Bulletin of Mathematical Biology

, Volume 41, Issue 3, pp 343–356 | Cite as

Analysis of potassium conductance in squid giant axons

  • Douglas K. McIlroy
Article

Abstract

Assuming a model of facilitated ionic transport across axonal membranes proposed by McIlroy (1975) and extended by McIlroy and Hahn (1978), it is shown that if the selectivity coefficient, πK, of the potassium conducting system ≃59 the permeabilityP Ks, of the periaxonal barrier of the squid giant axon for K+ ions≃(1.2±0.44)×10−4 cm sec−1 and the thickness of the periaxonal space ≃477±168 Å. Using a value (10−4 cm sec−1) ofP Ks in the foregoing range the experimental curves for the steady state membrane ionic conductance versus measured membrane potential difference (p.d.), ϕ, of Gilbert and Ehrenstein (1969) are corrected for the effect of accumulation of K+ in the periaxonal space. This correction is most marked for the axon immersed in a natural ionic environment, whose conductance curve is shifted ≃70mV along the voltage axis in the hyperpolarization direction.

By assuming that the physico-chemical connection between a depolarization of the axonal membrane and the consequent membrane conductance changes is a Wien dissociative effect of the membrane's electric field on a weak electrolyte situated in the axolemma, the position of the peaks of the corrected conductance versus ϕ curves can be identified with zero membrane electric field and hence with zero p.d.across the axolemma. A set of values for the double-layer p.d.s at the axonal membrane interfaces with the external electrolytes in the vicinity of the K+ conducting pores can therefore be deduced for the various external electrolytes employed by Gilbert and Ehrenstein. A model of these double-layer p.d.s in which the membrane interfaces are assumed to possess fixed monovalent negatively charged sites, at least in the neighbourhood of the K+ conducting pores, is constructed. It is shown that, using the previously deduced values for the doublelayer p.d.s, such a model has a consistent, physically realistic solution for the distance between the fixed charged sites and for the dissociation constants of these sites in their interaction with the ions of the extramembrane electrolytes.

Keywords

Giant Axon Potassium Conductance Conductance Curve Membrane Interface Axonal Membrane 

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Literature

  1. Adam, G. 1973. “The Effect of Potassium Diffusion through the Schwann Cell Layer on Potassium Conductance of the Squid Giant Axon.”J. Membrane Biol.,13, 353–386.CrossRefGoogle Scholar
  2. Adelman, W. A. Jr., Y. Palti and J. P. Senft. 1973. “Potassium Ion Accumulation in a Periaxonal Space and Its Effect on the Measurement of Membrane Potassium Ion Conductance”.J. Membrane Biol.,13, 387–410.CrossRefGoogle Scholar
  3. Bezanilla F. and C. M. Armstrong. 1972. “Negative Conductance Caused by Entry of Sodium and Cesium Ions into the Potassium Channels of Squid Axons”.J. Gen. Physiol.,60, 588–608.CrossRefGoogle Scholar
  4. Cole, K. S. 1969. “Zeta Potential and Discrete Versus Uniform Surface Charges”.Biophys. J.,9, 465–469.Google Scholar
  5. Ehrenstein, G. 1977. Private communication.Google Scholar
  6. Frankenhaeuser, B. and A. L. Hodgkin. 1956. “The After-Effects of Impulses in the Giant Nerve Fibres ofLoligo”.J. Physiol.,131, 341–376.Google Scholar
  7. — and —. 1957. “The Action of Calcium on the Electrical Properties of Squid Axons”.J. Physiol.,137, 218–244.Google Scholar
  8. Gilbert, D. L. and G. Ehrenstein. 1969. “Effect of Divalent Cations on Potassium Conductance of Squid Axons: Determination of Surface Charge”.Biophys. J. 9, 447–463.CrossRefGoogle Scholar
  9. Grahame, D. C. 1947. “The Electrical Double Layer and the Theory of Electrocapillarity”.Chem. Rev.,41, 441–501.CrossRefGoogle Scholar
  10. Hodgkin, A. L. 1964.The Conduction of the Nervous Impulse, p. 23. Liverpool: Liverpool University Press.Google Scholar
  11. — and A. F. Huxley. 1952a. “Currents Carried by Sodium and Potassium Ions Through the Membrane of the Giant Axon ofLoligo”.J. Physiol.,116, 449–472.Google Scholar
  12. — and —. 1952b. “The Components of Membrane Conductance in the Giant Axon ofLoligo”.J. Physol.,116, 473–496.Google Scholar
  13. — and —. 1952c. “The Dual Effect of Membrane Potential on Sodium Conductance in the Giant Axon ofLoligo”.J. Physiol.,116, 497–506.Google Scholar
  14. — and —. 1952d. “A Quantitative Description of Membrane Current and Its Applications to Conductance and Excitation in Nerve”.J. Physiol.,117, 500–554.Google Scholar
  15. Kay, G. W. C. and T. H. Laby, 1975.Tables of Physical and Chemical Constants and Some Mathematical Functions, 14th ed., p. 223. London and New York: Longman.Google Scholar
  16. Mason, D. P. and D. K. McIlroy. 1978. “A Perturbation Solution to the Problem of Wien Dissociation in Weak Electrolytes”.Proc. R. Soc. Lond. A. 359, 303–317.CrossRefGoogle Scholar
  17. McIlroy, D. K. 1970a. “A Mathematical Model of the Nerve Impulse at the Molecular Level”.Math. Biosci.,7, 313–328.CrossRefGoogle Scholar
  18. — 1970b. “Analysis of the Enzyme Model of the Nerve”.Math. Biosci.,8, 109–129.CrossRefGoogle Scholar
  19. — 1970c. “Deductions from the Enzyme Model of the Nerve”.Math. Biosci.,9, 135–146.CrossRefGoogle Scholar
  20. — 1975. “Electric Field Distributions in Neuronal Membranes”.Math. Biosci. 26, 191–206.MATHCrossRefGoogle Scholar
  21. — and B. D. Hahn. 1978. “Electric Field Distribution, Ionic Selectivity and Permeability in Nerve”.Bull. Math. Biol.,40, 637–649.MATHCrossRefGoogle Scholar
  22. Onsager, L. 1934. “Deviations from Ohm's Law in Weak Electrolyte”.J. Chem. Phys.,2, 599–615.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1979

Authors and Affiliations

  • Douglas K. McIlroy
    • 1
  1. 1.Department of Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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