Frequency modulation of stochastic gene expression bursts by strongly interacting small RNAs

The sporadic nature of gene expression at the single-cell level—long periods of inactivity punctuated by bursts of mRNA or protein production—plays a critical role in diverse cellular processes. To elucidate the cellular role of bursting in gene expression, synthetic biology approaches have been used to design simple genetic circuits with bursty mRNA or protein production. Understanding how such genetic circuits can be designed with the ability to control burst-related parameters requires the development of quantitative stochastic models of gene expression. In this work, we analyze stochastic models for the regulation of gene expression bursts by strongly interacting small RNAs. For the parameter range considered, results based on mean-field approaches are significantly inaccurate and alternative analytical approaches are needed. Using simplifying approximations, we obtain analytical results for the corresponding steady-state distributions that are in agreement with results from stochastic simulations. These results indicate that regulation by small RNAs, in the strong interaction limit, can be used to effectively modulate the frequency of bursting. We explore the consequences of such regulation for simple genetic circuits involving feedback effects and switching between promoter states.

Figure 1

Schematic representation of the model. Bursty synthesis of mRNAs is shown by green lines, blue hexagons represent proteins, and red circles denote sRNA.

Figure 2

Comparison of mean-field predictions with results from stochastic simulations. The figure shows variation of $\langle p_s\rangle /\langle {\tilde{p}}_s\rangle$ with $n_s{q}_m$, where $q_m=1/\langle m_b\rangle$ and $\langle m_b\rangle$ is the mean burst size. $\langle {\tilde{p}}_s\rangle =\left(k_m{k}_p\langle m_b\rangle \right)/\left({\mu }_m{\mu }_p\right)$ is the unregulated steady-state protein level and $n_s=k_s/{\mu }_s$ is the mean sRNA level in the absence of interaction with mRNAs. Lines are results from the mean-field predictions for three different values of ${\mu }_s$: $n_s{q}_m$ is varied by changing $q_m$, and the other parameters are fixed at $k_m=0.001$, ${\mu }_m=1$, $k_p=50$, ${\mu }_p=0.01$, $k_s=0.1$, and $\gamma =10$. Corresponding simulation results are shown as points.

Figure 3

Exponential suppression of protein levels due to sRNA regulation and its dependence on interaction strength $\gamma$: Scaled mean steady-state protein levels ($\langle p_s\rangle /\langle {\tilde{p}}_s\rangle$) as a function of $n_s{q}_m$ are plotted, where $\langle {\tilde{p}}_s\rangle$ is the unregulated steady-state protein level. Points are simulation results for three different values of $\gamma$ and the line represents the analytic result using Eq. (10). Other parameters are $k_m=0.001$, ${\mu }_m=1$, $k_p=50$, ${\mu }_p=0.01$, $k_s=0.1$, and ${\mu }_s=0.01$, and $n_s{q}_m$ was varied by changing $q_m$.

Figure 4

Leakage suppression due to strong sRNA-mRNA interaction. (a) Kinetic scheme for the bursty production of mRNAs from the two states of a promoter. (b) Simulation results for the steady-state probability distribution of proteins for such a kinetic scheme for the no-regulation case $\gamma =0$ (dashed line) and for $\gamma =10$ (solid line) with other parameters: $\alpha =0.003$, $\beta =0.001$, $k_{m0}=k_{m1}=0.01$, $q_{m0}=1$, $q_{m1}=0.01$, ${\mu }_m=0.5$, $k_p=0.01$, ${\mu }_p=0.0005$, $k_s=1$, and ${\mu }_s=0.1$. Points correspond to the steady-state protein distribution for the case when mRNAs are produced only from state 1 (i.e., $k_{m0}=0$) with $\gamma =10$ and keeping all the other parameters the same.

Figure 5

Effects of strong sRNA-mRNA interaction on the effective promoter switching rate in (a) a model with feedback. (b) Simulation results for the probability distribution of proteins in the steady state has been shown (line) for $\alpha =0.003$, $\tilde{\alpha }=0.1$, $\beta =0.001$, $k_{m0}=0.01$, $k_{m1}=0.05$, $q_{m0}=1$, $q_{m1}=0.01$, $k_p=0.05$, $k_s=1$, ${\mu }_m=0.5$, ${\mu }_p=0.005$, ${\mu }_s=0.1$, and $\gamma =10$. Solid circles correspond to the effective or reduced model with $\gamma =0$ but with $k_{mi}$ scaled by the factor $exp(-n_s{q}_{mi})$, $i=0,1$. (c) Variation of the ratio of effective switching rate in the presence of sRNAs, ${\alpha }_{\text{eff}}\left(n_s\right)$, to that in the absence of sRNAs, ${\alpha }_{\text{eff}}\left(0\right)$, with $n_s$ for different values of $q_{m1}$: 0.1 (solid), 0.05 (dashed), and 0.02 (dotted). (d) Variation of the ratio ${\alpha }_{\text{eff}}\left(n_s\right)/{\alpha }_{\text{eff}}\left(0\right)$ as a function of $q_{m1}$ for three different values of $n_s$: 5 (solid), 10 (dashed), and 20 (dotted). In both (c) and (d) other parameters are $\alpha =0.003$, $\tilde{\alpha }=0.1$, $\beta =0.001$, $k_{m0}=0$, $k_{m1}=0.05$, $k_p=0.05$, ${\mu }_m=0.5$, ${\mu }_p=0.005$, and $\gamma =10$.

Figure 6

Effects of burst frequency modulation due to sRNA regulation on a simple two-state switch. (a) Schematic figure showing a bursty input signal characterized by its mean burst size $\langle p_b\rangle$ and modulated burst frequency due to strong sRNA interaction $k_m\to k_{m}exp(-n_s{q}_m)$. (b) Variance of the switch plotted as a function of $n_s$ for three different values of $\langle p_b\rangle$: 10 (blue), 20 (red), and 30 (green). Other parameters are $\tilde{\alpha }=\beta ={\mu }_p=k_m=1$.