Enzyme Kinetics and Metabolic Control Analysis
209
When r, given by (6.23a) is equated with
r2
one obtains:
x
=
y
1
+ b
y + c
or
y
=
cx
1 +
b - x
(6.27)
We note that, as expected
J
=
k2
------ = k2
------ = โ€”
c
an be calculated both as r, and as
r..
r
y + c
\+b
\+b
When (6.23b) is used for r, then
y
is determined from
k\
1
+b
1+^
k2y
y + c
or
2y = -
(6.28)
Fig. 6.8 shows
T(x)
for the two expressions (6.23a and 6.23b) for r,. There is an important
difference between the two expressions. With (6.23a)
f
is proportional to x, and / is completely
independent of the parameters of
r2.
With (6.23b)
.Fix)
has the typical downwards concave shape of
the hyperbolic functions which determine the kinetics for individual enzymes. An increase in
k{
is
profitable for small x (i.e. small
k{)
but has little effect for large x, since d /โ€™/dx becomes zero as
f
increases towards 1.
For small x a power series expansion o f
y{x)
in (6.28) yields
c x
x
y
ยป -โ€” :
and
J
b +
1
1
+ b
(6 29)
Consequently, for small values of
k
, the flux
J
depends only on the kinetic parameters
k,
and
of
r,. The value of .v, never becomes large enough to give any influence of the denominator term
s,/K^q
in (6.23b). For larger
k,
values 5, increases, and the parameters
K2
and
Kcq
start to influence the flux
/through the pathway.
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