Robustness of a Bistable Biomolecular Positive Feedback Circuit to Global Parametric Perturbations

Achieving robustness to global perturbations where all parameters can change at the same time is challenging because the controller would also face the same disturbance as the plant. For nonlinear positive feedback, an important mechanism for cell fate determination in biomolecular contexts, quantitative aspects of robustness to such perturbations are generally unclear. Here we used mathematical methods of control and dynamical systems, interval analysis, and a benchmark model of a bistable biomolecular positive feedback circuit to address this. We confirmed that such perturbations can change the qualitative behaviour of the system extinguishing bistability. We obtained a quantitative relation between the relative variation in the stable steady state and the unstable steady state in terms of the relative changes in the parameters. We showed how the deviation in the trajectories near the unstable steady state due to global perturbations could diverge almost exponentially, after an initial transient, which could have a significant impact on the bistable switching dynamics. We found that the size of the eigenvalue for the unstable steady state was greater than that for the stable steady state, and proved this for certain parameters using a rigorous numerical construction. We noted a tradeoff between enhancing the parameter space of bistability and the increased sensitivity in the bistable dynamics due to parametric perturbations. We obtained rigorous bounds on the entire transient response for global parametric perturbations. These results provide a quantitative insight into the robustness of a bistable biomolecular positive feedback circuit to global parametric perturbations.