Biomedical Engineering Reference
In-Depth Information
Einstein Relationship
The relationship between the drift of particles in an electric field under osmotic pressure
described by Einstein in 1905 is given by
D
¼
KT
m
ð
12
:
3
Þ
q
where
is the diffusivity constant, m is mobility,
is Boltzmann's constant,
is the abso-
D
K
T
lute temperature in degrees Kelvin, and
is the magnitude of the electric charge (i.e.,
q
10
19
coulombs).
1.60186
12.4.2 Resting Potential of a Membrane Permeable to One Ion
The flow of ions in response to concentration gradients is limited by the selectively
permeable nerve cell membrane and the resultant electric field. As described, ions
pass through channels that are selective for that ion only. For clarity, the case of a mem-
brane permeable to one ion only is considered first and then the case of a membrane perme-
able to more than one ion follows. It is interesting to note that neuroglial cells are permeable
to only
Cl
. As will be shown, the
normal ionic gradient is maintained if the membrane is permeable only to
K
þ
and that nerve cells are permeable to
K
þ
,
Na
þ
, and
K
þ
as in the
neuroglial cell.
Consider the cell membrane shown in Figure 12.7 that is permeable only to
K
þ
, and
K
þ
is higher in the intracellular fluid than the extracellular
fluid. For this situation, the flow due to diffusion (concentration gradient) tends to push
assume that the concentration of
K
þ
outside of the cell and is given by
diffusion
Þ¼
D
d
½
K
þ
dx
J
K
(
ð
12
:
4
Þ
K
þ
inside the cell and is given by
The flow due to drift (electric field) tends to push
Z
½
K
þ
dv
dx
J
K
(
drift
Þ¼
m
ð
12
:
5
Þ
which results in a total flow
drift
Þ¼
D
d
½
K
þ
Z
½
K
þ
dv
dx
J
K
¼
J
K
(
diffusion
Þþ
J
K
(
dx
m
ð
12
:
6
Þ
D
¼
KT
m
Using the Einstein relationship
, the total flow is now given by
q
m
d
½
K
þ
J
K
¼
KT
q
Z
½
K
þ
dv
dx
dx
m
ð
12
:
7
Þ
K
þ
is found at any time for any given set of initial conditions. In
the special case of steady state—that is, at steady state when the flow of
From Eq. (12.7), the flow of
K
þ
into the cell is
exactly balanced by the flow out of the cell or
J
K
¼
0—Eq. (12.7) reduces to
