Stochastic analysis of bistability in coherent mixed feedback loops combining transcriptional and posttranscriptional regulations

Mixed feedback loops combining transcriptional and posttranscriptional regulations are common in cellular regulatory networks. They consist of two genes, encoding a transcription factor and a small noncoding RNA (sRNA), which mutually regulate each other's expression. We present a theoretical and numerical study of coherent mixed feedback loops of this type, in which both regulations are negative. Under suitable conditions, these feedback loops are expected to exhibit bistability, namely, two stable states, one dominated by the transcriptional repressor and the other dominated by the sRNA. We use deterministic methods based on rate equation models, in order to identify the range of parameters in which bistability takes place. However, the deterministic models do not account for the finite lifetimes of the bistable states and the spontaneous, fluctuation-driven transitions between them. Therefore, we use stochastic methods to calculate the average lifetimes of the two states. It is found that these lifetimes strongly depend on rate coefficients such as the transcription rates of the transcriptional repressor and the sRNA. In particular, we show that the fraction of time the system spends in the sRNA-dominated state follows a monotonically decreasing sigmoid function of the transcriptional repressor transcription rate. The biological relevance of these results is discussed in the context of such mixed feedback loops in Escherichia coli. It is shown that the fluctuation-driven transitions and the dependence of some rate coefficients on the biological conditions enable the cells to switch to the state which is better suited for the existing conditions and to remain in that state as long as these conditions persist.

Figure 1

A schematic representation of the mixed feedback loop. Gene $a$ is transcribed into mRNA molecules $\left(m\right)$, which are translated into transcriptional repressor proteins $\left(A\right)$. Gene $b$ is transcribed into sRNA molecules $\left(s\right)$. The $A$ repressors negatively regulate the transcription of gene $b$ by binding to its promoter (the bound repressor is denoted by $r)$. The sRNA molecules transcribed from gene $b$ bind to the mRNA molecules of gene $a$ and inhibit their translation. Truncated arrows represent negative regulation.

Figure 2

Bifurcation diagrams showing the levels of $s$ and $A$ as functions of the parameters $g_m$ (a),(c) and $g_s$ (b),(d). Solid lines represent stable solutions and dashed lines represent unstable solutions.

Figure 3

The MFL probability distribution $P(A,s)$ (a) for the state dominated by the sRNA regulator (obtained for $g_m=0.0058{\mathrm{s}}^{-1})$ and (b) for the state dominated by the transcriptional repressor (obtained for $g_m=0.0071{\mathrm{s}}^{-1})$.

Figure 4

The levels of the sRNA molecules (a), mRNA molecules (b), free $A$ repressors (c), and bound repressors (d) vs time in the MFL, obtained from Monte Carlo simulations. Failed attempts to transition from the $s$- to the $A$-dominated state are observed, following periods in which the $s$ promoter is occupied by the transcriptional repressor.

Figure 5

The average lifetime of the sRNA-dominated state, ${\tau }_s$, as a function of the parameters $g_m$ (a) and $g_s$ (b), obtained from MC simulations. The lifetime ${\tau }_s$ monotonically decreases with $g_m$ and monotonically increases with $g_s$. Each data point was averaged over 1000 MC runs.

Figure 6

(a) The sRNA level $s$ as a function of the transcription rate $g_m$, obtained from the rate equations. (b) The fraction of time $P_s$ that the system resides in the sRNA-dominated state vs $g_m$, obtained from MC simulations. The dependence of $P_s$ on $g_m$ is of the form of a decreasing sigmoid function. Two vertical lines mark the $g_m$ values, as shown in Fig. 3, for the state dominated by the sRNA regulator for which $P_s>0.5$ (obtained for $g_m=0.0058{\mathrm{s}}^{-1}$, marked by a dashed line), and for the state dominated by the transcriptional repressor for which $P_s<0.5$ (obtained for $g_m=0.0071{\mathrm{s}}^{-1}$, marked by a dash-dotted line).

Figure 7

Stability analysis for different $c_g,u_g$, and $g_s$ values. (a)–(f) Bifurcation diagrams showing the levels of $s$ and $A$ as a function of the parameters $c_g$ (a),(d), $u_g$ (b),(e), and $g_s$ (c),(f). (g)–(i) $P_s$ curves as a function of $c_g$ (g), $u_g$ (h), and $g_s$ (i). Solid lines represent stable solutions and dashed lines represent unstable solutions.

Figure 8

Stability analysis for different $u_c$ values. (a)–(f) Bifurcation diagrams showing the levels of $s$ and $A$ as a function of the parameter $g_m$, for different values of $u_c$. (g)–(i) $P_s$ curves as a function of $g_m$ for different values of $u_c$. The results are presented for different values of $u_c$: $u_c=0$ (a),(d),(g), $u_c=0.01$ (b),(e),(h), and $u_c=0.02$ (c),(f),(i). Solid lines represent stable solutions and dashed lines represent unstable solutions.

Figure 9

Stability analysis for different $\alpha$ values. (a)–(f) Bifurcation diagrams showing the levels of $s$ and $A$ as a function of the parameter $g_m$, for different values of $\alpha$. (g)–(i) $P_s$ curves as a function of $g_m$ for different values of $\alpha$. The results are presented for different values of $\alpha$: $\alpha =1$ (a),(d),(g), $\alpha =2$ (b),(e),(h), and $\alpha =5$ (c),(f),(i). Solid lines represent stable solutions and dashed lines represent unstable solutions.