Fig. 4.

Mathematical modeling of the various circuits. (A) A chemical reaction network compactly modeling the various circuits presented in Figs. 2 and 3. The sense mRNA, Z1, is constitutively produced at a rate $\mu (G_1)$ that depends on the gene (plasmid) concentration, G₁. Then, Z1 is translated into a fusion of a synthetic transcription factor, fluorescent protein, and inducible-degradation tag, referred to as X1, at a rate k. X1 is either actively degraded by the ASV disturbance D at a rate $\lambda (X_1;D)$ or converted to X2 at a rate c by releasing the SMASh tag. The protein X2 dimerizes to form A, which activates the transcription of the antisense RNA, Z2. The transcription rate, denoted by θ, is a function of A and the gene concentration G₂. The antithetic integral control, shown in the blue box, is modeled by the sequestration of Z1 and Z2 at a rate η. Note that the open-loop circuit is obtained by removing the feedback from the regulated output A. The proportional controller (orange box) is modeled by producing the protein $\mathbf{X}{\prime }_{\mathbf{1}}$, also at a rate k, in parallel with X1 to serve as its proxy. A negative feedback is then achieved by the (un)binding reaction between the proxy $\mathbf{X}{\prime }_{\mathbf{1}}$ and Z1. Finally, the network perturbation (purple box) is modeled by introducing an additional gene $\mathbf{G}{\prime }_{\mathbf{2}}$. This gene is activated by A to transcribe the mRNA $\mathbf{Z}{\prime }_{\mathbf{2}}$ at a rate θ_p which is a function of A and $G_2\prime$. $\mathbf{Z}{\prime }_{\mathbf{2}}$ is then translated into the protein $\mathbf{X}{\prime }_{\mathbf{1}}$ that has, once again, a negative feedback on the production of X1 by binding to Z1. See SI Appendix, Figs. S1, S3, and S4 for a detailed mathematical explanation for each separate circuit. (B and C) Model calibrations to experimental data. (Left) The model fits for the open-loop circuits with/without disturbance (B) and with/without network perturbation (C). (Right) Similarly, the model fits for the closed-loop circuits. The model fits for proportional control are reported in SI Appendix, Fig. S4C. The solid lines denote model fits, while dashed lines denote model predictions. The model fits and predictions show a very good agreement with the experiments over a wide range of plasmid ratios (setpoints) $G_1/G_2$, for all scenarios. (D) Stochastic simulations demonstrating the variance reduction property of the proportional controller. The calibrated steady-state parameter groups of the PI closed-loop circuit, given in SI Appendix, Eq. S42, are fixed, while the time-related parameters are set as follows: $\gamma =\gamma \prime ,k=c=d=1{ \text{min}}^{-1}$ to demonstrate the variance reduction property that is achieved when a proportional controller is appended to the antithetic integral motif. Note that $G_1=0.002 \text{pmol}$, and $G_2=0.004 \text{pmol}$.