# Enzyme kinetics and MCA: the elasticity coefficients

### Enzyme kinetics and MCA

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.

### Enzyme kinetics

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.

### Elasticity Coefficients

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:

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.

Last updated: March 28, 2013 at 16:22 pm