214
Chapter 6
l c v = 0 ;
/ = i.
..../ - i
(6.36)
This implies that for each of the / - 1
intermediates, at least one of the enzymes must exert a
negative control, i.e., when the level of one of the enzymes increases then the metabolite
concentration decreases.
The control coefficients specify the influence of the enzyme level on the overall flux, but it is also
valuable to examine the sensitivity of the net rate of individual enzymatic reactions to variations in
the size of each of the / - 1 metabolite pools. Thus we define
elasticity coefficients
for the
i
th
enzyme by
where
r,
is the net rate of the z'th enzymatic reaction. A negative value of £j, implies that when the
level of the y'th intermediate is increased the net rate of the z'th reaction decreases, and
vice versa.
For a simple, reversible enzymatic reaction (see Example 6.4) the elasticity coefficient a is
negative, whereas the elasticity coefficient gW
l, will be positive. Furthermore, if the y'th intermediate
does not influence the rate of the z'th enzymatic reaction (the rate of this reaction may be
independent of the concentration of y), then
e,,=
0. The elasticity coefficients are connected to the
flux control coefficients through
the flux-control connectivity theorem
:
for which the proof is given in Note 6.4. From Eq. (6.38) it is seen that enzymes that have high
elasticities tend to have low flux-control coefficients. The increase or decrease of an internal
reaction rate
r,
with an increase in the internal metabolite concentration y is not in itself of much
interest; the object of MCA is to identify the enzyme an increase in the activity of which will
maximize the increase of the productivity of the whole pathway, i.e., maximize the change in ./.
Thus
C‘
is of prime importance. But experimentally it may be much more difficult to vary the
activity level of the enzymes than it is to study the influence of metabolite concentrations on the
rates of the individual intracellular reactions
r,
for a fixed set of enzyme activities.
Note 6.4 Proof of the flux-control connectivity theorem
The differential change in the z'th reaction caused by a differential change in the y'th interm ediate y is
(6.37)
(6.38)
U)
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