Fig 1.
Regulation of colicin E2 expression and release.
The interaction scheme is a generalized adaption of that presented by Yang [28]. Under normal conditions, the SOS response system (yellow box) maintains a constant level of LexA dimers, which repress the SOS promoter of the colicin E2 system (gray box). In the event of DNA damage, RecA is activated and promotes auto-cleavage of LexA. This permits the transcription of two different mRNAs: Short mRNA codes for components of colicin immunity complexes (colicin gene cea, immunity gene cei), whereas long mRNA additionally encodes the protein that triggers cell lysis. Translation of long mRNA is regulated by binding of the protein CsrA to its Shine-Dalgarno sequence (SD). CsrA itself is regulated by the two sRNAs CsrB and CsrC.
Other elements: Psos: SOS promoter; T1 and T2: transcriptional terminators.
Fig 2.
Simplified interaction scheme for post-transcriptional regulation of long mRNA.
M,A,S: molecule numbers of free long mRNA, free CsrA dimers and the free effective sRNA; α: production rates; δ: degradation rates; k: effective rate of coupled degradation. The interaction network (see S1 Fig) of the regulatory system depicted in Fig 1 was reduced to a three component system. In both figures, the corresponding components have the same colors. In particular, we combined the complex dynamics (binding, dissociation, degradation) into an effective coupled degradation. The dynamics of sRNA complexes with N binding sites for CsrA and production rate αS were simplified to the dynamics of an effective sRNA with one CsrA binding site but N-times higher production rate (S1 Text).
Fig 3.
Approximate stationary solutions for (A) long mRNA, (B) CsrA dimers and (C) sRNA.
The stationary solutions are given as a function of the effective transcription rate αM of long mRNA and the production rate αS of sRNA.
The production rate of CsrA dimers was set to αA = 58.52. All other system parameters are given in S1 Table. For values of αM and αS 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 (Materials and Methods).
Fig 4.
Fluctuations in long mRNA abundance.
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 (NαS), 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 αA = 58.52, and all other system parameters are given in S1 Table.
Fig 5.
Dynamical behavior before and after a realistic SOS response.
We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from cp = 0.0 to cp = 6.0 between the two dashed vertical lines at t = 200 min and t = 500 min. The parameter cp 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 cp at t = 200 min, the abundance of short mRNA rose and is expressed in large bursts. After some time delay, during which all newly produced long mRNAs immediately sequestered CsrA dimers, discrete bursts of free long mRNA are seen, which were followed by periods of no production at all. The timing of the bursts varied considerably between different realizations. A comparison with (C) shows that the abundance of free long mRNA is anti-correlated with the molecule number of all free CsrA. Hence, free long mRNA is only present if the number of free CsrA dimers is low. In the simulation, the production rate of CsrA dimers was set to αA = 58.52 and the transcription rate of sRNA to αS = 57.5. All other parameters are given in S1 and S2 Tables.
Fig 6.
Probability distribution of the first peak in long mRNA abundance and survival function.
We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from cp = 0.0 to cp = 6.0 between the two dashed vertical lines at t = 200 min and t = 500 min (see also Fig 5). The parameter cp gives the rate at which LexA dimers degrade due to the presence of RecA. (A) With the parameters defined in S1 and S2 Tables, the timing of the first peak in long mRNA abundance is broadly distributed with maximal probability approximately 60 min after induction of the SOS signal. (B) The survival function is defined as the fraction of E. coli cells in a population that exhibited no peak in long mRNA abundance, and thus would not release colicin. The fraction of cells releasing colicin increased smoothly after induction up to 100%. This heterogeneous response of a bacterial population to an SOS signal is also observed in nature. In the simulation, the production rate of CsrA dimers was set to αA = 58.52 and the transcription rate of sRNA to αS = 57.5. All other parameters are given in S1 and S2 Tables.