Paradoxical control properties of enzymes within pathways: can activation cause an enzyme to have increased control?

It is widely assumed that within a metabolic pathway inhibition of an enzyme causes the control exerted by that enzyme over the flux through its own reaction to increase, whereas activation causes its control to decrease. This assumption forms the basis of a number of experimental methods. For a pathway conceptually divided into two enzyme groups connected via a single metabolite we have derived a general condition under which this assumption is false, and thus the pathway shows paradoxical control behaviour, i.e. increased control with activation and decreased control with inhibition of an enzyme or group of enzymes. Paradoxical control behaviour occurs widely when enzyme activity is altered by changing Km (if an enzyme is already close to saturation by its substrate), but may also occur with changes in Vmax. when the elasticity to the linking metabolite increases with its concentration (as in some cases of sigmoidal and exponential kinetics or for reactions catalysed by isoenzymes). These findings suggest that enzymes with sigmoidal kinetics may have low control in the absence of activation, but may gain control with activation, and thus have beneficial regulatory properties.