# American Institute of Mathematical Sciences

June 2018, 23(4): 1623-1643. doi: 10.3934/dcdsb.2018064

## Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions

 1 School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received  May 2017 Revised  September 2017 Published  February 2018

A quasi-one-dimensional steady-state Poisson-Nernst-Planck model with Bikerman's local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel is analyzed. Of particular interest is the qualitative properties of ionic flows in terms of individual fluxes without the assumption of electroneutrality conditions, which is more realistic to study ionic flow properties of interest. This is the novelty of this work. Our result shows that ⅰ) boundary concentrations and relative size of ion species play critical roles in characterizing ion size effects on individual fluxes; ⅱ) the first order approximation $\mathcal{J}_{k1} = D_kJ_{k1}$ in ion volume of individual fluxes $\mathcal{ J}_k = D_kJ_k$ is linear in boundary potential, furthermore, the signs of $\partial_V \mathcal{ J}_{k1}$ and $\partial^2_{Vλ} \mathcal{J}_{k1}$, which play key roles in characterizing ion size effects on ionic flows can be both negative depending further on boundary concentrations while they are always positive and independent of boundary concentrations under electroneutrality conditions (see Corollaries 3.2-3.3, Theorems 3.4-3.5 and Proposition 3.7). Numerical simulations are performed to identify some critical potentials defined in (2). We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

Citation: Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji 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 & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064
##### References:
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##### 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), 1507-1526. doi: 10.1137/070691322. [2] V. Barcilon, Ion flow through narrow membrane channels: Part Ⅰ, SIAM J. Appl. Math., 52 (1992), 1391-1404. doi: 10.1137/0152080. [3] D. Boda, D. Busath, B. Eisenberg, D. 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. [4] V. Barcilon, D.-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. 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), 631-648. doi: 10.1137/S0036139995312149. [6] M. Burger, R. S. Eisenberg and H. W. Engl, Inverse problems related to ion channel selectivity, SIAM J. Appl. Math., 67 (2007), 960-989. doi: 10.1137/060664689. [7] D. Boda, D. Gillespie, W. Nonner, D. Henderson and B. Eisenberg, Computing induced charges in inhomogeneous dielectric media: application in a Monte Carlo simulation of complex ionic systems, Phys. Rev. E, 69 (2004), 046702 (1-10). doi: 10.1103/PhysRevE.69.046702. [8] J. J. Bikerman, Structure and capacity of the electrical double layer, Philos. Mag., 33 (1942), 384-397. doi: 10.1080/14786444208520813. [9] P. W. Bates, Y. Jia, G. Lin, H. 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. Bates, W. Liu, H. Lu and M. Zhang, Ion size and valence effects on ionic flows via Poisson-Nernst-Planck models, Commu. Math. Sci., 15 (2017), 881-901. doi: 10.4310/CMS.2017.v15.n4.a1. [11] J. H. Chaudhry, S. D. Bond and L. 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Xu, Reversal permanent charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models, Nonlinearity, 28 (2015), 103-127. [23] A. Ern, R. Joubaud and T. Leliévre, Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes, Nonlinearity, 25 (2012), 1635-1652. doi: 10.1088/0951-7715/25/6/1635. [24] 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), 1909-1913. doi: 10.1063/1.454114. [25] 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. [26] D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels, Ph. D Dissertation, Rush University at Chicago, 1999. [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), 12129-12145. doi: 10.1088/0953-8984/14/46/317. [28] D. Gillespie, W. Nonner and R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids Phys. Rev. E, 68 (2003), 0313503 (1-10). doi: 10.1103/PhysRevE.68.031503. [29] D. Gillespie, W. Nonner and R. S. Eisenberg, Crowded charge in biological ion channels, Nanotech, 3 (2003), 435-438. [30] E. Gongadze, U. van Rienen, V. Kralj-lglič and A. lglič, Spatial variation of permittivity of an electrolyte solution in contact with a charged metal surface: A mini review, Computer Method in Biomechanics and Biomedical Engineering, 16 (2013), 463-480. [31] 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), 15598-15610. doi: 10.1021/jp052471j. [32] U. Hollerbach, D.-P. 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Numerical detection of critical values for $\lambda$ (left graph) and $m$ (right one) in Theorem 3.2.
Numerical detection of critical values for $\sigma$, which corresponds to statement (Ⅰ) in Theorem 3.2. The left graph is for $\lambda<\lambda_1^* = 0.072$, and the right one is for $\lambda>\lambda_2^* = 13.93.$
Numerical detection of critical values $\sigma$ corresponding to statement (Ⅱ) in Theorem 3.2 with $\lambda_1^*<\lambda<\lambda_2^*$. The left graph is for $0<m<m^* = 0.4934$, and the right one is for $m^*<m<\frac{1}{2}.$
Numerical identification of six critical potentials in (15) with $z_1 = -z_2 = 1$. In the left column, the vertical axis actually represents, from top to bottom, ${\mathcal I}(V;\nu, \lambda)-{\mathcal I}_0(V), \ {\mathcal J}_1(V;\nu, \lambda)-{\mathcal J}_{10}(V)$ and ${\mathcal J}_2(V;\nu, \lambda)-{\mathcal J}_{20}(V)$, respectively. In particular, the x-axis for all figures actually represents $\frac{e}{k_BT}V$.
Numerical identification of critical potentials $V_{1c}$ and $V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.
Numerical approximations of critical potentials $V_1^c$ for individual flux ${\mathcal J}_1$ with $z_1 = -z_2 = 1$ and nonzero permanent charge as illustrated in Proposition 3.14. The x-axis for all figures actually represents $\frac{e}{k_BT}V$.
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