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 and Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022
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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), 1507-1526. 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]

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.

[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), 631-648. doi: 10.1137/S0036139995312149.

[5]

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.

[6]

J. J. Bikerman, Structure and capacity of the electrical double layer, Philos. Mag., 33 (1942), 384-397. doi: 10.1080/14786444208520813.

[7]

P. W. Bates, W. Liu, H. Lu and M. Zhang, Ion size and valence effaces on ionic flows via Poisson-Nernst-Planck systems, Commun. Math. Sci., to appear.

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

[9]

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.

[10]

R. D. Coalson, Discrete-state model of coupled ion permeation and fast gating in ClC chloride channels, J. Phys. A, 41 (2009), 115001, 15pp. doi: 10.1088/1751-8113/41/11/115001.

[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), 81-93. doi: 10.1109/TNB.2004.842495.

[12]

B. Deng, Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal., 21(1990), 693-720. doi: 10.1137/0521037.

[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), 104104. doi: 10.1063/1.3476262.

[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), p515a.

[15]

B. Eisenberg, Ion channels as devices, J. Comp. Electro., 2 (2003), 245-249. doi: 10.1023/B:JCEL.0000011432.03832.22.

[16]

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

[17]

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

[18]

R. S. Eisenberg, Atomic biology, electrostatics and ionic channels, In New Developments and Theoretical Studies of Proteins, R. Elber, Editor, World Scientific, Philadelphia, 7 (1996), 269-357. doi: 10.1142/9789814261418_0005.

[19]

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

[20]

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.

[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), 103-127. doi: 10.1088/0951-7715/28/1/103.

[22]

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.

[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), 1909-1913. doi: 10.1063/1.454114.

[24]

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

[25]

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.

[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]

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