Enzyme Kinetics and Metabolic Control Analysis
229
y-y(t=0)=y-y(
0) = -(y(0) - / )( 1
- ex p (-(f12 -
£
,, )/))
(
24
)
4 r
(P
" /»(°)) =
tJ'(xa
) +
0X0) - / )(1 - exp(-(f12
-
s
x, )0)
S
£ n
E\\
But from (17):
C
j
i
£
12
£
,*>
£,s
Consequently if the first equation in (24) is multiplied by C / and added to the second equation one
obtains:
c {b -y m + \< j> -p m = n x *)t
(
25
)
s
In a series of experiments the internal metabolite concentration
y(t)
and
p(t)/s°
are measured at
t= l, t2
h,.
..tn
, and by linear regression the two coefficients a , and a
2
of (26) are determined:
a l(y-y(0 )) + a 2
P
The flux control coefficients are now constructed from a ! and a 2 using
(26)
[c/
c /H «
a 2]AJ'(Jc°) = [al
j
-/’O 0)
(27)
The stoichiometric matrix
A
has metabolite 1 (here
y)
in the first row and metabolite 2 (here
p/s!i)
in the
second row. The steady state flux
J'(x°,
1) is simply found as
0-2' ■
Fig. 6.10 shows
y(t)
and
p/s°(
t) for
fixed 5 =
in an
in silica
experiment with no error in the two dependent variables. There are 12
t
values,
t=
0.5, 1, 1.5,.
....6. The flux control coefficient and the steady state flux are of course exactly retrieved.
In reality there are experimental errors in all the concentration measurements. Fig. 6.11 shows another
in
silico
experiment where the exact values of the dependent variables are randomly overlaid by 2 % noise
taken from a Gauss distribution with mean equal to the exact concentration values. Now CV is calculated
to 0.8671 and
f
=0.3131.
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