Enzyme Kinetics and Metabolic Control Analysis
223
ri= Y ,aj‘sj +bi
>z
(6-46)
y=i
The coefficients ay and
b,
change from one steady state to the next, but as pointed out by Small and
Kacser (1993) it is not certain that the flux-control coefficients change dramatically. By using the
flux-control connectivity theorem [Eq. (6.38)] it is possible (see Note 6.5) to derive Eq. (6.47),
which correlates the control coefficients with the changes in internal reaction rates immediately
after a perturbation in the system. Thus, by measuring the transient responses of the pathway
intermediates after, e.g., changing the concentration
s
of the substrate, one can calculate the
variations in the fluxes and thereby the control coefficients:
/
X c/ Ar<=°
(6-47)
i=l
In two further papers, Delgado and Liao (1992a,b) have reshaped their original method into a
practical procedure for obtaining control coefficients from measurements of transient metabolite
concentrations. We will end our introduction to MCA with an extensive example (Example 6,5)
that will deal with these methods and discuss their drawbacks.
/-i
Note 6.5 Derivation of Eq. (6.47)
After linearization of the reaction rates, we find from the definition of the elasticity coefficients
£ a
= ~
a
j-
;
; = 1,.
..,/
and
y=l,.
..,/-l
(
1
)
and consequently by Eq. (6.38):
=Zc/«y;= o;
j =
(
2
)
since
S j
*
0 and all reaction rates
r,
are equal at any steady state. We now make a perturbation of the
system and observe each Av;
=s .(t) - s i (t -
0). Multiply the last part of Eq. (2) by As,:
X (VWV
0:
/
1
.....
/
!
(3)
and add the / - 1 equations: