A kinetic model for the bifunctional enzyme, phosphofructo-2-kinase:fructose-2,6-bisphosphatase, is analysed by application of the graph-theoretical method, considering comparable levels for all participants. Certain elementary reactions, distributed on the enzyme surface, are considered to be co-ordinated in a single conformational transition (a model of parallel molecular operations). The method allows us to identify in the kinetic scheme its destabilising sub-scheme as a branched cycle of elementary reactions. Under certain conditions this sub-scheme induces critical phenomena (bistability or oscillations). The computer calculations for the estimated parameter values fit well the experimental observations for this system. The model explains the periodic or bistable counterphase changes of the two opposing activities of this enzyme, observed after glucose perfusion of rat hepatic enzyme samples, and predicts drastic critical changes in kinetic behaviour induced by small external signals. The model also shows the necessity of the phosphoryl intermediate in the mechanism of the bisphosphatase for the critical kind of kinetic behaviour.