Enzyme Kinetics and Metabolic Control Analysis
203
The slope of r/rmax at
P
is
(n
-1)(« + 1)
4
ns
in fle x
(
6
.
20
)
Independently of the value of
K
an increase of r/r„iax from 0.5 at
s = K
to 0.9 will be mediated by
the same relative change in
s,
e.g for
n
= 4 by changing
s
from
K
to 1.7
K.
A corresponding
change in
would require an increase of
s
from
K
to 9
K
described the functionality between
r
and
s.
Fig. 6.5 shows
r/rma%
for
n
= 4 in eq (6.17) and for two values o f
K.
It is easily seen that the
formulas (6.18) to (6.20) can be used to calculate the major features of the two curves.
If desired a linear plot can be constructed to determine
n
and
K
from experimental data.
Reordering (6.17) gives
m ax
r
s"+ K n
ln-
- =
n
In
s - n
In
K
(
6
.
21
)
The linearity is most pronounced around
s = K
and deviations are likely to occur both at very
small s and when s is much larger than
K.
The reason is that eq. (6.17) is only an approximation
to more complete models of cooperativity as will be discussed below.
Naturally a lot of speculation has been devoted to find a mechanistic explanation for the Hill
equation. The simplest mechanism was already proposed by Hill himself: The enzyme is thought
to consist of
n
subunits, and each subunit must bind one substrate molecule before the enzyme
can function. Later research has shown that haemoglobin is an aggregate o f fixed size with four
binding sites for oxygen, but H ill’s experiments showed that the binding data corresponded to
n
= 2.7. Apparently the enzyme does not abide by the musketeers oath (“one for all, and all for
one”) but the binding of the first substrate molecule facilitates the binding of the next molecule,
and so on. This is why the binding is called cooperative. As shown in modem models of
cooperativity both the numerator and the denominator of (6.17) should consist of a sum of
powers of v up to j", and the approximation (6.17) is only accurate when the dominant power is
5
"
in both the numerator and the denominator. The derivation of models for cooperativity is
complicated, and the reader is referred to textbooks on enzymology, e.g. Comish-Bowden (1995)
and Fell (1997) for an adequate treatment of the subject.
As described above the cooperation may involve only the enzyme and the substrate, or it could
involve cooperation between enzyme, substrate and an effector (either an inhibitor or an
activator) of the enzyme. This last interaction is of great importance for the regulation of
metabolic pathways. Thus, in lactic acid bacteria fructose 1,6 bisphosphate is an effector of both
lactate dehydrogenase (LDH) that converts pyruvate to lactic acid and of pyruvate formate lyase
(PFL) that directs pyruvate into the mixed acids pathways, (see Fig. 2.6). A high glucose flux