435.   Google Scholar

[30]

P. Graf, M. G. Kurnikova, R. D. Coalson and A. Nitzan, Comparison of dynamic lattice monte-carlo simulations and dielectric self energy poisson-nernst-planck continuum theory for model ion channels,, J. Phys. Chem. B, 108 (2004), 2006.  doi: 10.1021/jp0355307.  Google Scholar

[31]

L. J. Henderson, The Fitness of the Environment: An Inquiry into the Biological Significance of the Properties of Matter,, Macmillan, (1927).   Google Scholar

[32]

U. Hollerbach, D.-P. Chen and R. S. Eisenberg, Two and Three-Dimensional Poisson-Nernst-Planck simulations of current flow through gramicidin-A,, J. Comp. Science, 16 (2002), 373.   Google Scholar

[33]

U. Hollerbach, D. Chen, W. Nonner and B. Eisenberg, Three-dimensional Poisson-Nernst-Planck theory of open channels,, Biophys. J., 76 (1999).   Google Scholar

[34]

Y. Hyon, B. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions,, Commun. Math. Sci., 9 (2010), 459.  doi: 10.4310/CMS.2011.v9.n2.a5.  Google Scholar

[35]

Y. Hyon, J. Fonseca, B. Eisenberg and C. Liu, A new Poisson-Nernst-Planck equation (PNP-FS-IF) for charge inversion near walls,, Biophys. J., 100 (2011).  doi: 10.1016/j.bpj.2010.12.3342.  Google Scholar

[36]

Y. Hyon, J. Fonseca, B. Eisenberg and C. Liu, Energy variational approach to study charge inversion (layering) near charged walls,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2725.  doi: 10.3934/dcdsb.2012.17.2725.  Google Scholar

[37]

Y. Hyon, C. Liu and B. Eisenberg, PNP equations with steric effects: A model of ion flow through channels,, J. Phys. Chem. B, 116 (2012), 11422.   Google Scholar

[38]

W. Im, D. Beglov and B. Roux, Continuum solvation model: Electrostatic forces from numerical solutions to the Poisson-Boltzmann equation,, Comp. Phys. Comm., 111 (1998), 59.  doi: 10.1016/S0010-4655(98)00016-2.  Google Scholar

[39]

W. Im and B. Roux, Ion permeation and selectivity of OmpF porin: A theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory,, J. Mol. Biol., 322 (2002), 851.  doi: 10.1016/S0022-2836(02)00778-7.  Google Scholar

[40]

J. W. Jerome, Mathematical Theory and Approximation of Semiconductor Models,, Springer-Verlag, (1995).   Google Scholar

[41]

J. W. Jerome and T. Kerkhoven, A finite element approximation theory for the drift-diffusion semiconductor model,, SIAM J. Numer. Anal., 28 (1991), 403.  doi: 10.1137/0728023.  Google Scholar

[42]

S. Ji and W. Liu, Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part I: Analysis,, J. Dyn. Diff. Equat., 24 (2012), 955.  doi: 10.1007/s10884-012-9277-y.  Google Scholar

[43]

S. Ji, W. Liu and M. Zhang, Effects of (small) permanent charges and channel geometry on ionic flows via classical Poisson-Nernst-Planck models,, SIAM J. on Appl. Math., 75 (2015), 114.  doi: 10.1137/140992527.  Google Scholar

[44]

C. Jones, Geometric singular perturbation theory,, Dynamical systems (Montecatini Terme, 1609 (1994), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[45]

C. Jones, T. Kaper and N. Kopell, Tracking invariant manifolds up tp exponentially small errors,, SIAM J. Math. Anal., 27 (1996), 558.  doi: 10.1137/S003614109325966X.  Google Scholar

[46]

C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems,, J. Differential Equations, 108 (1994), 64.  doi: 10.1006/jdeq.1994.1025.  Google Scholar

[47]

M. S. Kilic, M. Z. Bazant and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations,, Phys. Rev. E, 75 (2007).  doi: 10.1103/PhysRevE.75.021503.  Google Scholar

[48]

M. G. Kurnikova, R. D. Coalson, P. Graf and A. Nitzan, A Lattice Relaxation Algorithm for 3D Poisson-Nernst-Planck Theory with Application to Ion Transport Through the Gramicidin A Channel,, Biophys. J., 76 (1999), 642.   Google Scholar

[49]

B. Li, Minimizations of electrostatic free energy and the Poisson-Boltzmann equation for molecular solvation with implicit solvent,, SIAM J. Math. Anal., 40 (2009), 2536.  doi: 10.1137/080712350.  Google Scholar

[50]

B. Li, Continuum electrostatics for ionic solutions with non-uniform ionic sizes,, Nonlinearity, 22 (2009), 811.  doi: 10.1088/0951-7715/22/4/007.  Google Scholar

[51]

G. Lin, W. Liu, Y. Yi and M. Zhang, Poisson-Nernst-Planck systems for ion flow with density functional theory for local hard-sphere potential,, SIAM J. on Appl. Dyn. Syst., 12 (2013), 1613.  doi: 10.1137/120904056.  Google Scholar

[52]

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points,, J. Differential Equations, 167 (2000), 134.  doi: 10.1006/jdeq.2000.3778.  Google Scholar

[53]

W. Liu, Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems,, SIAM J. Appl. Math., 65 (2005), 754.  doi: 10.1137/S0036139903420931.  Google Scholar

[54]

W. Liu, Geometric singular perturbations for multiple turning points: Invariant manifolds and exchange lemmas,, J. Dynam. Differential Equations, 18 (2006), 667.  doi: 10.1007/s10884-006-9020-7.  Google Scholar

[55]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species,, J. Differential Equations, 246 (2009), 428.  doi: 10.1016/j.jde.2008.09.010.  Google Scholar

[56]

W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow,, J. Differential Equations, 258 (2015), 1192.  doi: 10.1016/j.jde.2014.10.015.  Google Scholar

[57]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels,, J. Dynam. Differential Equations, 22 (2010), 413.  doi: 10.1007/s10884-010-9186-x.  Google Scholar

[58]

W. Liu, X. Tu and M. Zhang, Poisson-Nernst-Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I-V relations and Critical Potentials. Part II: Numerics,, J. Dynam. Differential Equations, 24 (2012), 985.  doi: 10.1007/s10884-012-9278-x.  Google Scholar

[59]

M. S. Mock, An example of nonuniqueness of stationary solutions in device models,, COMPEL, 1 (1982), 165.  doi: 10.1108/eb009970.  Google Scholar

[60]

B. Nadler, Z. Schuss, A. Singer and B. Eisenberg, Diffusion through protein channels: From molecular description to continuum equations,, Nanotech., 3 (2003), 439.   Google Scholar

[61]

W. Nonner and R. S. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type Calcium channels,, Biophys. J., 75 (1998), 1287.  doi: 10.1016/S0006-3495(98)74048-2.  Google Scholar

[62]

S. Y. Noskov, S. Berneche and B. Roux, Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands,, Nature, 431 (2004), 830.  doi: 10.1038/nature02943.  Google Scholar

[63]

S. Y. Noskov, W. Im and B. Roux, Ion Permeation through the $z_1$-Hemolysin Channel: Theoretical Studies Based on Brownian Dynamics and Poisson-Nernst-Planck Electrodiffusion Theory,, Biophys. J., 87 (2004), 2299.   Google Scholar

[64]

S. Y. Noskov and B. Roux, Ion selectivity in potassium channels,, Biophys. Chem., 124 (2006), 279.  doi: 10.1016/j.bpc.2006.05.033.  Google Scholar

[65]

J.-K. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study,, SIAM J. Appl. Math., 57 (1997), 609.  doi: 10.1137/S0036139995279809.  Google Scholar

[66]

J. K. Percus, Equilibrium state of a classical fluid of hard rods in an external field,, J. Stat. Phys., 15 (1976), 505.  doi: 10.1007/BF01020803.  Google Scholar

[67]

J. K. Percus, Model grand potential for a nonuniform classical fluid,, J. Chem. Phys., 75 (1981), 1316.  doi: 10.1063/1.442136.  Google Scholar

[68]

A. Robledo and C. Varea, On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids,, J. Stat. Phys., 26 (1981), 513.  doi: 10.1007/BF01011432.  Google Scholar

[69]

Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing,, Phys. Rev. Lett., 63 (1989), 980.  doi: 10.1103/PhysRevLett.63.980.  Google Scholar

[70]

Y. Rosenfeld, Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas,, J. Chem. Phys., 98 (1993), 8126.  doi: 10.1063/1.464569.  Google Scholar

[71]

R. Roth, Fundamental measure theory for hard-sphere mixtures: A review,, J. Phys.: Condens. Matter, 22 (2010).  doi: 10.1088/0953-8984/22/6/063102.  Google Scholar

[72]

B. Roux, T. W. Allen, S. Berneche and W. Im, Theoretical and computational models of biological ion channel,, Quat. Rev. Biophys., 37 (2004), 15.  doi: 10.1017/S0033583504003968.  Google Scholar

[73]

B. Roux, Theory of transport in ion channels: From molecular dynamics simulations to experiments,, in Comp. Simul. In Molecular Biology, (1995), 133.   Google Scholar

[74]

B. Roux and S. Crouzy, Theoretical studies of activated processes in biological ion channels,, in Classical and quantum dynamics in condensed phase simulations, (1997), 445.  doi: 10.1142/9789812839664_0019.  Google Scholar

[75]

I. Rubinstein, Multiple steady states in one-dimensional electrodiffusion with local electroneutrality,, SIAM J. Appl. Math., 47 (1987), 1076.  doi: 10.1137/0147070.  Google Scholar

[76]

I. Rubinstein, Electro-Diffusion of Ions,, SIAM Studies in Applied Mathematics, (1990).  doi: 10.1137/1.9781611970814.  Google Scholar

[77]

M. Saraniti, S. Aboud and R. Eisenberg, The simulation of ionic charge transport in biological ion channels: an introduction to numerical methods,, Rev. Comp. Chem., 22 (2005), 229.  doi: 10.1002/0471780367.ch4.  Google Scholar

[78]

S. Schecter, Exchange Lemmas 1: Deng's lemma,, J. Differential Equations, 245 (2008), 392.  doi: 10.1016/j.jde.2007.08.011.  Google Scholar

[79]

S. Schecter, Exchange lemmas. II. General exchange lemma,, J. Differential Equations, 245 (2008), 411.  doi: 10.1016/j.jde.2007.10.021.  Google Scholar

[80]

Z. Schuss, B. Nadler and R. S. Eisenberg, Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.036116.  Google Scholar

[81]

A. Singer and J. Norbury, A Poisson-Nernst-Planck model for biological ion channels-an asymptotic analysis in a three-dimensional narrow funnel,, SIAM J. Appl. Math., 70 (2009), 949.  doi: 10.1137/070687037.  Google Scholar

[82]

A. Singer, D. Gillespie, J. Norbury and R. S. Eisenberg, Singular perturbation analysis of the steady-state Poisson-Nernst-Planck system: applications to ion channels,, European J. Appl. Math., 19 (2008), 541.  doi: 10.1017/S0956792508007596.  Google Scholar

[83]

H. Steinrück, Asymptotic analysis of the current-voltage curve of a $pnpn$ semiconductor device,, IMA J. Appl. Math., 43 (1989), 243.  doi: 10.1093/imamat/43.3.243.  Google Scholar

[84]

H. Steinrück, A bifurcation analysis of the one-dimensional steady-state semiconductor device equations,, SIAM J. Appl. Math., 49 (1989), 1102.  doi: 10.1137/0149066.  Google Scholar

[85]

S.-K. Tin, N. Kopell and C. Jones, Invariant manifolds and singularly perturbed boundary value problems,, SIAM J. Numer. Anal., 31 (1994), 1558.  doi: 10.1137/0731081.  Google Scholar

[86]

X.-S. Wang, D. He, J. Wylie and H. Huang, Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems,, Phys. Rev. E, 89 (2014).  doi: 10.1103/PhysRevE.89.022722.  Google Scholar

[87]

M. Zhang, Asymptotic expansions and numerical simulations of I-V relations via a steady-state Poisson-Nernst-Planck system,, Rocky Mountain J. Math., 45 (2015), 1681.  doi: 10.1216/RMJ-2015-45-5-1681.  Google Scholar

[88]

Q. Zheng and G. W. Wei, Poisson-Boltzmann-Nernst-Planck model,, J. Chem. Phys., 134 (2011).  doi: 10.1063/1.3581031.  Google Scholar

[89]

S. Zhou, Z. Wang and B. Li, Mean-field description of ionic size effects with nonuniform ionic sizes: A numerical approach,, Phy. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.021901.  Google Scholar

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