Stoichiometric balance and sustained rhythms

We investigate periodic solutions of a genetic negative feedback loop model incorporating protein-sequestration-based repression. Stoichiometry is defined as the average ratio of the concentration of repressors to that of activators over one period of a periodic solution. We examine how stoichiometric balance plays a critical role in the emergence of oscillations. Using Hopf bifurcation analysis combined with numerical simulations, we establish the existence of periodic solutions. We further analyze the stoichiometric conditions associated with oscillation generation in the system. To this end, we approximate the stoichiometry by a quantity evaluated at the bifurcation point and justify that this quantity decreases with increasing activator concentration. Our results precisely characterize the stoichiometric range in which sustained oscillations occur, as inferred from the approximating quantity. The validity of this approximation is supported by the properties of Hopf bifurcation and is illustrated by numerical computations. Finally, we address the effects of differential degradation rates on the existence and stability of periodic solutions, as well as on stoichiometric balance.