Proportional-Integral Degradation (PI-Deg) control allows accurate tracking of biomolecular concentrations with fewer chemical reactions

We consider the design of synthetic embedded feedback circuits that can implement desired changes in the concentration of the output of a biomolecular process (reference tracking in control terminology). Such systems require the use of a "subtractor", to generate an error signal that captures the difference between the current and desired value of the process output. Unfortunately, standard implementations of the subtraction operator using chemical reaction networks are one-sided, i.e. they cannot produce negative error signals. Previous attempts to deal with this problem by representing signals as the difference in concentrations of two different biomolecular species lead to a doubling of the number of chemical reactions required to generate the circuit, hence sharply increasing the difficulty of experimental implementations and limiting the complexity of potential designs. Here we propose an alternative approach that introduces a degradation term into the classical proportion-integral control scheme. The extra tuning flexibility of the resulting PI-Deg controller compensates for the limitations of the one-sided subtraction operator, providing robust high-performance tracking of concentration changes with a minimal number of chemical reactions.