Past exam of the biological physics course of the University of Oxford / 2018 / 4 / c

At the final steady state, the ligands control activity as well as the amount of each factor. Assuming active X exceeds its thresholds and the high Y level exceeds $K_{yz}$, the normalized output production rates areFor I1 incoherent feed-forward loop, active X makes Y abundant, but Y can repress Z only if its own activating ligand is present. Without $S_y$, Y is inactive and the output remains high. For I4 incoherent feed-forward loop, active X represses Y to below its threshold, so the output is low regardless of $S_y$. If X is inactive, the direct permissive arm is absent and both outputs are zero even when Y is active.

Thus I1 incoherent feed-forward loop can retain two-input combinatorial control at steady state as well as generate the transient in part (a). I4 incoherent feed-forward loop loses steady-state sensitivity to $S_y$ once X has suppressed Y, under these particular assumptions. That gives a functional reason the first motif may be more useful in many regulatory contexts. It is not a proof of an evolutionary frequency: promoter implementation, thresholds, noise, costs and other functions also influence how common a motif is. The argument follows the supplied rate table, including its nonzero ${\beta }_{\mathrm{low}}$, rather than a generic Boolean AND truth table.