Abstract
The transport equation of Kedem and Katchalsky for the flux of ions through a membrane is generalized to demonstrate explicitly the role of impermeant ions in determining its mathematical form. Whereas the Kedem-Katchalsky equation is linear in the salt concentrations in the bathing solutions, the more general equation is bilinear (and symmetric) in the ionic concentrations of the permeant species. The Kedem-Katchalsky flux equation is further generalized to include explicitly a term for ion-exchange in systems having more than a single permeant salt. This additional term is also bilinear (and antisymmetric) in the concentrations of the exchanging ionic species. Flux equations are derived for systems having (1) a single mono-monovalent salt, (2) two mono-monovalent salts and (3) an arbitrary number of salts with no restriction upon the valencies of the ionic components. Since it has no effect upon the form of concentration-dependent terms in the flux equations, coupling to volume flow is neglected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature
Bearman, R. J. and J. G. Kirkwood. 1958. “Statistical Mechanics of Transport Processes: XI. Equations of Transport in Multi-component Systems”.J. chem. Phys. 28, 136–145.
Katchalsky, A. and P. F. Curran. 1965.Nonequilibrium Thermodynamics in Biophysics. Harvard Univ. Press, Cambridge.
Kedem, O. and A. Katchalsky. 1958. “Thermodynamic, Analysis of the Permeability of Biological Membranes to Non-electrolytes.”Biochem. Biophys. Acta 27, 229–246.
— and — 1963. “Permeability of Composite Membranes: 1. Electric Current, Volume Flow and Flow of Solute through Membranes”.Trans. Faraday Soc. 59, 1918–1930.
Kirkwood, J. G. 1954. “Transport of Ions through Biological Membranes from the Standpoint of Irreversible Thermodynamics”. InIon Transport Across Membranes, Ed. H. T. Clark, pp. 119–127. New York: Academic Press.
Mikulecky, D. C. 1969. “Biological Aspects of Transport”. InTransport Phenomena in Fluids, Ed. H. J. M. Hanley, pp. 433–494. New York: Marcel Dekker.
— 1977. “A Simple Network Thermodynamic Method for Series-Parallel Coupled Flows: II. The Non-linear Theory, With Applications to Coupled Solute and Volume Flow in a Series Membrane”.J. theor. Biol. 69, 511–541.
— 1979. “A Network Thermodynamic Two-port Element to Represent the Coupled Flow of Salt and Current: Improved Alternative for the Equivalent Circuit”.Biophys. J. 25, 323–339.
Richardson, I. W. 1971. “Ionic flows through a Single Homogeneous Membrane”.J. Membrane Biol. 4, 3–15.
— 1973. “Bi-ionic Flows through a Single Homogeneous Membrane”.Bull. math. Biol. 35: 95–100.
Spiegler, K. S. 1958. “Transport Processes in Ionic Membranes”.Trans. Faraday Soc. 54, 1408–1428.
Thomas, R. S. and D. C. Mikulecky. 1978. Transcapillary Solute Exchange: a Comparison of the Kedem-Katchalsky Convection-Diffusion Equations with the Rigorous Nonlinear Equations for this Special Case”.Microvasc. Res. 15, 207–220.
Truesdell, C. 1969.Rational Thermodynamics. McGraw-Hill, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Richardson, I.W., Foster, E.A.D. & Miekisz, S. Nonlinear generalizations of the Kedem-Katachalsky equations for ionic fluxes. Bltn Mathcal Biology 44, 761–775 (1982). https://doi.org/10.1007/BF02465179
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02465179