Time-optimal control of piecewise affine bistable gene-regulatory networks

We study the minimal-time problem for a piecewise affine bistable switch. Motivated by applications in synthetic biology and biotechnology, the aim is to minimize the time needed for this system to achieve transitions between its two stable steady states. The latter represents the two possible states of a genetic toggle switch, a synthetic flip-flop device playing a fundamental role in biocomputing and gene therapy. Results show that a time-optimal transition between states should pass by an undifferentiated state, which is well known in cell biology for its importance in fate differentiation of cells. In order to characterize the capacity of the system to achieve transitions, we provide a lower bound on the minimal time, whose knowledge becomes relevant when considering realistic systems involving subsystems evolving on different time scales. Then, we show numerical simulations of optimal trajectories illustrating the structure of the bang-bang optimal control for different scenarios.