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Center for Theoretical Biology and Department of Physics, Peking University, Beijing, China
Correspondence: Address reprint requests to Qi Ouyang, E-mail: qi{at}pku.edu.cn.
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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–3). It provides a paradigm for studying how cells survive and recover from DNA damage. Many significant principles in the field of DNA damage repair are first discovered from this model (4).
The SOS response is mediated by two key proteins: RecA and LexA. DNA replication is blocked when it encounters a UV-induced lesion. RecA is then activated. The activated form of RecA (referred to as RecA*) acts as a coprotease to catalyze the autocleavage of LexA. LexA is a transcriptional repressor that binds to the operator region (often referred to as SOS box) and represses the expression of more than 40 genes (SOS genes) (5), including recA and lexA themselves. The SOS genes have different functions, including DNA repairs, DNA recombinations, replication restarts, inhibitions of cell premature division, and inducible mutagenesis (reviewed in Crowley and Courcelle (6)). Most of these functions contribute to the replication restart. When the DNA replication returns to normal, RecA* is eliminated, allowing LexA to reaccumulate. Thus the SOS genes expression is repressed again.
A series of mathematical models have been established to simulate the regulation of SOS response (7–9). These works are based on experiments that investigate the dynamics of responses at population level, using Western blot (10) or mRNA microarray (5) technology. Recently, Friedman et al. investigated the dynamics of SOS genes promoter activity after UV irradiation at single cell level (11). They found that the promoter activity shows a digital oscillator-like characteristic. As SOS genes are controlled by the transcription repressor LexA, their experimental results suggest that LexA and RecA* have fine-tuned dynamics during SOS response. Analogously, in eukaryotic cells, the tumor suppressor protein p53 level also exhibits undamped oscillations in response to DNA damage (12). Ma et al. have established a model to explain the oscillations (13). According to their explanation, the oscillations of p53 is due to a time-delay negative feedback between p53 and Mdm2. p53 activates MDM2 gene transcription, whereas Mdm2 binds to p53 to inhibit its transcriptional activity and ubiquitinates it, so that p53 is recognized by proteasome. Furthermore, because the inducing signal, which sustains the oscillations, is considered to be a sharp and all-or-nothing switch (13), the amplitude of the modulation does not decrease before it disappears. However, there are significant differences between the eukaryotic p53-Mdm2 control mechanism and the bacterial LexA-RecA response feedback. There is no strong negative feedback regulation between RecA (or RecA*) and LexA in Escherichia coli. On the other hand, LexA is a transcriptional repressor, whereas p53 is an activator. Therefore, the response timescales of the two systems, i.e., the accumulation rate of p53 and the degradation rate of LexA, are very different.
In this article, we present a model to describe the SOS response process in E. coli. Different from previously models for this system (7–9), our model describes the dynamics at single cell level, and takes into account the stochastic and discrete characteristic of the inducing signal, RecA*, which shows up only when the replication fork encounters a DNA damage point. We show that the digital oscillator-like modulation is mainly due to the fluctuations of RecA*. RecA* goes up and down as the replication fork travels through the lesions one by one. We conclude that for UV irradiation, the SOS inducing signal is strongly coupled with the DNA replication. The regulatory network for the SOS response influences the modulation of the SOS gene expression dynamics in certain frequency range.
Model description
The model that we present here can be divided into three modules. First, DNA damage causes a replication interruption and hence RecA* generation. The activated RecA* then eliminates LexA. Second, the elimination of LexA induces an elevated level of the SOS gene expression. Third, the inducible nucleotide excision repair (NER) system and the DNA translesion synthesis (TLS) system, together with the noninducible recombination repair (RR) system, work on DNA damage. They seek to remove the lesions or complete DNA replication. In the following we discuss these modules one by one.
SOS inducing signal: RecA activation and LexA cleavage
UV radiation produces DNA lesions; the cyclobutane pyrimidine dimers (CPD) and the pyrimidine(6-4)pyrimidone photoproducts (6-4PP) twist the DNA structure and block DNA replication (3). The DNA polymerase III (Pol III) cannot replicate the damaged DNA template. As shown in Fig. 1 a, when the enzyme encounters a lesion on the leading strand, the replication process stalls (14). If the lesion is on the lagging strand, replication may simply reinitiate downstream and leave a single daughter strand gap (DSG) behind (Fig. 1 b) (reviewed in Courcelle et al. (15)). Afterward, single strand DNA (ssDNA) appears at the stalled replication fork (reviewed in Michel et al. (16)). With the help of RecFOR, RecA is loaded onto the ssDNA, forming a nucleoprotein filament (17), as shown in Fig. 1. So far, whether RecA is loaded onto the leading (18) or lagging (19) strand is still in debate. RecA binds to ssDNA orderly in the 5' to 3' direction, and also disassociates from 5' to 3' (reviewed in Cox (20)). Every RecA molecule binds to three nucleotides. The ssDNA bounded RecA has multiple functions: it rearranges and stabilizes the stalled replication fork structure (21); it catalyzes DNA strand exchange as the first step of DSG repair (reviewed in Kreuzer (22)); it catalyzes autocleavage of LexA (23,24) and UmuD (25); it assists TLS (reviewed in Schlacher et al. (26)).
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). In our model, the lesions are randomly distributed on the E. coli chromosome with a total length of 4.6 x 106 basepair (see Appendix for simulation details). The number of nucleotides (NNC) that have been replicated is described in the following reaction:
 | (1) |
)). On the other hand if the lesion is on the lagging strand, the replication does not stop; it instead leaves a single strand gap behind. In both cases, ssDNA appears, therefore RecA* increases:
 | (2) |
), the total amount of RecA* molecules on one fragment of ssDNA is restricted to no more than 500. RecA*, in turn, mediates LexA cleavage:
 | (3) |
The parameters for this module are listed in Table 1.
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). S is the LexA binding region (SOS box), which can bind two LexA monomers. As LexA tends to form a stable dimer in solution (30), the LexA binding process is considered to be a single step. SL indicates S region occupied by one LexA dimer. M is mRNA, which is transcribed mainly from the unoccupied S region. We assume that the transcription from SL is negligible. P is the protein product of the corresponding mRNA. The degradation rate constants for mRNA and protein are denoted by kdm and kdp, respectively. In our model, five SOS genes are concerned: lexA, recA, uvrB, umuD, and umuC. uvrB is responsible for NER, and umuD and umuC are responsible for TLS. Note that the recFOR genes, which are responsible for RR, are not controlled by LexA. The parameters for this module are listed in Table 2.
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) is responsible for removing the UV-produced CPDs and 6-4PPs. It is composed of uvrA, uvrB, and uvrC. uvrA and uvrB are repressed by LexA, whereas uvrC is not. The NER genes have a significant basal level expression in the normal condition to deal with the naturally occurring DNA damages. During SOS response, uvr genes are induced to speed up removing the lesions (32). The NER process is considered to be vital for cells to resume replication from UV-induced DNA damage. It has been shown that in uvrA defective cells, LexA proteins are unable to reaccumulate long after UV irradiation (10), and the DNA replication do not resume either (33). This indicates that the NER system is crucial to SOS recovery. uvrA deficiency also causes elevated sensitivity against UV irradiation (34). These experimental evidences lead to the hypothesis (15,21) that, in response to UV irradiation, the most efficient way to overcome a DNA replication arrest is to remove the lesion via NER, which is followed by a replication restart. As shown in Fig. 1 a, ssDNA bound RecA may facilitate replication fork regression (35), so that the NER proteins can gain access to excise the lesion-containing DNA fragment (Fig. 1 a, NER pathway). When the damage is repaired, replication starts again.
Because uvrA and uvrB present similar expression profile in transcription (5) and translation level (32), only uvrB is included in the model to represent the NER system. The repair process is considered to be first order:
 | (4) |
When this reaction takes place, a lesion is randomly selected and removed from the chromosome. If the lesion chosen to be removed is the one right in front of the stalled replication fork, in other words, it is the one causing a replication arrest, the removal will lead to ssDNA-RecA filament disassembly and DNA replication restart. In the simulation, this process is realized by decreasing RecA* at a rate of kreca_off. When RecA* on the ssDNA involved decreases to zero, krep is set to be 900 s–1, indicating replication resumption.
TLS
TLS is another important SOS activity, which is thought to be responsible for SOS mutagenesis (reviewed in Tippin et al. (36)), replication restart, and DSG filling (37). In E. coli, there are three error-prone (EP) polymerases (36), which are all repressed by LexA: Pol II (encoded by polB), Pol IV (dinB), and Pol V (encoded by umuD and umuC). They are capable of passing through the lesion and continuing replication, yet with a low fidelity, thus raising the mutation rate. Pol V is the only EP polymerase that has evident phenotype during UV-induced SOS response (37). Deleting Pol V will dramatically reduce SOS mutagenesis, and will render the cell more sensitive to high dose UV exposure (37). For simplicity, among the three EP polymerases, only Pol V is considered in our model.
The cell has evolved an elaborate mechanism to control the in vivo concentrations of UmuD and UmuC, at both the transcriptional and the posttranscriptional levels (reviewed in Gonzalez and Woodgate (38)). The umu operon is tightly repressed by LexA. UmuD and UmuC are rapidly degraded by Lon protease (39,40). As shown in Fig. 3, UmuD can form homodimer UmuD2. UmuD and UmuD2 undergo autocleavage catalyzed by RecA*, similar to LexA cleavage. The 24 residues at the N-terminal of UmuD are then removed, so that UmuD turns into UmuD'. UmuD' can form homodimer UmuD'2, and form heterodimer UmuDD' with UmuD as well. UmuD'2 is more stable than UmuD2. UmuD and UmuD' prefer to form heterodimer rather than homodimer. When forming a heterodimer, UmuD subunit presents UmuD' substrate to the ClpXP protease for degradation (41), limiting the level of UmuD' to a minimum. UmuD'2 and UmuC finally form UmuD'2C, namely Pol V.
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). Pol V can effectively replicate across the lesion and continues to synthesize 
20 nucleotides with low fidelity (42). Afterward, Pol III takes the place again and resumes normal replication. Because neither Pol III nor Pol V can replicate on a RecA coated template (36,43), RecA should disassemble from ssDNA before the replication resumes.
In our model, the TLS process is similar to the NER process.
 | (5) |
) does not happen before the length of ssDNA-RecA filament reaches a certain threshold ThrTLS. When reaction (5) takes place on the leading strand lesion, RecA* at the replication fork falls to zero before replication resumption. If the reaction takes place on a DSG, RecA* on that DSG decreases and the lesion becomes accessible for the NER system again.
Besides TLS, we consider a checkpoint-like function of UmuD and UmuC. After DNA damage, the induced UmuD and UmuC are thought to slow down DNA replication (44), allowing the cell to perform more excision repair. Once UmuD turns into UmuD', it can no longer block replication. Therefore, we consider UmuD2 to be the key effecter:
 | (6) |
When UmuD2 binds to Pol III and forms a complex, PolIII_UmuD, DNA replication is interfered. As a result, krep is reduced by a factor of rslow. Namely, the replication rate drops to krep/rslow.
RR
RR is found to be the major pathway filling the DSG (45). As shown in Fig. 1 b of the RR pathway, after the RecA-ssDNA filament is formed, RecA* can catalyze strand exchange between the lesion-containing DSG and the homologous double strand. Afterwards, the DSG is lesion free so that it can be filled up by normal replication. RR is also supposed to be involved in replication restart (reviewed in Michel et al. (16)). As shown in Fig. 1 a of the RR pathway, when the replication on the leading strand is blocked by a lesion, the replication on the lagging strand will continue for a certain distance (14). After the fork regression, the leading nascent strand may extend using the lagging nascent strand as template. The nascent strands then reanneal with the mother strands, and thereby the lesion is covered.
In our model, the DSG filling and replication restart processes by RR are described as follows:
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 | (8) |
We assume that the RR process would be stimulated only when the length of ssDNA-RecA filament reaches a certain threshold ThrRR. Two SOS genes, ruvA and ruvB, can catalyze branch migration (46) and may contribute to the RR process. However, so far the detailed mechanism of the RR pathway is not clear (16); we assume the RR rate constants krrDSG and krrRF are not induced by SOS response.
The parameters for this module are listed in Table 1.
We assume that an E. coli cell is a well-mixed biochemical reaction system. The SOS response process can be described with a series of elemental reactions. We use the exact stochastic method with Gillespie's algorithm (47) to do the simulation.
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RESULTS |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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), whereas the protein level only increases 10-fold (48). As a result, we only choose one result between the two to fit the parameters (see Appendix). Note that this kind of contradiction derives from different experimental conditions and different strains. Fortunately, it does not have significant contribution to the core results in our model.
We now concentrate our study on single cell behavior. In single cell simulation the dynamics of recA mRNA is chosen to compare with the experimental results from Friedman et al. (11), in which the promoter activity of recA gene is found to show oscillatory-like behavior. Fig. 5 c shows the recA mRNA dynamics in one simulation. If we filter out the high frequency noise, the profile can be smoothed out, as shown in Fig. 5, d–f. In Fig. 5, d–f, the recA mRNA dynamics shows three peaks. As shown in Fig. 5, a and b, the modulation is derived from the fluctuation of RecA*. As RecA* goes up, LexA decreases, hence releasing the recA mRNA expression.
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). We choose 1/30 min–1 as the cut-off frequency in the following simulation, which accords with the filtering frequency that is used in Friedman et al. (11) (1/32 min–1).
The distributions of the time and amplitude of the three peaks are illustrated in Fig. 6, a–c, which indicates the first, the second, and the third peaks, respectively. As shown in the histogram of the peak distribution in time (Fig. 6, d–f), the average interval between the first and second peaks is 35 min, with the standard deviation (SD) as 16 min. The average and SD of the interval between the second and third peaks are 47 and 23 min. This distribution is consistent with the experimental observation from Friedman et al., which also exhibits a peak interval of 40 min (11). The average amplitudes of the first and second peaks are 90 and 73, respectively. The average of the third peak drops to 33. This is not consistent with Friedman et al. (11), in which the modulation shows undamped characteristic. However, as discussed in the Appendix, if the NER rate kner is cut to half, undamped modulation shows up. As kner is determined by Crowley and Hanawalt (32), we suppose that the inconsistency is due to different NER efficiency between the strains used in Crowley, D. J., and P. C. Hanawalt (32) and Friedman et al. (11).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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The fluctuation of RecA* activity leads to modulation in SOS gene promoter activity. The RecA* activity is in turn determined by the state of replication. When replication is blocked by a lesion, RecA* appears. On the other side, induced SOS genes would speed up removing the lesions or help the blocked replication to resume. This feedback is linked by the NER and TLS system. The RR system, which is not inducible in our model, is not considered. With the set of parameters that we use in the model, we are able to roughly compute the characteristic time of the accurate repair and the lesion-approved replication feedbacks, which are shown in Table 3. As many steps of the feedbacks actually overlap each other, the true feedback delay should be shorter than the total time we estimate. To reduce this inaccuracy, each overlapping step is considered to be the 70–90% of its maximum level (minimum level for LexA), with a reaction speed of 70–90% of the maximum. The "characteristic function time" of UvrB and Pol V are the time required that ensure them a 50% possibility to bind on the lesion that causes the replication stop. It is shown that the accurate repair feedback has a characteristic delay of 27 min, which can roughly explain why the first peak appears 20 min after radiation. The delay of lesion-approved replication is estimated to be 50 min, which is consistent with the experimental results that the mutations on the PolV-related genes mainly lead to changes of the second and the third peaks (11).
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The model we present has noticeable limitations though. Details of molecular mechanisms in DNA replication, replication arrest, and restart are still unclear (50). The process we described above is based on the recent knowledge, but there are still some missing links. For example, there are several proteins that function in regulating RecA filamentation process, such as DinI, RecX, RecFOR, and SSB (17,51,52). Because it is not clear how RecA would disassemble from ssDNA before replication restart, we just let RecA to do so automatically. The model can be enriched as long as new experimental findings come up.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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). Consider there are N chemical reactions taking place. A common formula of the reaction is
 | (A1) |
In deterministic method, the dynamics of [A] can be described by an ordinary differential equation,
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
Then go to step 1 for next iteration.
Initially, the number of lesions (Nle) is determined by the incident UV dose (UVdose) in unit of J/m2 (27).
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As shown in Fig. 7, two arrays of Nle/2 random numbers are generated evenly between 1 and 2.3 x 106. These two arrays of random numbers (ARN) denote the position of the lesions that are distributed. When the NER reaction happens, a lesion is randomly chosen and removed from ARN.
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UmuDC expression
The umuD and umuC mRNA and protein levels are shown in Fig. 8. According to Woodgate and Ennis (48), after SOS induction, the UmuD and UmuD' protein levels are 1000 and 2000 molecules per cell, the UmuC protein level is 200 molecules per cell. PolV reaches about 60 molecules per cell (54).
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FOOTNOTES |
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Submitted on June 13, 2006; accepted for publication February 28, 2007.
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