Incoherent feed-forward loop

In an incoherent feed-forward loop, the direct path from an input to an output and the indirect path through a second transcription factor have opposite net signs. Their different response times can generate a pulse or adaptation. A nonzero basal production rate makes the final adapted output nonzero, so its logic and response time must be calculated from the actual gene regulation rule rather than assumed from a Boolean label.

In the I4 incoherent feed-forward loop, an input activates the output but represses an intermediate activator of that output. Decay of the intermediate can interrupt an initial output pulse. Residual production and signal dependence must be retained when specifying the gene regulation rule.

In the I1 incoherent feed-forward loop, an input activates both an intermediate regulator and the output, while the intermediate represses the output. Delayed accumulation of the intermediate can therefore interrupt an initial output pulse. The precise steady state depends on the actual gene regulation rule and thresholds.

Let an output start at zero, degrade at rate $\alpha$, and switch from production $F\beta$ to $\beta$ at time $d$, with $F>1$. Its final level is $Z_s=\beta /\alpha$. Writing $\delta =1-e^{-\alpha d}$, its first crossing of $Z_s/2$ occurs at $t_{1/2}={\alpha }^{-1}\ln [2F/(2F-1)]$ if $F\delta \ge 1/2$; otherwise it occurs at $d+{\alpha }^{-1}\ln [2(1-F\delta )]$. This follows by solving the two linear differential equations before and after the switch. A large F alone does not make the first formula uniform in arbitrarily short pulse durations.