New insights into the effects of small permanent charge on ionic flows: A higher order analysis

Hamid Mofidi; Hamid Mofidi

doi:10.3934/mbe.2024266

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 5: 6042-6076. doi: 10.3934/mbe.2024266

Previous Article Next Article

Research article Special Issues

New insights into the effects of small permanent charge on ionic flows: A higher order analysis

Hamid Mofidi ^{1,2
,
,}

1.
Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing 101408, China
2.
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Academic Editor: Zhongwei Shen

Received: 05 April 2024 Revised: 13 May 2024 Accepted: 20 May 2024 Published: 24 May 2024

This study investigated how permanent charges influence the dynamics of ionic channels. Using a quasi-one-dimensional classical Poisson–Nernst–Planck (PNP) model, we investigated the behavior of two distinct ion species—one positively charged and the other negatively charged. The spatial distribution of permanent charges was characterized by zero values at the channel ends and a constant charge $ Q_0 $ within the central region. By treating the classical PNP model as a boundary value problem (BVP) for a singularly perturbed system, the singular orbit of the BVP depended on $ Q_0 $ in a regular way. We therefore explored the solution space in the presence of a small permanent charge, uncovering a systematic dependence on this parameter. Our analysis employed a rigorous perturbation approach to reveal higher-order effects originating from the permanent charges. Through this investigation, we shed light on the intricate interplay among boundary conditions and permanent charges, providing insights into their impact on the behavior of ionic current, fluxes, and flux ratios. We derived the quadratic solutions in terms of permanent charge, which were notably more intricate compared to the linear solutions. Through computational tools, we investigated the impact of these quadratic solutions on fluxes, current-voltage relations, and flux ratios, conducting a thorough analysis of the results. These novel findings contributed to a deeper comprehension of ionic flow dynamics and hold potential implications for enhancing the design and optimization of ion channel-based technologies.
- permanent charge,
- ionic flows,
- PNP,
- flux ratios,
- Ⅰ-Ⅴ relations
Citation: Hamid Mofidi. New insights into the effects of small permanent charge on ionic flows: A higher order analysis[J]. Mathematical Biosciences and Engineering, 2024, 21(5): 6042-6076. doi: 10.3934/mbe.2024266

Related Papers:

Abstract

This study investigated how permanent charges influence the dynamics of ionic channels. Using a quasi-one-dimensional classical Poisson–Nernst–Planck (PNP) model, we investigated the behavior of two distinct ion species—one positively charged and the other negatively charged. The spatial distribution of permanent charges was characterized by zero values at the channel ends and a constant charge $ Q_0 $ within the central region. By treating the classical PNP model as a boundary value problem (BVP) for a singularly perturbed system, the singular orbit of the BVP depended on $ Q_0 $ in a regular way. We therefore explored the solution space in the presence of a small permanent charge, uncovering a systematic dependence on this parameter. Our analysis employed a rigorous perturbation approach to reveal higher-order effects originating from the permanent charges. Through this investigation, we shed light on the intricate interplay among boundary conditions and permanent charges, providing insights into their impact on the behavior of ionic current, fluxes, and flux ratios. We derived the quadratic solutions in terms of permanent charge, which were notably more intricate compared to the linear solutions. Through computational tools, we investigated the impact of these quadratic solutions on fluxes, current-voltage relations, and flux ratios, conducting a thorough analysis of the results. These novel findings contributed to a deeper comprehension of ionic flow dynamics and hold potential implications for enhancing the design and optimization of ion channel-based technologies.

References

[1]	D. Boda, W. Nonner, M. Valisko, D. Henderson, B. Eisenberg, D. Gillespie, Steric selectivity in Na channels arising from protein polarization and mobile side chains, Biophys. J., 93 (2007), 1960–1980. https://doi.org/10.1529/biophysj.107.105478 doi: 10.1529/biophysj.107.105478
[2]	B. Hille, Ion Channels of Excitable Membranes, 3rd Edition, Sinauer Associates, Inc., Sunderland, Massachusetts, USA, 2001.
[3]	B. Eisenberg, Ion channels as devices, J. Comput. Electron., 2 (2003), 245–249. https://doi.org/10.1023/B:JCEL.0000011432.03832.22 doi: 10.1023/B:JCEL.0000011432.03832.22
[4]	B. Eisenberg, Proteins, channels, and crowded ions, Biophys. Chem., 100 (2003), 507–517. https://doi.org/10.1016/S0301-4622(02)00302-2 doi: 10.1016/S0301-4622(02)00302-2
[5]	A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764 doi: 10.1113/jphysiol.1952.sp004764
[6]	A. L. Hodgkin, R. D. Keynes, The potassium permeability of a giant nerve fibre, J. Physiol., 128 (1955), 61–88. https://doi.org/10.1113/jphysiol.1955.sp005291 doi: 10.1113/jphysiol.1955.sp005291
[7]	A. Malasics, D. Gillespie, W. Nonner, D. Henderson, B. Eisenberg, D. Boda, Protein structure and ionic selectivity in calcium channels: Selectivity filter size, not shape, matters, Biochim, J. Biophys. Acta, 1788 (2009), 2471–2480. https://doi.org/10.1016/j.bbamem.2009.09.022 doi: 10.1016/j.bbamem.2009.09.022
[8]	A. L. Hodgkin, A. F. Huxley, Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo, J. Physol., 116 (1952), 449–472. https://doi.org/10.1113/jphysiol.1952.sp004717 doi: 10.1113/jphysiol.1952.sp004717
[9]	A. L. Hodgkin, A. F. Huxley, The components of membrane conductance in the giant axon of Loligo, J. Physiol., 116 (1952), 473–496. https://doi.org/10.1113/jphysiol.1952.sp004718 doi: 10.1113/jphysiol.1952.sp004718
[10]	S. Ji, B. Eisenberg, W. Liu, Flux ratios and channel structures, J. Dyn. Differ. Equations, 31 (2019), 1141–1183. https://doi.org/10.1007/s10884-017-9607-1 doi: 10.1007/s10884-017-9607-1
[11]	X. Deng, Y. Jia, M. Zhang, Studies on current-voltage relations via Poisson-Nernst-Planck systems with multiple cations and small permanent charges, J. Appl. Anal. Comput., 12 (2022), 932–951. https://doi.org/10.11948/20210003 doi: 10.11948/20210003
[12]	Y. Wang, L. Zhang, M. Zhang, Mathematical analysis on current-voltage relations via classical Poisson-Nernst-Planck systems with nonzero permanent charges under relaxed electroneutrality boundary conditions, Membranes, 13 (2023), 131. https://doi.org/10.3390/membranes13020131 doi: 10.3390/membranes13020131
[13]	J. Chen, Y. Wang, L. Zhang, M. Zhang, Mathematical analysis of Poisson-Nernst-Planck models with permanent charges and boundary layers: studies on individual fluxes, Nonlinearity, 34 (2021), 3879–3906. https://doi.org/10.1088/1361-6544/abf33a doi: 10.1088/1361-6544/abf33a
[14]	B. Eisenberg, W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (2007), 1932–1966. https://doi.org/10.1137/060657480 doi: 10.1137/060657480
[15]	B. Eisenberg, W. Liu, H. Xu, Reversal permanent charge and reversal potential: Case studies via classical Poisson-Nernst-Planck models, Nonlinearity, 28 (2015), 103–128. https://doi.org/10.1088/0951-7715/28/1/103 doi: 10.1088/0951-7715/28/1/103
[16]	S. Ji, W. Liu, M. Zhang, Effects of (small) permanent charge and channel geometry on ionic flows via classical Poisson-Nernst-Planck models, SIAM J. Appl. Math., 75 (2015), 114–135. https://doi.org/10.1137/140992527 doi: 10.1137/140992527
[17]	W. Liu, Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math., 65 (2005), 754–766. https://doi.org/10.1137/S0036139903420931 doi: 10.1137/S0036139903420931
[18]	W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differ. Equations, 246 (2009), 428–451. https://doi.org/10.1016/j.jde.2008.09.010 doi: 10.1016/j.jde.2008.09.010
[19]	J. K. Park, J. W. Jerome, Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math., 57 (1997), 609–630. https://doi.org/10.1137/S0036139995279809 doi: 10.1137/S0036139995279809
[20]	M. Zhang, Competition between cations via classical Poisson-Nernst-Planck models with nonzero but small permanent charges, Membranes, 11 (2021), 236. https://doi.org/10.3390/membranes11040236 doi: 10.3390/membranes11040236
[21]	M. Zhang, Existence and local uniqueness of classical Poisson-Nernst-Planck systems with multi-component permanent charges and multiple cations, Discrete Contin. Dyn. Syst. - Ser. S, 16 (2023), 725–752. https://doi.org/10.3934/dcdss.2022134 doi: 10.3934/dcdss.2022134
[22]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems. Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1609 (1995), 44–118. https://doi.org/10.1007/BFb0095239
[23]	C. Kuehn, Multiple time scale dynamics, in Applied Mathematical Sciences, Springer, Cham, 191 (2015). https://doi.org/10.1007/978-3-319-12316-5
[24]	R. S. Eisenberg, Atomic biology, electrostatics and ionic channels, in New Developments and Theoretical Studies of Proteins, World Scientific, Philadelphia, (1996), 269–357.
[25]	Y. Fu, W. Liu, H. Mofidi, M. Zhang, Finite ion size effects on ionic flows via Poisson-Nernst-Planck systems: Higher order contributions, J. Dyn. Differ. Equations, 35 (2022), 1585–1609. https://doi.org/10.1007/s10884-021-10114-1 doi: 10.1007/s10884-021-10114-1
[26]	P. W. Bates, Z. Wen, M. Zhang, Small permanent charge effects on individual fluxes via Poisson-Nernst-Planck models with multiple cations, J. Nonlinear Sci., 31 (2021), 55. https://doi.org/10.1007/s00332-021-09715-3 doi: 10.1007/s00332-021-09715-3
[27]	W. Liu, A flux ratio and a universal property of permanent charges effects on fluxes, Comput. Math. Biophys., 6 (2018), 28–40. https://doi.org/10.1515/cmb-2018-0003 doi: 10.1515/cmb-2018-0003
[28]	Z. Wen, P. W. Bates, 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. https://doi.org/10.1088/1361-6544/abfae8 doi: 10.1088/1361-6544/abfae8
[29]	B. Eisenberg, W. Liu, H. Mofidi, Effects of diffusion coefficients on reversal potentials in ionic channels, preprint, arXiv: 2311.02895.
[30]	H. Mofidi, Geometric mean of concentrations and reversal permanent charge in Zero-Current ionic flows via Poisson-Nernst-Planck models, preprint, arXiv: 2009.09564.
[31]	H. Mofidi, Reversal permanent charge and concentrations in ionic flows via Poisson-Nernst-Planck models, Q. Appl. Math., 79 (2021), 581–600. https://doi.org/10.1090/qam/1593 doi: 10.1090/qam/1593
[32]	H. Mofidi, Bifurcation of flux ratio in ionic flows via a PNP model, preprint, arXiv: 2311.02895.
[33]	H. Mofidi, 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. https://doi.org/10.1137/19M1269105 doi: 10.1137/19M1269105
[34]	W. Liu, X. Tu, 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. Equations, 24 (2012), 985–1004. https://doi.org/10.1007/s10884-012-9278-x doi: 10.1007/s10884-012-9278-x
[35]	H. Mofidi, B. Eisenberg, W. Liu, Effects of diffusion coefficients and permanent charge on reversal potentials in ionic channels, Entropy, 22 (2020), 325. https://doi.org/10.3390/e22030325 doi: 10.3390/e22030325
[36]	L. Zhang, B. Eisenberg, W. Liu, An effect of large permanent charge: Decreasing flu with increasing transmembrane potential, Eur. Phys. J. Spec. Top., 227 (2019), 2575–2601. https://doi.org/10.1140/epjst/e2019-700134-7 doi: 10.1140/epjst/e2019-700134-7
[37]	W. M. Lee, Python Machine Learning, John Wiley & Sons, Inc., (2019), 1–296. https://doi.org/10.1002/9781119557500
[38]	Z. Schuss, B. Nadler, R. S. Eisenberg, Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model, Phys. Rev. E, 64 (2001), 1–14. https://doi.org/10.1103/PhysRevE.64.036116 doi: 10.1103/PhysRevE.64.036116
[39]	V. Barcilon, Ion flow through narrow membrane channels: Part Ⅰ, SIAM J. Appl. Math., 52 (1992), 1391–1404. https://doi.org/10.1137/0152080 doi: 10.1137/0152080
[40]	Y. Hyon, B. Eisenberg, C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci., 9 (2010), 459–475. https://doi.org/10.4310/CMS.2011.v9.n2.a5 doi: 10.4310/CMS.2011.v9.n2.a5
[41]	Y. Hyon, C. Liu, B. Eisenberg, PNP equations with steric effects: A model of ion flow through channels, J. Phys. Chem., 116 (2012), 11422–11441. https://doi.org/10.1021/jp305273n doi: 10.1021/jp305273n
[42]	P. M. Biesheuvel, Two-fluid model for the simultaneous flow of colloids and fluids in porous media, J. Colloid Interface Sci., 355 (2011), 389–395. https://doi.org/10.1016/j.jcis.2010.12.006 doi: 10.1016/j.jcis.2010.12.006
[43]	D. Chen, R. Eisenberg, J. Jerome, C. Shu, Hydrodynamic model of temperature change in open ionic channels, Biophys. J., 69 (1995), 2304–2322. https://doi.org/10.1016/S0006-3495(95)80101-3 doi: 10.1016/S0006-3495(95)80101-3
[44]	V. Sasidhar, E. Ruckenstein, Electrolyte osmosis through capillaries, J. Colloid Interface Sci., 82 (1981), 1439–1457. https://doi.org/10.1016/0021-9797(81)90386-6 doi: 10.1016/0021-9797(81)90386-6
[45]	R. J. Gross, J. F. Osterle, Membrane transport characteristics of ultra fine capillary, J. Chem. Phys., 49 (1968), 228–234. https://doi.org/10.1063/1.1669814 doi: 10.1063/1.1669814
[46]	W. Im, 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. https://doi.org/10.1016/S0022-2836(02)00778-7 doi: 10.1016/S0022-2836(02)00778-7
[47]	B. Roux, T. W. Allen, S. Berneche, W. Im, Theoretical and computational models of biological ion channels, Q. Rev. Biophys., 37 (2004), 15–103. https://doi.org/10.1017/S0033583504003968 doi: 10.1017/S0033583504003968
[48]	W. Nonner, 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. https://doi.org/10.1016/S0006-3495(98)74048-2 doi: 10.1016/S0006-3495(98)74048-2
[49]	W. Liu, B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dyn. Differ. Equations, 22 (2010), 413–437. https://doi.org/10.1007/s10884-010-9186-x doi: 10.1007/s10884-010-9186-x
[50]	Y. Wang, L. Zhang, M. Zhang, Studies on individual fluxes via Poisson-Nernst-Planck models with small permanent charges and partial electroneutrality conditions, J. Appl. Anal. Comput., 12 (2022), 87–105. https://doi.org/10.11948/20210045 doi: 10.11948/20210045
[51]	D. Gillespie, A Singular Perturbation Analysis of the Poisson-Nernst-Planck System: Applications to Ionic Channels, Ph.D thesis, Rush University at Chicago, 1999.
[52]	S. Ji, 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. Equations, 24 (2012), 955–983. https://doi.org/10.1007/s10884-012-9277-y doi: 10.1007/s10884-012-9277-y
[53]	W. Liu, H. Mofidi, Local Hard-Sphere Poisson-Nernst-Planck models for ionic channels with permanent charges, preprint, arXiv: 2203.09113.
[54]	W. Huang, W. Liu, Y. Yu, Permanent charge effects on ionic flow: a numerical study of flux ratios and their bifurcation, Commun. Comput. Phys., 30 (2021), 486–514. https://doi.org/10.4208/cicp.OA-2020-0057 doi: 10.4208/cicp.OA-2020-0057

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)