Retroactivity to the Input and Thevenin’s Theorem for Complex Gene Transcription Networks

Synthetic biology is a bottom-up engineering discipline: modules are systematically designed with predefined behavior and then combined to build up larger biomolecular circuits. Although the modules produce the desired behavior in isolation, they fail to operate properly when they are connected due to retroactivity, an effect which extends the notion of impedance to biomolecular systems. Despite playing a central role, retroactivity is not yet characterized in complex gene transcription networks. In this paper, we mathematically describe and quantify this effect. This result is obtained by applying singular perturbation on the finite time interval. We identify the biomolecular counterpart of impedance and introduce the effective retroactivity to the input of a gene. Furthermore, we provide a theorem similar to Thevenin’s, describing how modules affect each other when connected. We illustrate the implications of the results by investigating crosstalk in a simple genetic system.