On the optimal control of birhythmic oscillatory PWA systems: an application to the p53-Mdm2 network

In this work, we tackle the problem of inducing optimal transfers between the two oscillatory regimes of a birhythmic genetic network, represented through a piecewise affine dynamical system. For that, we resort to an adaptation of Pontryagin’s Maximum Principle to the hybrid setting, with a cost function that combines the transfer time and an L<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> - control cost. We focus on a two-dimensional PWA model of the p53-Mdm2 network, a well-known tumor suppressor module that represents a key example of birhythmicity naturally found in mammalian cells. The resulting optimal control can be expressed in feedback form, and is able to remove an oscillatory mode of the system, allowing selection between low or high frequency oscillations of the bimodal genetic network.