Qualitative control of undesired oscillations in a genetic negative feedback loop with uncertain measurements

In the context of gene regulatory networks, a negative feedback loop is modeled by N-coupled ordinary differential equations. The resulting system is highly non-linear due to the use of smooth Hill functions. This classical dynamical system properly captures the two main biological behaviors arising from this type of recurrent network motif: homeostasis under global stability of the unique fixed point, and biochemical oscillations otherwise. When homeostatic conditions are disrupted, undesired sustained oscillations can appear. In this context, a biologically relevant control strategy is designed in order to suppress these undesirable oscillations. As biological measurement techniques do not provide a quantitative knowledge of the system, the control law is chosen piecewise constant and dependent on specific regions of the state space. Moreover, due to biological devices inaccuracies, the measurements are considered uncertain leading to regions in which the control law is undefined. Under appropriate conditions on the control inputs, successive repelling regions of the state space are determined in order to prove the global convergence of the system towards an adjustable zone around the fixed point. These results are illustrated with the well-known p53-Mdm2 genetic feedback loop. (C) 2019 Elsevier Ltd. All rights reserved.