Guidelines for designing the antithetic feedback motif

Integral feedback control is commonly used in mechanical and electrical systems to achieve zero steady-state error following an external disturbance. Equivalently, in biological systems, a property known as robust perfect adaptation guarantees robustness to environmental perturbations and return to the pre-disturbance state. Previously, Briat et al proposed a biomolecular design for integral feedback control (robust perfect adaptation) called the antithetic feedback motif. The antithetic feedback controller uses the sequestration binding reaction of two biochemical species to record the integral of the error between the current and the desired output of the network it controls. The antithetic feedback motif has been successfully built using synthetic components in vivo in Escherichia coli and Saccharomyces cerevisiae cells. However, these previous synthetic implementations of antithetic feedback have not produced perfect integral feedback control due to the degradation and dilution of the two controller species. Furthermore, previous theoretical results have cautioned that integral control can only be achieved under stability conditions that not all antithetic feedback motifs necessarily fulfill. In this paper, we study how to design antithetic feedback motifs that simultaneously achieve good stability and small steady-state error properties, even as the controller species are degraded and diluted. We provide simple tuning guidelines to achieve flexible and practical synthetic biological implementations of antithetic feedback control. We use several tools and metrics from control theory to design antithetic feedback networks, paving the path for the systematic design of synthetic biological controllers.