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doi: 10.3934/dcdsb.2022013
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Studies on reversal permanent charges and reversal potentials via classical Poisson-Nernst-Planck systems with boundary layers

1. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266510, China

2. 

School of Mathematics and Physics, University of South China, Hengyang, 421001, China

*Corresponding author: Chaohong Pan

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: L. Zhang and X. Liu are supported by the National Natural Science Foundation of China (No. 12172199, 12011530062, 11672270)

We consider a one-dimensional classical Poisson-Nernst-Planck model with two ion species, one positively charged and one negatively charged, and a simple profile of nonzero permanent charges. Of particular interest is to examine the effect from boundary layers on zero-current ionic flows in terms of reversal potentials and reversal permanent charges through membrane channels. This is important because boundary layers of charge are particularly likely to create artifacts over long distances, and this could dramatically affect the behavior of ionic flows. With boundary layers, the existence of reversal potentials and reversal permanent charges is established. Particularly, the reversal potentials are further compared with those identified under electroneutrality boundary conditions, and their orders are provided, which sensitively depends on the complicated nonlinear interaction among system parameters, particularly, boundary layers, boundary concentrations and channel geometry.

Citation: Lijun Zhang, Xiangshuo Liu, Chaohong Pan. Studies on reversal permanent charges and reversal potentials via classical Poisson-Nernst-Planck systems with boundary layers. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022013
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]

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

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

P. W. BatesZ. Wen and M. Zhang, Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations, J. Nonlinear Sci., 33 (2021), 1-62.  doi: 10.1007/s00332-021-09715-3.

[10]

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

[11]

B. Eisenberg, Ion Channels as Devices, J. Comput. Electro., 2 (2003), 245-249.  doi: 10.1023/B:JCEL.0000011432.03832.22.

[12]

R. S. Eisenberg, Channels as enzymes, J. Memb. Biol., 115 (1990), 1-12.  doi: 10.1007/BF01869101.

[13]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels, Recent Developments in Theoretical Studies of Proteins, World Scientific, Philadelphia, 1996,269–357. doi: 10.1142/9789814261418_0005.

[14]

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.

[15]

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.

[16]

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

[17]

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

[18]

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.

[19]

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

[20]

A. L. Hodgkin and R. D. Keynes, The potassium permeability of a giant nerve fibre, J. Physiol., 128 (1955), 61-88.  doi: 10.1113/jphysiol.1955.sp005291.

[21]

Y. HyonB. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci., 9 (2010), 459-475.  doi: 10.4310/CMS.2011.v9.n2.a5.

[22]

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.

[23]

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. 

[24]

S. JiB. Eisenberg and W. Liu, Flux ratios and channel structures, J. Dyn. Differ. Equ., 31 (2019), 1141-1183.  doi: 10.1007/s10884-017-9607-1.

[25]

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. Differ. Equ., 24 (2012), 955-983.  doi: 10.1007/s10884-012-9277-y.

[26]

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. Appl. Math., 75 (2015), 114-135.  doi: 10.1137/140992527.

[27]

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.

[28]

C.-C. LeeH. LeeY. HyonT.-C. Lin and C. Liu, New Poisson-Boltzmann type equations: one-dimensional solutions, Nonlinearity, 24 (2011), 431-458.  doi: 10.1088/0951-7715/24/2/004.

[29]

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.

[30]

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.

[31]

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

[32]

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. Differ. Equ., 24 (2012), 985-1004.  doi: 10.1007/s10884-012-9278-x.

[33]

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

[34]

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

[35]

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.

[36]

S. MafeJ. A. Manzanares and and J. Pellicer, On the introduction of the pore wall charge in the space-charge model for microporous membranes, Journal of Membrane Science, 51 (1990), 161-168.  doi: 10.1016/S0376-7388(00)80899-6.

[37]

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.

[38]

W. Nooner 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.

[39]

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.

[40]

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

[41]

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), 036116.  doi: 10.1103/PhysRevE.64.036116.

[42]

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.

[43]

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.

[44]

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 (1–14).

[45]

R. M. Jr. Warner, Microelectronics: Its unusual origin and personality, IEEE Trans. Electron. Devices, 48 (2001), 2457-2467.  doi: 10.1109/16.960368.

[46]

Z. WenP. 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.

[47]

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

[48]

E. B. Zambrowicz and M. Colombini, Zero-current potentials in a large membrane channel: A simple theory accounts for complex behavior, Biophysical Journal, 65 (1993), 1093-1100.  doi: 10.1016/S0006-3495(93)81148-2.

[49]

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.

[50]

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.

[51]

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

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]

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

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

P. W. BatesZ. Wen and M. Zhang, Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations, J. Nonlinear Sci., 33 (2021), 1-62.  doi: 10.1007/s00332-021-09715-3.

[10]

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

[11]

B. Eisenberg, Ion Channels as Devices, J. Comput. Electro., 2 (2003), 245-249.  doi: 10.1023/B:JCEL.0000011432.03832.22.

[12]

R. S. Eisenberg, Channels as enzymes, J. Memb. Biol., 115 (1990), 1-12.  doi: 10.1007/BF01869101.

[13]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels, Recent Developments in Theoretical Studies of Proteins, World Scientific, Philadelphia, 1996,269–357. doi: 10.1142/9789814261418_0005.

[14]

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.

[15]

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.

[16]

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

[17]

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

[18]

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.

[19]

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

[20]

A. L. Hodgkin and R. D. Keynes, The potassium permeability of a giant nerve fibre, J. Physiol., 128 (1955), 61-88.  doi: 10.1113/jphysiol.1955.sp005291.

[21]

Y. HyonB. Eisenberg and C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci., 9 (2010), 459-475.  doi: 10.4310/CMS.2011.v9.n2.a5.

[22]

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.

[23]

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. 

[24]

S. JiB. Eisenberg and W. Liu, Flux ratios and channel structures, J. Dyn. Differ. Equ., 31 (2019), 1141-1183.  doi: 10.1007/s10884-017-9607-1.

[25]

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. Differ. Equ., 24 (2012), 955-983.  doi: 10.1007/s10884-012-9277-y.

[26]

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. Appl. Math., 75 (2015), 114-135.  doi: 10.1137/140992527.

[27]

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.

[28]

C.-C. LeeH. LeeY. HyonT.-C. Lin and C. Liu, New Poisson-Boltzmann type equations: one-dimensional solutions, Nonlinearity, 24 (2011), 431-458.  doi: 10.1088/0951-7715/24/2/004.

[29]

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.

[30]

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.

[31]

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

[32]

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. Differ. Equ., 24 (2012), 985-1004.  doi: 10.1007/s10884-012-9278-x.

[33]

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

[34]

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

[35]

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.

[36]

S. MafeJ. A. Manzanares and and J. Pellicer, On the introduction of the pore wall charge in the space-charge model for microporous membranes, Journal of Membrane Science, 51 (1990), 161-168.  doi: 10.1016/S0376-7388(00)80899-6.

[37]

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.

[38]

W. Nooner 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.

[39]

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.

[40]

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

[41]

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), 036116.  doi: 10.1103/PhysRevE.64.036116.

[42]

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.

[43]

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.

[44]

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 (1–14).

[45]

R. M. Jr. Warner, Microelectronics: Its unusual origin and personality, IEEE Trans. Electron. Devices, 48 (2001), 2457-2467.  doi: 10.1109/16.960368.

[46]

Z. WenP. 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.

[47]

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

[48]

E. B. Zambrowicz and M. Colombini, Zero-current potentials in a large membrane channel: A simple theory accounts for complex behavior, Biophysical Journal, 65 (1993), 1093-1100.  doi: 10.1016/S0006-3495(93)81148-2.

[49]

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.

[50]

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.

[51]

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

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