Enzyme Kinetics and Metabolic Control Analysis
231
The general formula corresponding to (27) for I enzymes in the linear pathway is:
( CY, CV, C , \ .
.....
C,J )
= (en.anta .
........
t * ) A J ’(x°)
(29)
Besides the sad issue of the influence of experimental errors on the calculated flux control coefficients it
should also be noted that (29) holds
only
when each reaction
R,
in the pathway is independent of the
concentration of the substrate which feeds the pathway and of the concentration of the final product of
the pathway. This is very inconvenient since experimental determination of
s(t)
and of
p(t)
is much easier
than determination of -Si(t) , I = 1, 2 ,3 .
...I - 1, and we would like to start the transient experiment by
S
(e.g. glucose). It is inconceivable that /f, should in general be independent of s -
perhaps with the exception that the first enzyme could be saturated with its substrate.
The transient method can, however be reformulated into a procedure that has a much better relation to
experimental practice - and as a bonus appears to give much better estimates of the flux control
coefficients.
Consider in a thought experiment a continuous, stirred tank reactor that is fed with
S
while
P
is
withdrawn with the effluent from the reactor.
S
is converted to
P via
an intermediate
S-,
that cannot leave
the reactor - we might imagine that Sj is only present in the immobilized enzyme particles that remain in
the reactor. There is no
P
in the feed to the reactor. Mass balances for
S, S,
and
P
are:
ds
ds'
dp
- i " * * D
<
n-r,:
f - r . - D ,
(30)
With the kinetics given in (4) and linearized around the point (£,=0.5, £y= 1, .?=.9°) one obtains (14)-( 15) :
~Z
=
r" - Sn(s,-s,°) - e„,(s-s°) + D (Sf-s°)
-
D (s-s°)
~ T
=
r° + Ell (s, -s")
+
E0I (s -s°) - r:" - el: (s,
-,
s,°)
(31)
— 7 =
r / +
e i: (s i - s,") - D p " - D ( p - p " )
dt
In (31) Eoi is the elasticity of r with respect to changes in j. ,v,0 and
p
are steady state values of 5, and
P,
respectively at the point of expansion. In the steady state r,° = r2°, and the terms
D (s, - s°)
and
D p°
just
balance these steady state rates. Consequently after scaling the equations with
one obtains the
following differential equations for the deviations from the steady state values of s, s, and p.
d{s'~
1
)
dt
-E,,(y-y°) - e0l(s '~ I)