Fig. 6. Reduced models for the ZF, intraDD, TetR and Gal4 controllers.

a Reduced Motif. For simplicity, we assume that the protein degradation rates are negligible compared to the dilution rate δ; however, this assumption can be easily relaxed (see Supplementary Information Section 3). Note that δ is assumed to be non-zero here to capture the more realistic scenario. The model reduction recipe presented in Theorem 2 and Fig. 5 can be straightforwardly applied to all of the four controller topologies in Fig. 3, where the “charge” vectors q⁺, q⁻ and q⁰ are shown explicitly. Observe that all four controllers reduce to the same motif comprised of the three effective species Z⁺, Z⁻ and Z⁰. The difference between them appears only in the effective control action $u=\mathcal{U}\left(z^{+},z^0\right)$. b Effective control actions. The control actions $u=\mathcal{U}\left(z^{+},z^0\right)$ is given separately for each controller as a function of the effective species concentrations. For the intraDD controller, the control action u is a strictly monotonically increasing function of z⁺ only, and hence the control structure is a standalone integrator. In contrast, for the ZF and Gal4 controllers, it is shown (see Supplementary Information Sections 3.D and 3.B) that the control action u is strictly monotonically increasing (resp. decreasing) in z⁺ (resp. z⁰). This gives rise to a filtered proportional-integral (PI) control structure³⁸. Finally, for the TetR controller, it is shown that the control action u is stricly monotonically increasing in z⁺; whereas, its monotonicity switches from increasing to decreasing as the levels of (z⁺, z⁰) rise (see Supplementary Information Section 3.A). This gives rise to a filtered PI control structure where the P-component switches sign. Note that the algebraic equations presented in Fig. 5 are solved explicitly for the ZF and intraDD controllers; however, they are kept in their implicit form for the TetR and Gal4 controllers.