How to control a biological switch: a mathematical framework for the control of piecewise affine models of gene networks

This article introduces preliminary results on the control of gene networks, in the context of piecewise-affine models. We propose an extension of this well-documented class of models, where some input variables can affect the main terms of the equations, with a special focus on the case of affine dependence on inputs. This class is illustrated with the example of two genes inhibiting each other. This example has been observed on real biological systems, and is known to present a bistable switch for some parameter values. Here, the parameters can be controlled. Some generic control problems are proposed, which are qualitative, respecting the coarse-grained nature of piecewise-affine models. Piecewise constant feedback laws that solve these control problems are characterized in terms of affine inequalities, and can even be computed explicitly for a subclass of inputs. The latter is characterized by the condition that each state variable of the system is affected by at most one input variable. These general feedback laws are then applied to the two dimensional example, showing how to control this system toward various behaviours, including the usual bi-stability, as well as situations involving a unique global equilibrium.