Enzyme Kinetics and Metabolic Control Analysis
215
or
dr
ds
i
= 1,.
..,/
and
j =
1
-
1
(2)
This fractional change in the rate of the
i
th reaction can also be accomplished by changing the level of the
ith enzyme, i.e.,
dr,
det
n
e.
(3)
If we balance the change in the reaction rate due to the differential change of the intermediate by a change
in the enzyme level, we find from Eqs. (2) and (3)
de,
ds
,
;
/ =
and y' = l ,.
..,/- l
(4)
If we change the internal rates without changing the total flux
J
through the pathway, then by Eq. (4):
d£_
_
X' pJ dei
j ~ h
1
e,
(5)
By combination of Eq. (5) with Eq. (4) we get
(ds
Ï
ds, JL
X C ,J^ ,= 0 ;
y = l ,.
..,/ - l
(
6
)
and since we imposed a fractional change on each of the
I -
1
intermediates, i.e.,
ds j
*= 0, we obtain the
flux-control connectivity theorem, Eq. (6.38).___________________________________________________
In addition to the flux-control connectivity theorem, Westerhoff and Chen (1984) introduced two
other connectivity theorems:
Z c , ^ , = - 1 ;
J
i....
/
i
(6.39)
/«i
/
y = l,.
..,/- l
and
k * j
(6.40)
1=1
In matrix notation we can summarize all the theorems given in Eqs. (6.34), (6.36), and (6.38)-(6.40)
by
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