Hierarchical Post-transcriptional Regulation of Colicin E2 Expression in <i>Escherichia coli</i>

<div><p>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 <i>cea</i> gene encoding the toxin colicin E2, and the <i>cel</i> gene that codes for the induction of cell lysis and release of colicin. Although previous studies have uncovered the system’s basic regulatory interactions, its dynamical behavior is still unknown. Here, we develop a simple, yet comprehensive, mathematical model of the colicin E2 regulatory network, and study its dynamics. Its post-transcriptional regulation can be reduced to three hierarchically ordered components: the mRNA including the <i>cel</i> gene, the mRNA-binding protein CsrA, and an effective sRNA that regulates CsrA. We demonstrate that the stationary state of this system exhibits a pronounced threshold in the abundance of free mRNA. As post-transcriptional regulation is known to be noisy, we performed a detailed stochastic analysis, and found fluctuations to be largest at production rates close to the threshold. The magnitude of fluctuations can be tuned by the rate of production of the sRNA. To study the dynamics in response to an SOS signal, we incorporated the LexA-RecA SOS response network into our model. We found that CsrA regulation filtered out short-lived activation peaks and caused a delay in lysis gene expression for prolonged SOS signals, which is also seen in experiments. Moreover, we showed that a stochastic SOS signal creates a broad lysis time distribution. Our model thus theoretically describes Colicin E2 expression dynamics in detail and reveals the importance of the specific regulatory components for the timing of toxin release.</p></div

Dynamical behavior before and after a realistic SOS response.

<p>We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from <i>c</i>_<i>p</i> = 0.0 to <i>c</i>_<i>p</i> = 6.0 between the two dashed vertical lines at <i>t</i> = 200 min and <i>t</i> = 500 min. The parameter <i>c</i>_<i>p</i> 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 <i>c</i>_<i>p</i> at <i>t</i> = 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 <i>α</i>_<i>A</i> = 58.52 and the transcription rate of sRNA to <i>α</i>_<i>S</i> = 57.5. All other parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s001" target="_blank">S1</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s002" target="_blank">S2</a> Tables.</p

Simplified interaction scheme for post-transcriptional regulation of long mRNA.

<p>M,A,S: molecule numbers of free long mRNA, free CsrA dimers and the free effective sRNA; <i>α</i>: production rates; <i>δ</i>: degradation rates; <i>k</i>: effective rate of coupled degradation. The interaction network (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s003" target="_blank">S1 Fig</a>) of the regulatory system depicted in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.g001" target="_blank">Fig 1</a> 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 <i>α</i>_<i>S</i> were simplified to the dynamics of an effective sRNA with one CsrA binding site but N-times higher production rate (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s008" target="_blank">S1 Text</a>).</p

Regulation of colicin E2 expression and release.

<p>The interaction scheme is a generalized adaption of that presented by Yang [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.ref028" target="_blank">28</a>]. 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 <i>cea</i>, immunity gene <i>cei</i>), 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.</p> <p>Other elements: <i>P</i>_sos: SOS promoter; <i>T</i>₁ and <i>T</i>₂: transcriptional terminators.</p

Fluctuations in long mRNA abundance.

<p>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 (<i>Nα</i>_<i>S</i>), 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 <i>α</i>_<i>A</i> = 58.52, and all other system parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s001" target="_blank">S1 Table</a>.</p

Probability distribution of the first peak in long mRNA abundance and survival function.

<p>We simulated an SOS signal by temporarily up-regulating the LexA auto-cleavage parameter from <i>c</i>_<i>p</i> = 0.0 to <i>c</i>_<i>p</i> = 6.0 between the two dashed vertical lines at <i>t</i> = 200 min and <i>t</i> = 500 min (see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.g005" target="_blank">Fig 5</a>). The parameter <i>c</i>_<i>p</i> gives the rate at which LexA dimers degrade due to the presence of RecA. (A) With the parameters defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s001" target="_blank">S1</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s002" target="_blank">S2</a> 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 <i>E. coli</i> 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 <i>α</i>_<i>A</i> = 58.52 and the transcription rate of sRNA to <i>α</i>_<i>S</i> = 57.5. All other parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s001" target="_blank">S1</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s002" target="_blank">S2</a> Tables.</p

Approximate stationary solutions for (A) long mRNA, (B) CsrA dimers and (C) sRNA.

<p>The stationary solutions are given as a function of the effective transcription rate <i>α</i>_<i>M</i> of long mRNA and the production rate <i>α</i>_<i>S</i> of sRNA.</p> <p>The production rate of CsrA dimers was set to <i>α</i>_<i>A</i> = 58.52. All other system parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#pcbi.1005243.s001" target="_blank">S1 Table</a>. For values of <i>α</i>_<i>M</i> and <i>α</i>_<i>S</i> 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 (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005243#sec008" target="_blank">Materials and Methods</a>).</p

Time series for the simulated distribution of the population composition <i>x</i>.

<p>The distribution initially broadens, then freezes to a steady state (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s001" target="_blank">S1 Video</a>). The fraction of populations that have <i>x</i> = 0 or <i>x</i> = 1 (indicated by the black bins) remains constant during the time evolution, as expected for a Pólya urn process, and in contrast to expectations from genetic drift (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s008" target="_blank">S1 Table</a>). In each well the population follows a stochastic path and reaches a (random) limit composition, and the distribution freezes only when all populations reached their limit. The parameter values used in the simulation are <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>10</mn></mrow></math> and <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>33</mn></mrow></math> The inset shows the mean, standard deviation and skewness as a function of the number of generations, with symbols denoting numerical simulations, and the solid lines corresponding to the theoretical prediction of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.e014" target="_blank">Eq (2)</a> (and also those in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s003" target="_blank">S2 Text</a>). Analytical and numerical values agree. The mean ⟨<i>x</i>⟩ remains constant throughout the evolution, as expected for a non-selective process; standard deviation and skewness saturate to limit values, confirming the freezing of the distribution.</p

Initial distributions for population size <i>N</i>₀ and composition <i>x</i>₀ (parameter values N‾0=2.55, x‾0=0.45).

<p>The experimental distributions (bars) for <i>N</i>₀ (panel <b>(a)</b>) and <i>x</i>₀ (panel <b>(b)</b>) are measured from 120-well ensembles. The average <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub></mrow></math> and <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub></mrow></math> calculated from the measured values determine the parameters for the simulated distributions. The theoretical average distribution (solid blue line) is the average of the same distributions generated for 84 sets of 120 wells. Using that average we calculate the Wilson binomial confidence intervals (gray areas) for 68% (between dashed lines), 95% (between dotted lines) and 99.73% confidence. The measured and simulated distributions agree well within statistical error, confirming our assumption that individuals of strain <i>A</i> and <i>B</i> in the experiments start Poisson-distributed with mean <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub></mrow></math> and <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></math>, respectively. The ragged distribution of <i>x</i>₀ derives from a small-number effect, and disappears at larger values of <i>N</i>₀ (see main text, and also <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s007" target="_blank">S3 Fig</a>).</p

Steady-state distributions of population composition <i>x</i> for different initial conditions.

<p>The experimental distribution (bars) is the result of growth on 120 independent wells. We use the measured average <i>x</i>₀ and <i>N</i>₀ from the experiments to initialize the simulations of several 120-well ensembles. After growth, we compute the histogram for each of these ensembles and obtain the average theoretical distribution (blue line). Using the values from this distribution, we compute the three confidence intervals (shaded gray areas) for each bin for 68% (between dashed lines), 95% (between dotted lines) and 99.73% confidence. The two sets of data match: most experimental data agree with the first prediction confidence region, practically all with the second one. The limit distributions are clearly different from the initial ones (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s004" target="_blank">S1 Fig</a>). The importance of growth in changing the distributions depends on the initial size <i>N</i>₀ (see main text, and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0134300#pone.0134300.s004" target="_blank">S1 Fig</a>). Parameter values: <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>2</mn><mo>.</mo><mn>9</mn></mrow></math>, <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>32</mn></mrow></math> (panel <b>(a)</b>); <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>18</mn><mo>.</mo><mn>4</mn></mrow></math>, <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>22</mn></mrow></math> (panel <b>(b)</b>); <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>19</mn><mo>.</mo><mn>6</mn></mrow></math>, <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>52</mn></mrow></math> (panel <b>(c)</b>); <math><mrow><msub><mi>N</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>14</mn><mo>.</mo><mn>5</mn></mrow></math>, <math><mrow><msub><mi>x</mi><mo>‾</mo><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>71</mn></mrow></math> (panel <b>(d)</b>).</p