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doi: 10.3934/dcdsb.2021312
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Boundary layer effects on 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

*Corresponding author: Mingji Zhang

Received  January 2021 Revised  December 2021 Early access January 2022

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

We study a one-dimensional Poisson-Nernst-Planck model with two oppositely charged particles, zero permanent charges and nonuniform finite ion sizes through a local hard-sphere model. Of particular interest is to examine the boundary layer effects on ionic flows systematically in terms of individual fluxes, the total flow rate of charges (current-voltage relations) and the total flow rate of matter. This is particularly 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. Several critical potentials are identified, which play unique and critical roles in examining the dynamics of ionic flows. Some can be estimated experimentally. Numerical simulations are performed for a better understanding and further illustrating our analytical results. We believe the analysis can provide complementary information of the qualitative properties of ionic flows and help one better understand the mechanism of ionic flow through membrane channels.

Citation: Jianing Chen, Mingji Zhang. Boundary layer effects on ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021312
References:
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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.

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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.

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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$^{rd}$ edtioon, 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. BazantK. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Review E, 70 (2004), 1-24.  doi: 10.1103/PhysRevE.70.021506.

[13]

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.

[14]

D. BodaD. BusathB. EisenbergD. Henderson and W. Nonner, Monte Carlo simulations of ion selectivity in a biological Na+ channel: Charge-space competition, Phys. Chem. Chem. Phys., 4 (2002), 5154-5160.  doi: 10.1039/B203686J.

[15]

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.

[16]

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.

[17]

B. Eisenberg, Proteins, channels, and crowded ions, Biophysical Chemistry, 100 (2003), 507-517. 

[18]

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

[19]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels, Advanced Series in Physical ChemistryRecent Developments in Theoretical Studies of Proteins, (1996), 269–357. doi: 10.1142/9789814261418_0005.

[20]

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.

[21]

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.

[22]

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.

[23]

D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic CDhannels, Thesis (Ph.D.)¨CRush University, College of Nursing. 1999.

[24]

D. Gillespie and R. S. Eisenberg, Modified Donnan potentials for ion transport through biological ion channels, Phys. Rev. E, 63 (2001), 061902.  doi: 10.1103/PhysRevE.63.061902.

[25]

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.

[26]

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.

[27]

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

[28]

D. GillespieW. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids, Phys. Rev. E, 68 (2003), 0313503.  doi: 10.1103/PhysRevE.68.031503.

[29]

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

[30]

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.

[31]

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), pp. 578a. doi: 10.1016/j.bpj.2010.12.3342.

[32]

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. 

[33]

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.

[34]

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

[35]

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. Dynam. Differential Equations, 24 (2012), 955-983.  doi: 10.1007/s10884-012-9277-y.

[36]

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.

[37]

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.

[38]

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.

[39]

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.

[40]

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.

[41]

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.

[42]

J. L. Liu and B. Eisenberg, Poisson-Nernst-Planck-Fermi theory for modeling biological ion channels, J. Chem. Phys., 141 (2014), 12B640. 

[43]

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Figure 1.  Graph of function $ F(x) $ as $ (\sigma, \rho)\to (1^+,1^{-}) $ with $ \sigma+\rho<2 $ helps understand the result stated in Lemma 3.2. To be specific, the graph corresponds to the statement (ii) of the lemma with $ \lambda>1 $
Figure 2.  Function $ J_{11}(V) $ (solid line) for $ x>x_2^* $ (left figure) and $ 1<x<x_2^* $ (right figure) with $ \sigma = 1.001 $ and $ \rho = 0.998 $, where $ x_2^* = 1.01978 $; and function $ J_{11}^{EN}(V) $ (dashed line with star) with $ \sigma = \rho = 1 $. In the left figure, the slope of $ J_{11} $ is $ 0.291 $ while the one of $ J_{11}^{EN} $ is $ 0.29 $
Figure 3.  Function $ J_{21}(V) $ (solid line) for $ x>x_2^* $ (left figure) and $ 1<x<x_2^* $ (right figure) with $ \sigma = 1.001 $ and $ \rho = 0.998 $, where $ x_2^* = 1.01978 $; and function $ J_{21}^{EN}(V) $ (dashed line with star) with $ \sigma = \rho = 1 $. In the left figure, the slope of $ J_{21} $ is $ -0.291 $ while the one of $ J_{21}^{EN} $ is $ -0.29 $
Figure 4.  Function $ I_{1}(V) $ (solid line) for $ x>x_2^{*} $ (left figure) and $ 1<x<x_2^{*} $ (right figure) with $ \sigma = 1.001 $ and $ \rho = 0.998 $, where $ x_2^* = 1.01978 $; and function $ I_{1}^{EN}(V) $ (dashed line with star) with $ \sigma = \rho = 1 $. In the left figure, the slope of $ I_{1} $ is $ 9.7945\times10^{-10} $ while the one of $ I_{1}^{EN} $ is $ 9.7617\times10^{-10} $
Figure 5.  Function $ T_{1}(V) $ for $ x>x_2^{*} $ (left figure) and $ 1<x<x_2^{*} $ (right figure)with $ \sigma = 1.001 $ and $ \rho = 0.998 $, where $ x_2^* = 1.01978 $; and function $ T_{1}^{EN}(V) $ (dashed line with star) with $ \sigma = \rho = 1 $. In the left figure, the slope of $ T_{1} $ is $ -2.0311\times 10^{-10} $ while the one of $ T_{1}^{EN} $ is $ -2.0243\times 10^{-10} $
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