## 3 Coherent FFLs  
### C1-FFL: A Sign-Sensitive Delay Device  
#### 3.1.1 AND Logic C1-FFL  
In its canonical representation, a C1-FFL is made of three activation edges. Therefore, X activates Z expression both directly and indirectly through Y. Let us suppose that X encodes for an activator whose wild type is inactive (X\({}^{i}\)), i.e., unable to bind the DNA. Upon binding the inducer Sx, X\({}^{i}\) is activated (X\({}^{\alpha}\)) and is thus able to bind the promoter upstream of both Y (Py) and Z (Pz) (see Figure 2). If X\({}^{i}\) is constitutively expressed, we can let the system reach a steady state, characterized by a high concentration of X\({}^{i}\), and then add Sx, to the C1-FFL in a sufficiently high amount. By assuming a high affinity between Sx and X\({}^{i}\), X\({}^{i}\) switches rapidly to X\({}^{\alpha}\) (step-like activation). X\({}^{\alpha}\), then, binds simultaneously Py and Pz. Z expression varies depending on the logic function implemented by Pz. If Pz mimics an AND gate, after the step-like activation, there will be a delay in the synthesis of Z due to the time Y takes to reach the concentration threshold to activate, together with X\({}^{\alpha}\), the transcription of Z. A delay is present only upon induction with Sx (ON step) but not after Sx removal (OFF step). By washing away the inducer Sx, X\({}^{\alpha}\) switches rapidly to X\({}^{i}\), which prevents any further production of Z. This behavior is known as sign-sensitive delay since it takes place only at the ON sign of the steps in Sx. The extent of delay depends on the kinetic parameters that regulate the interactions involving X\({}^{\alpha}\), Y, and the DNA. This delay is functional for filtering out fluctuations and spurious pulses, such that only persistent signals lead to the expression of Z [9]. Thus, the C1-FFL is said to work as a _persistence detector_[6]. In _E. coli_, the L-arabinose utilization system, which permit the bacterial cells to grow in the absence of glucose, has the structure of a C1-FFL (see Figure 3A). Upon glucose starvation, cAMP (Sx) is produced by the cell and activates CRP. CRP, then, initiates the synthesis of AraC (which, in its turn, is activated by L-arabinose) and binds at the araBAD and araFGH operons where transcription starts upon arrival of AraC, i.e., after 13 and 20 min, respectively. At this point, the consumption of L-arabinose can start [37].  
Figure 2: The dynamics of the coherent type I-FFL with AND logic gate. Sx and Sy activates X\({}^{i}\) and Y\({}^{i}\), respectively. Hence, both signals are required to turn on Pz that leads the expression of gene Z [6]. K\({}_{ij}\) represents the affinity between \(i\) activator and \(j\) promoter.  
In a recent work, [PERSON] et al. [38] implemented, in an E. coli cell-free transcription-translation (TXTL) system, an AND logic C1-FFL that made use of two RNA molecules (named trigger and toehold switch) and the \(\sigma^{28}\) factor as regulatory elements (see Figure 3B). Artificial noise signals, generated via microfluidic flow reactors, were used to mimic disturbances on the input that can be met in vivo. This C1-FFL variant decreased, on one hand, the background activity--in comparison to a reference circuit--thus enhancing the fold-change detection [29] but, on the other hand, showed some limitations as a temporal filter.