doi: 10.3934/dcdsb.2022030
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Qualitative properties of zero-current ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes

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

Received  August 2021 Revised  January 2022 Early access February 2022

Fund Project: This work was supported by MPS Simons Foundation (No. 628308)

We consider a one-dimensional Poisson-Nernst-Planck system with two oppositely charged particles and nonuniform finite ion sizes modeled through a local hard-sphere potential. The existence and local uniqueness result is established under the framework of geometric singular perturbation theory. Treating the fi- nite ion size as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to further study the qualitative properties of zero-current ionic flows, a special state among the range of the value for ionic current, which is significant for physiology. Of particular interest are the effects on the zero-current ionic flows from finite ion sizes, diffusion coefficients and ion valences. Critical potentials are identified and their important roles played in the study of ionic flow properties are characterized. Those non-intuitive observations from mathematical analysis of the system provide better understandings of the mechanism of ionic flows through membrane channels, particularly the internal dynamics of ionic flows, which cannot be detected via current technology. Numerical simulations are performed to provide more intuitive illustrations of the analytical results.

Citation: Mingji Zhang. Qualitative properties of zero-current ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022030
References:
[1]

N. AbaidR. S. Eisenberg and W. Liu, Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1507-1526.  doi: 10.1137/070691322.

[2]

R. AitbayevP. W. BatesH. LuL. Zhang and M. Zhang, Mathematical studies of Poisson-Nernst-Planck systems: Dynamics of ionic flows without electroneutrality conditions, J. Comput. Appl. Math., 362 (2019), 510-527.  doi: 10.1016/j.cam.2018.10.037.

[3]

B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, Molecular Biology of the Cell, $3^{rd}$ edition., Garland, New York, 1994.

[4]

V. Barcilon, Ion flow through narrow membrane channels: Part I, SIAM J. Appl. Math., 52 (1992), 1391-1404.  doi: 10.1137/0152080.

[5]

V. BarcilonD.-P. Chen and R. S. Eisenberg, Ion flow through narrow membrane channels: Part II, SIAM J. Appl. Math., 52 (1992), 1405-1425.  doi: 10.1137/0152081.

[6]

V. BarcilonD.-P. ChenR. 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), 631-648.  doi: 10.1137/S0036139995312149.

[7]

J. Barthel, H. Krienke and W. Kunz, Physical Chemistry of Electrolyte Solutions: Modern Aspects, Springer-Verlag, New York, 1998.

[8]

P. W. BatesJ. Chen and M. Zhang, Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations, Math. Biosci. Eng., 17 (2020), 3736-3766.  doi: 10.3934/mbe.2020210.

[9]

P. W. BatesY. JiaG. LinH. Lu and M. Zhang, Individual flux study via steady-state Poisson-Nernst-Planck systems: Effects from boundary conditions, SIAM J. Appl. Dyn. Syst., 16 (2017), 410-430.  doi: 10.1137/16M1071523.

[10]

P. W. BatesW. LiuH. Lu and M. Zhang, Ion size and valence effects on ionic flows via Poisson-Nernst-Planck systems, Commun. Math. Sci., 15 (2017), 881-901.  doi: 10.4310/CMS.2017.v15.n4.a1.

[11]

P. W. Bates, Z. Wen and M. Zhang, Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations, J. Nonlinear Sci., 31 (2021), Paper No. 55, 62 pp. doi: 10.1007/s00332-021-09715-3.

[12]

M. Z. BazantK. T. Chu and B. J. Bayly, Current-voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484.  doi: 10.1137/040609938.

[13]

M. Z. BazantK. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Review E, 70 (2004), 021506.  doi: 10.1103/PhysRevE.70.021506.

[14]

S. BhattacharyaJ. MuzardL. PayetJ. Math$\acute{e}$U. BockelmannA. Aksimentiev and V. Viasnoff, Rectification of the current in $\alpha$-hemolysin pore depends on the cation type: The alkali series probed by molecular dynamics simulations and experiments, J. Phys. Chem. C, 115 (2011), 4255-4264. 

[15]

M. Burger, Inverse problems in ion channel modelling, Inverse Problems, 27 (2011), 083001, 34 pp. doi: 10.1088/0266-5611/27/8/083001.

[16]

M. BurgerR. S. Eisenberg and H. Engl, Inverse problems related to ion channel selectivity, SIAM J. Appl. Math., 67 (2007), 960-989.  doi: 10.1137/060664689.

[17]

A. E. CardenasR. 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), 80-93.  doi: 10.1016/S0006-3495(00)76275-8.

[18]

D.-P. Chen and R. S. Eisenberg, Charges, currents and potentials in ionic channels of one conformation, Biophys. J., 64 (1993), 1405-1421.  doi: 10.1016/S0006-3495(93)81507-8.

[19]

J. ChenY. WangL. Zhang and M. Zhang, Mathematical analysis of Poisson- Nernst-Planck models with permanent charge and boundary layers: Studies on individual fluxes, Nonlinearity, 34 (2021), 3879-3906.  doi: 10.1088/1361-6544/abf33a.

[20]

B. Dworakowska and K. Dołowy, Ion channels-related diseases, Acta Biochim Pol., 47 (2000), 685-703. 

[21]

B. Eisenberg, Proteins, Channels, and Crowded Ions, Biophys. Chem., 100 (2002), 507-517.  doi: 10.1016/S0301-4622(02)00302-2.

[22]

B. Eisenberg, Ions in fluctuating channels: Transistors alive, Fluctuation and Noise Letters., 11 (2012), 76-96.  doi: 10.1142/S0219477512400019.

[23]

B. Eisenberg, Crowded charges in ion channels, In Advances in Chemical Physics; Rice, S. A. Ed.;, John Wiley & Sons: Hoboken, NJ, USA, (2011), 77–223.

[24]

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), 104104, 1–23. doi: 10.1063/1.3476262.

[25]

B. Eisenberg and W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (2007), 1932-1966.  doi: 10.1137/060657480.

[26]

B. Eisenberg and W. Liu, Relative dielectric constants and selectivity ratios in open ionic channels, Mol. Based Math. Biol., 5 (2017), 125-137.  doi: 10.1515/mlbmb-2017-0008.

[27]

B. EisenbergW. Liu and H. Xu, Reversal charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models, Nonlinearity, 28 (2015), 103-127.  doi: 10.1088/0951-7715/28/1/103.

[28]

R. S. Eisenberg, From structure to function in open ionic channels, J. Memb. Biol., 171 (1999), 1-24.  doi: 10.1007/s002329900554.

[29]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[30]

D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels, Ph.D Thesis, Rush University at Chicago, Chicago, IL, USA, 1999.

[31]

D. Gillespie, Energetics of divalent selectivity in a calcium channel: The Ryanodine receptor case study, Biophys. J., 94 (2008), 1169-1184.  doi: 10.1529/biophysj.107.116798.

[32]

D. Gillespie and R. S. Eisenberg, Physical descriptions of experimental selectivity measurements in ion channels, European Biophys. J., 31 (2002), 454-466.  doi: 10.1007/s00249-002-0239-x.

[33]

D. GillespieW. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter, 14 (2002), 12129-12145.  doi: 10.1088/0953-8984/14/46/317.

[34]

D. GillespieW. Nonner and R. S. Eisenberg, Crowded charge in biological ion channels, Nanotech., 3 (2003), 435-438. 

[35]

D. GillespieL. XuY. 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), 15598-15610.  doi: 10.1021/jp052471j.

[36]

D. E. Goldman, Potential, impedance, and rectification in membranes, J. Gen. Physiol., 27 (1943), 37-60.  doi: 10.1085/jgp.27.1.37.

[37]

P. GrafM. G. KurnikovaR. 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), 2006-2015.  doi: 10.1021/jp0355307.

[38]

L. J. Henderson, The Fitness of the Environment: An Inquiry Into the Biological Significance of the Properties of Matter, Macmillan, New York, 1927.

[39]

A. L. HodgkinA. Huxley and B. Katz, Ionic Currents underlying activity in the giant axon of the squid, Arch. Sci. Physiol., 3 (1949), 129-150. 

[40]

A. L. Hodgkin and B. Katz, The effect of sodium ions on the electrical activity of the giant axon of the squid, J. Physiol., 108 (1949), 37-77.  doi: 10.1113/jphysiol.1949.sp004310.

[41]

Y. HyonB. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci., 9 (2011), 459-475.  doi: 10.4310/CMS.2011.v9.n2.a5.

[42]

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), 100 pp. doi: 10.1016/j.bpj.2010.12.3342.

[43]

Y. HyonJ. FonsecaB. Eisenberg and C. Liu, Energy variational approach to study charge inversion (layering) near charged walls, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2725-2743.  doi: 10.3934/dcdsb.2012.17.2725.

[44]

Y. HyonC. Liu and B. Eisenberg, PNP equations with steric effects: A model of ion flow through channels, J. Phys. Chem. B, 116 (2012), 11422-11441. 

[45]

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), 851-869.  doi: 10.1016/S0022-2836(02)00778-7.

[46]

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), 955-983.  doi: 10.1007/s10884-012-9277-y.

[47]

S. Ji and W. Liu, Flux ratios and channel structures, J. Dynam. Differ. Equations, 31 (2019), 1141-1183.  doi: 10.1007/s10884-017-9607-1.

[48]

S. JiW. 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), 114-135.  doi: 10.1137/140992527.

[49]

Y. JiaW. Liu and M. Zhang, Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1775-1802.  doi: 10.3934/dcdsb.2016022.

[50]

C. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), Lect. Notes in Math., Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.

[51]

C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-88.  doi: 10.1006/jdeq.1994.1025.

[52]

M. S. KilicM. 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), 021503.  doi: 10.1103/PhysRevE.75.021503.

[53]

G. LinW. LiuY. Yi and M. Zhang, Poisson-Nernst-Planck systems for ion flow with density functional theory for local hard-sphere potential, SIAM J. Appl. Dyn. Syst., 12 (2013), 1613-1648.  doi: 10.1137/120904056.

[54]

J. Liu and B. Eisenberg, Molecular mean-field theory of ionic solutions: A Poisson-Nernst-Planck-Bikerman model, Entropy, 22 (2020), Paper No. 550, 39 pp. doi: 10.3390/e22050550.

[55]

W. Liu, Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math., 65 (2005), 754-766.  doi: 10.1137/S0036139903420931.

[56]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.  doi: 10.1016/j.jde.2008.09.010.

[57]

W. Liu, A flux ratio and a universal property of permanent charges effects on fluxes, Comput. Math. Biophys., 6 (2018), 28-40.  doi: 10.1515/cmb-2018-0003.

[58]

W. LiuX. 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. Dyn. Diff. Equat., 24 (2012), 985-1004.  doi: 10.1007/s10884-012-9278-x.

[59]

W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228.  doi: 10.1016/j.jde.2014.10.015.

[60]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dyn. Diff. Equat., 22 (2010), 413-437.  doi: 10.1007/s10884-010-9186-x.

[61]

H. LuJ. LiJ. ShackelfordJ. Vorenberg and M. Zhang, Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1623-1643.  doi: 10.3934/dcdsb.2018064.

[62]

M. S. Mock, An example of nonuniqueness of stationary solutions in device models, COMPEL, 1 (1982), 165-174.  doi: 10.1108/eb009970.

[63]

H. Mofidi, B. Eisenberg and W. Liu, Effects of diffusion coefficients and permanent charge on reversal potentials in ionic channels, Entropy, 22 (2020), Paper No. 325, 23 pp. doi: 10.3390/e22030325.

[64]

H. Mofidi and W. Liu, Reversal potential and reversal permanent charge with unequal diffusion coefficients via classical Poisson-Nernst-Planck models, SIAM J. Appl. Math., 80 (2020), 1908-1935.  doi: 10.1137/19M1269105.

[65]

N. F. Mott, The theory of crystal rectifiers, World Scientific Series in 20th Century PhysicsSir Nevill Mott-65 Years in Physics, (1995), 153–165. doi: 10.1142/9789812794086_0013.

[66]

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), 1287-1305.  doi: 10.1016/S0006-3495(98)74048-2.

[67]

S. Y. NoskovS. Berneche and B. Roux, Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands, Nature, 431 (2004), 830-834.  doi: 10.1038/nature02943.

[68]

S. Y. Noskov and B. Roux, Ion selectivity in potassium channels, Biophys. Chem., 124 (2006), 279-291.  doi: 10.1016/j.bpc.2006.05.033.

[69]

J.-K. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630.  doi: 10.1137/S0036139995279809.

[70]

Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Phys. Rev. Lett., 63 (1989), 980-983.  doi: 10.1103/PhysRevLett.63.980.

[71]

Y. Rosenfeld, Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas, J. Chem. Phys., 98 (1993), 8126-8148.  doi: 10.1063/1.464569.

[72]

D. J. Rouston, Bipolar Semiconductor Devices, McGraw-Hill, New York, 1990.

[73]

B. Roux, Theory of transport in ion channels: From molecular dynamics simulations to experiments, In Comp. Simul. In Molecular Biology, J. Goodefellow ed., VCH Weinheim, Ch., 6 (1995), 133–169.

[74]

B. RouxT. W. AllenS. Berneche and W. Im, Theoretical and computational models of biological ion channels, Quat. Rev. Biophys., 37 (2004), 15-103.  doi: 10.1017/S0033583504003968.

[75]

I. Rubinstein, Electro-Diffusion of Ions, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1990. doi: 10.1137/1.9781611970814.

[76]

M. Saraniti, S. Aboud, S and R. Eisenberg, The simulation of ionic charge transport in biological ion channels: An introduction to numerical methods, Rev. Comp. Chem., 22 (2006), 229-294. doi: 10.1002/0471780367.ch4.

[77]

Z. SchussB. 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), 1-14.  doi: 10.1103/PhysRevE.64.036116.

[78]

A. SingerD. GillespieJ. 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), 541-560.  doi: 10.1017/S0956792508007596.

[79]

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), 949-968.  doi: 10.1137/070687037.

[80]

B. G. Streetman, Solid State Electronic Devices, 4$^{th}$ edition, Prentice-Hall, Englewood Cliffs, NJ, 1972.

[81]

L. Sun and W. Liu, Non-localness of excess potentials and boundary value problems of Poisson-Nernst-Planck Systems for ionic flow: A case study, J. Dyn. Diff. Equat., 30 (2018), 779-797.  doi: 10.1007/s10884-017-9578-2.

[82] C. Tanford and J. Reynolds, Nature's Robots: A History of Proteins, Oxford University Press, New York, 2001. 
[83]

N. Unwin, The structure of ion channels in membranes of excitable cells, Neuron, 3 (1989), 665-676.  doi: 10.1016/0896-6273(89)90235-3.

[84]

J. H. Vera and G. Wilczek-Vera, Classical Thermodynamics of Fluid Systems: Principles and Applications, Crc Press, 2016.

[85]

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), 022722, 14 pp. doi: 10.1103/PhysRevE.89.022722.

[86]

Z. WenP. W. Bates and M. Zhang, Effects on I-V relations from small permanent charge and channel geometry via classical Poisson-Nernst-Planck equations with multiple cations, Nonlinearity, 34 (2021), 4464-4502.  doi: 10.1088/1361-6544/abfae8.

[87]

Z. WenL. Zhang and M. Zhang, Dynamics of classical Poisson-Nernst-Planck systems with multiple cations and boundary layers, J. Dyn. Diff. Equat., 33 (2021), 211-234.  doi: 10.1007/s10884-020-09861-4.

[88]

L. ZhangB. Eisenberg and W. Liu, An effect of large permanent charge: Decreasing flux with increasing transmembrane potential, Eur. Phys. J. Special Topics, 227 (2019), 2575-2601.  doi: 10.1140/epjst/e2019-700134-7.

[89]

L. Zhang and W. Liu, Effects of large permanent charges on ionic flows via Poisson-Nernst-Planck models, SIAM J. Appl. Dyn. Syst., 19 (2020), 1993-2029.  doi: 10.1137/19M1289443.

[90]

M. Zhang, Asymptotic expansions and numerical simulations of I-V relations via a steady-state Poisson-Nernst-Planck system,, Rocky MT. J. Math., 45 (2015), 1681-1708.  doi: 10.1216/RMJ-2015-45-5-1681.

[91]

M. Zhang, Competition between cations via Poisson-Nernst-Planck systems with nonzero but small permanent charges, Membranes, 11 (2021), 236. 

[92]

M. Zhang, Boundary layer effects on ionic flows via classical Poisson-Nernst-Planck systems, Comput. Math. Biophys., 6 (2018), 14-27.  doi: 10.1515/cmb-2018-0002.

[93]

Q. Zheng and G. W. Wei, Poisson-Boltzmann-Nernst-Planck model, J. Chem. Phys., 134 (2011), 1-17.  doi: 10.1063/1.3581031.

[94]

S. ZhouZ. Wang and B. Li, Mean-field description of ionic size effects with nonuniform ionic sizes: A numerical approach, Phy. Rev. E, 84 (2011), 1-13.  doi: 10.1103/PhysRevE.84.021901.

show all references

References:
[1]

N. AbaidR. S. Eisenberg and W. Liu, Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1507-1526.  doi: 10.1137/070691322.

[2]

R. AitbayevP. W. BatesH. LuL. Zhang and M. Zhang, Mathematical studies of Poisson-Nernst-Planck systems: Dynamics of ionic flows without electroneutrality conditions, J. Comput. Appl. Math., 362 (2019), 510-527.  doi: 10.1016/j.cam.2018.10.037.

[3]

B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, Molecular Biology of the Cell, $3^{rd}$ edition., Garland, New York, 1994.

[4]

V. Barcilon, Ion flow through narrow membrane channels: Part I, SIAM J. Appl. Math., 52 (1992), 1391-1404.  doi: 10.1137/0152080.

[5]

V. BarcilonD.-P. Chen and R. S. Eisenberg, Ion flow through narrow membrane channels: Part II, SIAM J. Appl. Math., 52 (1992), 1405-1425.  doi: 10.1137/0152081.

[6]

V. BarcilonD.-P. ChenR. 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), 631-648.  doi: 10.1137/S0036139995312149.

[7]

J. Barthel, H. Krienke and W. Kunz, Physical Chemistry of Electrolyte Solutions: Modern Aspects, Springer-Verlag, New York, 1998.

[8]

P. W. BatesJ. Chen and M. Zhang, Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations, Math. Biosci. Eng., 17 (2020), 3736-3766.  doi: 10.3934/mbe.2020210.

[9]

P. W. BatesY. JiaG. LinH. Lu and M. Zhang, Individual flux study via steady-state Poisson-Nernst-Planck systems: Effects from boundary conditions, SIAM J. Appl. Dyn. Syst., 16 (2017), 410-430.  doi: 10.1137/16M1071523.

[10]

P. W. BatesW. LiuH. Lu and M. Zhang, Ion size and valence effects on ionic flows via Poisson-Nernst-Planck systems, Commun. Math. Sci., 15 (2017), 881-901.  doi: 10.4310/CMS.2017.v15.n4.a1.

[11]

P. W. Bates, Z. Wen and M. Zhang, Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations, J. Nonlinear Sci., 31 (2021), Paper No. 55, 62 pp. doi: 10.1007/s00332-021-09715-3.

[12]

M. Z. BazantK. T. Chu and B. J. Bayly, Current-voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (2005), 1463-1484.  doi: 10.1137/040609938.

[13]

M. Z. BazantK. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Review E, 70 (2004), 021506.  doi: 10.1103/PhysRevE.70.021506.

[14]

S. BhattacharyaJ. MuzardL. PayetJ. Math$\acute{e}$U. BockelmannA. Aksimentiev and V. Viasnoff, Rectification of the current in $\alpha$-hemolysin pore depends on the cation type: The alkali series probed by molecular dynamics simulations and experiments, J. Phys. Chem. C, 115 (2011), 4255-4264. 

[15]

M. Burger, Inverse problems in ion channel modelling, Inverse Problems, 27 (2011), 083001, 34 pp. doi: 10.1088/0266-5611/27/8/083001.

[16]

M. BurgerR. S. Eisenberg and H. Engl, Inverse problems related to ion channel selectivity, SIAM J. Appl. Math., 67 (2007), 960-989.  doi: 10.1137/060664689.

[17]

A. E. CardenasR. 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), 80-93.  doi: 10.1016/S0006-3495(00)76275-8.

[18]

D.-P. Chen and R. S. Eisenberg, Charges, currents and potentials in ionic channels of one conformation, Biophys. J., 64 (1993), 1405-1421.  doi: 10.1016/S0006-3495(93)81507-8.

[19]

J. ChenY. WangL. Zhang and M. Zhang, Mathematical analysis of Poisson- Nernst-Planck models with permanent charge and boundary layers: Studies on individual fluxes, Nonlinearity, 34 (2021), 3879-3906.  doi: 10.1088/1361-6544/abf33a.

[20]

B. Dworakowska and K. Dołowy, Ion channels-related diseases, Acta Biochim Pol., 47 (2000), 685-703. 

[21]

B. Eisenberg, Proteins, Channels, and Crowded Ions, Biophys. Chem., 100 (2002), 507-517.  doi: 10.1016/S0301-4622(02)00302-2.

[22]

B. Eisenberg, Ions in fluctuating channels: Transistors alive, Fluctuation and Noise Letters., 11 (2012), 76-96.  doi: 10.1142/S0219477512400019.

[23]

B. Eisenberg, Crowded charges in ion channels, In Advances in Chemical Physics; Rice, S. A. Ed.;, John Wiley & Sons: Hoboken, NJ, USA, (2011), 77–223.

[24]

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), 104104, 1–23. doi: 10.1063/1.3476262.

[25]

B. Eisenberg and W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (2007), 1932-1966.  doi: 10.1137/060657480.

[26]

B. Eisenberg and W. Liu, Relative dielectric constants and selectivity ratios in open ionic channels, Mol. Based Math. Biol., 5 (2017), 125-137.  doi: 10.1515/mlbmb-2017-0008.

[27]

B. EisenbergW. Liu and H. Xu, Reversal charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models, Nonlinearity, 28 (2015), 103-127.  doi: 10.1088/0951-7715/28/1/103.

[28]

R. S. Eisenberg, From structure to function in open ionic channels, J. Memb. Biol., 171 (1999), 1-24.  doi: 10.1007/s002329900554.

[29]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[30]

D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels, Ph.D Thesis, Rush University at Chicago, Chicago, IL, USA, 1999.

[31]

D. Gillespie, Energetics of divalent selectivity in a calcium channel: The Ryanodine receptor case study, Biophys. J., 94 (2008), 1169-1184.  doi: 10.1529/biophysj.107.116798.

[32]

D. Gillespie and R. S. Eisenberg, Physical descriptions of experimental selectivity measurements in ion channels, European Biophys. J., 31 (2002), 454-466.  doi: 10.1007/s00249-002-0239-x.

[33]

D. GillespieW. Nonner and R. S. Eisenberg, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter, 14 (2002), 12129-12145.  doi: 10.1088/0953-8984/14/46/317.

[34]

D. GillespieW. Nonner and R. S. Eisenberg, Crowded charge in biological ion channels, Nanotech., 3 (2003), 435-438. 

[35]

D. GillespieL. XuY. 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), 15598-15610.  doi: 10.1021/jp052471j.

[36]

D. E. Goldman, Potential, impedance, and rectification in membranes, J. Gen. Physiol., 27 (1943), 37-60.  doi: 10.1085/jgp.27.1.37.

[37]

P. GrafM. G. KurnikovaR. 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), 2006-2015.  doi: 10.1021/jp0355307.

[38]

L. J. Henderson, The Fitness of the Environment: An Inquiry Into the Biological Significance of the Properties of Matter, Macmillan, New York, 1927.

[39]

A. L. HodgkinA. Huxley and B. Katz, Ionic Currents underlying activity in the giant axon of the squid, Arch. Sci. Physiol., 3 (1949), 129-150. 

[40]

A. L. Hodgkin and B. Katz, The effect of sodium ions on the electrical activity of the giant axon of the squid, J. Physiol., 108 (1949), 37-77.  doi: 10.1113/jphysiol.1949.sp004310.

[41]

Y. HyonB. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci., 9 (2011), 459-475.  doi: 10.4310/CMS.2011.v9.n2.a5.

[42]

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), 100 pp. doi: 10.1016/j.bpj.2010.12.3342.

[43]

Y. HyonJ. FonsecaB. Eisenberg and C. Liu, Energy variational approach to study charge inversion (layering) near charged walls, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2725-2743.  doi: 10.3934/dcdsb.2012.17.2725.

[44]

Y. HyonC. Liu and B. Eisenberg, PNP equations with steric effects: A model of ion flow through channels, J. Phys. Chem. B, 116 (2012), 11422-11441. 

[45]

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), 851-869.  doi: 10.1016/S0022-2836(02)00778-7.

[46]

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), 955-983.  doi: 10.1007/s10884-012-9277-y.

[47]

S. Ji and W. Liu, Flux ratios and channel structures, J. Dynam. Differ. Equations, 31 (2019), 1141-1183.  doi: 10.1007/s10884-017-9607-1.

[48]

S. JiW. 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), 114-135.  doi: 10.1137/140992527.

[49]

Y. JiaW. Liu and M. Zhang, Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1775-1802.  doi: 10.3934/dcdsb.2016022.

[50]

C. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), Lect. Notes in Math., Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.

[51]

C. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-88.  doi: 10.1006/jdeq.1994.1025.

[52]

M. S. KilicM. 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), 021503.  doi: 10.1103/PhysRevE.75.021503.

[53]

G. LinW. LiuY. Yi and M. Zhang, Poisson-Nernst-Planck systems for ion flow with density functional theory for local hard-sphere potential, SIAM J. Appl. Dyn. Syst., 12 (2013), 1613-1648.  doi: 10.1137/120904056.

[54]

J. Liu and B. Eisenberg, Molecular mean-field theory of ionic solutions: A Poisson-Nernst-Planck-Bikerman model, Entropy, 22 (2020), Paper No. 550, 39 pp. doi: 10.3390/e22050550.

[55]

W. Liu, Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math., 65 (2005), 754-766.  doi: 10.1137/S0036139903420931.

[56]

W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations, 246 (2009), 428-451.  doi: 10.1016/j.jde.2008.09.010.

[57]

W. Liu, A flux ratio and a universal property of permanent charges effects on fluxes, Comput. Math. Biophys., 6 (2018), 28-40.  doi: 10.1515/cmb-2018-0003.

[58]

W. LiuX. 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. Dyn. Diff. Equat., 24 (2012), 985-1004.  doi: 10.1007/s10884-012-9278-x.

[59]

W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow, J. Differential Equations, 258 (2015), 1192-1228.  doi: 10.1016/j.jde.2014.10.015.

[60]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dyn. Diff. Equat., 22 (2010), 413-437.  doi: 10.1007/s10884-010-9186-x.

[61]

H. LuJ. LiJ. ShackelfordJ. Vorenberg and M. Zhang, Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1623-1643.  doi: 10.3934/dcdsb.2018064.

[62]

M. S. Mock, An example of nonuniqueness of stationary solutions in device models, COMPEL, 1 (1982), 165-174.  doi: 10.1108/eb009970.

[63]

H. Mofidi, B. Eisenberg and W. Liu, Effects of diffusion coefficients and permanent charge on reversal potentials in ionic channels, Entropy, 22 (2020), Paper No. 325, 23 pp. doi: 10.3390/e22030325.

[64]

H. Mofidi and W. Liu, Reversal potential and reversal permanent charge with unequal diffusion coefficients via classical Poisson-Nernst-Planck models, SIAM J. Appl. Math., 80 (2020), 1908-1935.  doi: 10.1137/19M1269105.

[65]

N. F. Mott, The theory of crystal rectifiers, World Scientific Series in 20th Century PhysicsSir Nevill Mott-65 Years in Physics, (1995), 153–165. doi: 10.1142/9789812794086_0013.

[66]

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), 1287-1305.  doi: 10.1016/S0006-3495(98)74048-2.

[67]

S. Y. NoskovS. Berneche and B. Roux, Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands, Nature, 431 (2004), 830-834.  doi: 10.1038/nature02943.

[68]

S. Y. Noskov and B. Roux, Ion selectivity in potassium channels, Biophys. Chem., 124 (2006), 279-291.  doi: 10.1016/j.bpc.2006.05.033.

[69]

J.-K. Park and J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609-630.  doi: 10.1137/S0036139995279809.

[70]

Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Phys. Rev. Lett., 63 (1989), 980-983.  doi: 10.1103/PhysRevLett.63.980.

[71]

Y. Rosenfeld, Free energy model for the inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas, J. Chem. Phys., 98 (1993), 8126-8148.  doi: 10.1063/1.464569.

[72]

D. J. Rouston, Bipolar Semiconductor Devices, McGraw-Hill, New York, 1990.

[73]

B. Roux, Theory of transport in ion channels: From molecular dynamics simulations to experiments, In Comp. Simul. In Molecular Biology, J. Goodefellow ed., VCH Weinheim, Ch., 6 (1995), 133–169.

[74]

B. RouxT. W. AllenS. Berneche and W. Im, Theoretical and computational models of biological ion channels, Quat. Rev. Biophys., 37 (2004), 15-103.  doi: 10.1017/S0033583504003968.

[75]

I. Rubinstein, Electro-Diffusion of Ions, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA, 1990. doi: 10.1137/1.9781611970814.

[76]

M. Saraniti, S. Aboud, S and R. Eisenberg, The simulation of ionic charge transport in biological ion channels: An introduction to numerical methods, Rev. Comp. Chem., 22 (2006), 229-294. doi: 10.1002/0471780367.ch4.

[77]

Z. SchussB. 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), 1-14.  doi: 10.1103/PhysRevE.64.036116.

[78]

A. SingerD. GillespieJ. 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), 541-560.  doi: 10.1017/S0956792508007596.

[79]

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), 949-968.  doi: 10.1137/070687037.

[80]

B. G. Streetman, Solid State Electronic Devices, 4$^{th}$ edition, Prentice-Hall, Englewood Cliffs, NJ, 1972.

[81]

L. Sun and W. Liu, Non-localness of excess potentials and boundary value problems of Poisson-Nernst-Planck Systems for ionic flow: A case study, J. Dyn. Diff. Equat., 30 (2018), 779-797.  doi: 10.1007/s10884-017-9578-2.

[82] C. Tanford and J. Reynolds, Nature's Robots: A History of Proteins, Oxford University Press, New York, 2001. 
[83]

N. Unwin, The structure of ion channels in membranes of excitable cells, Neuron, 3 (1989), 665-676.  doi: 10.1016/0896-6273(89)90235-3.

[84]

J. H. Vera and G. Wilczek-Vera, Classical Thermodynamics of Fluid Systems: Principles and Applications, Crc Press, 2016.

[85]

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), 022722, 14 pp. doi: 10.1103/PhysRevE.89.022722.

[86]

Z. WenP. W. Bates and M. Zhang, Effects on I-V relations from small permanent charge and channel geometry via classical Poisson-Nernst-Planck equations with multiple cations, Nonlinearity, 34 (2021), 4464-4502.  doi: 10.1088/1361-6544/abfae8.

[87]

Z. WenL. Zhang and M. Zhang, Dynamics of classical Poisson-Nernst-Planck systems with multiple cations and boundary layers, J. Dyn. Diff. Equat., 33 (2021), 211-234.  doi: 10.1007/s10884-020-09861-4.

[88]

L. ZhangB. Eisenberg and W. Liu, An effect of large permanent charge: Decreasing flux with increasing transmembrane potential, Eur. Phys. J. Special Topics, 227 (2019), 2575-2601.  doi: 10.1140/epjst/e2019-700134-7.

[89]

L. Zhang and W. Liu, Effects of large permanent charges on ionic flows via Poisson-Nernst-Planck models, SIAM J. Appl. Dyn. Syst., 19 (2020), 1993-2029.  doi: 10.1137/19M1289443.

[90]

M. Zhang, Asymptotic expansions and numerical simulations of I-V relations via a steady-state Poisson-Nernst-Planck system,, Rocky MT. J. Math., 45 (2015), 1681-1708.  doi: 10.1216/RMJ-2015-45-5-1681.

[91]

M. Zhang, Competition between cations via Poisson-Nernst-Planck systems with nonzero but small permanent charges, Membranes, 11 (2021), 236. 

[92]

M. Zhang, Boundary layer effects on ionic flows via classical Poisson-Nernst-Planck systems, Comput. Math. Biophys., 6 (2018), 14-27.  doi: 10.1515/cmb-2018-0002.

[93]

Q. Zheng and G. W. Wei, Poisson-Boltzmann-Nernst-Planck model, J. Chem. Phys., 134 (2011), 1-17.  doi: 10.1063/1.3581031.

[94]

S. ZhouZ. Wang and B. Li, Mean-field description of ionic size effects with nonuniform ionic sizes: A numerical approach, Phy. Rev. E, 84 (2011), 1-13.  doi: 10.1103/PhysRevE.84.021901.

Figure 1.  Schematic picture of a singular orbit $ \Gamma^0\cup\Lambda\cup \Gamma^1 $ over $ [0, 1] $ projected to the space of $ (u, z_1c_1+z_2c_2, \tau) $: a boundary layer $ \Gamma^0 $ at $ \tau = 0 $, a regular layer $ \Lambda $ on $ \mathcal{ Z} $ from $ \tau = 0 $ to $ \tau = 1 $, and a boundary layer $ \Gamma^1 $ at $ \tau = 1 $. Two boundary layers disappear if electroneutrality boundary conditions are assumed
Figure 2.  Numerical simulations for I-V relations with cations Li$ ^{+} $, Na$ ^{+} $ and K$ ^{+} $ and anion Cl$ ^{-} $ for the cases with $ L>R $ and $ L<R $, respectively
Figure 3.  Numerical simulations to investigate the effects on the individual flux from both the finite ion size and the diffusion coefficients under the zero-current condition
Figure 4.  Numerical simulations to investigate the effects on the individual flux from ion valences under the zero-current condition. The left column is with $ L>R $ while the right one is with $ L<R $
Table 1.  The values of the ionic size in this table is from Database of Ionic Radii from Imperial College London. The diffusion coefficients are taken from the thermodynamics database "phreeqc.dat"
ion species ionic radius (m) Diffusion coefficient (m$ ^2 $/s) ion valence
Li$ ^{+} $ $ 0.076\times 10^{-9} $ $ 1.03 \times 10^{-9} $ 1
Na$ ^{+} $ $ 0.102 \times 10^{-9} $ $ 1.33\times 10^{-9} $ 1
K$ ^{+} $ $ 0.138\times 10^{-9} $ $ 1.96\times 10^{-9} $ 1
Ca$ ^{2+} $ $ 0.114\times 10^{-9} $ $ 0.792\times 10^{-9} $ 2
Cl$ ^{-} $ $ 0.181\times 10^{-9} $ $ 2.03 \times 10^{-9} $ -1
ion species ionic radius (m) Diffusion coefficient (m$ ^2 $/s) ion valence
Li$ ^{+} $ $ 0.076\times 10^{-9} $ $ 1.03 \times 10^{-9} $ 1
Na$ ^{+} $ $ 0.102 \times 10^{-9} $ $ 1.33\times 10^{-9} $ 1
K$ ^{+} $ $ 0.138\times 10^{-9} $ $ 1.96\times 10^{-9} $ 1
Ca$ ^{2+} $ $ 0.114\times 10^{-9} $ $ 0.792\times 10^{-9} $ 2
Cl$ ^{-} $ $ 0.181\times 10^{-9} $ $ 2.03 \times 10^{-9} $ -1
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