Extending the dynamic range of transcription factor action by translational regulation

A crucial step in the regulation of gene expression is binding of transcription factor (TF) proteins to regulatory sites along the DNA. But transcription factors act at nanomolar concentrations, and noise due to random arrival of these molecules at their binding sites can severely limit the precision of regulation. Recent work on the optimization of information flow through regulatory networks indicates that the lower end of the dynamic range of concentrations is simply inaccessible, overwhelmed by the impact of this noise. Motivated by the behavior of homeodomain proteins, such as the maternal morphogen Bicoid in the fruit fly embryo, we suggest a scheme in which transcription factors also act as indirect translational regulators, binding to the mRNA of other regulatory proteins. Intuitively, each mRNA molecule acts as an independent sensor of the input concentration, and averaging over these multiple sensors reduces the noise. We analyze information flow through this scheme and identify conditions under which it outperforms direct transcriptional regulation. Our results suggest that the dual role of homeodomain proteins is not just a historical accident, but a solution to a crucial physics problem in the regulation of gene expression.

Figure 1

Schematic comparison of direct transcriptional regulation (DTR) and indirect translational regulation (ITR) schemes. (a) In direct transcriptional regulation (DTR), activator (or repressor) TFs, depicted as green squares and present at concentration $c$, interact with (potentially multiple, not necessarily identical) TF binding regions to activate (repress) expression of the regulated gene $g$. (b) In the indirect translational regulation (ITR) scenario, input molecules (green squares) bind to mRNAs $m$ of protein $y$ (red chain) to make the mRNA unaccessible for translation (gray oval). Translation can proceed from unbound mRNA molecules, giving rise to proteins $y$ (red stars). These proteins act as repressors (or activators) for gene $g$; the overall mapping from $c$ to $g$ is thus activating (repressing) in both scenarios.

Figure 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 ${log}_2\tilde{Z}$ 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, $\left(\text{ii}\right)+\left(\text{iii}\right)+\left(\text{iv}\right)$ 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 $M$.

Figure 3

Scaling of the optimal capacity with “ITR noise” parameters $y_{\max }$ and $1/F$. The optimal capacity ${log}_2\tilde{Z}$ for different fixed values of the mRNA number $M$ in the ITR model is plotted against “ITR noise” parameters properly rescaled by $M$, to observe the compensation between translational noise components and the input noise. (a) For the diffusion noise due to the intermediary protein $y$, the relevant parameter is $M\times y_{\max }$, where $y_{\max }$ is the maximal concentration of $y$. (b) For the shot noise due to the expression of intermediary mRNA and protein $y$, the relevant parameter is $M\times (1/F)$, where $1/F$ is the inverse Fano factor. In both cases, we set the remaining noise sources to be as small as possible: $C=1000$ and $F=1$ for (a), $C=1000$ and $y_{\max }=1000$ for (b). A perfect collapse in (a), comparable to that in (b), could only be achieved for Fano factors $F\ll 1$, which are biologically unrealistic.

Figure 4

Optimal regulatory curves. (a) Optimal regulatory curves, $\overline{g}\left(c\right)$, for $C=0.1$ (left), $C=1$ (middle), and $C=10$ (right), and different values of $M$ in the indirect translational regulation (ITR) model (colored curves): $M=10$ (top), $M=100$ (middle), $M=1000$ (bottom). The direct transcriptional regulation (DTR) model is plotted in black for reference. Colors correspond to the vertical lines in Fig. 2. Here, $F=1$ and $y_{\max }=10$. $\mathrm{\Delta }I$, the difference in capacity between ITR and DTR in bits, is shown in each panel. (b) and (c) Optimal regulation threshold, $K_{\mathrm{eff}}^{*}$, and the optimal Hill coefficient, $H^{*}$, that characterize the regulatory curves in (a), shown as a function of $C$. Since the overall regulation is repressive in either scheme, Hill coefficients are negative. In the DTR model, $K_{\mathrm{eff}}\equiv K$; for ITR, $K_{\mathrm{eff}}=(K_c/K)M{y}_{\max }-K_c$.

Figure 5

Direct transcriptional regulation (DTR) vs indirect translational regulation (ITR) at equal resources. (a) Comparison of the maximal capacity as a function of $C$ between indirect translational regulation (ITR, red curves) and direct transcriptional regulation (DTR, black curves) scenarios, for three values of total resource cost (total mRNA number), $\mathcal{M}$. Here, $y_{\max }=10,F=1$. Compared to Fig. 2, ITR stops being beneficial over DTR at lower $C$. (b) Capacity difference between the ITR and DTR models as a function of $C$ and $\mathcal{M}$. The thick red line marks the $(C,\mathcal{M})$ values that lead to equal capacities; in the regime above the red line ITR, in spite of intermediary regulatory steps, is beneficial over DTR.