## 4 Incoherent FFLs
### Synthetic II-FFLs in E. coli  
In a pioneering work back in 2004, [PERSON] et al. [49] designed an II-FFL in order to engineer cell-cell communication between the two subpopulations (sender and receiver cells) of an _E. coli_ consortium (see Figure 6A). The II-FFL was fully contained into the receiver cells and needed a signal from the sender cells to generate a pulse in green fluorescence. In the presence of tetracycline, the sender cells could synthesize Luxl synthase that catalyzed the production of AHL (acyl-homoserine lactone). AHL diffused through the cell membranes and reached the spatially nearby receiver cells. Here, AHL worked as the Sr input since it bound and activated LuxR (X), an activator protein that was constitutively expressed. The activated LuxR triggered the expression of the cl repressor (Y), and GFP (Z), with the consequent generation of a pulse in fluorescence. The amplitude and the time-location of the peak depended on both AHL concentration and increasing rate. Moreover, no communication was possible between too distant (4.5 mm) sender and receiver cells on a solid medium. One year later, [PERSON] et al. [50] realized pattern formation in a similar bacterial consortium by placing the receiver cells at different distances from the sender ones. The receiver cells contained a non-canonical incoherent FFL made of five nodes (see Figure 6B). AHL, produced by the sender cells, activated LuxR in a concentration-dependent fashion. In its turn, the activated LuxR triggered the synthesis of two repressor proteins: cl, which downregulated strongly the expression of LacI, and LacI\({}_{\text{MI}}\)--a variant of LacI--that exerted  
Figure 5: The II-FFL in the _E. coli_ galactose system. GalR is a repressor protein that establish the connection between the FFL and GalP. Moreover, GalR is able to bind and downregulate the _galE_ promoter (upstream of _GalP_). a mild repression on GFP production. Finally, LacI was able to switch OFF fluorescence expression as well. Receivers close to the senders obtain a high quantity of AHL such that both LacI\({}_{\text{M1}}\) and cl were highly expressed. The former protein was the responsible for the absence of a fluorescent signal. At lower concentrations of AHL, LacI\({}_{\text{M1}}\) and cI were rather poorly expressed. However, while LacI\({}_{\text{M1}}\) failed to lower GFP synthesis, cl completely shut down the production of LacI, which could no longer inhibit GFP expression. Finally, at very low concentration of AHL LuxR was not activated. This resulted in a high amount of LacI in the receivers, which implied absence of fluorescence again. This spatial arrangement of sender and receiver cells led to a so-called bullseye pattern on plate. In a more recent work, [PERSON] et al. [51] employed the same I1-FFL as in [49] in order to construct an arsenic biosensor. Here, LuxI synthesis was prevented by the action of the arsenic repressor, ArsR, that was constitutively expressed. Molecules of arsenic, thus, triggered AHL production that led to a short-delayed red fluorescence pulse that resented of the arsenic concentration. Three II-FFLs, regulated by IPTG and using the T7-RNAP as X node, were presented by [PERSON] et al. [52]. Even though each of them was made of three TUs, their designs were different. The first I1-FFL showed the usual scheme with a competition, at Pz, between T7-RNAP (which worked as an activator) and the MetJ (Y) repressor. Mutations in the met operator allowed to move the fluorescent peak towards lower concentration of IPTG and achieve full output repression around 100 mM IPTG. The second I1-FFL variant (termed gfp:anti-gfp) mixed transcriptional activation and translational repression of GFP. The Y gene corresponded to a fragment of the reverse GFP sequence that, upon transcription, sequestered the mRNA of GFP (Z) and lowered its translation. Stronger fluorescent repression was achieved by transcribing the full GFP reverse sequence. Finally, in the last design, Y encoded for the T7 lysozyme that bound and inhibited T7-RNAP.