ANALYSIS OF MEMBRANE PERMEABILITY COEFFICIENTS OF AMPHIBIAN SKIN BY MEANS OF ELECTRONIC DATA PROCESSING

II. ABSOLUTE VALUES OF THE PERMEABILITY AND CROSS COEFFICIENTS FROM THE CURRENT-VOLTAGE RELATION

Abstract

The ionic current across a membrane is obtained from the extended membrane equation applicable to carrier-mediated active transport as follows:
I=-FβΔE [P_K (Cⁱ_Ke^βΔE/2-C^o_Ke^-βΔE/2) +P_Na (Cⁱ_Nae^-βΔE/2) +P_C1 (C^o_C1eβΔE/2)-φ(eβΔE/2-e^-βΔE/2)-ψe^βΔE/2]/2sinh βΔE/2, where I is current, and ΔE is voltage difference across the membrane; β=F/RT, F, R, and I have their usual thermodynamic meanings; P_jj represents the permeability coefficient of the j-th ion; φ is carrier flux, and ψ is active-transport flux. Short circuit current is given by I_SCC=Fψ.
From the phenomenological equations in irreversible thermodynamics, the cross coefficients p_ij are represented as P_j=Σⁿ>_i=1P_ij, (j=1, 2, ., n).
The current-voltage curve was observed on the abdominal skin of frogs and toads and simulated using a digital computer system. Applying the two-membrane theory, the permeability coefficients of the outer membrane of the frog skin obtained are P_Na=1.53×10^-6cm/sec, P_K=0.158×10^-6cm/sec, and P_C1=1.24×10^-6cm/sec.
Straight and cross coefficients are computed for Na⁺, K⁺, and Cl^- ions.
The relation between current carried by ions of z_j valence and that of membrane potential was represented by GOLDMAN (1943) and HODGKIN and KATZ (1949) using the following assumption concerning the constant electric field: I=-FβΔEΣⁿ_j=1P_i (C^ije^z_j^βΔE/2-C^o_j^βΔE/2) /2 sinh Z_jβΔE/2=-FβΔEΣⁿ_j=1P_jCⁱ_j+C^o_j/2-FβΔE coth Z_jβΔE/2Σⁿ_j=1Z_jP_jCⁱ_j-C^o_j/2, (1) where β=F/RT; F, R, and T have their usual thermodynamic meanings; the origin is set at the halfway point through the membrane; I and ΔE are the ionic current and the transmembrane potential, respectively; P_j represents the permeability coefficient of the j-th ion; C_jj is the concentration; and the superscripts o and i stand for the bulk phase of the outer and inner sides of the membrane, respectively.
It is obvious that the linear current-voltage relation according to Ohm's law is obtained from Eq.(1) when the ionic concentrations in the bathing solution of both sides of the membrane are equal C^o_j=Cⁱ_j (j=1, 2, ., n). In such bulk solution, e.g., in Ringer's solution, the nonlinear current-voltage curve of isolated frog skin has been described by FINKELSTEIN (1964).