From simple to complex patterns of oscillatory behavior in a model for the mammalian cell cycle containing multiple oscillatory circuits

We previously proposed an integrated computational model for the network of cyclin-dependent kinases (Cdks) that controls the dynamics of the mammalian cell cycle [C. Gérard and A. Goldbeter, "Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle," Proc. Natl. Acad. Sci. U.S.A. 106, 21643 (2009)]. The model contains four Cdk modules regulated by reversible phosphorylation, Cdk inhibitors, protein synthesis or degradation, and the balance between antagonistic effects of the tumor suppressor pRB and the transcription factor E2F. Increasing the level of a growth factor above a critical threshold triggers the transition from a quiescent, stable steady state to self-sustained oscillations in the Cdk network. These oscillations correspond to the repetitive, transient activation of cyclin D/Cdk4-6 in G1, cyclin E/Cdk2 at the G1/S transition, cyclin A/Cdk2 in S and at the S/G2 transition, and cyclin B/Cdk1 at the G2/M transition. This periodic, ordered activation of the various cyclin/Cdk complexes can be associated with cell proliferation. The multiplicity of feedback loops within the Cdk network is such that it contains at least four distinct circuits capable of producing oscillations. The tight coupling of these oscillatory circuits generally results in simple periodic behavior associated with repetitive cycles of mitosis or with endoreplication. The latter corresponds to multiple passages through the phase of DNA replication without mitosis. We show here that, as a result of the interaction between the multiple oscillatory circuits, particularly when attenuating the strength of the oscillatory module involving cyclin B/Cdk1, the model for the Cdk network can also produce complex periodic oscillations, quasiperiodic oscillations, and chaos. Numerical simulations based on limited explorations in parameter space nevertheless suggest that these complex modes of oscillatory behavior remain less common than the evolution to simple periodic oscillations of the limit cycle type, holding with the view that simple periodic oscillations in the Cdk network correspond to its physiological mode of dynamic behavior.