Half-final response of a graded incoherent feed-forward loop ($t_{1/2}$)

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.