224
Chapter 6
u
(4)
/
/-I
£ c / 2 > , ^ / = 0
Since the change in the ith internal rate (or flux) is found from Eq. (6.46) to be given by
/-i
Ar. = Z
aaAsj ’
,' = 1- - 7
(5)
Eq. (6.47) is obtained by inserting Eq. (5) in Eq. (4).
Example 6.5
Approximate calculation and experimental determination of flux control coefficients.
Consider the two-step pathway (6.23b) and (6.24). If
K>, K2
and
Krq
are constant there is only one
variable left to characterize the architecture of the pathway. This could be
k}
or it could be x =
k,/k2
since
with only two control coefficients one of them can always be calculated from the flux control summation
theorem. Consequently a study of
J(x)
or of
T(x) = J(x)/k2
will show how the control is distributed
between the two reaction steps when the enzyme concentration
e,
(considered to be proportional to
k,)
is
varied relative to
e2.
There is one environmental variable and this is the concentrafion s of the substrate to the first reaction in
the pathway. In our first treatment of the problem s was held constant, and in (6.25) the variable s, and
the parameters could be normalized by this constant. If we wish to study the influence of 5 on the flux
and on the flux control distribution we must normalize Si with a reference value 5° of the exterior
substrate concentration.
Consequently we have the following dimensionless variables:
The variable v is, however, a function of the other two variables
x
x = kt/k2
s' = s/s°
and
y
= .^/.s0
(
1
)
and the parameters :
a
= ATeq/.v° ,
b
=
K]/
,v° and
c = K2! s"
2
y =
a
,
a
,
an— 5 —
s x
b
b
a
,
a
,
a + —s — s x
b
b
2
4
ac
H
-------
s x
b
(
3
)
The rates r, and
r2
of the two reaction steps are normalized by
k2: