22
Chapter 2
At the two surfaces
de _ dcme
dz
dz
(
7
)
since neither carrier nor carrier-substrate complex can leave the membrane. Hence the arbitrary constant
k\
in Eq. (
6
) is zero, and on further integration one obtains
e + cme = k 2
(
8
)
Comparison with Eq. (4) shows that the arbitrary constant
ki
=
e,
and hence that the sum of the two carrier
species is
e,
at all positions in the membrane. Adding Eqs. (1) and (3) and integrating yields
- J =
(9)
where
J
is the desired total flux of substrate from z = 0 to
z = d
through each unit area of the membrane.
Since
c me = et - e
~
K e -e-cm
one obtains
Ci
II
f
dcm
d
'
e
tK ,
^
+ —
---
e—~ c
dz
dz
1
J J
(10)
Separation of variables and integration from z = 0 to z =
d
yields
J = -K (c a-cb) + — K-
------
e'K '( c° -
----------r
d
d
(\ + K tK c J \ + K eKcb)
(
11
)
The first term on the right-hand side of Eq. (11) is obviously the free diffusion term corresponding to Eq.
(2.1). If the partition coefficient
K
is so small that
K eKca
and
K eK cb
are small compared with 1, it is seen
that the last term is larger by a factor
e, K e
than the free diffusion term. This is the effect of the facilitated
diffusion. It is seen that the facilitated diffusion, just like free diffusion is dependent on the concentration
gradient, but also it is a function of the concentration of the carrier protein.____________________________
2.1.23 Active Transport
Active transport resembles facilitated diffusion since specific membrane-located proteins mediate
the transport process. In contrast to facilitated diffusion, the transport can be in the uphill direction
of a concentration gradient, and active transport is therefore a free energy consuming process. The
free energy required for the transport process may be supported by consumption of high-energy
phosphate bonds in ATP (primary active transport), or the process may be coupled to another
transport process with a downhill concentration gradient (secondary active transport). Finally, in a
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