Retroactivity analysis of a chemical reaction network module for the subtraction of molecular fluxes

The availability of a well-characterised subtraction module is a key step towards the realisation of modular embedded feedback controllers in synthetic biological systems. A well-known problem when dealing with complex biosystems is represented by the retroactivity effect, which can significantly modify the dynamics of interconnected subsystem, with respect to the behaviour they exhibit when disconnected from each other. In this paper, we illustrate a minimal CRN that implements a subtraction operation between two input molecular fluxes. In order to assess its effectiveness as a module of a more complex system, we analyse its retroactivity upon interconnection. More specifically, we connect the subtraction module with an upstream module, which determines the dynamics of the inputs species, and with a downstream transcriptional module, which acts as a load. By comparing the dynamics of the loaded and unloaded subtractor, we show that the retroactivity can be attenuated when the dynamics of the subtractor and of the load system evolve over different timescales. This result, obtained through a singular perturbation analysis, is confirmed by means of numerical simulations.