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Kinetics of Multi-enzyme Systems

12.4 Control coefficients

12.4.1 Definitions

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 X0 and X5 are constant, as are the kinetic properties of the enzymes, but the individual enzyme rates vi and the concentrations of the internal metabolites Sj are free to vary. Even if these concentrations are initially arbitrary, they will tend to vary so that each approaches a steady state. (A steady state does not necessarily exist, and if one does exist it is not necessarily unique: for simplicity, however, I shall assume that there is a unique steady state). Taking S1 as an example, a steady state implies that the rate v1 at which it is supplied must be equal to the rate v2 at which it is consumed. A steady state in s2 likewise implies v2 = v3 and so on; when all the metabolites are in steady state all the enzyme rates in a pathway as simple as that in Figure 12.1 must be equal to one another, with a value J that is called the flux. If there are branches or other complications there can be several different fluxes in the same pathway, and the relationships are more complicated. However, the principles remain straightforward and obvious: the total flux into each branch-point metabolite is equal to the total flux out of it.

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 vi when the enzyme Ei is isolated, what is the corresponding effect on the system flux J when Ei is embedded in the system? This is not known a priori, and the ith flux control coefficient is defined by the following ratio of derivatives:

C(J/i) = partial d ln J by d ln vi (12.12)

The simpler form shown at the right is not strictly correct, because vi 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 u even if it is not shown explicitly. This definition corresponds to the way Heinrich and Rapoport (1974) defined their control strength; in apparent contrast, the sensitivity coefficient of Kacser and Burns (1973) was defined in terms of the effect of changes in enzyme concentration on flux (both of these terms have been superseded in current work by control coefficient):

C(J/i) = partial d ln J by d ln ei (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 CJi = 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 CJi = 1 and one could call it the rate-limiting enzyme for glycolysis. In fact, however, Heinisch (1986) found experimentally that increasing the activity of phosphofructokinase 3.5-fold in fermenting yeast had no detectable effect on the flux to ethanol. Similar experiments have subsequently been carried with other supposedly rate-limiting enzymes, with similar results, and there are theoretical reasons for expecting it to be very rare for any enzyme to have complete flux control (Section 12.5).

A concentration control coefficient is the corresponding quantity that defines effects on metabolite concentrations, for example, for a metabolite Sj with concentration sj:

C(sj/i) = partial d ln sj by d ln ei (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.

12.4.2 The perturbing parameter

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 e0, and so the most obvious interpretation of the perturbing parameter u that appeared in equation 12.12 is that it is identical to e0. 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.