Fig. 1: The synthetic biological integral controller.

a Block diagram of a typical closed-loop controller. b Design of the synthetic biological integral controller, where the color coding corresponds to the blocks shown in a. The reference is set by the input DNA P_X and the plant signal V (mRNA of gene y) is measured through the expression of another gene z, which encodes for a protein Z. Error computation is achieved through a molecular sequestration reaction between the proteins X and Y. Here the red-color cross represents the open-loop configuration of the controller, which results when the feedback signal Y is absent. c In the closed-loop configuration, the error criterion is the difference between scaled versions of the input P_X and the plant signal to be regulated, V. In the actual controller implementation, this role is played by the free amount of the protein X, which we denote as X_R because we think of it as what remains after binding to Y (and to promoter sites). To produce an output (Z) that is independent of the disturbances in \(P_Y^{{\mathrm{tot}}}\) and \(P_Z^{{\mathrm{tot}}}\), the error signal is integrated (see Supplementary Note 1). The assumptions made to derive this expression are mentioned in Supplementary Note 1 where it is also shown that the controller output (Z) linearly depends on the plant signal at the steady state. In the absence of the feedback the output depends nonlinearly on P_X and \(P_Z^{{\mathrm{tot}}}\) such that any disturbance in \(P_Z^{{\mathrm{tot}}}\) may perturb the output. d Overview of the E. coli cell-free toolbox for prototyping and executing parts and circuits in vitro. e Experimental implementation of the integral controller. Three plasmids are used, P_70a-σ₂₈, expressing the E. coli σ₂₈ (X) from a σ₇₀ promoter (P_X), P_28a-FlgM, expressing the anti-σ₂₈ factor (Y) from a σ₂₈ promoter \((P_Y^{{\mathrm{tot}}})\), and P_28a-deGFP, expressing the reporter deGFP (G) from a σ₂₈ promoter \((P_Z^{{\mathrm{tot}}})\). In the open-loop controller, instead of the anti-σ₂₈ factor (FlgM), mSA is expressed, which is not sequestered by σ₂₈, nor does it directly affect any reaction rates (see Methods). The mSA control gene promoter is denoted as \(P_{YC}^{{\mathrm{tot}}}\).