## 4 Incoherent FFLs  
### I1-FFL as a Pulse Generator and a Response-Time Accelerator  
The behavior of genetic circuits and their response to external perturbations depend highly on their connectivity. The dynamics of a circuit consists of a transient phase, during which the species concentrations change at various rates, and one (or more) steady states, the equilibria, at which concentrations are constant since production and degradation cancel out for each species. Often, circuit analysis focuses on steady states such that the experiments neglect the transient phase. In this way, however, an output signal limited in time, such as a pulse, is missed. Moreover, the steady-state analysis does not permit to estimate the circuit response time, i.e., the time a circuit takes to reach one half of the output level at the equilibrium.  
Incoherent FFLs--especially type II--are both pulse generators and response-time accelerators [9]. Hence, these network motifs cannot be understood completely via the sole steady-state analysis.  
The height and the amplitude of the peak, as well as the duration of the response time, depend on several FFL kinetic parameters (e.g., the strength of the three promoters and the decay rate of the three proteins) [45]. Since the cells in a population are not synchronized, the peak detection and the response time estimation can be difficult tasks and, in general, require single-cell analysis. This can be carried out via flow cytometry (FACS--fluorescence-activated cell sorting), fluorescent microscopy, or by recording live movies of the synthetic cells [46, 47].  
The galactose system in \(E\). _coli_ contains an AND logic I1-FFL [31] that has permitted to verify a further feature of this network motif, i.e., the non-monotonic dependence on the input signal [48]. As shown in the simplified scheme in Figure 5, X is the CRP activator  
Figure 4: The dynamics of the coherent type I FFL with OR logic. Gene Z is expressed once a single transcription factor is activated by the corresponding inducer [6]. K\({}_{\text{ij}}\) represents the affinity between \(i\) activator and \(j\) promoter.  
that responds to cAMP (Sx), Y corresponds to the repressor galS that senses D-galactose (S\(y\)), Z is the galETK operon that presides over galactose consumption. As we have seen in Section 3.1.1, in the absence of glucose cAMP accumulates in _E. coli_ cells and binds CRP. As a consequence, CRP assumes its active configuration and stimulates the expression of galS and the genes in the galETK operon that make use of galactose. Upon reaching a sufficiently high amount, galS (which is activated by D-galactose) binds the _galE_ promoter and contrasts the expression of the galETK operon leading to a pulse in the concentration of the galETK genes followed by the steady state. Both _galE_ and _galP_ promoter (galP, which is connected to the II-FFL, acts as a pump to transport galactose into the cell) show the highest activity (a peak) at intermediate concentrations of cAMP, i.e., a non-monotonic response to the input. The II-FFL confers a rapid response time (one-third of the cell generation time [31]) to the galactose system such that the cells switch quickly to galactose in the absence of glucose.