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MCA |
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In the previous section the the control
coefficients were described. These are properties of the system as a
whole, that quantify the control exerted by each enzyme on a certain
pathway flux (or metabolite concentration). What about the large body of
data gathered by enzyme kineticists throughout the whole of this
century? It would be useful if one could merge those data with the MCA
formalism to characterize the control properties of metabolic pathways.
This section of the MCA Web describes how the kinetic data of
isolated enzymes can be converted to elasticity coefficients.
The next section deals with how one can use
the elasticity coefficients to calculate the control
coefficients.
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Enzyme Kinetics |
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In enzyme kinetics one is interested in characterizing the progress
of enzyme catalysed reactions in the time dimension. One way of doing so
is by following the appearence of one of the products of the reaction or
the disappearence of one of the substrates. Curves such as that of figure 2 are typical.
More frequently, however, the kinetic behaviour of isolated enzymes
is studied through the study of the dependence of the initial rates of
reaction with the concentration of the substrate(s).
Figure 3 shows a typical example of such a relation.
Enzyme kinetic studies are centered on the determination of kinetic
parameters such as Michaelis constants or limiting-rates (see figure 3) or (more rarely) on the elementary rate constants of a
specific reaction mechanism. See Cornish-Bowden
(1979) or Keleti (1986) for
more information on enzyme kinetics.
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Elasticity Coefficients |
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In metabolic control analysis the properties of each (isolated)
enzyme are measured in a way very similar to how the flux-control
properties were: using a sensitivity, known as the elasticity
coefficient (Kacser & Burns 1973,
Heinrich & Rapoport 1974,
Burns et al. 1985).
In this case, one has to consider the effect of perturbations of a
reaction parameter on the local reaction rate. By local one
means that this sensitivity refers to the isolated reaction which
has the same characteristics (effector and enzyme concentrations,
temperature, and so on) as in the whole system at the operating point
(steady state) of interest. The elasticity coefficients are
defined as the ratio of relative change in local rate to the relative
change in one parameter (normally the concentration of an effector ).
Infinitesimally, this is written as:
 (5)
where vi is the rate of the enzyme in question and p
is the parameter of the perturbation. Each enzyme has as many
elasticity coefficients as the number of parameters that
affect it. One can immediatly recognise the concentration of the
reaction substrates, products and effectors as parameters of the reaction.
Unlike control coefficients, elasticity coefficients are not
systemic properties but reather measure how isolated enzymes are
sensitive to changes in parameters. The elasticity coefficients can
be obtained from the initial-rate kinetic functions illustrated in
figure 3 by partial derivation. Again like
the control coefficients, the elasticity coefficients are not constants,
they are dependent on the value of the relevant parameter and so are
different for each steady-state.
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