# Control coefficients and rate-limiting steps

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### The control coefficients

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:

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):

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.

### The Summation Theorems

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:

and for concentration-control coefficients:

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).

### Rate-Limiting Steps

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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