Biomedical Engineering Reference
In-Depth Information
where
J
is the flow of ions due to diffusion,
½
is the ion concentration,
dx
is the membrane
is the diffusivity constant in m
2
/s. The negative sign indicates that the
flow of ions is from higher to lower concentration, and
d
½
I
dx
thickness, and
D
represents the concentration
gradient. Fick's Law of diffusion was described in Section 7.3 involving a first-order
differential equation that describes change in concentration as a function of time. Here,
we are only interested in steady state.
Ohm's Law
Charged particles in a solution experience a force resulting from other charged particles
and electric fields present. The flow of ions across a membrane is given by
Z
½
dv
dx
J
ð
drift
Þ¼
m
ð
12
:
2
Þ
!
, m
mobility in m
2
/sV,
where
J
is the flow of ions due to drift in an electric field
E
¼
Z
¼
ionic valence,
½
is the ion concentration,
v
is the voltage across the membrane, and
!
). Note that
Na
þ
and
dv
dx
is (
E
Z
is positive for positively charged ions (e.g., Z
¼
1 for
Ca
þ2
) and negative for negatively charged ions (e.g., Z
Cl
). Positive ions
Z
¼
2 for
¼
1for
drift down the electric field and negative ions drift up the electric field.
Figure 12.7 illustrates a cell membrane that is permeable to only
K
þ
and shows the
K
þ
isthatofaneuronwithahigher
concentration inside than outside and that the membrane resting potential is negative
from inside to outside. Clearly, only
K
þ
. Assume that the concentration of
forces acting on
K
þ
can pass through the membrane, and
Na
þ
,
Cl
,
A
cannot move through it, since there are no channels for them to pass through.
Depending on the actual concentration and membrane potential,
and
K
þ
will pass through
themembraneuntiltheforcesduetodriftanddiffusion are balanced. The chemical force
due to diffusion from inside to outside decreases as
K
þ
moves through the membrane,
K
þ
accumulates outside the cell until the two forces
and the electric force increases as
are balanced.
K
+
Cl
−
K
+
K
+
Na
+
Outside
Cl
−
Inside
K
+
K
+
K
+
A
−
FIGURE 12.7
K
þ
due to drift and diffusion across a cell membrane that is only
The direction of the flow of
K
þ
.
permeable to
