2 1 2
Chapter 6
shall see in Example 6.3 that there is an easier way to calculate the flux control coefficients, but
from the calculations above we can already draw the main conclusion that
( J
decreases while Cf
increases when x increases. Since
J(x)
is a monotonous function of
x
so must also In
J(x)
be a
monotonous function of In x and consequently
dJ
/(:,
8J' x
dlnJ'
1
8k{
J
dx J'
d\nx
must decrease monotonously to zero when x (or
k f
increases to infinity. Since CJ
*-
= 1
the
value of
Cf2
must correspondingly increase towards 1.
The conclusion is that the control shifts from the first reaction step to the second for increasing
activity of the first enzyme. The rate of change varies with the environmental conditions as was
seen for the case of increasing 5. For large
s
the concentration of the first enzyme (represented
by the parameter
k f
must be relatively large to change the relative size of C1, and C,2 since the
competitive inhibition of the first enzyme by
St
is less pronounced when ,v is large.
Having introduced the flux control coefficients and explained their significance by means of a
simple example we can now turn to a general unbranched pathway with I reaction steps:
S
S, S S2 .
......
S , , f t
P
(6.32)
There are 7 - 1
intermediates 5 ,.
...........
S
, , and
I
pathway reactions mediated by the enzymes
£ ,.
.........
In the steady state the rates of all the pathway reactions
r,
are equal and equal to the
flux
J
through the pathway. We wish to calculate the flux J of substrate of known concentration
that can be processed by the pathway, and we wish to investigate how the pathway architecture
can be improved to allow a larger flux to be processed.
The flux control coefficients C1
, are defined by
c / = £ L a /
J de,
(6.33)
In Note 6.3 it is shown that irrespective of the kinetics of the individual enzyme catalyzed
reactions the sum of the flux control coefficients is 1
for the unbranched pathway.
I C / = 1
(6.34)
i =
l
This result is named the
flux control summation theorem
N ote 6.3 Proof o f the flux-control sum m ation theorem
If in a thought experim ent w e increase all the enzym e activities sim ultaneously by the sam e fractional
am ount
a,
i.e.
.