Week 6 Review
What was covered:
- Ion transport
- Active transport
- Steady state, rest, quasi- equilibrium, and equilibrium
- Indirect and direct effect of active transport
- Circuit review
Ion transport:
inside outside
f
The membrane is only permeable to ion n that has a valence, z
n
. (Valence is the amount
of charge on one ion.)
So if there is a difference in concentration between the inside and the outside, the ions are
going to move down the concentration gradient because of diffusion . But this will begin
to make charg e build up on the membrane, that effect, drift , will cause the ions to flow in
the opposite direction. Therefore, the flux will be due to both diffusion and drift.
Flux is due to 2 phenomena:
Diffusion Drift
f = -D
n
?c ( x , t )
- m
n
z
n
Fc
n
( x , t )
?y ( x , t )
?x ?x
Because the movement of charged transport is easier to think about in terms of current,
we convert flux to current density.
J z F x
n
( , )
n n
n
( , )
n n
2
( , )
( ,
Of course there is still the Continuity Equation:
?J
n
( x , t )
= - z F
?c
n
( x , t )
n
?x ?t
If the membrane is permeable to more than one ionic species, you will end up with a
Nernst- Planck Equation and a Continuity Equation for each ion. So, if there are k
permeable ionic species that makes 2k equations. (In this example, there is onl y 1
permeable ionic species and therefore there are 2 equations)
However, we have 2k+1 unknowns (in this example, there are 3: J
n
, c
n
, and y ). So we
need one more equation. Remember, 8.02? Well you can use it here! We can use Gauss’s
Law and the definitio n of potential and get:
?
2
y
2
= -
1
( total charge)
?x e
Poisson’s Equation:
?
2
y
2
= -
1
∑
z
n
Fc
n
( x , t )
?x e
n
Electroneutrality approximation
To simplify the equations:
On the length scale of cells and on the time scale of biological processes, we can make
the approximation that the total charge is 0. That is:
∑
z
n
Fc
n
( x , t ) = 0
n
That’s because all the charge build up on the membrane occurs very fast (charge
relaxation time is on the order of nanoseconds), is on the order of nanometers (“Debye
length” is ~1nm for physi ological conditions, and the amount is very small compared to
the amount of ions in the bath.
Steady State Electrodiffusion through membranes
Steady State still has the same definition (
?
→ 0 ). This means that:
?t
?J
n
( x , t )
= - z
n
F
?c
n
( x , t )
= 0 fr om the continuity equation and so it implies that J
n
is
?x ?t
constant.
Then you can solve for J
n
through the membrane from the Nernst- Planck equation (see
the lecture notes):
J
n
= G
n
( V
m
- V
n
)
where Vm is the total membrane voltage ( V
m
= y(0)- y(d) ). G
n
i s the conductivity (units
of S=1/W ) of the ion species n through the membrane. It’s a function of the ion mobility,
valence and concentration in the membrane.
Finally, V
n
is defined as the Nernst Equilibrium Potential:
V =
RT
ln(
c
n
( outside )
)
n
z
n
F c
n
( inside )
From this w e define a circuit model for the conductance of one ionic species through the
membrane:
+
_
V
m
J
m
_
+ V
m
J
n _
G
n
+ V
n
So, if we have more that one permeable ionic species you will get one branch in the
circuit model:
What is the Nernst equilibrium potential?
C)25(Tretemperaturoomat)log(
60
)
)(
)(
ln( °=
?
?
?
?
?
?
?
?
≈= mV
c
c
zinsidec
outsidec
Fz
RT
V
i
n
o
n
nn
n
n
n
This means that if V
m
=V
n
then ionic species n is not moving across the membrane.
…
G
1 G
2
G
3
G
k
+
V
1
?
+
V
2
?
+
V
3
?
+
V
k
?
+
V
m
?
J
m
What’s the problem with this model?
Well, since all the transport is passive, you will eventually make the cellular
concentrations change but this doesn ’t happen…
The cell uses active transport to maintain its internal concentration of ions constant. So to
account for this we add active transport as current sources in the model:
…
G
1 G
2
G
3
G
k
V
1
+
V
2
?
+
V
3
?
+
V
k
?
J
m
…
J
p
J
a
a
J
1
a
J
2
a
k
J
+
V
m
?
+
?
Steady State, Rest, Equil ibrium, and Quasi- Equilibrium: What’s the difference?
(Steady State and Equilibrium still have the same definitions)
Steady State : nothing changes with time (i.e.
?
→ 0 )
?t
Rest : the net flux of charge particles is zero (i.e. J
m
=
∑
J
n
p
+ J
n
a
= 0 . This is only one
n
equation…) This implies that: V
m
=
1 ?
?
∑
G
n
V
n
- J
n
a
?
?
∑
G
n ?
n ?
n
p a
Quasi- Equilibrium : The net flux of each species is 0 (i.e. J + J = 0 for all n . This is a
n n
set of k equations: one equation for each permeable ionic species…)
Equilibri um : There is no flux of anything (i.e. all the J = 0). This can only happen if the
active transport is blocked.
Indirect vs. Direct Effect of Active Transport
Indirect Effect:
If you turn off the active transport, then if you wait long enough the concent ration in the
cell will begin to change. (over a long time at least several hours)
Direct Effect:
If the active transport is electrogenic, then turning it off will cause an immediate change
in V
m
(see equation for V
m
at rest).
a
Electrogenic active transport means that
∑
J
n
≠ 0 .
n
Example: The Na/K ATPase covered in lecture pumps more Na+ than K+ every iteration:
3Na
+
inside
outside
2K
+
Review Circuits
Remember our friends KVL and KCL. If you don’t, come talk to a TA for a quick
refresher course. (Also, G
n
is conductance so now the constitutive relationship is
J
n
=G
n
V
n
)