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@V
where E D
@x is the electric field and z is the valence of the ion, that is ˙1; ˙2,
etc. and parameter is the so-called mobility and is measured in cm 3 /V s. The
aggregate flow through the membrane is given by the relation
J total DD @ŒC
@x
z ŒC @ @x
(1.5)
The diffusion coefficient that appears in the aforementioned equations is given by
kT
q
D D
(1.6)
where k is the Boltzmann constant, T denotes the absolute temperature, and q is the
electric charge (measured in Coulombs). Therefore the aggregate flux can be written
as
kT
q
@ŒC
@x
z ŒC @ @x
J total D
(1.7)
Multiplying this current flux with the valence and Faraday's constant and using
RT =F in place of kT =q one has [ 16 , 65 ]
zRT @ŒC
@x
z 2 FŒC @ @x
I D
(1.8)
This is the Nernst-Planck equation describing current flow through the membrane.
Nernst's equation is obtained setting the current in Nernst-Planck equation to be
equal to zero (I D 0). That is, at equilibrium condition of the diffusion and electric
effects is reached. Thus, after intermediate operations one has the Nernst potential
equation
z F ln ŒC in
ŒC out
RT
V eq D V in V out D
(1.9)
1.1.2
The Goldman-Hodgkin-Katz Equation
Under the assumption that the electric field across the lipid membrane remains
constant one has [ 16 , 65 ]
V M
l
V M
l
dV
dx
E D
and
D
(1.10)
The mobility of ions across the membrane is denoted by u . The aqueous concen-
tration of ions is denoted by ŒC and the concentration of ions on the membrane is
denoted as ˇŒC. The Nernst-Planck equation for current flow across the membrane
becomes
u zRT ˇ dŒC
dx
I D u z 2 FˇŒC V l
0<x<l
(1.11)
 
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