The ionic current across a membrane is obtained from the extended membrane equation applicable to carrier-mediated active transport as follows:
I=-
FβΔ
E [
PK (
CiKeβΔE/2-
CoKe-βΔE/2) +
PNa (
CiNae-βΔE/2) +
PC1 (
CoC1eβΔE/2)-φ(
eβΔ
E/2-
e-βΔE/2)-ψ
eβΔE/2]/2sinh βΔ
E/2, where
I is current, and Δ
E is voltage difference across the membrane; β=
F/RT, F, R, and
I have their usual thermodynamic meanings;
Pjj 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
ISCC=
Fψ.
From the phenomenological equations in irreversible thermodynamics, the cross coefficients
pij are represented as
Pj=Σ
n>
i=1Pij, (
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
PNa=1.53×10
-6cm/sec,
PK=0.158×10
-6cm/sec, and
PC1=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
zj 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=-
FβΔ
EΣ
nj=1Pi (
CijezjβΔE/2-
CojβΔE/2) /2 sinh
ZjβΔ
E/2=-
FβΔ
EΣ
nj=1PjCij+Coj/2-
FβΔ
E coth
ZjβΔ
E/2Σ
nj=1
ZjPjCij-Coj/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 Δ
E are the ionic current and the transmembrane potential, respectively;
Pj represents the permeability coefficient of the
j-th ion;
Cjj 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
Coj=
Cij (
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).
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