Small RNA-driven feed-forward loop: fine-tuning of protein synthesis through sRNA-mediated crosstalk

Abstract

Often in bacterial regulatory networks, small non-coding RNAs (sRNA) interact with several mRNA species. The competition among mRNAs for binding to the common pool of sRNA might lead to crosstalk between the mRNAs. This is similar to the competing endogenous RNA effect that leads to complex gene regulation with stabilized gene expression in Eukaryotes. Here, we study an sRNA-driven feed-forward loop (sFFL) where the top-tier regulator, an sRNA, translationally activates the target protein (TP) as well as a transcriptional activator of the TP through binding to the respective mRNAs. We show that the sRNA-mediated crosstalk between the two mRNA species enables the sFFL to function in three different regimes depending on the synthesis rate of the transcriptional activator mRNA. Of these three regimes, there exists a sensitive regime where the TP level shows interesting features depending on the precise mechanism of target translation. In the case of translation entirely from sRNA–mRNA bound complexes, the TP level becomes maximum around the sensitive regime. Through stochastic analysis and simulations, we show that relative fluctuations in the TP level is minimized here. For translation both from mRNA and sRNA–mRNA bound complexes, the target expression shows a threshold response across the sensitive regime.

Graphic abstract

Appendix

1.1 Appendix A: Deterministic analysis

1.1.1 1. sRNA-driven cascade network (sCN)

In this section, we consider the sRNA-driven cascade network (sCN) where sRNA translationally activates protein $p_1$ which then transcriptionally activates protein $p_2$ (see Fig. 7b). Thus, it is a cascade network where no sharing of free sRNA takes place. The variation in concentrations of various components of sCN with time is as follows:

$$\begin{aligned} \dot{s}&=r_s-\gamma _s s- k_1^+ s\, m_1 + (k_1^- + \kappa _1 ) c_1, \end{aligned}$$

(A1)

$$\begin{aligned} \dot{m}_1&=r_{m_1} -\gamma _{m_1} m_1 -k_1^+ s\, m_1 +k_1^- c_1,\end{aligned}$$

(A2)

$$\begin{aligned} \dot{c}_1&=k_1^+ m_1\, s - (k_1^- + \gamma _1 + \kappa _1 ) c_1,\end{aligned}$$

(A3)

$$\begin{aligned} \dot{p}_1&=r_{p_1} c_1 -\gamma _{p_1} p_1,\end{aligned}$$

(A4)

$$\begin{aligned} \dot{m}_2&=\frac{r_{m_2} {k_c}\, p_1}{1+ k_c\, p_1} -\gamma _{m_2} m_2\ \ \mathrm{and}\end{aligned}$$

(A5)

$$\begin{aligned} \dot{p}_2&=r_{{p_2}_0} m_2 -\gamma _{p_2} p_2. \end{aligned}$$

(A6)

Fig. 7

Schematic diagrams for a sFFL and b sCN

The steady-state concentrations of mRNAs $m_1$ and $m_2$ in terms of functions $F_1(s)$ and $F_2(s)$ are

$$\begin{aligned} m_1= & {} m_1^*\, F_1(s),\end{aligned}$$

(A7)

$$\begin{aligned} m_2= & {} \frac{m_2^*\, k_c\, a\, s\, F_1(s)}{1+ k_c\, a\, s\, F_1(s)}, \ \ \mathrm{where} \end{aligned}$$

(A8)

$m_1^* =\frac{r_{m_1}}{\gamma _{m_1}},\quad m_2^*= \frac{r_{m_2}}{\gamma _{m_2}},\ \ \quad a= \frac{r_{p_1}\, k_1^+\, m_1^*}{\gamma _{p_1} (k_1^-+ \gamma _1 + \kappa _1 )},$ $F_1(s)=\frac{1}{1+s/s_{01}}$ and $s_{01}=\frac{\gamma _{m_1}}{k_1^+}\frac{k_1^-+\gamma _1 +\kappa _1}{\gamma _1 +\kappa _1}$. In the steady-state, the concentration of sRNA, can be found by solving $r_s - \gamma _s\, s - r_{m_1}\, \zeta _1\, s\, F_1(s) =0$, where $\zeta _1=\frac{k_1^+ \gamma _1}{\gamma _{m_1}(k_1^-+ \gamma _1+\kappa _1 )}$. The target protein concentration for sCN is $p_2=\frac{r_{{p_2}_0} m_2}{\gamma _{p_2}}$ where $m_2$ is as shown in Eq. (A8).

1.1.2 2. Response functions for protein

The response functions for the intermediate and target proteins can be calculated, to understand the sensitivity of proteins, ($p_1$ and $p_2$) with respect to the change in the synthesis rate of mRNA1 ($m_1$), $r_{m_1}$. We introduce response functions as

$$\begin{aligned} {\mathcal {Y}}_{ij}=\frac{\partial p_i}{\partial r_{m_j}},\ \ \mathrm{where }\ \ i,\ j=1,\ 2 \ \ \mathrm{with}\ \ i\ne j. \end{aligned}$$

(A9)

At steady state, the solutions for $p_1$ and $p_2$ are

$$\begin{aligned}&p_1=a\, s\, F_1(s)\ \ \ \mathrm{and} \end{aligned}$$

(A10)

$$\begin{aligned}&p_2=\frac{r_{p_2}}{\gamma _{p_2}}\frac{m_2^*\, k_c\, k_2^+ a\, s^2\,F_1\, F_2}{{(k_2^- +\gamma _2+\kappa _2)(1 + k_c\, a\, s\, F_1)}}. \end{aligned}$$

(A11)

Using steady-state solutions for $m_1$, $m_2$ along with Eqs. (A10) and (A11), we find

$$\begin{aligned} {\mathcal {Y}}_{12}&= a\, \chi _{s2} \left[ F_1+ s\,F_1' \right] , \end{aligned}$$

(A12)

$$\begin{aligned} {\mathcal {Y}}_{21}&= \frac{r_{p_2}}{\gamma _{p_2}} \frac{m_2^*\, k_c\, k_2^+ a\, s}{{(k_2^- +\gamma _2+\kappa _2)(1 + k_c a s F_1)}^2}\left\{ \chi _{s1} \left[ 2 F_1\, F_2\, + s(F_1'F_2 + F_1 F_2')+ k_c\, a\, s\, F_1^2\,(F_2 + s\, F_2') \right] +\frac{s F_2 F_1}{r_{m_1}} \right\} . \end{aligned}$$

(A13)

Figure 8 shows a sharp response in the target protein concentration in the sensitive region.

Fig. 8

Response function of the target protein, ${\mathcal {Y}}_{21}$ for sFFL plotted with $r_{m_1}$, the transcription rate of mRNA, $m_1$. The same parameter values as in Fig. 2 are used here

Fig. 9

a Changes in various concentrations with $r_{m_1}$ with $r_{{p_2}_0} = 0.01$. b The variation in concentration of $p_2$ with $r_{m_1}$ for different values of $r_{{p_2}_0}$. The other parameter values are the same as those in Fig. 2

1.1.3 3. sFFL with the direct synthesis of the target protein

In this section, we consider the sRNA-driven feed-forward loop (sFFL) with the translation of $p_2$ from $m_2$ without the intervention of sRNA. This direct synthesis of $p_2$ from $m_2$ is in addition to translation facilitated by sRNA through formation of sRNA–mRNA2 complex ($c_2$). The variation of concentrations of various regulatory molecules with time are

$$\begin{aligned} \dot{s}= & {} r_s-\gamma _s s-k_1^+ s\ m_1-k_2^+ s\, m_2+\nonumber \\&(k_1^-+\kappa _1)c_1 +(k_2^-+\kappa _2)c_2,\end{aligned}$$

(A14)

$$\begin{aligned} \dot{m}_1= & {} r_{m_1}-\gamma _{m_1} m_1-k_1^+ s\ m_1+k_1^- c_1,\end{aligned}$$

(A15)

$$\begin{aligned} \dot{c}_1= & {} k_1^+ s\, m_1-(k_1^-+\gamma _1+\kappa _1) c_1,\end{aligned}$$

(A16)

$$\begin{aligned} \dot{p}_1= & {} r_{p_1} c_1-\gamma _{p_1} p_1,\end{aligned}$$

(A17)

$$\begin{aligned} \dot{m}_2= & {} \frac{r_{m_2} k_c \, p_1}{1+k_c\, p_1}-\gamma _{m_2} m_2-k_2^+ s \, m_2+k_2^- c_2, \ \end{aligned}$$

(A18)

$$\begin{aligned} \dot{c}_2= & {} k_2^+ s\, m_2-(k_2^-+\gamma _2+\kappa _2)c_2 \ \ \text {and} \end{aligned}$$

(A19)

$$\begin{aligned} \dot{p}_2= & {} r_{p_2} c_2 + r_{{p_2}_0} m_2-\gamma _{p_2} p_2, \end{aligned}$$

(A20)

where $r_{{p_2}_0}$ represents the rate of synthesis of the target protein, $p_2$ from $m_2$ without the involvement of sRNA.

Figure 9a shows that there is similar crosstalk behaviour as shown in Fig. 2. However, unlike $r_{{p_2}_0}=0$ case, here, due to direct synthesis of the target protein, the $p_2$ concentration remains high even beyond the peak (see Fig. 9b). This region appears to be $r_{{p_2}_0}$ dominated. The overlap of $p_2$ plots for different $r_{{p_2}_0}$ values before the sensitive region indicates that the target protein synthesis happens here predominantly due to bound complexes.

Fig. 10

Variation in the target protein concentration ($p_2$) of sFFL with $r_{m_1}$ a for different $r_{m_2}$ values with $\gamma _1 = \gamma _2 = \kappa _1 = \kappa _2 = 0.01$, b for different $r_{m_2}$ values with $\gamma _1 = \gamma _2 =0$ and $\kappa _1 = \kappa _2 = 0.01$, c for different $r_{m_2}$ values with $\gamma _1 = \gamma _2 =0.01$ and $\kappa _1 = \kappa _2 = 0$ and d for $r_{m_2} = 1$ under different conditions like with crosstalk i.e., $\gamma _1 = \gamma _2 = 0.01$ and $\kappa _1 = \kappa _2 = 0.01$, without crosstalk i.e., $\gamma _1 = \gamma _2 = 0$ and $\kappa _1 = \kappa _2 = 0.01$ and without recycling of sRNA i.e., $\kappa _1 = \kappa _2 = 0$ and $\gamma _1 = \gamma _2 =0.01$. The other parameter values are the same as those in Fig. 2

1.1.4 4. Effect of different factors on target protein synthesis

In this section, we study the change in the target protein ($p_2$) concentration with $r_{m_1}$ for different synthesis rates ($r_{m_2}$) of mRNA2 with (i) nonzero stoichiometric as well as catalytic decay rates (Fig. 10a) (ii) only catalytic decay i.e., $\gamma _1=\gamma _2=0$ and $\kappa _1$,  $\kappa _2 \ne 0$ (Fig. 10b) and (iii) with only stoichiometric decay and no catalytic decay i.e., $\kappa _1=\kappa _2=0$ and $\gamma _1,\, \gamma _2 \ne 0$ (Fig. 10c). For all the figures, $r_{{p_2}_0}=0$. As expected, all the figures show a strong dependence on $r_{m_2}$. With the increase in $r_{m_2}$, the target protein concentration increases naturally. Further, the increase in $m_2$ concentration leads to an increased competition for the available free sRNAs which finally leads to reduced plateau region. Figure 10d displays a comparison of the three cases, namely (i) no crosstalk i.e., $\gamma _1=\gamma _2=0$ and $\kappa _1,\, \kappa _2 \ne 0$, (dotted line) (ii) with crosstalk i.e., $\gamma _1,\, \gamma _2 \ne 0$ and $\kappa _1,\, \kappa _2 \ne 0$ (solid line) and (iii) with crosstalk but without sRNA recycling i.e., $\kappa _1=\kappa _2=0$ and $\gamma _1,\, \gamma _2 \ne 0$ (dashed line).

1.2 Appendix B: Noise analysis for sFFL

1.2.1 1. Equations and notations

The reactions incorporated into the master equation, Eq. (21), are based on the following effective equations.

$$\begin{aligned}&{\dot{s}}=r_s-\gamma _s\, s -g_1\, s\, m_1-g_2\, s\, m_2,\end{aligned}$$

(B1)

$$\begin{aligned}&\dot{m_1}=r_{m_1}-\gamma _{m_1}\, m_1-d_1 \, s\, m_1-g_1\, s\, m_1,\end{aligned}$$

(B2)

$$\begin{aligned}&\dot{p_1}=r_{p_1}' s\, m_1-\gamma _{p_1}\, p_1,\end{aligned}$$

(B3)

$$\begin{aligned}&\dot{m_2}=\frac{r_{m_2}\, k_c\, p_1}{1+k_c\, p_1}-\gamma _{m_2} m_2-d_2\, s\, m_2-g_2\, s\, m_2 \ \ \mathrm{and} \ \ \qquad \end{aligned}$$

(B4)

$$\begin{aligned}&\dot{p_2}=r_{p_2}' s\, m_2 -\gamma _{p_2}\, p_2. \end{aligned}$$

(B5)

As mentioned in the main text, a quasi-steady-state approximation has been used in obtaining these equations. In this approximation, we assume that the process of complex formation attains steady state faster compared to other processes. Using $c_1=\frac{k_1^+\, s\, m_1}{k_1^-+\gamma _1 +\kappa _1}$ and $c_2=\frac{k_2^+\,s\, m_1}{k_2^-+\gamma _2 + \kappa _2}$ in Eqs. (1)–(7), we find the effective equations mentioned above [Eqs. (B1)–(B5)]. Here, $g_1 = \frac{k_1^+\, \gamma _1}{k_1^- + \gamma _1 +\kappa _1} \qquad \mathrm{and} \qquad g_2 = \frac{k_2^+\, \gamma _2}{k_2^- + \gamma _2 + \kappa _2}$ correspond to combined degradation of sRNA and mRNA and $d_1 = \frac{k_1^+\, \kappa _1}{k_1^- + \gamma _1 + \kappa _1} \qquad \mathrm{and} \qquad d_2 = \frac{k_2^+\, \kappa _2}{k_2^- + \gamma _2 + \kappa _2}$ indicate degradation of mRNA alone, while sRNAs are recycled back. With the quasi-steady-state approximation, the effective protein synthesis rates are $r_{p_1}'= \frac{r_{p_1}k_1^+}{k_1^- + \gamma _1 + \kappa _1} \ \mathrm{and}\ r_{p_2}'= \frac{r_{p_2}k_2^+}{k_2^- + \gamma _2 + \kappa _2}$.

1.2.2 2. Moments

In this section, we list the results for first- and second-order moments obtained from the generating function approach. The results show that moments of a given order involve higher-order moments. In order to simplify our calculations, we consider moments up to second order and express the third-order moments in terms of lower-order moments using Gaussian approximation.

$$\begin{aligned}&G_1 = \frac{r_s - g_1\,G_{12} - g_2\,G_{14}}{\gamma _s}\end{aligned}$$

(B6)

$$\begin{aligned}&G_2 = \frac{r_{m_1} - (g_1+ d_1) G_{12}}{\gamma _{m_1}}\end{aligned}$$

(B7)

$$\begin{aligned}&G_3 = \frac{r_{p_1}' G_{12}}{\gamma _{p_1}}\end{aligned}$$

(B8)

$$\begin{aligned}&G_4 = \frac{r_{m_2}^0 + r_{m_2}^1\, G_3 - (g_2+ d_2) G_{14} }{\gamma _{m_2}}\end{aligned}$$

(B9)

$$\begin{aligned}&G_5 = \frac{r_{p_2}' G_{14}}{\gamma _{p_2}}\end{aligned}$$

(B10)

$$\begin{aligned}&G_{11} = \frac{r_s\, G_1 - g_1\,G_{112} - g_2\,G_{114}}{\gamma _s}\end{aligned}$$

(B11)

$$\begin{aligned}&G_{22} = \frac{r_{m_1} G_2 - (g_1+ d_1) G_{122} }{\gamma _{m_1}}\end{aligned}$$

(B12)

$$\begin{aligned}&G_{33} = \frac{r_{p_1}' G_{123}}{\gamma _{p_1}}\end{aligned}$$

(B13)

$$\begin{aligned}&G_{44} = \frac{r_{m_2}^0 G_4 + r_{m_2}^1 G_{34} - (g_2+ d_2) G_{144} }{\gamma _{m_2}}\end{aligned}$$

(B14)

$$\begin{aligned}&G_{55} = \frac{r_{p_2}' G_{145}}{\gamma _{p_2}}\end{aligned}$$

(B15)

$$\begin{aligned}&G _{12} = \frac{r_s G_2 + r_{m_1} G_1 - g_1(G_{112} + G_{122}) - g_2G_{124} - d_1G_{112}}{\gamma _s + \gamma _{m_1} + g_1+ d_1}\end{aligned}$$

(B16)

$$\begin{aligned}&G_{13} = \frac{r_s G_3 + r_{p_1}' ( G_{12} + G_{112}) - g_1\,G_{123} - g_2G_{134}}{\gamma _s + \gamma _{p_1}}\end{aligned}$$

(B17)

$$\begin{aligned}&G_{14} = \frac{r_s G_4 + r_{m_2}^0 G_1 + r_{m_2}^1 G_{13} - g_1\,G_{124} -( g_2 + d_2)G_{114} -g_2 G_{144}}{\gamma _s + \gamma _{m_2} + g_2+ d_2 } \end{aligned}$$

(B18)

$$\begin{aligned}&G_{15} = \frac{r_s G_5 + r_{p_2}' ( G_{14} + G_{114}) - g_1\,G_{125} - g_2\,G_{145}}{\gamma _s + \gamma _{p_2}}\end{aligned}$$

(B19)

$$\begin{aligned}&G_{23} = \frac{r_{m_1} G_3 + r_{p_1}' ( G_{12} + G_{122}) - (g_1+ d_1) G_{123}}{\gamma _{m_1} + \gamma _{p_1}}\end{aligned}$$

(B20)

$$\begin{aligned}&G_{24} = \frac{r_{m_1} G_4 + r_{m_2}^0 G_2 + r_{m_2}^1 G_{23} - (g_1+ g_2+ d_1+ d_2 ) G_{124} }{\gamma _{m_1} + \gamma _{m_2}}\end{aligned}$$

(B21)

$$\begin{aligned}&G_{25} = \frac{r_{m_1} G_5 + r_{p_2}' G_{124} - (g_1+ d_1) G_{125}}{\gamma _{m_1} + \gamma _{p_2} }\end{aligned}$$

(B22)

$$\begin{aligned}&G_{34} = \frac{r_{p_1}' G_{124} + r_{m_2}^0 G_3 + r_{m_2}^1 ( G_3 + G_{33}) - (g_2+ d_2) G_{134}}{\gamma _{p_1} + \gamma _{m_2}}\end{aligned}$$

(B23)

$$\begin{aligned}&G_{35} = \frac{r_{p_1}' G_{125} + r_{p_2}' G_{134}}{\gamma _{p_1} + \gamma _{p_2} }\end{aligned}$$

(B24)

$$\begin{aligned}&G_{45} = \frac{r_{m_2}^0 G_5 + r_{m_2}^1 G_{35} + r_{p_2}'(G_{14} + G_{144}) - (g_2+ d_2 ) G_{145} }{\gamma _{m_2} + \gamma _{p_2}} \end{aligned}$$

(B25)

Using Gaussian approximation, we express the third-order moments in terms of the lower-order moments as shown below.

$$\begin{aligned}&G_{112} = G_{11}G_{2} + G_{1} G_{2} + 2 G_{1}G_{12} - 2 G_{1}^2G_{2}-G_{12}\end{aligned}$$

(B26)

$$\begin{aligned}&G_{113} = G_{11}G_{3} + G_{1} G_{3} + 2 G_{1}G_{13} - 2 G_{1}^2G_{3}-G_{13} \end{aligned}$$

(B27)

$$\begin{aligned}&G_{114} = G_{11}G_{4} + G_{1} G_{4} + 2 G_{1}G_{14} - 2 G_{1}^2G_{4}-G_{14}\end{aligned}$$

(B28)

$$\begin{aligned}&G_{122} = G_{22}G_{1} + G_{1} G_{2} + 2 G_{2}G_{12} - 2 G_{2}^2G_{1}-G_{12} \end{aligned}$$

(B29)

$$\begin{aligned}&G_{123} = G_{12}G_{3} + G_{13} G_{2} + G_{23}G_{1} - 2 G_{1}G_{2}G_{3}\end{aligned}$$

(B30)

$$\begin{aligned}&G_{124} = G_{12}G_{4} + G_{14} G_{2} + G_{24}G_{1} - 2 G_{1}G_{2}G_{4}\end{aligned}$$

(B31)

$$\begin{aligned}&G_{125} = G_{12}G_{5} + G_{15} G_{2} + G_{25}G_{1} - 2 G_{1}G_{2}G_{5}\end{aligned}$$

(B32)

$$\begin{aligned}&G_{133} = G_{33}G_{1} + G_{1} G_{3} + 2 G_{3}G_{13} - 2 G_{3}^2G_{1}-G_{13}\end{aligned}$$

(B33)

$$\begin{aligned}&G_{134} = G_{13}G_{4} + G_{14} G_{3} + G_{34}G_{1} - 2 G_{1}G_{3}G_{4}\end{aligned}$$

(B34)

$$\begin{aligned}&G_{135} = G_{13}G_{5} + G_{15} G_{3} + G_{35}G_{1} - 2 G_{1}G_{3}G_{5}\end{aligned}$$

(B35)

$$\begin{aligned}&G_{144} = G_{44}G_{1} + G_{1} G_{4} + 2 G_{4}G_{14} - 2 G_{4}^2G_{1}-G_{14}\end{aligned}$$

(B36)

$$\begin{aligned}&G_{145} = G_{14}G_{5} + G_{15} G_{4} + G_{45}G_{1} - 2 G_{1}G_{4}G_{5} \end{aligned}$$

(B37)

Fig. 11

Absolute fluctuations, ($(\langle p_2^2\rangle -\langle p_2\rangle ^2)^{1/2}$ normalized values) of the target protein for sFFL with crosstalk and without crosstalk. Parameter values used here are the same as those of Fig. 5 and for $r_s$ = 1

Fig. 12

Coefficient of variation for the target protein in the case of sFFL with $\gamma _1=\gamma _2=0$. $r_{m_1}$ represents the synthesis rate $m_1$. Here, $r_s=1$ and the other parameter values are the same as those in Fig. 5

In order to see the effect of crosstalk, we have plotted the coefficient of variation for the target protein with $r_{m_1}$ in the main text. As Fig. 11 shows, the absolute fluctuations in the target protein level i.e., $(\langle p_2^2 \rangle -\langle p_2\rangle ^2)^{1/2}$ increase in the sensitive regime. In the absence of crosstalk (i.e., for $\gamma _1=\gamma _2=0$), the coefficient of variation is significantly different showing no indication of an optimum in noise attenuation (see Fig. 12).

1.3 Appendix C: Noise analysis for sCN

In the case of sCN, we follow the same master equation approach as done for sFFL. The master equation for the probability of a given state is based on the following effective differential equations

$$\begin{aligned}&{\dot{s}}=r_s-\gamma _s\, s -g_1 s\, m_1,\end{aligned}$$

(C1)

$$\begin{aligned}&\dot{m_1}=r_{m_1}-\gamma _{m_1} m_1-d_1 s\, m_1-g_1 s\, m_1,\end{aligned}$$

(C2)

$$\begin{aligned}&\dot{p_1} =r_{p_1}' s\, m_1-\gamma _{p_1} p_1,\end{aligned}$$

(C3)

$$\begin{aligned}&\dot{m_2}=\frac{r_{m_2} k_c\, p_1}{1+k_c\, p_1}-\gamma _{m_2} m_2 \ \mathrm{and}\end{aligned}$$

(C4)

$$\begin{aligned}&\dot{p_2}=r_{{p_2}_0} m_2 -\gamma _{p_2} p_2. \end{aligned}$$

(C5)

Fig. 13

Coefficient of variation for the target protein plotted with $r_{m_1}$ in sCN. Here, $r_s=1$, $r_{{p_2}_0} = 0.01$ and all other parameter values are the same as those in Fig. 5. Further, in the case of sCN, $k_2^+=k_2^-=\gamma _2=\kappa _2=0$

In Fig. 13, we plot the coefficient of variation $CV_{p_2}=(G_{55}+G_5-G_5^2)^{1/2}/G_5$ with $r_{m_1}$. No minimum in the coefficient of variation is found in this case. Here, $r_s=1$ and all other parameter values are the same as those in Fig. 5. In the following, we present first and second moments necessary for obtaining the coefficient of variation for the sRNA-driven cascade network. As before, we use Gaussian approximation to express the third moments in terms of various first and second moments.

$$\begin{aligned} G_1= & {} \frac{r_s - g_1\,G_{12}}{\gamma _s}\end{aligned}$$

(C6)

$$\begin{aligned} G_2= & {} \frac{r_{m_1} - (g_1+ d_1) G_{12}}{\gamma _{m_1}}\end{aligned}$$

(C7)

$$\begin{aligned} G_3= & {} \frac{r_{p_1}' G_{12}}{\gamma _{p_1}}\end{aligned}$$

(C8)

$$\begin{aligned} G_4= & {} \frac{r_{m_2}^0 + r_{m_2}^1\, G_3 }{\gamma _{m_2}}\end{aligned}$$

(C9)

$$\begin{aligned} G_5= & {} \frac{r_{{p_2}_0} G_{4}}{\gamma _{p_2}}\end{aligned}$$

(C10)

$$\begin{aligned} G_{11}= & {} \frac{r_s\, G_1 - g_1\,G_{112}}{\gamma _s}\end{aligned}$$

(C11)

$$\begin{aligned} G_{22}= & {} \frac{r_{m_1} G_2 - (g_1+ d_1) G_{122} }{\gamma _{m_1}}\end{aligned}$$

(C12)

$$\begin{aligned} G_{33}= & {} \frac{r_{p_1}' G_{123}}{\gamma _{p_1}}\end{aligned}$$

(C13)

$$\begin{aligned} G_{44}= & {} \frac{r_{m_2}^0 G_4 + r_{m_2}^1 G_{34} }{\gamma _{m_2}}\end{aligned}$$

(C14)

$$\begin{aligned} G_{55}= & {} \frac{r_{{p_2}_0} G_{45}}{\gamma _{p_2}}\end{aligned}$$

(C15)

$$\begin{aligned} G_{12}= & {} \frac{r_s G_2 + r_{m_1} G_1 - g_1(G_{112} + G_{122}) - d_1G_{112}}{\gamma _s + \gamma _{m_1} + g_1+ d_1} \nonumber \\ \end{aligned}$$

(C16)

$$\begin{aligned} G_{13}= & {} \frac{r_s G_3 + r_{p_1}' ( G_{12} + G_{112}) - g_1\,G_{123}}{\gamma _s + \gamma _{p_1}}\end{aligned}$$

(C17)

$$\begin{aligned} G_{14}= & {} \frac{r_s G_4 + r_{m_2}^0 G_1 + r_{m_2}^1 G_{13} - g_1\,G_{124}}{\gamma _s + \gamma _{m_2}} \end{aligned}$$

(C18)

$$\begin{aligned} G_{15}= & {} \frac{r_s G_5 + r_{{p_2}_0} G_{14} - g_1\,G_{125} }{\gamma _s + \gamma _{p_2}}\end{aligned}$$

(C19)

$$\begin{aligned} G_{23}= & {} \frac{r_{m_1} G_3 + r_{p_1}' ( G_{12} + G_{122}) - (g_1+ d_1) G_{123}}{\gamma _{m_1} + \gamma _{p_1}}\nonumber \\\end{aligned}$$

(C20)

$$\begin{aligned} G_{24}= & {} \frac{r_{m_1} G_4 + r_{m_2}^0 G_2 + r_{m_2}^1 G_{23} - (g_1+ d_1) G_{124} }{\gamma _{m_1} + \gamma _{m_2}}\nonumber \\\end{aligned}$$

(C21)

$$\begin{aligned} G_{25}= & {} \frac{r_{m_1} G_5 + r_{{p_2}_0} G_{24} - (g_1+ d_1) G_{125}}{\gamma _{m_1} + \gamma _{p_2} }\end{aligned}$$

(C22)

$$\begin{aligned} G_{34}= & {} \frac{r_{p_1}' G_{124} + r_{m_2}^0 G_3 + r_{m_2}^1 ( G_3 + G_{33})}{\gamma _{p_1} + \gamma _{m_2}}\end{aligned}$$

(C23)

$$\begin{aligned} G_{35}= & {} \frac{r_{p_1}' G_{125} + r_{{p_2}_0} G_{34}}{\gamma _{p_1} + \gamma _{p_2} }\end{aligned}$$

(C24)

$$\begin{aligned} G_{45}= & {} \frac{r_{m_2}^0 G_5 + r_{m_2}^1 G_{35} + r_{{p_2}_0} ( G_{4} + G_{44})}{\gamma _{m_2} + \gamma _{p_2}} \end{aligned}$$

(C25)

1.4 Appendix D: Stochastic simulations

The reactions considered for the stochastic simulations and the corresponding rates are listed below.