Skip to main content
SpringerLink
Log in

Description of interacting channel gating using a stochastic Markovian model

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Single-channel recordings from membrane patches frequently exhibit multiple conductance levels. In some preparations, the steady-state probabilities of observing these levels do not follow a binomial distribution. This behavior has been reported in sodium channels, potassium channels, acetylcholine receptor channels and gap junction channels. A non-binomial distribution suggests interaction of the channels or the presence of channels with different open probabilities. However, the current trace sometimes exhibits single transitions spanning several levels. Since the probability of simultaneous transitions of independent channels is infinitesimally small, such observations strongly suggest a cooperative gating behavior. We present a Markov model to describe the cooperative gating of channels using only the all-points current amplitude histograms for the probability of observing the various conductance levels. We investigate the steady-state (or equilibrium) properties of a system ofN channels and provide a scheme to express all the probabilities in terms of just two parameters. The main feature of our model is that lateral interaction of channels gives rise to cooperative gating. Another useful feature is the introduction of the language of graph theory which can potentially provide a different avenue to study ion channel kinetics. We write down explicit expressions for systems of two, three and four channels and provide a procedure to describe the system ofN channels.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Ansari, A., J. Berendzeu, S. Bowne, H. Frauenfelder, I. Ibeu, T. Sanke, E. Shyamsunder and R. Young. 1985. Protein states and protein quakes.Proc. Natl. Acad. Sci. U.S.A. 82 5000–5004.

    Article  Google Scholar 

  • Bezanilla, F., E. Perozo, D. M. Papazian and E. Stefani. 1991. Molecular basis of gating charge immobolization in shaker potassium channels.Science 254, 679–683.

    Google Scholar 

  • Brink, P. R. and S-F. Fan. 1989. Patch clamp recordings from membranes which contain gap junction channels.Biophys. J. 56, 579–593.

    Google Scholar 

  • Clay, J. R. and L. J. DeFelice. 1983. Relationship between membrane excitability and single channel open-close recording.Biophys. J.,42, 151–157.

    Google Scholar 

  • Colquhoun, D. and A. G. Hawkes. 1977. Relaxation and fluctuations of membrane currents that flow through drug-operated ion channels.Proc. Roy. Soc. London Ser. B 199, 231–262.

    Google Scholar 

  • Colquhoun, D. and A. G. Hawkes. 1982. On the stochastic properties of bursts of single ion channel openings and of clusters of bursts.Phil. Trans. Roy. Soc. London Ser. B 300, 1–59.

    Google Scholar 

  • Colquhoun, D. and A. G. Hawkes. 1987. A note on correlations in single ion channel records.Proc. Roy. Soc. London Ser. B 230, 15–52.

    Google Scholar 

  • Colquhoun, D. and A. G. Hawkes. 1990. Stochastic properties of ion channel openings and bursts in a membrane patch that contains two channels: evidence concerning the number of channels present when a record containing only single openings is observed.Proc. Roy. Soc. London Ser. B 240, 453–477.

    Google Scholar 

  • Draber, S., R. Schultze and U.-P. Hansen. 1993. Cooperative behavior ofK + channels in the tonoplast ofChara corallina.Biophys. J. 65, 1553–1559.

    Google Scholar 

  • Edeson, R. D., G. F. Yeo, R. K. Milne and B. W. Madson. 1990. Graphs random sums and sojourn time distributions with application to ion-channel modeling.Math. Biosci. 102, 75–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Fredkin, D. R., M. Montal and J. A. Rice. 1985. Identification of aggregated Markovian models: Application to the nicotinic acetylcholine receptor. InProceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. M. Le Cam and R. A. Olshen (Eds), Vol. 1, pp. 269–289. Belmont, CA: Wadsworth.

    Google Scholar 

  • Fredkin, D. R. and J. A. Rice. 1986. On aggregated Markov process.J. Appl. Probab. 23, 208–214.

    Article  MATH  MathSciNet  Google Scholar 

  • Grigolini, P. and F. Marchesoni. 1985. Memory function approach to stochastic problems in condensed matter.Advances in Chemical Physics, M. W. Evans, P. Grigolini, G. Pastori Parravicini, I. Prigogine and S. A. Rice (Eds), Vol. 62. New York: Wiley.

    Google Scholar 

  • Hamill, O. P. and B. Sakmann. 1981. Multiple conductance states of single acetylcholine receptor channels in embryonic muscle cells.Nature 294, 462–464.

    Article  Google Scholar 

  • Harary, F. 1972.Graph Theory. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Hille, B. 1992.Ionic Channels of Excitable Membranes. Sunderland, MA: Sinauer.

    Google Scholar 

  • Hodgkin, A. L. and A. F. Huxley. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve.J. Physiol. (London) 117, 500–544.

    Google Scholar 

  • Hoh, J. H., R. Lal, S. A. John, J.-P. Revel and M. F. Arnsdorf. 1991. Atomic force microscopy and dissection of gap junctions.Science 253, 1405–1408.

    Google Scholar 

  • Hoh, J. H., G. E. Sosinsky, J.-P. Revel and P. K. Hansma. 1993. Structure of the extracellular surface of the gap junction by atomic force microscopy.Biophys. J. 65, 149–163.

    Google Scholar 

  • Horn, R. 1991. Estimating the number of channels in patch recordings.Biophys. J. 60, 433–439.

    Google Scholar 

  • Horn, R. and S. Korn. 1989. Model selection: reliability and bias.Biophys. J. 55, 379–381.

    Google Scholar 

  • Horn, R. and K. Lange, 1983. Estimating kinetic constants from single channel data.Biophys. J. 43, 207–223.

    Google Scholar 

  • Horn, R. and C. A. Vandenberg. 1989. Statistical properties of single sodium channels.J. Gen. Physiol. 84, 505–534.

    Article  Google Scholar 

  • Iwasa K., G. Ehrenstein, N. Moran and M. Jia. 1986. Evidence for interactions between batrachotoxin-modified channels in hybrid neuroblastoma cells.Biophys. J. 50, 531–537.

    Google Scholar 

  • Jackson M. B. 1985. Stochastic behavior of a many channel membrane system.Biophys. J. 47, 129–137.

    Google Scholar 

  • Kerry, C. J., R. L. Ramsey, M. S. P. Sansom and P. N. R. Usherwood. 1988. Glutamate receptor channel kinetics. The effect of glutamate concentration.Biophys. J. 53, 39–52.

    Google Scholar 

  • Kiss, T. and K. Nagy. 1985. Interactions between sodium channels in mouse neuroblastoma cells.Eur. Biophys. J. 12, 13–18.

    Article  Google Scholar 

  • Liebovitch, L. S. 1989. Testing fractal and Markov models of ion channel kinetics.Biophys. J. 55, 373–377.

    Google Scholar 

  • Liu, Q-Y., F. A. Lai, E. Rousseau, R. V. Jones and G. Meissner. 1989. Multiple conductance states of the purified calcium release channel complex from skeletal sarcoplasmic reticulum.Biphys. J. 55, 415–424.

    Google Scholar 

  • Liu, Y. and J. P. Dilger. 1993. Application of the one- and two-dimensional Ising model to studies of cooperativity between ion channels.Biophys. J. 64, 26–35.

    Google Scholar 

  • Magleby, K. L. 1992. Preventing artifacts and reducing errors in single-channel analysis.Methods in Enzymology,207, 763–791.

    Article  Google Scholar 

  • Makowski, L., D. L. D. Caspar, W. C. Phillips and D. A. Goodenough. 1977. Gap junction structures II. Analysis of the X-ray diffraction data.J. Cell. Biol. 74, 629–645.

    Article  Google Scholar 

  • Manivannan, K., S. V. Ramanan, R. T. Mathias and P. R. Brink, 1990. Single and multichannel recordings from gap junction membranes.Biophys. J. 57, 224A.

    Google Scholar 

  • Manivannan, K., S. V. Ramanan, R. T. Mathias and P. R. Brink. 1992. Multichannel recordings from membranes which contain gap junctions.Biophys. J. 61, 216–227.

    Google Scholar 

  • Mathias, R. T. and J. L. Rae. 1989. Cell to cell communication in lens. InCell Interactions and Gap Junctions, N. Sperelakis and C. C. Cole (Eds), Vol. 1, pp. 29–50. Boca Raton, FL: CRC Press.

    Google Scholar 

  • McGeoch, M. W. and J. E. McGeogh. 1994. Power spectra and cooperativity of a calcium-regulated cation channel.Biophys. J. 66, 161–168.

    Google Scholar 

  • McManus, O. B., C. E. Spivak, A. L. Blatz, D. S. Weiss and K. L. Magleby. 1989. Fractal models, markov models, and channel kinetics.Biophys. J. 55, 383–385.

    Article  Google Scholar 

  • McManus, O. B. and K. L. Magleby. 1989. Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activatesK channel.J. Gen. Physiol. 94, 1037–1070.

    Article  Google Scholar 

  • Monod, J., J. Wyman and J.-P. Changeux. 1965. On the nature of allosteric transitions: a sensible model.J. Mol. Biol. 12, 88–118.

    Article  Google Scholar 

  • Neher, E. and B. Sakmann. 1976. Single-channel currents recorded from membrane of denervated frog muscle fibres.Nature 260, 799–801.

    Article  Google Scholar 

  • Neumcke, B. and R. Stampfli. 1983. Alteration of the conductance of Na+ channels in the nodal membrane of frog nerve by holding potential and tetrodotoxin.Biochem. Biophys. Acta 272, 177–184.

    Google Scholar 

  • Pallota, B. S. 1985. Calcium-activated potassium channels in rat muscle inactivate from a short-duration open state.J. Physiol. (London) 363, 501–516.

    Google Scholar 

  • Patlak, J. and R. Horn. 1982. The effect ofN-bromoacetamide on single sodium channel currents in excised membrane patches.J. Gen. Physiol. 79, 333–351.

    Article  Google Scholar 

  • Pauling, L. 1935. The oxygen equilibrium of hemoglobin and its structural interpretation.Proc. Natl. Acad. Sci. U.S.A. 21, 186–191.

    Article  Google Scholar 

  • Ramanan, S. V., K. Manivannan, R. T. Mathias and P. R. Brink. 1993. Evidence for heterogeneous channel behavior in gap junctions. InProgress in Cell Research, J. E. Hall, G. A. Zampighi and R. M. Davis, Eds. Vol. 3, pp. 121–125. Amsterdam: Elservier.

    Google Scholar 

  • Richard, E. A. and C. Miller. 1990. Steady-state coupling of ion-channel conformations to a transmembrane ion gradient.Science 247, 1208–1210.

    Google Scholar 

  • Sakmann, B. and E. Neher (Eds). 1983.Single-Channel Recordings. New York: Plenum Press.

    Google Scholar 

  • Steinberg, I. Z. 1987. Relationship between statistical properties of single ionic channel recordings and the thermodynamic state of the channels.J. Theor. Biol. 124, 71–87.

    Article  Google Scholar 

  • Tytgat, J. and P. Hess. 1992. Evidence for cooperative interactions in potassium channel gating.Nature 359, 420–423.

    Article  Google Scholar 

  • Tytgat, J., K. Nakazawa and P. Hess. 1993. Cooperative and non-cooperative subunit interactions determine voltage-dependentK + channel gating.Biophys. J. 64, A226.

    Google Scholar 

  • Unwin, P. N. T. and G. Zampighi. 1980. Structure of the gap junction between communicating cells.Nature,283, 545–549.

    Article  Google Scholar 

  • Yeramian, E., A. Trautmann and P. Claverie. 1986. Acetylcholine receptors are not functionally independent.Biophys. J. 50, 253–263.

    Google Scholar 

Download references

Author information

Author notes

  1. K. Manivannan

    Present address: Department of Chemistry and Physics, Southeastern Louisiana University, 70402, Hammond, LA

Authors and Affiliations

  1. Department of Physiology and Biophysics, State University of New York, 11794, Stony Brook, New York, U.S.A.

    K. Manivannan & R. T. Mathias

  2. Department of Applied Science and Department of Chemistry, Brookhaven National Laboratory, 11973, Upton, New York, U.S.A.

    E. Gudowska-Nowak

Authors
  1. K. Manivannan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. T. Mathias
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Gudowska-Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Manivannan, K., Mathias, R.T. & Gudowska-Nowak, E. Description of interacting channel gating using a stochastic Markovian model. Bltn Mathcal Biology 58, 141–174 (1996). https://doi.org/10.1007/BF02458286

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02458286

Keywords

Search

Navigation