Week 2 Review
What was covered:
- Dissolve and diffuse
- 2 compartment model
- thin vs. thick membrane
- measurements in cells
Dissolve and diffuse:
Assume: dissolve is much faster than diffuse
k = partition coefficient (dissolve)
D = diffusivity (diffuse)
f = -kD
?c
-
?f
=
?c
?tx ? x ?
2
?? c c
= kD
?t ?x
2
2 compartment model
A= cross- sectional area
V
1
c
1
(t)
V
2
c
2
(t)
d
M e mbrane: only lets
Assume:
solute through
- well stirred baths (in baths c(x,t)=c (t))
- solute is conserved (nothing is eating it up or producing it)
- baths big compared to membrane
- thin membrane
d
2
Steady State (SS) time constant: t
SS
=
2
p D
I f at SS then:
Dk
f = ( c
1
( t ) - c
2
( t )) = P ( c
1
( t ) - c
2
t )) Fick’ s law for membranes
d
P is the membrane permeability
1 d
f ( x , t ) = - ? ( n ( x , t )) (where n is number of solutes) Definition of flux
A dt
From Fick’ s law for membranes, can get the equilibrium time constant, t
EQ
:
1
t
EQ
= (see supplement for derivation)
? 1 1 ?
AP
?
?
?
V
1
+
V
2 ?
?
?
Thin vs. thick membranes
Wh en does this theory break down?
C ompare t
SS
to t
EQ
: I f t
EQ
>>t
SS
then thin membrane…
However, if t
EQ
is on the order of t
SS
then not thin membrane:
W hat does this mean:
1. time to get to SS cannot be ignored
2. concentration in baths will change significantly before reaching SS
3. amount of solute in membrane might not be negligible
4. overall time profiles of concentration/flux are NOT exponentials (can’t
reduce to Fick ’s law for membranes so profiles are not solutions to 1
st
order linear differential equation)
Measurements:
( To m easure time constant of exponential curve: extend a line at initial time and
intersecting with the asymptote… see problem set 1)
c
o r
f
t
e xponential
curve
t
How to measure t
SS
?
O n SMALL time scale:
1. look at plot of concentration profile in membrane (remember: on short
time scale, only membrane concentration is changing; bath
concentrations are not changing significantly at this point.)
2. look at plot of f ( t)
How to measure t
EQ
?
O n LARGE time scale:
1. look at plots of concentration. (in bath or membrane)
2. look at plot of f (t)
I f y o u a r e n ’t c o m f o r t a b l e w i t h f i gur i n g o ut t i m e c o n s t a n t s a n d s t uf f l i ke t h a t f r o m
c o n c e n t r a t i o n a n d f l ux pl o t s r e v i e w pr o b l e m 4 and5 o n ps e t #2 a n d pr a c t i c e w i t h t h e
simulation s o f t w a r e … ( a n d i f y o u a r e s t i l l c o n f us e d, f e e l f r e e t o a s k us ( t h e T a s )
que s t i o n s ! ! ? )
More measurements:
B e comfortable w i t h t h e pl o t s P r o f . F r e e m a n put up i n l e c t ur e w h i c h ki n d o f l o o k l i ke
t h i s :
See pg. 145 in the text for nicer graph:
w here P is the permeability of a solute and k
P
is the partitioning coefficient.
k
*
*
*
*
* M<60
60<M<160
M>160
What do these show:
1. since linearly P is linearly dependent on partition coefficient (which was
measured in oil), membrane is lipid
2. bigger solutes (M larger) d iffuse more slowly (plot above assumed D was the
same for all solute)
3. if there is a solute that is really off from line (even when you take M into
account), probably has specialized transport mechanism in the cell