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:
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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]

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[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]

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[6]

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[7]

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[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]

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[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]

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[16]

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[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

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