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Global asymptotic stability of a genetic negative feedback loop with an affine control

Genetic negative feedback loops are essential and recurrent biological motifs. They are traditionally described with N-dimensional competitive dynamical systems, composed of highly non-linear Hill functions. The stability property of their unique steady state usually determines the global dynamical behavior: homeostasis under stability or emergence of oscillations. When homeostasis conditions are disrupted, undesired oscillations can emerge and may lead to various diseases. This paper presents a classical affine control strategy that is able to stabilize the unstable steady state of the disrupted system and suppress undesired oscillations. For biological purpose, this control is designed as simple as possible in order to reduce the use of devices and the complexity of the biological set-up. For this reason, the control law only depends on the measurement of a unique gene and only acts on its own expression. Due to the complexity of this controlled dynamical system, a new methodology, based on the construction of successive hyperrectangles of the state space that act as Lyapunov function level-sets, is proposed in order to prove global convergence and global stability results. Despite its apparent simplicity, this affine control law is shown to globally stabilize the disrupted system and recover homeostasis.