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MCA |
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The Control Coefficients |
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A control coefficient is a relative measure of how much a
perturbation affects a system variable (e.g. fluxes or concentrations).
It is defined (Kacser & Burns 1973,
Heinrich & Rapoport 1974,
Burns et al. 1985) as:
 (1)
where A is the variable, i the step (enzyme) and vi
its steady-state rate of the step perturbed. The most common control
coefficients are those for fluxes and metabolite concentrations. However,
any variable of the system can be analysed with MCA and have control
coefficents defined by equations analogous to equation 1. In fact, there
is no need even for the system to be in a steady state. Any limit set
can have control coefficients defined for any of its variables which are
invariant (such as the period of an oscillation; Markus & Hess 1990). They were also defined for ordinary points of
a trajectory (Khon & Chiang 1982,
Acerenza et al. 1990) in which
case there are additional time components that make the analysis more
difficult.
Since the rate of reaction cannot be perturbed directly, control
coefficients must be determined by perturbations in parameters that
affect the rate linearly. Because many enzyme-catalysed reaction
rates are linear in terms of the enzyme concentration (at least in a
certain range of enzyme concentrations), control coefficients can be
written using enzyme concentrations
(Kacser & Burns 1973):
 (2)
Figure 1 shows one example of the
relation between the flux of a pathway to the concentration of one of
the enzymes of that pathay. The flux-control coefficient of the enzyme
is the slope of the tangent to the curve. It is easy to see that the
flux-control coefficients are different from one steady state to
another.
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The Summation Theorems |
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A very important property of steady-state metabolic systems was
uncovered with the MCA formalism. This concerns the summation of all the
flux control coefficients of a pathway. By various procedures
(Kacser & Burns 1973,
Heinrich & Rapoport 1975,
Giersch 1988,
Reder 1988) it can be demonstrated
that for a given flux the sum of its flux-control coefficients of all
steps in the system is equal to unity. The simplest way in which this
property can be derived is by considering a simultaneous small relative
increase (a) in all reaction rates of a metabolic system.
Because for each metabolite the relative rates of its production
increase exactly by the same amount (a) as the relative rates of
its consumption, the metabolite concentrations remain unchanged. The
flux of the system increases exactly by a. In mathematical terms
this means that the flux is a homogeneous function of degree one and the
metabolite concentrations homogeneous functions of degree zero. The
summation theorems follow from this by applying a corollary of the Euler
theorem for homogeneous functions (Giersch
1988). For flux-control coefficients:
 (3)
and for concentration-control coefficients:
 (4)
where the summations are over all the steps of the system. This may
include not only the steps of the pathway of interest but also of other
pathways (as long as there are links between them). In principle for a
whole cell, the summation would have to be over all metabolic steps of
that cell.
According to equation 3, increases in some of
the flux-control coefficients imply decreases in the others so that the
total remains unity. As a consequence of the summation theorems, one
concludes that the control coefficients are global properties
and that in metabolic systems,control is a systemic property,
dependent on all of its elements (steps).
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Rate-Limiting Steps |
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In the past many authors referred to some enzymes as rate-limiting
(or bottlenecks, or even pace-makers). These enzymes
were mostly after a branch point and catalyse essentially irreversible
reactions (with very high equilibrium constants). The thought was that
these enzymes operated at a lower velocity than the others (downstream)
in the pathway and so they "controlled" the pathway; if one wanted to
increase the throughput of the pathway it would be enough to increase
the amount of that enzyme. Applying the MCA reasoning, such rate-limiting
enzymes must have a flux-control coefficient equal to 1, and
consequently (see equation 3) all other enzymes have
flux-control coefficients of 0. While this is theoretically not
impossible, it is very improbable and most experimental studies of
enzyme over-expression by cloning have revealed that large increases of
enzyme concentrations are not accompanied by equivalent increases in
pathway flux. Furthermore, while one is increasing the amount of the
hypothetical rate-limiting enzyme, its control over the pathway
flux would decrease until it eventually approached 0 (as is illustrated
in figure 1. The moral is that one must be
extremely careful to use the expression "rate-limiting" as enzymes are
almost never such.
A more reallistic picture of the way that a pathway flux is
controlled by its enzymes emanates from the flux-control summation
theorem (equation 3). The magnitude of the
flux-control coefficients can be seen as a percentage of control exerted
by the individual step over the flux of interest. Control is shared
between all enzymes in different proportions.
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