The animal cell cycle is controlled by the periodic variation of two cyclin-dependent protein kinases, cdk1 and cdk2, which govern the entry into the M (mitosis) and S (DNA replication) phases, respectively. The ordered progression between these phases is achieved thanks to the existence of checkpoint mechanisms based on mutual inhibition of these processes. Here we study a simple theoretical model for oscillations in cdk1 and cdk2 activity, involving mutual inhibition of the two oscillators. Each minimal oscillator is described by a three-variable cascade involving a cdk, together with the associated cyclin and cyclin-degrading enzyme. The dynamics of this skeleton model of coupled oscillators is determined as a function of the strength of their mutual inhibition. The most common mode of dynamic behavior, obtained under conditions of strong mutual inhibition, is that of alternating oscillations in cdk1 and cdk2, which correspond to the physiological situation of the ordered recurrence of the M and S phases. In addition, for weaker inhibition we obtain evidence for a variety of dynamic phenomena such as complex periodic oscillations, chaos, and the coexistence between multiple periodic or chaotic attractors. We discuss the conditions of occurrence of these various modes of oscillatory behavior, as well as their possible physiological significance.

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The conditions for temporal self‐organization in the form of sustained oscillations are determined in a minimal cascade model previously proposed (A. Goldbeter, Proc. Natl. Acad. Sci. USA 88, 9107–9111 (1991)) for the mitotic oscillator driving the embryonic cell division cycle. The model is based on a phosphorylation‐dephosphorylation cascade involving cyclin and cdc2 kinase. In the first cycle of the cascade, cdc2 kinase is activated through dephosphorylation triggered by the accumulation of cyclin, while in a second cycle the activation of a cyclin protease is brought about through phosphorylation by cdc2 kinase. The fact that cyclin promotes the activation of cdc2 kinase while the latter enzyme triggers cyclin degradation introduces a negative feedback loop which is at the core of the periodic operation of the cascade. We analyze the mechanism of oscillatory behavior by constructing stability diagrams as a function of some of the main parameters of the model. Investigated in turn are the roles of negative feedback and of phosphorylation‐dephosphorylation thresholds. Such thresholds arise from the phenomenon of zero‐order ultrasensitivity associated with the kinetics of covalent modification cycles. An extension of the minimal model allows one to address the effect of additional phosphorylation‐dephosphorylation cycles and the possible role of autocatalysis by cdc2 kinase in the generation of periodic behavior. The analysis shows that the existence of thresholds as well as an increase in the number of cycles in the cascade favor the occurrence of sustained oscillations. The results further indicate that negative and positive feedback may both contribute to the repetitive activation of cdc2 kinase.