Bounding the effect of retroactivity in the presence of parameter uncertainty

As the number of synthetic genetic modules grows, the issue of reliably predicting their behavior upon interconnection becomes more pressing. The trajectory of an upstream module changes once connected to a downstream module due to retroactivity. Here, we employ dissipativity analysis to provide an upper bound on the L2 measure of this difference. To obtain this upper bound we formulate a Sum of Squares (SOS) optimization problem which we then solve using semi-definite programming. One particular strength of this approach is the ability to successfully handle parameter uncertainties while providing guaranteed upper bounds on the difference between the trajectories. We illustrate how to apply our method in the case of the most recurrent motif in gene networks.