Transcriptional leakage versus noise: A simple mechanism of conversion between binary and graded response in autoregulated genes

We study the response of an autoregulated gene to a range of concentrations of signal molecules. We show that transcriptional leakage and noise due to translational bursting have the opposite effects. In a positively autoregulated gene, increasing the noise converts the response from graded to binary, while increasing the leakage converts the response from binary to graded. Our findings support the hypothesis that, being a common phenomenon, leaky expression may be a relatively easy way for evolutionary tuning of the type of gene response without changing the type of regulation from positive to negative.

Figure 1

The model of an autoregulated gene with leaky transcription [(a), positive regulation]. The intersections of the straight line $L\left(P\right)$ and the transfer function $h\left(P\right)$ (b1) and (c1) indicate the extrema of the protein distributions (b2) and (c2).

Figure 2

Transcriptional leakage converts binary response into graded response and counteracts the effect of noise. Binary response occurs for the leakage below the threshold ${\epsilon }_{\mathrm{thr}}$, when $1/\alpha <1$ (dashed line). Otherwise, graded response occurs. (Dots) Mesoscopic simulations of the gene regulation kinetics using the Gillespie algorithm [38]. (a) $\epsilon =0.1$. (b) $\epsilon =0.3$. Parameters are as follows: $n=-3,\alpha =30,\beta =2,{\epsilon }_{\mathrm{thr}}=0.275$. (Arrows) Mean values of the distributions.

Figure 3

When bursts are rare, $\alpha <1$, then the maximum of the protein distribution $p\left(P\right)$ is always at $P_{\max }=0$ independently of the leakage rate $\epsilon$ and the regulation strength $c$. This is because there is no intersection between the straight line $L\left(P\right)$ and the transfer function $h\left(P\right)$. Parameter values are as follows: $n=2,\alpha =0.8,\beta =60,\epsilon =0.3$. Arrows mark the mean values of the shown distributions.

Figure 4

Transcriptional leakage $\epsilon >1/\alpha$ narrows the range of maxima and mean values of protein distributions. (a) No leakage, $\epsilon =0$. (b) Leakage $\epsilon =0.3$. (Dotted lines) Range of maxima. (Dashed lines) Range of mean values. (Arrows) Mean values of the shown distributions. Parameters are as follows: $n=4,\alpha =30,\beta =2$.

Figure 5

When the transcriptional noise is sufficiently high, $1/\alpha >\epsilon$, the leakage does not narrow the range of maxima (dotted lines). However, the range of mean values (dashed lines) is narrower for nonzero leakage. (Arrows) Mean values of the shown distributions. (a) No leakage, $\epsilon =0$. (b) Leakage $\epsilon =0.3$. Parameters are as follows: $n=4,\alpha =2,\beta =30$.

Figure 6

(a) On the contrary to the deterministic model, binary response is possible in the stochastic model due to transcriptional bursting, even when there is just one transcription factor binding site. Here, only a very small leakage is present $\left(\epsilon =0.01\right)$. (b) Transcriptional leakage $\left(\epsilon =0.3\right)$ counteracts the stochastic effect of bursting by converting binary response into graded response. Parameters are as follows: $n=-1,\alpha =10,\beta =6$. (Arrows) Mean values of the shown distributions.