An allosteric model of an open monosubstrate enzyme reaction is analyzed for the case where the enzyme, containing two protomers, is activated by the product. It is shown that this system can lead to instabilities beyond which a new state organized in time or in space (dissipative structure) can be reached. The conditions for both types of instabilities are presented and the occurrence of a temporal structure, consisting of a limit cycle behavior, is determined numerically as a function of the important parameters involved in the system. Sustained oscillations in the product and substrate concentrations are shown to occur for acceptable values of the allosteric and kinetic constants; moreover, they seem to be favored by substrate activation. The model is applied to phosphofructokinase, which is the enzyme chiefly responsible for glycolytic oscillations and which presents the same pattern of regulation as the allosteric enzyme appearing in the model. A qualitative and quantitative agreement is obtained with the experimental observations concerning glycolytic self-oscillations.
Goldbeter, A., & Lefever, R. (1972). Dissipative Structures for an Allosteric Model. Biophysical Journal, 12(10), 1302–1315. https://doi.org/10.1016/s0006-3495(72)86164-2