To this point we have only been discussing the ordinary kinetic behaviour of isolated enzymes, albeit in terminology rather different from
that used in mechanistic studies. The objective of metabolic control analysis is now to determine how the kinetic behaviour of a sequence of
enzymes composing a pathway can be explained in terms of the properties of the individual isolated enzymes. If a system such as the one
defined in Figure 12.1 is set up, the concentrations of the reservoirs X_{0} and X_{5} are constant, as are the kinetic
properties of the enzymes, but the individual enzyme rates *v _{i}* and the concentrations of the internal metabolites
S

Enzyme rates are local properties, because they refer to enzymes isolated from the system. Steady-state fluxes and metabolite concentrations,
by contrast, are systemic properties. Elasticities (defined in Section 12.3.1)
are also local properties, but there are systemic properties analogous
to them that are called *control coefficients.* Suppose that some change in an external parameter * u* (undefined for the moment,
but see Section 12.4.2)
brings about a change in a local rate

(12.12) |

The simpler form shown at the right is not strictly correct, because *v _{i}* is not a true independent variable of the system,
but it is acceptable as long as it is remembered that there is always an implied external parameter

(12.13) |

These definitions may appear to be different, but provided that equation 12.7 is true, so that each enzyme rate is proportional to the total enzyme concentration, equations 12.12 and 12.13 are equivalent. Equation 12.12 has the advantage of avoiding the widespread misunderstanding that metabolic control analysis is limited to effects brought about by changes in enzyme concentration. Initially it was usual to follow Kacser and Burns in using definitions similar to equation 12.13, but there is now a widespread view that control coefficients ought not to be defined in terms of any specific parameter (though see Section 12.4.2), and that equation 12.12 should be regarded as the fundamental definition of a control coefficient. The quantity defined by equation 12.13 is then better regarded as an example of a response coefficient, which happens to be numerically equal to the corresponding control coefficient only because the connecting elasticity is assumed to be unity (see Section 12.7 below).

These definitions of a flux control coefficient now allow a precise definition of the characteristics of an enzyme could be considered to catalyse
the rate-limiting step of a pathway. Such a description would be reasonable if any variation in the activity of the enzyme produced a proportional
variation in the flux through the pathway, and in terms of equations 12.12-13 this would mean that such an enzyme had
*C ^{J}_{i}* = 1. For example, if
increasing the activity of phosphofructokinase two-fold in a living cell caused a two-fold increase in the glycolytic flux then phosphofructokinase
would have

A concentration control coefficient is the corresponding quantity that defines effects on metabolite concentrations, for example, for a metabolite
S_{j} with concentration *s _{j}*:

(12.14) |

In this equation the simpler form at the right is subject to the same reservations as the corresponding form in equation 12.12, implying the existence of a parameter *u* even if this is not explicit.

As noted already, the easiest way to perturb the activity of an enzyme in a system without perturbing anything else is to vary its concentration
*e*_{0}, and so the most obvious interpretation of the perturbing parameter *u* that appeared in equation 12.12 is that it is identical
to *e*_{0}. At first sight this may seem to be just one of many possibilities, but in practice it is virtually the only one, because most
inhibitors and activators change not only the activity of an enzyme but also its sensitivity to its substrates and products: in other words, they alter
not only the apparent limiting rate of an enzyme but also its Michaelis constants, as extensively discussed in Chapters 5 and 6 of
Fundamentals of Enzyme Kinetics. Pure non-competitive inhibitors would be the exception, but, as
discussed in those chapters, these are not available for almost any enzyme, apart from highly unspecific effectors such as protons that would not
fulfil the required role of perturbing just one enzyme activity. Finding a pure non-competitive inhibitor for just one enzyme in a system would be
difficult enough; finding a whole series to allow perturbation of any enzyme at will is likely to be a fantasy for the foreseeable future. In practice,
therefore, identifying the parameter *u* with the enzyme concentration corresponds closely with experimental reality, and varying enzyme
concentrations by genetic or other means remains the only practical way of perturbing one enzyme activity at a time.

None of this means that ordinary inhibitors and activators cannot be used to probe the control structure of a pathway. On the contrary, Groen and co-workers (1982) used them very effectively for this purpose in a pioneering study of mitochondrial respiration, but the analysis was more complicated than simply treating each inhibitor titration as a perturbation of the activity of one enzyme.