Feedforward Loops

Recall the different types of feedforward loops. Last lecture we discussed the behavior of the C1 loop. The advantage of the C1 loop is that it protects Z from turning on due to fluctuations in X. This is called a persistence detector.

Now we will examine the behavior of some of the incoherent feedforward loops.

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I1 Loop

Assume that the signals from X and Y combine in an AND NOT logic fashion to create output Z.

Transcription Rate	X	Y
\(0\)	\(0\)	\(0\)
\(0\)	\(0\)	\(>0\)
\(k\)	\(>0\)	\(0\)
\(k'<< k\)	\(>0\)	\(>0\)

As in the previous lecture, assume that the concentration of X quickly turns on at time \(t=0\).

\[y(t) = y^* \left( 1 - e^{-r_yt} \right)\] \[z(t) = \begin{cases} z_\text{max} \left( 1 - e^{-r_zt} \right) , & t < \tau \\ z^*+(z_y-z^*)e^{-r_z(t-\tau)}, & t > \tau \end{cases}\] \[\tau = \frac{1}{r_y}\ln\left( \frac{1}{1-K/y^*} \right) \\ \frac{z^*}{z_\text{max}} = \frac{k'}{k} = \frac{1}{F} \\ F \equiv \text{Repression Factor}\] \[\tau_{1/2} = \frac{1}{r_z}\ln\left( \frac{2F}{2F-1} \right) \\ \tau_{1/2} = \begin{cases} 0, & F>>1 \\ \frac{1}{r_z}\ln 2, & F = 1 \end{cases}\]

In summary, there are three ways to speed up response.

1. Auto-repression
2. Incoherent feedforward loop
3. Increase the degradation rate r and increase the production rate k.

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