Dynamic Behavior of an Isolated Repressilator with Feedback

Dynamic behavior of the model describing the literature-known synthetic genetic oscillator (repressilator), which is constructed as a three-gene ring, is studied by the methods of numerical analysis of bifurcations. The protein products of each gene inhibit activity of the subsequent neighbor gene, which ensures the limit-cycle existence. One feedback caused by the production of the signal molecules activating expression of a ring gene is added to the repressilator ring. Such a system of four ordinary differential equations is shown to demonstrate diversified and unusual multistable behavior, i.e., coexistence of the stable limit cycle and stable stationary state. Homoclinic saddle–node bifurcations occur at the boundaries of parameter ranges where such a coexistence is observed. The number of homoclinic transitions increases with increasing cycle amplitude if the characteristic times of the protein and signal-molecule dynamics converge. In this case, the limit cycle undergoes forward and backward period-doubling bifurcations.