Enzyme Kinetics and Metabolic Control Analysis
211
the slope of J ’(x). This is where the flux control coefficients are introduced.
Define
(6.30)
In (6.30) the enzyme concentrations
e,
and
e2
are used instead of the rate constants
k,
and
k2.
It
would be more correct to use the enzyme activities determined through the rate constants than
the enzyme concentrations
(or enzyme “dosages”) since the activity o f an enzyme is not
necessarily proportional with the enzyme concentration. Here we will, however, assume that
there is proportionality between enzyme concentration and enzyme activity. Flux and enzyme
concentrations are measured in units that are not commensurable, and consequently the
sensitivity of the flux with respect to changes in
e,
is scaled by
eJJ
in order to give dimensionless
flux control coefficients.
In the example we assume that the rate constants are proportional to the enzyme concentrations
and we obtain the following for the two cases of r,.
(6.26a)
since
kx
(or
x)
is the independent variable while
k2
is a constant.
To obtain the result for (6.23b) the following “chain-rule” for differentiation is used:
dJ kl
dJ' x
_
ÔJ' dy x
ôkl J
8x J'
dy dx J'
(6.31)
which means that also in general CJ, +
C‘2
= 1, just as in the simple case (6.23a).
Inserting
a/'
c
c f
dy
(c +
y
)2
y (c + y)
in (6.31) together with
y(x)
and dv/dx from (6.27) one obtains
C1,
as an explicit function of
x.
We
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