Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system

We analyze on a model biochemical system the effect of a coupling between two instability-generating mechanisms. The system considered is that of two allosteric enzymes coupled in series and activated by their respective products. In addition to simple periodic oscillations, the system can exhibit a variety of new modes of dynamic behavior; coexistence between two stable periodic regimes (birhythmicity), random oscillations (chaos), and coexistence of a stable periodic regime with a stable steady state (hard excitation) or with chaos. The relationship between these patterns of temporal self-organization is analyzed as a function of the control parameters of the model. Chaos and birhythmicity appear to be rare events in comparison with simple periodic behavior. We discuss the relevance of these results with respect to the regularity of most biological rhythms.