228
Chapter 6
This result is not far from the true value 0.9082, and in reality we have no way of obtaining the true value
of
C1
since this would demand that we either knew the exact relationship between the rates and the
variable r or the linear approximations (14),(15).
As mentioned earlier formula (19) is a consequence of a remarkable paper by Small and Kacser (1993)
who introduced the concept of flux control coefficients obtained by
large deviations.
Since for each
enzyme, even in long pathways, the relationship between flux
J
and enzyme activity usually has an
approximately hyperbolic form the large deviation approach is eminently practical in an experimental
study. An integral change in flux is measured in a series of experiments where, one at a time, each
enzyme dosage is integrally changed from a given initial level, e.g. that in the wild type strain. Formula
(19) is afterwards used to obtain approximate values of C,J.
The other coefficient 0.10697 in (14) and (15) is the
response coefficient
, i.e. the response of the pathway
flux to a change in the environmental variable s. If s is changed from
s"
to 2 ,v"at constant
x
= 0.5 then
J'txl'.s ')
increases by 0.1070 according to (15), while an exact calculation from (3) with
s' = 2
and
x =
0.5 givesy = 0.30278 and
/1
./' = 0.0671.
From (15) yapprox(0.5,2)
=
0.22474 + 0.11237 =
0.33711, and using (17) for the exact and the
approximate values of
y
yields:
C /exac,(0.5,2) =0.9160
and
C /approx(0.5,2) = 0.9044.
We shall finally revert to the method of Delgado and Liao (1992) which as shown in eq. (6.47) promises
to deliver the flux control coefficients based on the linearized rates, i.e. from the elasticities eu and e12
at
the point of linearization.
Let us therefore consider
Rt
and
R2
at
P =
(0.5,1), and perturb
y
from its value
at the steady state. From
(14) or (15):
R,
*
J \ x ° ) -
0.09616 (y(/)
-y ° )= J '(x ° ) + £ll (y
-
y °)
R2
«
J'(x°)
+ 0.9519 (
y (t)-y ° ) = J'(x°) + £l2( y - y 0)
dy
dt
dt
R ,- R 2
={eu - e n )(y{t)-y°)
= R2
= J'(x°)+ £l2(y(t)-y °)
(
23
)
The solution of (21) and (22) is:
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