American Institute of Mathematical Sciences

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September  2014, 19(7): 1837-1853. doi: 10.3934/dcdsb.2014.19.1837

Effect of intracellular diffusion on current--voltage curves in potassium channels

Daniele Andreucci 1, , Dario Bellaveglia 1, , Emilio N.M. Cirillo 1, and  Silvia Marconi 1,

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, I-00161, Roma, Italy, Italy, Italy, Italy

Received  April 2013 Revised  July 2013 Published  August 2014

We study the effect of intracellular ion diffusion on ionic currents permeating through the cell membrane. Ion flux across the cell membrane is mediated by specific channels, which have been widely studied in recent years with remarkable results: very precise measurements of the true current across a single channel are now available. Nevertheless, a complete understanding of this phenomenon is still lacking, though molecular dynamics and kinetic models have provided partial insights. In this paper we demonstrate, by analyzing the KcsA current-voltage currents via a suitable lattice model, that intracellular diffusion plays a crucial role in the permeation phenomenon. We believe that the interplay between the channel behavior and the ion diffusion in the cell is a key ingredient for a full explanation of the current-voltage curves.
Keywords: current--voltage curves, diffusion., KcsA, selection, Potassium channel.
Mathematics Subject Classification: Primary: 92B05, 60G50; Secondary: 92C3.
Citation: Daniele Andreucci, Dario Bellaveglia, Emilio N.M. Cirillo, Silvia Marconi. Effect of intracellular diffusion on current--voltage curves in potassium channels. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1837-1853. doi: 10.3934/dcdsb.2014.19.1837
References:
[1]

A. Abenavoli, M. L. Di Francesco, I. Schroeder, S. Epimoshko, S. Gazzarini, U. P. Hansen G. Thiel and A. Moroni, Fast and slow gating are inherent properties of the pore module of the K$^+$ channel Kcv,, J. Gen. Physiol., 134 (2009), 219.  doi: 10.1085/jgp.200910266.  Google Scholar

[2]

D. Andreucci, D. Bellaveglia, E. N. M. Cirillo and S. Marconi, Monte Carlo study of gating and selection in potassium channels,, Physical Review E, 84 (2011).  doi: 10.1103/PhysRevE.84.021920.  Google Scholar

[3]

D. Andreucci, D. Bellaveglia, E. N. M. Cirillo and S. Marconi, Flux through a time-periodic gate: Monte Carlo test of a Homogenization result,, Simultech 2013, (2013), 626.   Google Scholar

[4]

D. Andreucci and D. Bellaveglia, Permeability of interfaces with alternating pores in parabolic problems,, Asymptotic Analysis, 79 (2012), 189.   Google Scholar

[5]

J. rAqvist and V. Luzhkov, Ion permeation mechanism of the potassium channel,, Nature, 404 (2000), 881.   Google Scholar

[6]

S. Bernèche and B. Roux, A microscopic view of ion conduction through the K$^+$ channel,, PNAS, 100 (2003), 8644.   Google Scholar

[7]

S. Chowdhuri and A. Chandra, Molecular dynamics simulations of aqueous NaCl and KCl solutions: Effects of ion concentration on the single-particle, pair, and collective dynamical properties of ions and water molecules,, J. Chem. Phys., 115 (2001), 3732.  doi: 10.1063/1.1387447.  Google Scholar

[8]

E. N. M. Cirillo and A. Muntean, Dynamics of pedestrians in regions with no visibility - a lattice model without exclusion,, Physica A, 392 (2013), 3578.  doi: 10.1016/j.physa.2013.04.029.  Google Scholar

[9]

E. N. M. Cirillo and A. Muntean, Can cooperation slow down emergency evacuations?,, Comptes Rendus Mecanique, 340 (2012), 626.  doi: 10.1016/j.crme.2012.09.003.  Google Scholar

[10]

M. Ciszkowska, L. Zeng, E. O. Stejskal and J. G. Osteryoung, Transport of Thallium(I) Counterion in Polyelectrolyte Solution Determined by Voltammetry with Microelectrodes and by Pulsed-Field-Gradient, Spin-Echo NMR,, J. Phys. Chem., 99 (1995), 11764.  doi: 10.1021/j100030a022.  Google Scholar

[11]

E. L. Cussler, Diffusion,, Second Edition, (1997).  doi: 10.1017/CBO9780511805134.  Google Scholar

[12]

D. A. Doyle, C. J. Morais, R. A. Pfuetzner, A. kuo, J. M. Gulbis, S. L. Cohen, B. T. Chait and R. MacKinnon, The structure of the potassium channel: Molecular basis of K$^+$ conduction and selectivity,, Science, 280 (1998), 69.  doi: 10.1126/science.280.5360.69.  Google Scholar

[13]

J.-F. Dufréche, O. Bernard, S. Durand-Vidal and P. Turq, Analytical Theories of Transport in Concentrated Electrolyte Solutions from the MSA,, J. Phys. Chem. B, 109 (2005), 9873.   Google Scholar

[14]

D. Fedida and J. C. Hesketh, Gating of voltage-dependent potassium channels,, Prog. Bio. Mol. Biology, 75 (2001), 165.  doi: 10.1016/S0079-6107(01)00006-2.  Google Scholar

[15]

S. A. N. Goldstein, D. Bockenhauer, I. O'Kelly and N. Zilberberg, Potassium leak channels and the Kcnk family of two-p-domain subunits,, Nature Reviews Neuroscience, 2 (2001), 175.   Google Scholar

[16]

G. Grimmet and D. Stirzaker, Probability and Random Processes,, Oxford University Press Inc., (2001).   Google Scholar

[17]

H. S. Harned and J. A. Shropshire, The Diffusion Coefficietn at $25^o$ of Potassium Chloride at Low Concentrations in $0.25$ Molar Aqueous Sucrose Solutions,, J. of the American Chemical Society, 80 (1958), 5652.   Google Scholar

[18]

L. Heginbotham and R. MacKinnon, Conduction Properties of the Cloned Shaker K$^+$ Channel,, Biophysical Journal, 65 (1993), 2089.  doi: 10.1016/S0006-3495(93)81244-X.  Google Scholar

[19]

B. Hille, Ion Channels of Excitable Membranes,, Third Edition, (2001).   Google Scholar

[20]

A. L. Hodgkin and A. F. Huxley, The components of membrane conductance in the giant axon of Loligo,, J. Physiol., 116 (1952), 473.   Google Scholar

[21]

A. L. Hodgkin and R. D. Keynes, The potassium permeability of a giant nerve fibre,, J. Physiol., 128 (1955), 61.   Google Scholar

[22]

A. L. Hodgkin and R. D. Keynes, The mobility and diffusion coefficient of potassium in giant axons from SEPIA,, J. Physiol., 119 (1953), 513.   Google Scholar

[23]

S. Imai, M. Osawa, K. Takeichi and I. Shimada, Structural basis underlying the dual gate properties of KcsA,, PNAS, 107 (2010), 6216.  doi: 10.1073/pnas.0911270107.  Google Scholar

[24]

M. LeMasurier, L. Heginbotham and C. Miller, Kcsa: It's a Potassium Channel,, J. Gen. Physiol., 118 (2001), 303.  doi: 10.1085/jgp.118.3.303.  Google Scholar

[25]

S. Mafé and J. Pellicer, Ion conduction in the KcsA potassium channel analyzed with a minimal kinetic model,, Phy. Rev. E, 71 (2005).   Google Scholar

[26]

C. Miller, An overview of the potassium channel family,, Genome Biology, 1 (2000).   Google Scholar

[27]

C. Miller, Ionic hopping defended,, J. Gen. Physiol., 113 (1999), 783.  doi: 10.1085/jgp.113.6.783.  Google Scholar

[28]

E. Neher and B. Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibres,, Nature, 260 (1976), 799.  doi: 10.1038/260799a0.  Google Scholar

[29]

P. H. Nelson, A permeation theory for single-file ion channels: Corresponding occupancy states produce Michaelis-Menten behavior,, J. Chem. Phys., 117 (2002), 11396.  doi: 10.1063/1.1522709.  Google Scholar

[30]

P. H. Nelson, Modeling the concentration-dependent permeation modes of the KcsA potassium ion channel,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.061908.  Google Scholar

[31]

P. H. Nelson, A permeation theory for single-file ion channels: One- and two-step models,, J. Chem. Phys., 134 (2011).  doi: 10.1063/1.3580562.  Google Scholar

[32]

M. Recanatini, A. Cavalli and M. Masetti, Modeling hERG and its interactions with drugs: Recent advances in light of current potassium channel simulations,, Chem. Med. Chem., 3 (2008), 523.  doi: 10.1002/cmdc.200700264.  Google Scholar

[33]

M. L. Renart, E. Montoya, A. M. Fernández, M. L. Molina, J. A. Poveda, J. A. Encinar, J. L. Ayala, A. V. Ferrer-Montiel, J. Gómez, A. Morales and J. M. González Ros, Controbution of Ion inding Affinity to Ion Selectivity and Permeation in KcsA, a Model Potassium Channel,, Biochemistry, 51 (2012), 3891.   Google Scholar

[34]

I. Schroeder, U. P. Hansen, Saturation and Microsecond Gating of Current Indicate Depletion-induced Instability of the MaxiK Selectivity Filter,, J. Gen. Physiol., 130 (2007), 83.  doi: 10.1085/jgp.200709802.  Google Scholar

[35]

A. M. J. VanDongen, Structure and Function of Ion Channels: A Hole in Four?,, Comm. Theor. Biol., 2 (1992), 429.   Google Scholar

[36]

A. M. J. VanDongen, K channel gating by an affinity-switching selectivity filter,, PNAS, 101 (2004), 3248.  doi: 10.1073/pnas.0308743101.  Google Scholar

[37]

Y. Zhou and R. Mackinnon, The occupancy of ions in the K$^+$ Selectivity Filter: Charge balance and coupling of ion binding to a protein conformational change underlie high conduction rates,, J. Mol. Biol., 333 (2003), 965.  doi: 10.1016/j.jmb.2003.09.022.  Google Scholar

show all references

References:
[1]

A. Abenavoli, M. L. Di Francesco, I. Schroeder, S. Epimoshko, S. Gazzarini, U. P. Hansen G. Thiel and A. Moroni, Fast and slow gating are inherent properties of the pore module of the K$^+$ channel Kcv,, J. Gen. Physiol., 134 (2009), 219.  doi: 10.1085/jgp.200910266.  Google Scholar

[2]

D. Andreucci, D. Bellaveglia, E. N. M. Cirillo and S. Marconi, Monte Carlo study of gating and selection in potassium channels,, Physical Review E, 84 (2011).  doi: 10.1103/PhysRevE.84.021920.  Google Scholar

[3]

D. Andreucci, D. Bellaveglia, E. N. M. Cirillo and S. Marconi, Flux through a time-periodic gate: Monte Carlo test of a Homogenization result,, Simultech 2013, (2013), 626.   Google Scholar

[4]

D. Andreucci and D. Bellaveglia, Permeability of interfaces with alternating pores in parabolic problems,, Asymptotic Analysis, 79 (2012), 189.   Google Scholar

[5]

J. rAqvist and V. Luzhkov, Ion permeation mechanism of the potassium channel,, Nature, 404 (2000), 881.   Google Scholar

[6]

S. Bernèche and B. Roux, A microscopic view of ion conduction through the K$^+$ channel,, PNAS, 100 (2003), 8644.   Google Scholar

[7]

S. Chowdhuri and A. Chandra, Molecular dynamics simulations of aqueous NaCl and KCl solutions: Effects of ion concentration on the single-particle, pair, and collective dynamical properties of ions and water molecules,, J. Chem. Phys., 115 (2001), 3732.  doi: 10.1063/1.1387447.  Google Scholar

[8]

E. N. M. Cirillo and A. Muntean, Dynamics of pedestrians in regions with no visibility - a lattice model without exclusion,, Physica A, 392 (2013), 3578.  doi: 10.1016/j.physa.2013.04.029.  Google Scholar

[9]

E. N. M. Cirillo and A. Muntean, Can cooperation slow down emergency evacuations?,, Comptes Rendus Mecanique, 340 (2012), 626.  doi: 10.1016/j.crme.2012.09.003.  Google Scholar

[10]

M. Ciszkowska, L. Zeng, E. O. Stejskal and J. G. Osteryoung, Transport of Thallium(I) Counterion in Polyelectrolyte Solution Determined by Voltammetry with Microelectrodes and by Pulsed-Field-Gradient, Spin-Echo NMR,, J. Phys. Chem., 99 (1995), 11764.  doi: 10.1021/j100030a022.  Google Scholar

[11]

E. L. Cussler, Diffusion,, Second Edition, (1997).  doi: 10.1017/CBO9780511805134.  Google Scholar

[12]

D. A. Doyle, C. J. Morais, R. A. Pfuetzner, A. kuo, J. M. Gulbis, S. L. Cohen, B. T. Chait and R. MacKinnon, The structure of the potassium channel: Molecular basis of K$^+$ conduction and selectivity,, Science, 280 (1998), 69.  doi: 10.1126/science.280.5360.69.  Google Scholar

[13]

J.-F. Dufréche, O. Bernard, S. Durand-Vidal and P. Turq, Analytical Theories of Transport in Concentrated Electrolyte Solutions from the MSA,, J. Phys. Chem. B, 109 (2005), 9873.   Google Scholar

[14]

D. Fedida and J. C. Hesketh, Gating of voltage-dependent potassium channels,, Prog. Bio. Mol. Biology, 75 (2001), 165.  doi: 10.1016/S0079-6107(01)00006-2.  Google Scholar

[15]

S. A. N. Goldstein, D. Bockenhauer, I. O'Kelly and N. Zilberberg, Potassium leak channels and the Kcnk family of two-p-domain subunits,, Nature Reviews Neuroscience, 2 (2001), 175.   Google Scholar

[16]

G. Grimmet and D. Stirzaker, Probability and Random Processes,, Oxford University Press Inc., (2001).   Google Scholar

[17]

H. S. Harned and J. A. Shropshire, The Diffusion Coefficietn at $25^o$ of Potassium Chloride at Low Concentrations in $0.25$ Molar Aqueous Sucrose Solutions,, J. of the American Chemical Society, 80 (1958), 5652.   Google Scholar

[18]

L. Heginbotham and R. MacKinnon, Conduction Properties of the Cloned Shaker K$^+$ Channel,, Biophysical Journal, 65 (1993), 2089.  doi: 10.1016/S0006-3495(93)81244-X.  Google Scholar

[19]

B. Hille, Ion Channels of Excitable Membranes,, Third Edition, (2001).   Google Scholar

[20]

A. L. Hodgkin and A. F. Huxley, The components of membrane conductance in the giant axon of Loligo,, J. Physiol., 116 (1952), 473.   Google Scholar

[21]

A. L. Hodgkin and R. D. Keynes, The potassium permeability of a giant nerve fibre,, J. Physiol., 128 (1955), 61.   Google Scholar

[22]

A. L. Hodgkin and R. D. Keynes, The mobility and diffusion coefficient of potassium in giant axons from SEPIA,, J. Physiol., 119 (1953), 513.   Google Scholar

[23]

S. Imai, M. Osawa, K. Takeichi and I. Shimada, Structural basis underlying the dual gate properties of KcsA,, PNAS, 107 (2010), 6216.  doi: 10.1073/pnas.0911270107.  Google Scholar

[24]

M. LeMasurier, L. Heginbotham and C. Miller, Kcsa: It's a Potassium Channel,, J. Gen. Physiol., 118 (2001), 303.  doi: 10.1085/jgp.118.3.303.  Google Scholar

[25]

S. Mafé and J. Pellicer, Ion conduction in the KcsA potassium channel analyzed with a minimal kinetic model,, Phy. Rev. E, 71 (2005).   Google Scholar

[26]

C. Miller, An overview of the potassium channel family,, Genome Biology, 1 (2000).   Google Scholar

[27]

C. Miller, Ionic hopping defended,, J. Gen. Physiol., 113 (1999), 783.  doi: 10.1085/jgp.113.6.783.  Google Scholar

[28]

E. Neher and B. Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibres,, Nature, 260 (1976), 799.  doi: 10.1038/260799a0.  Google Scholar

[29]

P. H. Nelson, A permeation theory for single-file ion channels: Corresponding occupancy states produce Michaelis-Menten behavior,, J. Chem. Phys., 117 (2002), 11396.  doi: 10.1063/1.1522709.  Google Scholar

[30]

P. H. Nelson, Modeling the concentration-dependent permeation modes of the KcsA potassium ion channel,, Phys. Rev. E, 68 (2003).  doi: 10.1103/PhysRevE.68.061908.  Google Scholar

[31]

P. H. Nelson, A permeation theory for single-file ion channels: One- and two-step models,, J. Chem. Phys., 134 (2011).  doi: 10.1063/1.3580562.  Google Scholar

[32]

M. Recanatini, A. Cavalli and M. Masetti, Modeling hERG and its interactions with drugs: Recent advances in light of current potassium channel simulations,, Chem. Med. Chem., 3 (2008), 523.  doi: 10.1002/cmdc.200700264.  Google Scholar

[33]

M. L. Renart, E. Montoya, A. M. Fernández, M. L. Molina, J. A. Poveda, J. A. Encinar, J. L. Ayala, A. V. Ferrer-Montiel, J. Gómez, A. Morales and J. M. González Ros, Controbution of Ion inding Affinity to Ion Selectivity and Permeation in KcsA, a Model Potassium Channel,, Biochemistry, 51 (2012), 3891.   Google Scholar

[34]

I. Schroeder, U. P. Hansen, Saturation and Microsecond Gating of Current Indicate Depletion-induced Instability of the MaxiK Selectivity Filter,, J. Gen. Physiol., 130 (2007), 83.  doi: 10.1085/jgp.200709802.  Google Scholar

[35]

A. M. J. VanDongen, Structure and Function of Ion Channels: A Hole in Four?,, Comm. Theor. Biol., 2 (1992), 429.   Google Scholar

[36]

A. M. J. VanDongen, K channel gating by an affinity-switching selectivity filter,, PNAS, 101 (2004), 3248.  doi: 10.1073/pnas.0308743101.  Google Scholar

[37]

Y. Zhou and R. Mackinnon, The occupancy of ions in the K$^+$ Selectivity Filter: Charge balance and coupling of ion binding to a protein conformational change underlie high conduction rates,, J. Mol. Biol., 333 (2003), 965.  doi: 10.1016/j.jmb.2003.09.022.  Google Scholar

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