Biomedical Engineering Reference
In-Depth Information
In 1905 Einstein recognized that mobility (
μ
p
) and the diffusion coefficient (
D
) are related by
D
|
Z
|
F
μ
p
=
(2.28)
RT
where
R
is the ideal gas constant,
T
is the temperature, and
F
is Faraday's constant. We can therefore,
rewrite the equation for
R
m
as
dx
RT
DCFZ
·
R
m
=
.
(2.29)
Substitution back into Eq. (2.26) yields
[
φ
i
−
φ
e
]
DCFZ
dx
I
φ
=
(2.30)
·
RT
DCFZ
RT
dφ
dx
.
I
φ
=
(2.31)
2.4.6 The Nernst Equation
Using Eqs. (2.24) and (2.31)
I
ion
=
I
c
+
I
φ
(2.32)
I
ion
=−
D
dC
.
CFZ
RT
dφ
dx
dx
+
(2.33)
=
At rest we know that
I
ion
0
D
dC
ZCF
RT
dφ
dx
0
=−
dx
+
(2.34)
and after some rearranging
ZF
ln
C
e
.
RT
V
rest
m
=
E
rest
=
(2.35)
C
i
Equation (2.35) is the
Nernst
equation that relates
V
res
m
to the difference in intracellular and extracellular
concentrations. In our derivation we assumed that
C
was a positive ion. The only different in Eq. (2.35)
for negative ions is that
C
e
and
C
i
are switched in the numerator and denominator. Figure 2.8 is the
circuit analog for the passive membrane where the Nernst potential is represented by a battery,
E
rest
.
The ionic current that flows through the resistor is described by
1
R
m
[
I
ion
=
V
m
−
E
rest
]
.
(2.36)
It can now be understood why the membrane settles to
V
rest
m
E
rest
.If
V
m
>V
rest
=
, then
I
ion
will be
m
positive. If
V
m
<V
rest
then
I
ion
is negative. Therefore,
E
rest
is sometimes also referred to as the
reversal
m
potential
.
