[Qualitative study of a mathematical model of the open futile cycle fructose-6-P--fructose-1,6-P2].

A simple mathematical model of the open futile cycle fructose-6-P in equilibrium fructose-1,6-P2 in which the fructose bisphosphatase reaction is inhibited by excess of its substrate has been analysed. A detailed qualitative investigation of the model shows that it possesses all properties characteristic of any other dynamical system of the second order which has a hysteretic major null-cline, 1 to 3 steady states and is capable of generating self-oscillations.

[Comparative theoretic analysis of an open reaction S1 goes to and comes from S2 E(R,T) in which the oligomeric enzyme E(R,T) is isosterically or allosterically activated by the product S2].

In connection with evolutionary aspects of the mechanisms of allosteric regulation of cell metabolism, a mathematical model of an open reaction leads to S1 E(R,T) in equilibrium S2 leads to involving product activation of the olygomeric enzyme E(R,T) whose protomers undergo the concerted conformational transitions R in equilibrium T, has been analysed. Two activation mechanisms, isosteric and allosteric, were considered. Both mechanisms produce qualitatively the same effects: the input characteristic of the reaction possesses hysteresis, which causes multiple steady states and self-oscillations.

[Hysteresis, alternative stationary states and auto-oscillations in an open futile cycle one of the reactions of which is substrate inhibited].

In connection with the problem of regulation of futile (energy-dissipating) cycles in cell metabolism, a kinetic model has been investigated of an open cycle S1 (see article) S2, in which one of the enzymes (E-) is inhibited by the excess of its substrate S2. The quasi-stationary net velocity of the utilization of substrate S1 in the cycle as a function of its concentration is shown to be of a hysteretic character. Owing to this the alternative stationary states and self-oscillations may occur in the cycle.

[Hysteresis, multiplicity of stationary states and auto-oscillations in the reversible flow-through reaction].

A mathematical model has been investigated of a reversible flow-through reaction S1 reversible S2 catalyzed by an olygomeric enzyme E(R,T) the protomers of which undergo concerted conformational transitions R reversible T. The isosteric activation of olygomer E by product S2 binding preferably to protomer active sites in conformation R is shown to be a possible cause of hysteresis in the quasi-stationary input characteristic of the reaction, v (s2). The latter determines the rate law of the reaction, provided the concentration of S2 is a quasi-stationary one.