Fig. 1: Overview.

a The closed-loop system is comprised of the controller network (Z₁, Z₂) connected in a feedback configuration with an arbitrary regulated network. By examining the controller dynamics, it is straightforward to uncover the integral control action that endows the closed-loop system with RPA. That is, as long as the closed-loop system is stable, the concentration of the regulated output X_L converges to a prescribed value μ/θ, referred to as the setpoint, despite the presence of disturbances and uncertainties in the regulated network. b The heart of the basic AIF motif is the sequestration reaction. In this paper, we exploit the exquisite flexibility of split inteins to genetically implement a broad class of integral controllers that endow the closed-loop system with RPA. The flexibility of split inteins offers an easy-to-build biological framework at the price of (potentially) more complex mathematical models. To this end, we establish a set of simple reaction rules that enable RPA. c The shaded blue box schematically depicts the products of intein-splicing reactions starting from the educts. The first schematic (top left) describes the general split intein-splicing reaction where both split inteins are flanked by protein domains, labeled \((\,{{\mbox{N}}},{{{\mbox{N}}}}^{{\prime} })\) and \((\,{{\mbox{C}}},{{{\mbox{C}}}}^{{\prime} })\) for the N- and C-terminal protein sequences, respectively. The first product is a new protein containing the N and \({{{\mbox{C}}}}^{{\prime} }\) domains of the educts, while the second product is a heterodimer containing \({{{\mbox{N}}}}^{{\prime} }\) and C, which are held together by the two inactive split inteins (see Supplementary Information, Fig. 22). The remaining four schematics in the shaded blue box are instantiations of the general case and are labeled according to the perspective of the protein containing the Int^N segment. As the labels suggest, a part of the protein sequence is either exchanged for another one or removed through cleavage. Furthermore, it is possible to ligate another sequence to the protein of interest or make it non-responsive to future splicing reactions through intein removal. To illustrate the design modularity and flexibility, we list a selection of intein-based implementation examples of the antithetic sequestration motif (below the shaded blue box) based on the described possible splicing reactions. In the first example (bottom left), inteins are used to shuffle proteins between the nucleus and the cytoplasm due to the NLS and the NES which flank the protein sequences. In particular, the intein-splicing reaction exchanges the NLS with a NES which leads to the export of a transcription factor (TF) out of the nucleus where it cannot initiate transcription anymore. In the second example, inteins are used to exchange an AD by a RD, which inverts the function of the TF. In the third example, a split inteins is introduced within a functional domain without disturbing it. The splicing reaction results in the cleavage of the domain rendering it nonfunctional. In the fourth example, a protein is fused to the first split-intein and a split-degradation tag, while the second split-intein is fused to the other half of the split-degradation tag. The splicing reaction re-ligates the degradation tag rendering it functional and capable of degrading a POI. In the final example, a DBD can reversibly heterodimerize with an AD via its split inteins. Note that the split intein on the DBD is mutated so that it cannot perform the splicing reaction upon dimerization. A separate non-mutated intein is able to remove the intein from the AD through splicing. This renders the AD unable to heterodimerize with the DBD. AD: activation domain, RD: repressing domain, Int^C: intein C, Int^N: intein N, DBD: DNA binding domain, NLS: nuclear localization signal, NES: nuclear export signal, N-Deg: N terminus of split degradation domain, C-Deg: C terminus of split degradation domain. d A simple recipe is developed to reduce the otherwise mathematically complex controller models into simple motifs that resemble the basic AIF motif, but with a fundamental difference: the sequestration product is allowed to have a function that can be leveraged as a tuning knob to enhance the controller performance while maintaining RPA.