In this work we first explore the variability in LuxI and AHL species concentrations in a synthetic genetic circuit implementing a feedback loop and cell-to-cell communication.
Four different configurations: open and closed loop without and with cell to cell communication, allows us to see how feedback loop with cell to cell communication effectively reduces the variability across the population of cells.
To have a more analytical insight and a quantitative measure of how much the synthetic circuit improves the variability along the population, we approximate the sigmoidal repressible promoter response with a piecewise linear function with saturation, and we lump all the variability into the tightness of the promoters transcription rate.
With all this, we present a framework where under certain situation, the systems converge into the linear region of the controller, and then we obtain a closed expression for the coefficient of variation of the steady state distribution. This results in the possibility of fine-tune the parameters of the controller to obtain a desired output variability in the protein concentration.