Post-transcriptional bursting in genes regulated by small RNA molecules

Gene expression programs in living cells are highly dynamic due to spatiotemporal molecular signaling and inherent biochemical stochasticity. Here we study a mechanism based on molecule-to-molecule variability at the RNA level for the generation of bursts of protein production, which can lead to heterogeneity in a cell population. We develop a mathematical framework to show numerically and analytically that genes regulated post transcriptionally by small RNA molecules can exhibit such bursts due to different states of translation activity (on or off), mostly revealed in a regime of few molecules. We exploit this framework to compare transcriptional and post-transcriptional bursting and also to illustrate how to tune the resulting protein distribution with additional post-transcriptional regulations. Moreover, because RNA-RNA interactions are predictable with an energy model, we define the kinetic constants of on-off switching as functions of the two characteristic free-energy differences of the system, activation and formation, with a nonequilibrium scheme. Overall, post-transcriptional bursting represents a distinctive principle linking gene regulation to gene expression noise, which highlights the importance of the RNA layer beyond the simple information transfer paradigm and significantly contributes to the understanding of the intracellular processes from a first-principles perspective.

Figure 1

(a) Scheme of post-transcriptional gene regulation leading to the generation of bursts at the translation level. An sRNA interacts with the leader region of an mRNA, initially inactive, to release the RBS and then activate translation. Also shown are numerical simulations of protein dynamics at constant mRNA amounts: (b) $x=1$ mol and (c) $x=100$ mol. In both cases, $s=10$ mol (constant), $k_{\text{on}}=0.01$ ${\mathrm{mol}}^{-1}{\min }^{-1}$, $k_{\text{off}}=10{\min }^{-1}$, $k_p=10{\min }^{-1}$, and ${\gamma }_p=0.02{\min }^{-1}$ (parameter values characteristic of prokaryotes).

Figure 2

(a) and (b) Dynamics of protein expression with the full kinetic model. On the right, the dynamics within a time window is presented. The transcription rate is (a) $k_m=0.15$ mol/min and (b) $k_m=1$ mol/min. In both cases, $k_{\text{on}}=0.01$ ${\mathrm{mol}}^{-1}{\min }^{-1}$, $k_{\text{off}}=10{\min }^{-1}$, $k_s=1.5$ mol/min (transcription rate of sRNA), ${\gamma }_m=0.15{\min }^{-1}$, $k_p=10{\min }^{-1}$, and ${\gamma }_p=0.02{\min }^{-1}$. (c) Cumulative probability distributions of protein amount $\left[\mathrm{CDF}\right(y\left)\right]]$ for different transcription rates of mRNA, comparing the dynamics from the full kinetic model with the dynamics from the stochastic differential equations.

Figure 3

Noise in protein expression ${\eta }_y^2$ calculated analytically. The red line corresponds to the nominal parameter values of $k_{\text{on}}=0.01$ ${\mathrm{mol}}^{-1}{\min }^{-1}$, $k_{\text{off}}=10{\min }^{-1}$, $c=1$, $k_m=1$ mol/min, ${\gamma }_m=0.15{\min }^{-1}$, $k_p=10{\min }^{-1}$, and ${\gamma }_p=0.02{\min }^{-1}$ (characteristic of prokaryotes). Noise decreases by augmenting $k_m$ ($10\times$) or reducing $k_{\text{off}}$ ($0.1\times$), while noise increases by reducing $k_p$ ($0.2\times$) or augmenting ${\gamma }_p$ ($6\times$).

Figure 4

(a) Scheme of transcriptional gene regulation where a TF interacts with the promoter, initially inactive, to activate transcription. (b) Numerical simulation of protein dynamics according to this model, with $k_m=10$ mol/min and $k_p=1{\min }^{-1}$. (c) Scheme of post-transcriptional gene regulation and (d) numerical simulation of protein dynamics, with $k_m=1$ mol/min and $k_p=10{\min }^{-1}$. In both cases, $s=10$ mol (constant) and the rest of the parameters take the nominal values corresponding to Fig. 3. (e) Noise in protein expression ${\eta }_y^2$ as a function of the signal molecule $s$ in the case of transcriptional (dashed line and triangles) and post-transcriptional (solid line and circles) gene regulation. The lines correspond to analytical solutions, while points correspond to numerical simulations. (f) Relative information transfer from signal to response ${\mathcal{I}}_{s\to y}$, post-transcriptional vs transcriptional control, for different input dynamic ranges (given by $s_{\text{max}}$; $s_{\text{min}}=10$ mol).

Figure 5

(a) Scheme of multiple post-transcriptional gene regulations where, in addition to the sRNA that activates translation, a given asRNA can interact with the mRNA to induce its degradation through RNase and the protein can have a degradation tag to recruit a protease. (b) and (c) Probability distributions of protein amount $\left[P\right(y\left)\right]$. Red lines correspond to analytical solutions, while black lines correspond to numerical simulations. (b) Effect of the asRNA, with $k_m=1$ mol/min, ${\mu }_m=0.5{\min }^{-1}$, and ${\mu }_p=0$. (c) Effect of the degradation tag, with $k_m=0.15$ mol/min, ${\mu }_m=0$, and ${\mu }_p=0.1{\min }^{-1}$. The rest of the parameters take the nominal values corresponding to Fig. 3.

Figure 6

Dynamics and distribution of protein expression in the case of (a) and (f) $k_{\text{on}}s=0.1{\min }^{-1}$ and $k_{\text{off}}=10{\min }^{-1}$, (b) and (g) $k_{\text{on}}s=0.03{\min }^{-1}$ and $k_{\text{off}}=3{\min }^{-1}$, (c) and (h) $k_{\text{on}}s=0.01{\min }^{-1}$ and $k_{\text{off}}=1{\min }^{-1}$, (d) and (i) $k_{\text{on}}s=1{\min }^{-1}$ and $k_{\text{off}}=10{\min }^{-1}$, and (e) and (j) $k_{\text{on}}s=0.1{\min }^{-1}$ and $k_{\text{off}}=1{\min }^{-1}$ for (a)–(e) stable protein, ${\gamma }_p=0.02{\min }^{-1}$, and (f)–(j) unstable protein, ${\gamma }_p=0.12{\min }^{-1}$. In all cases, $k_m=1$ mol/min, ${\gamma }_m=0.15{\min }^{-1}$, and $k_p=10{\min }^{-1}$. This leads to (a)–(c) $p_{\text{on}}=0.01$ and $\langle y\rangle =33.33$ mol, (d) and (e) $p_{\text{on}}=0.1$ and $\langle y\rangle =333.33$ mol, (f)–(h) $p_{\text{on}}=0.01$ and $\langle y\rangle =5.56$ mol, and (i) and (j) $p_{\text{on}}=0.1$ and $\langle y\rangle =55.56$ mol. The analytical distribution for $y$ ($\text{Gamma}$) is shown in red.

Figure 7

(a) Energy landscape associated with post-transcriptional gene regulation with an sRNA. The reaction coordinate is given by the number of intermolecular base pairs between the mRNA and sRNA. The free energy of activation $\mathrm{\Delta }{G}^{†}$ determines the value of $k_{\text{on}}$, while the value of $k_{\text{off}}$ is determined by both the free energy of formation $\mathrm{\Delta }G$ and of activation. (b) and (c) Numerical simulations of protein dynamics for different energetics, with $k_{\text{on}}^0=3\times {10}^{-5}$ ${\mathrm{mol}}^{-1}{\min }^{-1}$, $k_{\text{off}}^0=3\times {10}^4{\min }^{-1}$, and (b) $\beta \mathrm{\Delta }{G}^{\#}=-6$ and $\beta \mathrm{\Delta }G=-8$ and (c) $\beta \mathrm{\Delta }{G}^{\#}=-4$ and $\beta \mathrm{\Delta }G=-10$. The rest of the parameters take the nominal values corresponding to Fig. 3.

Figure 8

(a) Scheme of negative post-transcriptional gene regulation. An sRNA interacts with the leader region of an mRNA, initially active, to block the RBS and then inactivate translation. (b) and (c) Numerical simulations for two different values of $s$. The parameters take the nominal values corresponding to Fig. 3.