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The authors have declared that no competing interests exist.

Post-transcriptional regulation of gene expression plays a crucial role in many bacterial pathways. In particular, the translation of mRNA can be regulated by trans-acting, small, non-coding RNAs (sRNAs) or mRNA-binding proteins, each of which has been successfully treated theoretically using two-component models. An important system that includes a combination of these modes of post-transcriptional regulation is the Colicin E2 system. DNA damage, by triggering the SOS response, leads to the heterogeneous expression of the Colicin E2 operon including the

Gene expression is a fundamental biological process, in which living cells use genetic information to synthesize functional products like proteins. To control this process, cells make use of many different mechanisms. A well-studied example is the binding of expression intermediates by a cellular component in order to delay the synthesis. This mechanism is known to regulate the stress-induced release of the toxin colicin E2 by

Regulation of gene expression occurs at transcriptional and post-transcriptional levels, and has been studied intensively both experimentally and theoretically [

Colicins are toxic proteins produced by certain

The interaction scheme is a generalized adaption of that presented by Yang [

Other elements: _{sos}: SOS promoter; _{1} and _{2}: transcriptional terminators.

Each of the three components is encoded by a single gene—the colicin by _{1} and _{2}, located upstream and downstream of the

Post-transcriptional regulation makes use of many different mechanisms. Recent studies emphasize the particular importance of non-coding sRNAs [

The basic interaction network that controls the ColE2 regulatory network has been studied in great detail in previous works [

M,A,S: molecule numbers of free long mRNA, free CsrA dimers and the free effective sRNA; _{S} were simplified to the dynamics of an effective sRNA with one CsrA binding site but N-times higher production rate (

For our theoretical analysis, we initially developed a detailed mathematical model for the post-transcriptional regulation of colicin E2 release. To this end, we derived a set of coupled, deterministic rate equations from the interaction network depicted in

As we wished to study the post-transcriptional regulation of colicin E2 expression, we included in the model only those components that are relevant at that stage. The model therefore omits the short mRNA and its products. However, the rate of transcription of the long mRNA is a crucial parameter, which is influenced by the kinetics of activation of the SOS promoter, and thus by the processing of its repressor LexA. Upon DNA damage, RecA promotes auto-cleavage of LexA dimers, thus removing inhibition of the SOS response (marked in red in _{M} (

CsrB and CsrC regulate CsrA by forming complexes with it. The two sRNAs each have several (on average: _{S} (see _{M}) and (1 − _{S}) are the probabilities for CsrA to survive the coupled degradation. A graphical illustration of this differential equation system is depicted in

For the analysis of our model, we had to determine production, degradation and binding rates. The particular values used are listed in

We analyzed the reduced post-transcriptional model by first calculating its steady state. In order to obtain a cleaner and simpler result, we derived an approximation (see _{M}) and sRNA (_{S}) on the levels of the three components. The results (see

The stationary solutions are given as a function of the effective transcription rate _{M} of long mRNA and the production rate _{S} of sRNA.

The production rate of CsrA dimers was set to _{A} = 58.52. All other system parameters are given in _{M} and _{S} below the threshold, the abundances of free long mRNA and sRNA are zero, as any newly produced component quickly forms a complex with the highly abundant CsrA. At sufficiently large production or transcription rates, sRNA and long mRNA titrate all available CsrA molecules and can thus attain non-zero molecule numbers, The white line gives the transition between two approximate analytical solutions (

From the aforementioned analytic solution we calculated the threshold position as a function of the system parameters (_{A} is equal to the sum of transcription rates for long mRNA _{M} and sRNA _{S} (_{M} + _{S} < _{A}, as shown in Figs

The fluctuations are quantified by the Fano factor (see main text) and depicted as heatmap in the plot. They are most pronounced at the threshold, and fade for parameter sets above the threshold. With an increase in sRNA production (_{S}), the fluctuations become smaller and more localized to the threshold. This illustrates how the third component sRNA acts as a means to reduce intrinsic fluctuations. The production rate of CsrA dimers was again set to _{A} = 58.52, and all other system parameters are given in

Apart from the threshold itself, we find that the levels of free CsrA and free sRNA predicted by our steady state analysis are consistent with experimental in-vivo values determined by previous studies [

So far, we have demonstrated that our three-component system is capable of producing a threshold behavior. However, it has been shown previously that a mutually exclusive production of sRNA and a target mRNA is possible with just two components [

After SOS signals, the sRNA controls and accelerates the degradation of CsrA (see section on expression dynamics below), eventually leading to the expression of the lysis protein.

In a next step, we analyzed the stochastic dynamics of the post-transcriptional regulation network. To this end, we switched to a stochastic description, calculated the Fano factor (Var

We found that fluctuations in mRNA were most pronounced close to the threshold position, with the largest fluctuations occurring slightly above the threshold (

To understand why the fluctuations are localized to the region near threshold, one must take the characteristics of this parameter regime into account. Around the threshold, molecule numbers are close to zero, which has a direct affect on the relative size of fluctuations: the lower the abundance, the larger the fluctuations (stochastic regime). Moreover, the threshold is the only regime in which all three components, CsrA, mRNA and sRNA, can coexist and interact with each other: An increase in the level of CsrA will lead to a decrease in the abundance of long mRNA and sRNA, owing to increased complex formation and subsequent degradation. Analogously, an increase in long mRNA and sRNA molecule numbers leads to a decrease in CsrA abundance. Therefore, the abundance of CsrA dimers is anti-correlated with the abundance of both long mRNA and sRNA. It has been shown for a two-component system, that anti-correlated components can create anomalously large fluctuations [

These results show that a third component can reduce intrinsic fluctuations of a hierarchically ordered regulatory network.

To study the dynamical response of the ColE2 system to an SOS signal, we extended the post-transcriptional network by including the LexA-RecA regulatory network [

In our analysis of the ColE2 post-transcriptional regulation network so far (see above), we have assumed the dynamics of SOS promoter activation to be so fast that we could use an effective transcription rate _{M} for long mRNA. To link the LexA regulatory network to the post-transcriptional regulation network, we must drop this assumption and explicitly model the dynamics of LexA dimers, which connect the two networks. In the biological system, this involves the binding and dissociation of LexA dimers to and from the SOS promoter in the ColE2 operon. Long mRNA and short mRNA are transcribed only from the derepressed promoter at rates _{Ml} and _{Ms}, respectively. Thus, the transcription rates of long mRNA and short mRNA are proportional to the number of open SOS promoters in the bacterium. The majority of transcripts are short mRNAs. The mathematical implementation of the integrated regulation network is again a system of coupled rate equations, which we describe in

We simulated the SOS signal by temporarily up-regulating the coupling parameter _{p}, which quantifies the ability of RecA to induce cleavage of LexA (_{p} = 0. Under SOS stress _{p} was increased to _{p} = 6. This increase in _{p} subsequently boosts the long mRNA production, and therefore relates to a transition from a sub-threshold state (gray area below the white line in _{Ml} of long mRNA is not constant, but fluctuates about a mean value. The production rate of sRNA was held constant at _{S} = 57.5.

We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from _{p} = 0.0 to _{p} = 6.0 between the two dashed vertical lines at _{p} gives the rate at which LexA dimers degrade due to the presence of RecA. During the simulation, we tracked the abundance of (A) free short mRNA, (B) free long mRNA, (C) free CsrA dimers and (D)free sRNA over time. In each panel, the fluctuating colored curve represents a single realization of the stochastic system as implemented by a Gillespie simulation. The smoother darker-colored curve shows the average of 500 different realizations. The black dashed curve depicts the results found by numerical integration of the deterministic rate equations, which neglects fluctuations. In general, the stochastic realizations deviated significantly from both the simulation average and the deterministic solution, as they exhibited large spontaneous bursts. As the short mRNA is not post-transcriptionally regulated, its abundance level can serve as a proxy for the SOS promoter activity. Comparing the free short mRNA abundance with free long mRNA shows that short promoter activity peaks were reliably filtered out by post-transcriptional regulation. After an up-regulation of the LexA auto-cleavage parameter _{p} at _{A} = 58.52 and the transcription rate of sRNA to _{S} = 57.5. All other parameters are given in

Focusing on the dynamics of mRNA transcription, we found that, due to initial simulation parameters, only small numbers of the short mRNA are produced in the uninduced state. After up-regulation of the LexA auto-cleavage parameter _{p} at

Studying the dynamics of a single stochastic realization, we observed that the number of long mRNA molecules underwent large fluctuations, which were followed by periods of no expression at all. Moreover, the timing of these bursts varied considerably between different realizations. This constitutes a significant qualitative difference compared to the average over 500 realizations and to the deterministic dynamics (

We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from _{p} = 0.0 to _{p} = 6.0 between the two dashed vertical lines at _{p} gives the rate at which LexA dimers degrade due to the presence of RecA. (A) With the parameters defined in _{A} = 58.52 and the transcription rate of sRNA to _{S} = 57.5. All other parameters are given in

Incorporation of the LexA-RecA regulatory network allowed us to model the colicin E2 expression dynamics in response to a realistic SOS signal, and the results presented above highlight the importance of CsrA for colicin release.

Gene expression is a process that allows for various forms of regulation at all levels. In theoretical studies of post-transcriptional regulation of several biological systems, modulation of mRNA production by proteins or sRNA has been shown to create, for instance, temporal thresholds for mRNA translation [

In our analysis of the model, we used rate constants that were determined from experimental systems (see chapter 2 of

Investigation of the dynamics revealed that the model exhibits a time delay in the production of free long mRNAs. This delay is due to the high abundance of CsrA in the non-SOS state of the cell, which causes CsrA to quickly bind to free long mRNA and thus prevents its transcription. Only during an SOS signal, which indicates external stress for the cell, the level of CsrA gets steadily reduced. The time this process takes to get CsrA levels so low that fluctuations in long mRNA production result in free long mRNA, causes a delay in colicin release. As colicin release is coupled to cell lysis, the delay is therefore a mechanism for filtering out transient SOS signals that might erroneously lead to synthesis of the lysis protein. Moreover, also intrinsic fluctuations, for instance in sRNA production, are filtered out by this mechanism: Even if a large and sudden burst in sRNA were strong enough to drop CsrA abundance close to zero, the CsrA buffer gets restored quickly due to the large production rate of CsrA. This rate is only effectively lowered during a SOS signal, which increases the production of the CsrA-sequestering long mRNA. The fact that lysis is regulated by a threshold mechanism of a global regulator protein like CsrA might also be a guarding mechanism for the cell: only prolongued extreme situations will cause the abundance of these regulators to drop to low molecule numbers.

However, delays and similar threshold behavior also emerge in two-component systems, raising the question why a third component is necessary here. Strikingly, we found that the third component (sRNA) in the post-transcriptional interaction network enables the cell to tune the duration of the delay by sequestering CsrA. In the case of the ColE2 system, this means that cells are able to adjust the (average) time between a SOS signal and the onset of cell lysis leading to colicin release.

Furthermore, previous studies of systems with slow, bursting promoter kinetics have also uncovered a major limitation of two-component sRNA-based regulation compared to regulation based on transcription factors: Two-component systems are subject to significantly higher levels of intrinsic noise [_{S}. The sRNA might therefore allow for significant dampening of these fluctuations. This idea is supported by the fact that the relatively high degradation rate of sRNA makes it less susceptible to induced fluctuations.

In bacteria, these mechanisms could have several functions: First, a comparison of different sRNA production rates (_{p}. Our predictions for lysis time distributions (

In order to focus on the interplay between the LexA-RecA system and the hierarchical regulation of long mRNA by CsrA and sRNA, we kept the plasmid number constant. If we considered random, Poisson-distributed plasmid numbers instead, the effect would be very small, as shown in

In conclusion, we have provided here the first detailed theoretical description of colicin E2 production and release, and used it to study the dynamical behavior of this system. Moreover, the general three-component model described here should be applicable to many other systems of toxin production in microorganisms.

In most models of prokaryotic gene expression, it is assumed that promoter kinetics are fast compared to RNA production and degradation rates. In that case, the promoter state is well approximated by its steady state [

The two sRNAs CsrB and CsrC regulate CsrA via complex formation. More specifically, each CsrB molecule has approximately 22 binding sites for CsrA, with 9 CsrA dimers being attached on average [_{S} (see

For the calculations of the abundances of the three components (for example, to obtain the plots of

We started the analysis of noise properties by reformulating the simplified three-component system as a Master equation. As Master equations are typically impossible to solve analytically, we performed a general van Kampen expansion in multiple variables (components). Our analysis included all higher orders, and not only lowest order terms as is commonly found in textbooks [

To verify how well our analytical results of the deterministic rate equations coincide with the actual mean molecule numbers, we set up a Gillespie simulation [

Rates are given in molecules per cell volume _{EC} = 0.65^{3} per minute. The number of ColE2 plasmids is _{sos} = 20. The literature values can be found in [

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Rates are given in molecules per cell volume _{EC} = 0.65^{3} per minute. The number of ColE2 plasmids is _{sos} = 20. _{l},_{r},_{s},_{l} _{r},_{sos}: number of LexA dimers bound to the

(PDF)

The interaction scheme is mathematically formulated as _{ma} give the numbers of long mRNA, CsrA dimers, sRNA, lysis protein and long mRNA-CsrA complexes. C_{n} gives the number of sRNA molecules with n CsrA dimers bound. The rates of a reaction is expressed by the formula next to the arrows. ^{−},^{−}: Complex dissociation rates; ^{+},^{+}: Complex formation rates. To illustrate the complex dynamics between CsrA dimers and sRNA we depict the reaction rates of CsrA with an sRNA that has already bound

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The stationary solution _{S} = 20 and _{S} = 40. The points indicate the result of Gillespie simulations, whereas the lines show the analytical result obtained from the approximated steady state equations. The production rate of CsrA dimers was chosen to be _{A} = 58.52, all other parameters are given in Table

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The production rate of CsrA dimers was set to _{A} = 58.52. All other parameters are given in _{S} = 20 and _{S} = 40, the analytic calculations using van Kampen’s system size expansion reproduced the shape of the fluctuations obtained by Gillespie simulations well. In the threshold regime the analytic result overestimated the fluctuations slightly.

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(A) shows the lysis time distribution as in _{SOS}, follows a Poisson distribution. (C) Lowering the sRNA production rate to _{S} = 56 shifts the lysis distribution towards later times, whereas (D) doubling it to _{S} = 58 causes several cells to lyse even before (and hence independent of) the SOS signal. This illustrates that the sRNA is a possible means of controlling cell lysis.

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To illustrate the predictive possibilities of our three component model, we compare the results of numerical simulations using our model with experimental data [_{p} (values: 1, 3, 6, 12, 15, 20, 30, 90) to emulate the stress levels. To fit the data, we only applied a scaling factor to map the Mitomycin concentration to values of _{p}, and shifted the theoretical delays by a constant value. The last step is necessary, as the numerical simulations also account for the constant time between SOS signal and first appearance of short mRNA, which is not the case in the experiments.

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Detailed derivations of the (simplified) rate equations and the linear noise approximation, as well as the detailed reaction scheme used in the Gillespie simulations.

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We thank A. Mader and K. Wienand for fruitful discussions and critical reading of the manuscript.