$4\mathrm{.}7.$ Amplifying intermediate stimuli. This problem highlights an additional possible

function of incoherent type$-1$ FFLs for subsaturating stimuli $S_x.$ Consider an I$1-$

FFL$,$ such that the activation threshold of Z by $x,K_{zx},$ is smaller than the activation

threshold of Y by $x,K_{yx}.$ That is$,$ Z is activated when $x^{*}>K_{zx},$ but it is repressed by

Y when $x^{*}>K_{yx}.$ Schematically plot the steady$-$state concentration of Z as a func$-$

tion of $x^{*}.$ Note that intermediate values of $x^{*}$ lead to the highest Z expression.