The cell cycle represents a key physiological process, both in normal and pathological conditions. To gain insight into the dynamics of this major cellular process we developed a detailed mathematical model for the enzymatic network of cyclin-dependent kinases (Cdks) driving progression along the successive phases M (mitosis), G1, S (DNA replication) and G2 of the mammalian cell cycle. The model is described by a system of 39 nonlinear, coupled differential equations. A reduced, skeleton model containing only 5 ordinary differential equations yields similar results. The analysis of the models shows how the balance between cell cycle arrest and cell proliferation is controlled by growth factors (GFs) or by the levels of activators (oncogenes) and inhibitors (tumor suppressors) of cell cycle progression. Supra-threshold changes in the level of any of these factors can trigger a switch in the dynamical behavior of the Cdk network corresponding to a Hopf bifurcation between a stable steady state, associated with cell cycle arrest, and sustained Cdk oscillations of the limit cycle type corresponding to cell proliferation. To further study the dynamics of the cell cycle we developed an automaton model that switches stochastically between the sequential phases of the cell cycle, which have durations distributed around mean values. This approach is well suited to studying in a cell population how the fractions of cells in a given cell cycle phase evolve in time. The automaton model can be used to probe the existence of optimal schedules of circadian administration of anticancer drugs that target cells in a particular phase of the cell cycle.

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