= Solution
Both motifs have a direct path $X\to Z$ and an indirect path $X\to Y\to Z$. <C1 coherent feed-forward loop> uses three activating edges. <I1 incoherent feed-forward loop> activates $Y$ and $Z$ from $X$, but lets $Y$ repress $Z$. The AND condition means that the direct input must be active and the intermediate input must permit output: present for an activator, absent for a repressor. <Structure and function of the feed-forward loop network motif by Shmoolik Mangan and Uri Alon>[2003, AND-gated coherent and incoherent motifs]{ref}
For a simple threshold model let
$$
\dot Y=\beta_Y H(X-K_{XY})-\alpha_Y Y,\qquad Y_\infty=\beta_Y/\alpha_Y.
$$
The coherent output obeys $\dot Z=\beta_ZH(X-K_{XZ})H(Y-K_{YZ})-\alpha_ZZ$. Starting from $Y=0$ under a sustained high input, $Y=Y_\infty(1-e^{-\alpha_Yt})$. Provided $0<K_{YZ}<Y_\infty$, the <gene expression> delay is
$$
\boxed{t_d=-\alpha_Y^{-1}\ln(1-K_{YZ}/Y_\infty).}
$$
Shorter input pulses do not turn on output <gene expression>. This is a <persistence detector>: it rejects brief positive inputs. Removing $X$ immediately removes the direct permission for <gene expression>, even if $Y$ remains high. The <concentration> of $Z$ subsequently decays with its own lifetime rather than vanishing instantly.
The incoherent output instead obeys $\dot Z=\beta_ZH(X-K_{XZ})H(K_{YZ}-Y)-\alpha_ZZ$. <Gene expression> begins immediately, then stops when accumulated repressor crosses threshold at $t_d$. With zero initial output,
$$
\boxed{Z(t)=\frac{\beta_Z}{\alpha_Z}(1-e^{-\alpha_Zt})\quad(0<t<t_d),\qquad
Z(t)=Z(t_d)e^{-\alpha_Z(t-t_d)}\quad(t>t_d).}
$$
A maintained stimulus therefore generates a pulse in this ideal Boolean model. Delayed inhibition can accelerate an early response and limit its duration. The ideal final zero output is conditional on crossing the repression threshold; graded response laws or basal <gene expression> can give a nonzero final level and need not adapt perfectly. If $Y_\infty\leq K_{YZ}$, the coherent circuit does not activate, while the incoherent circuit fails to terminate its <gene expression>. These boundary cases show why regulatory signs alone do not fix function.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2016-feed-forward-loop-responses.png]
{title=AND-gated coherent and incoherent type-1 feed-forward loops}
{description=Original regulatory graphs and threshold-model responses. The coherent motif delays production until an activator accumulates; the incoherent motif produces a pulse before a repressor accumulates. Output concentrations retain their finite decay times.}
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