Fractal, Chaotic, and Self-Organizing Critical System: Descriptions of the Kinetics of Cell Membrane Ion Channels

  • Larry S. Liebovitch
  • Ferenc P. Czegledy
Part of the NATO ASI Series book series (NSSB, volume 270)


Channels are proteins in the cell membrane that spontaneously fluctuate between conformational shapes that are closed or open to the passage of ions. The kinetics of these changes in conformational state can be described in different ways, that suggest different physical properties for the ion channel protein. We describe kinetic models based on: 1) random switching between a few independent states, 2) random switching between many states that are cooperatively linked together, 3) deterministic, chaotic, nonlinear oscillations, amplifying themselves until the channel switches states, and 4) deterministic local interactions that self-organize the fluctuations in channel structure near a phase transition, switching it between different states.


Conformational State Fractal Scaling Activation Energy Barrier Random Switching State Markov Model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alcala, J. R., Gratton, E., and Prendergast, F. G., 1987, Interpretation of fluorescence decays in proteins using continuous lifetime distributions, Biophys. J., 51:925–936.PubMedCrossRefGoogle Scholar
  2. Austin, R. H., Beeson, L., Eisenstein, H., Frauenfelder, H., and Gunsalus, I. C., 1975, Dynamics of ligand binding to myoglobin, Biochem., 14:5355–5373.CrossRefGoogle Scholar
  3. Bak, P., Chao, T., and Wiesenfeld, K., 1988, Scale invariant spatial and temporal fluctuations in complex systems, in “Random Fluctuations and Pattern Growth: Experiments and Models,” H. E. Stanley and N. Ostrowsky, eds., Kluwer, Boston.Google Scholar
  4. Blatz, A. L. and Magleby, K. L., 1986, Quantitative description of three modes of activity of fast chloride channels from rat skeletal muscle, J. Physiol. (Lond.)., 378:141–174.Google Scholar
  5. Croxton, T. L., 1988, A model of the gating of ion channels, Biochem. Biophys. Acta, 946:19–24.PubMedCrossRefGoogle Scholar
  6. French, A. S. and Stockbridge, L. L., 1988, Fractal and Markov behavior in ion channel kinetics, Can. J. Physiol. Pharm., 66:967–970.CrossRefGoogle Scholar
  7. Gleick, J., 1987, “Chaos, Making a New Science,” Viking, New York.Google Scholar
  8. Guckenheimer, J. and Holmes, P., 1983, “Non-linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,” Springer-Verlag, New York.Google Scholar
  9. Hodgkin, A. L. and Huxley, A. F., 1952, A quantitative description of membrane current and its application to conductance and excitation in nerve. J. Physiol. (Lond.)., 117:500–544.Google Scholar
  10. Karplus, M. and McCammon, J. A., 1981, The internal dynamics of globular proteins, CRC Crit. Rev. Biochem., 9:293–349.PubMedCrossRefGoogle Scholar
  11. Läuger, P., 1988, Internal motions in proteins and gating kinetics of ionic channels, Biophys. J., 53:877–884.PubMedCrossRefGoogle Scholar
  12. Levitt, D. G., 1989, Continuum model of voltage dependent gating, Biophys. J., 55:489–498.PubMedCrossRefGoogle Scholar
  13. Liebovitch, L. S., 1989, Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies, Math. Biosci., 93:97–115.PubMedCrossRefGoogle Scholar
  14. Liebovitch, L. S. and Czegledy, F. P., 1990, A model of ion channel kinetics based on deterministic chaotic motion in a potential with two local minima, submitted to Ann. Biomed. Engr. Google Scholar
  15. Liebovitch, L. S., Fischbarg, J., and Koniarek, J. P., 1987, Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes, Math. Biosci., 84:37–68.CrossRefGoogle Scholar
  16. Liebovitch, L. S. and Sullivan, J. M., 1987, Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons, Biophys. J., 52:979–988.PubMedCrossRefGoogle Scholar
  17. Liebovitch, L. S. and Toth, T. I., 1990a, Fractal activity in cell membrane ion channels, in “Mathematical Approaches to Cardiac Arryhthmias,” J. Jalife, ed., Ann. New York, Acad. Sci., 591:375–391.Google Scholar
  18. Liebovitch, L. S. and Töth, T. I., 1990b, Distributions of activation energy barriers that produce stretched exponential probability distributions for the time spent in each state of the two state reaction A⇋B, Bull. Math. Biol., in press.Google Scholar
  19. Liebovitch, L. S. and Toth, T. I., 1990c, A model of ion channel kinetics using deterministic rather than stochastic processes, J. Theor. Biol., in press.Google Scholar
  20. McGee Jr., R., Sansom, M. S. P., and Usherwood, P. N. R., 1988, Characterization of a delayed rectified K+ channel in NG108–15 neuroblastoma x glioma cells, J. Memb. Biol., 102:21–34.CrossRefGoogle Scholar
  21. Millhauser, G. L., Salpeter, E. E., and Oswald, R. E., 1988, Diffusion models of ion-channel gating and the origin of the power-law distributions from single-channel recording, Proc. Natl. Acad. Sci., U.S.A., 85:1503–1507.PubMedCrossRefGoogle Scholar
  22. Moon, F. C., 1987, “Chaotic Vibrations,” Wiley, New York.Google Scholar
  23. Osborne, A. R. and Provenzale, A., 1989, Finite correlation dimension for stochastic systems with power-law spectra, Physica D, 35:357–381.CrossRefGoogle Scholar
  24. Sakmann, B. and Neher, E., eds., 1983, “Single-Channel Recording,” Plenum, New York.Google Scholar
  25. Scanlan, R. H. and Vellozzi, J. W., 1980, Catastrophic and annoying responses of long-span bridges to wind action, in “Long-Span Bridges,” E. Cohen and B. Birdsall, eds., Ann. New York, Acad. Sci., 352:247–263.Google Scholar
  26. Welch, G. R., ed., 1986, “The Fluctuating Enzyme,” Wiley, New York.Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Larry S. Liebovitch
    • 1
  • Ferenc P. Czegledy
    • 1
  1. 1.Department of Ophthalmology, College of Physicians & SurgeonsColumbia UniversityNew YorkUSA

Personalised recommendations