Stochastic Description of a Bistable Frustrated Unit
Mixed positive and negative feedback loops are often found in biological systems which support oscillations. In this work we consider a prototype of such systems, which has been recently found at the core of many genetic circuits showing oscillatory behaviour. Our model consists of two interacting species A and B, where A activates not only its own production, but also that of its repressor B. While the self-activation of A leads already to a bistable unit, the coupling with a negative feedback loop via B makes the unit frustrated. In the deterministic limit of infinitely many molecules, such a bistable frustrated unit is known to show excitable and oscillatory dynamics, depending on the maximum production rate of A which acts as a control parameter. We study this model in its fully stochastic version and we find oscillations even for parameters which in the deterministic limit are deeply in the fixed-point regime. The deeper we go into this regime, the more irregular these oscillations are, becoming finally random excitations whenever fluctuations allow the system to overcome the barrier for a large excursion in phase space. The fluctuations can no longer be fully treated as a perturbation. The smaller the system size (the number of molecules), the more frequent are these excitations. Therefore, stochasticity caused by demographic noise makes this unit even more flexible with respect to its oscillatory behaviour.