Design and analysis of a biomolecular positive-feedback oscillator

Nonlinear relaxation oscillators in engineering rely on positive feedback to operate. One category of relaxation oscillators is given by astable multivibrators, that include a bistable component at the core of their architecture. Here we describe a molecular network motif that operates as an astable multivibrator, and relies on a bistable switch (the classical Gardner and Collins genetic switch) that is toggled between its stable steady states by a persistent input. We show that oscillations arise in the presence of two negative feedback loops that process the persistent input and influence the production of the molecular species forming the bistable subsystem. We perform a thorough stability analysis of this motif obtaining closed-form practical conditions for the emergence of oscillations, and we examine the sensitivity of the system to parameter variations.