Fig. 2.

Optimal capacity of direct transcriptional regulation (DTR, black), compared to indirect translational regulation (ITR, red), as a function of the maximal input concentration C. (a) Optimal capacity log₂ $\tilde{Z}$ vs C for various total amounts of translationally targeted mRNAs, M. Here and in subsequent panels we fix N_max, the maximal output copy number, to a reference value N_max = 1 and show the resulting capacity; larger values of N_max simply shift all capacities upwards by an additive amount, as in Eq. (36). The strength of “ITR noise” sources, (ii) + (iii) + (iv) of Eq. (29), is set by (Fano factor) F = 1 and (maximal concentration of the intermediary protein) y_max = 10. For M > 1, ITR outperforms DTR at low C, but at high C DTR can still reach higher capacities if M is not sufficiently large (e.g., dashed red line for M = 10). At M = 1 (dotted red line), the ITR scheme cannot benefit from input noise averaging, yet the intermediary regulatory steps still contribute the “ITR noise” absent in DTR, causing DTR to be superior to ITR at all C. (b) Capacity curves for different values of M in (a) collapse when plotted against the product of maximal input concentration and the number of mRNA targets, MC, as predicted by the scaling relation in Eq. (34). Increasing input noise by lowering C thus can be compensated for by increasing M. The collapse is not perfect at high MC and curves for different M saturate at different capacity values because the strength of “ITR noise” is not negligible and its effect on capacity depends on M.