Results

To study transcriptional elongation on highly transcribed genes and, in particular, to investigate the effect of RNAP pausing, we developed a stochastic model of RNAP translocation based on in vivo and in vitro data; see SI Text for a detailed description. The model is a variant of stochastic cellular automaton models, which have been studied extensively in statistical physics in the contexts of vehicular traffic (31), nonequilibrium phase transitions (32), and cytoskeletal motors (33, 34). Models of this type have also been used extensively to describe the dynamics of ribosomes on mRNA (35–37) and a recent study used a related model (but without pausing or termination) to study the stepping kinetics of RNAP (38). As depicted in Fig. 1B, the model allows each RNAP (represented by an oval) to be in the active (dark gray) or the paused (white) states. (An extended version of the model, which also incorporates backtracking, the backward translocation of paused RNAPs, is described in the SI Text.) Within the active state, each RNAP may step forward by a single base at the rate ε if the next base is not occupied by another RNAP. We refer to ε as the stepping rate or elongation attempt rate; it gives the maximal instantaneous elongation speed u₀ of a single RNAP, as u₀ = ε s with the step size s = 1 nt. The time-averaged elongation speed u of even a single RNAP is smaller than u₀ because of transcriptional pauses. RNAPs are taken to occupy L = 50 nt on the DNA template (see SI Text). If the first 50 nucleotides are not occupied, a new transcript is initiated with rate α. We call α the “initiation attempt rate;” the actual initiation rate depends on promoter clearance and is in general <α.

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Fig. 1.

Modeling transcription of rRNA: (A) Schematic structure of an rrn operon with the genes encoding the 16S, 23S, and 5S rRNAs (tRNA genes that are present in some rrn operons are not shown) showing the loading sequence boxA of the AT complex (black ovals), the approximate positions of rho-dependent terminators as inferred from studies of mutants defective in AT (T) (see SI Text), and the promoter pair P1–P2 (P). (B) Model for the traffic of RNA polymerases. Active RNAPs (dark gray) elongate an RNA transcript by making single-nucleotide forward steps along the DNA template. These elongation steps occur with rate ε, provided the site in front of the RNAP is not occupied by another RNAP. In addition, RNAPs can switch to an inactive or paused state (white). The transition rate to the paused state is given by the pause frequency f, and a paused RNAP returns to the active state with rate 1/τ, where τ is the average pause duration. Paused RNAPs are terminated with rate k_t mimicking Rho-dependent termination. At the promoter (represented by the first site to the left) transcripts are initiated with the initiation attempt rate α if the access of the initiating RNAP to the promoter is not blocked by another RNAP bound there.

In the following, we report results from analytical and numerical studies of the above model. We will describe the behavior of this system step by step, first just looking at the effect of RNAP–RNAP interaction, then including the effects of pausing, backtracking, pause suppression by the AT complex, incomplete AT, and finally, termination by Rho. For each case, we used model parameters given in Table 1, which are estimated either directly or indirectly from experimental results as described in SI Text. We monitored a number of observable transcriptional characteristics, including the average rate of transcription and elongation and the average density of RNAPs on the transcribed genes and explored the dependence of these results on key model parameters.

Dense RNAP Traffic Without Pauses.

We first considered the case of active transcription only, without pausing or termination. When transcription is infrequent, the average rate of transcription J, i.e., the number of complete transcripts synthesized per minute by 1 operon, is just given by the initiation attempt rate α and is independent of the elongation attempt rate ε. As the transcription frequency increases, the RNAP–RNAP interaction (i.e., mutual exclusion) must be taken into account. Naïvely, one might expect the transcription rate J to be given by α until a critical value α_c = ε s/L set by the clearance rate of the promoter of size L, with J taking on the maximal value J_max = α_c for α > α_c where the RNAPs are closely packed (thin solid lines in Fig. 2A). We refer to the regions at small α as the “initiation-limited regime,” and the one at large α as the “elongation-limited regime,” because they depend only on the initiation and elongation attempt rate, respectively.

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Fig. 2.

Effect of pauses during transcription. (A–C) Transcription rate J (A), RNAP density ρ (B), and elongation velocity u (C) as functions of the initiation attempt rate α. Thin solid lines indicate the naïve (and incorrect) estimate in which RNAPs move synchronously. Thick solid lines (analytical results as given by Eqs. 1–3 in SI Text) and squares (simulations) show results for the case without transcriptional pausing. Circles show the case including pauses with pause parameters from Table 1. (D) The maximal transcription rate J_max, which is attained for large initiation attempt rates α, as a function of the pause parameters τ and f. The arrow indicates the parameters used in A–C.

However, because of the stochastic, asynchronous nature of RNAP translocation, neighboring RNAPs affect each other's motion already at densities much below the closely packed limit. This interaction leads to a reduction in the density ρ, the average elongation speed u, and the transcription rate J compared with the naïve expectation. For the case without pauses, exact analytical results of our model are known (35–37); see SI Text. These results are plotted as thick solid lines in Fig. 2 A–C and have been verified by numerical simulations (symbols). Data over a larger range of α are shown in Fig. S1. Although the average transcription rate J is indistinguishable from the naïve expectation (thin solid line) at very low initiation attempt rate α, it becomes noticeably reduced as α increases. There still exist 2 distinct regimes of α separated by a critical value α_c (marked by the filled triangle in Fig. S1), with J independent of α in the elongation-dominated regime, α > α_c.^† In this regime, the maximal RNAP density is reduced to ≈87% of maximal coverage (compare the thick and thin solid lines in Fig. 2B). The average elongation speed, u(α) = J·L/ρ, is also diminished for increasing α (Fig. 2C), but only by a small amount (≈12% for the minimal speed u_min) compared with the maximal elongation speed u₀.

Simulations of our model also allow us to take “snapshots” of the spatial distribution of RNAPs on the operons, similar to the images of transcribed genes taken by electron microscopy (6, 7, 25). From these spatial distributions, we determined histograms of the distance between adjacent RNAPs (Fig. S2). In saturated (elongation-limited) traffic (Fig. S2A) the distribution is very narrow, with a strong peak at the minimal possible distance L between RNAPs and a small fraction of longer distances. In dilute traffic (Fig. S2B), the distribution is broader, and the maximum of the distribution is at a distance >L.

Effect of Transcriptional Pausing.

We next investigated the effect of transcriptional pausing, which is modeled as a stochastic process that occurs at a rate f and persists for a period τ, on average; see Fig. 1B. According to single-molecule experiments, there are 2 main types of pauses, the “ubiquitous” or “elemental” short pauses and prolonged backtracking pauses (18, 39). In our model, we assume the duration of the short pauses (ubiquitous or elemental pauses) to be independent of the density of RNAP traffic. This assumption is based on the observation that short pauses are unaffected by mechanical force exerted on the RNAP (18, 29), so that a paused RNAP cannot be pushed back into the active state by a trailing RNAP. In contrast, the duration of the less-frequent backtracking pauses, which we study in an extended version of our model in SI Text, are found to be significantly shortened in dense traffic (Fig. S3), in agreement with in vitro and in vivo experiments (11). Because the inclusion of backtracking hardly affects the results in the dense traffic regime (Fig. S3), we describe below results for the short pauses only. We also neglect the weak sequence dependence of the short pauses shown by recent single-molecule experiments (17), because it has almost no effect on the results presented here (Fig. S4).

Qualitatively, the results we obtained from simulations including pausing (dashed lines in Fig. 2 A–C and Fig. S1) are similar to the case without pausing (thick solid lines). In particular, the 2 different regimes can still be distinguished in the presence of pauses. In the initiation-limited regime (α < α_c), the transcription rate J is hardly affected by the pauses at all (Fig. 2A). The average elongation speed u is gradually reduced because of pausing (Fig. 2C), and the average density ρ is slightly increased (Fig. 2B), reflecting the fact that slower RNAPs remain on the operon for a longer time.

On the other hand, the results for the elongation-limited regime (α > α_c) depend strongly on the presence of pauses: Pauses with the parameters given in Table 1 reduced the maximal transcription rate J_max from 92 min¹ to 43 min¹ (plateau of the thick lines in Fig. 2A, the dependence of J_max on the pause parameters is shown in Fig. 2D). Similarly, the values ρ_max and u_min are substantially reduced (Fig. 2 B and C). In particular, the reduction of the elongation speed is much stronger in dense traffic than in the dilute traffic (40% and 12%, respectively, compared with the case without pauses, see arrows in Fig. S2C). This result can be attributed to traffic jams that build up behind paused RNAPs and amplify the effect of pausing (Fig. S5). These traffic jams are also reflected in the histograms of RNAP–RNAP distances (Fig. S2 C and D), where we observe a strong peak at the minimal distance both in saturated and dilute RNAP traffic.

Effect of Ribosomal AT.

As mentioned above, ribosomal RNA operons are transcribed with an elongation speed that is increased approximately 2-fold compared with protein-encoding genes because of the association of the AT complex with the RNAPs transcribing rRNA (10, 40, 41). This increase in speed is generally attributed to the suppression of pauses (see SI Text for a discussion). We used our model to see whether the increased elongation speed may be quantitatively explained by only a reduction in the pause duration and what effect this change has on the transcription rate. Our study is predicated on experiments carried out to examine transcriptional characteristics with and without AT in vivo. For example, in the experiments of ref. 10, the elongation speed u of an rrn operon was found to be 79 nt/s. This value decreased to 52 nt/s, similar to the elongation speed of mRNA, upon deletion of the AT sequence boxA. Similar reductions have been observed for various Nus factor mutations (40, 41).

We determined the transcriptional characteristics in the presence and absence of AT, using the measured elongation speeds as sole constraints; see the results as bold entries in Table 2. To do this, we estimated the values of the pause frequency f and the elongation attempt rate ε to be f = 0.1 s¹ and ε = 100 s¹ and took transcription to be in the elongation-limited regime (see SI Text and Fig. S6). We further assumed that the pause frequency is unchanged with and without AT as suggested by the related N-dependent AT system (42). We first obtained the value of the pause duration τ with and without AT by demanding the corresponding elongation speeds to match those measured in ref. 10, i.e., u_min = 79 nt/s with AT and u_min = 52 nt/s without AT. The pause duration is found to be τ = 1.17 s in the absence of AT, and 0.23 s in the presence of AT. The result of 1.17 s is in good agreement with the pause duration of ≈1 s observed in single-molecule experiments (15, 17, 18). Also, the predicted reduction in pause duration to 0.23 s (≈5-fold) agrees well with that of the related N-dependent AT system (≈5-fold) as estimated from the data of ref. 42. We further used our model to determine the expected maximal value of the transcription rate, J_max, both in the presence and absence of AT. Our simulations predict that AT increases the maximal achievable transcription rate from 38 min¹ to 76 min¹ (≈2-fold), see Table 2. With this increase, the transcription rate is boosted to the range needed to sustain the fastest growth [68 min¹ for 20-min doubling time (3)].

Effect of Partial AT.

So far, we have assumed ideal AT assembly and efficiency, i.e., in the presence of the box A sequence each RNAP is antiterminated and exhibits reduced pauses. It has, however, been shown that even in the nominal presence of AT, a fraction of RNAPs is stopped by Rho (30). Experiments with multiple termination sites in sequence further showed that essentially each RNAP that reads through 1 termination site also reads through additional termination further downstream (30). This observation suggests that AT is very effective and rather persistent once an RNAP has become antiterminated but also that not all RNAPs are antiterminated. Partial AT may have a detrimental effect on transcription because RNAPs that are not antiterminated will make longer pauses and may cause jamming even if the majority of RNAPs are antiterminated.

To model the possible consequences of partial AT on rrn transcription, we performed simulations for a mixture of RNAPs with and without the AT complex. This is implemented by randomly assigning to each RNAP a pause duration of either 0.2 s (with probability 1 − p) or 1 s (with probability p) at the initiation of transcription. Here, p indicates the fraction of nonantiterminated RNAPs, and the pause-duration values are chosen according to those in Table 2; they remain unchanged throughout the duration of transcription for each RNAP. In Fig. 3A, we show the transcription rate J_max obtained from this model for different fractions p of nonantiterminated RNAPs (filled circles). The transcription rate exhibits a rather steep decrease for small percentages of nonantiterminated RNAPs. E.g., if 10% of the RNAPs are not antiterminated, then the maximal transcription rate is predicted to decrease from 81 min¹ to 69 min¹, a ≈30% loss of the potential increase in transcription rate due to AT.

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Fig. 3.

Effect of partial AT. (A) The maximal transcription rate J_max as obtained from simulations with 2 populations of RNAPs with different pause durations τ = 0.2 s and τ = 1 s, which represent antiterminated and nonantiterminated RNAPs, respectively (filled symbols). The effect of AT is reduced by the presence of nonantiterminated RNAPs. Termination of the nonantiterminated RNAPs (with a termination efficiency of 80% during pauses, open symbols) can compensate this decrease for small fractions of nonantiterminated RNAPs. (B and C) Histograms of RNAP-RNAP distances for perfect (B) and defective (C) AT. Distances between adjacent RNAPs were determined from snapshots of the spatial distribution of RNAPs on simulated rrn operons as shown in Fig. 1A. Black and white bars show distributions for the 16S and 23S genes, respectively. For perfect AT, the distributions are almost the same; for defective AT, the distribution for the 23S gene is much broader than that for the 16S gene, similar to what has been observed for a NusB mutant (figure 3 in ref. 25).

Effect of Rho-Dependent Termination.

Because the decrease of the transcription rate (filled circles) shown in Fig. 3A is caused by the much slower, nonantiterminated RNAPs, we tested the effect of Rho-dependent termination in this context. We performed simulations by incorporating into our model features mimicking forced termination. We only allowed removal of paused nonantiterminated RNAPs.^‡ In one implementation (the “uniform termination model”), we assigned a rate k_t at which any paused nonantiterminated RNAP is removed from the system. In an alternative implementation (the “localized termination model”), we allowed termination (with the same rate k_t) only to occur within the first 1,000 nt of the operon, where RNAPs, on average, pause for the first time, to mimic a termination site located in the leader region of the transcript (Fig. 1A).

The simulation results obtained for the 2 models are shown as open symbols in Fig. 3A, for a choice of k_t that corresponds to a termination efficiency of 80%, as appropriate for a weak Rho-dependent termination site (30). Similar results are obtained for stronger termination (data not shown). These data show that forced termination increases the maximal transcription rate in both models if the fraction p of nonantiterminated RNAPs is not too large. For the uniform termination model (open circles in Fig. 3A), we observe an increase of the transcription rate for p below ≈40% (where the curves with the filled and open circles cross each other). For higher fractions of nonantiterminated RNAPs (and thus also in the complete absence of AT, p = 100%), the transcription rate is, however, found to be strongly reduced, as one would usually expect in the presence of strong indiscriminant termination. For the localized termination model (open squares in Fig. 3A), the transcription rate is increased for up to 90% of nonantiterminated RNAPs. Thus, forced termination near the start of transcription initiation can have a substantially more beneficial effect in speeding up transcription. We attribute this effect to the fact that removing nonantiterminated RNAPs soon after initiation not only reduces the number of traffic obstacles but that these nonantiterminated RNAPs are also quite likely to be replaced by antiterminated RNAPs, whereas forced termination downstream of the transcription start has the adverse effect of reducing the density of the transcribing RNAP. For both models, the transcription rate can be maintained close to the level of perfect AT with as much as 10% RNAPs without AT. This conclusion remains valid if we consider a more realistic rrn operon (Fig. 1A) where the AT complex is reloaded in the spacer region between the 16S and 23S genes with an additional termination site immediately downstream (data not shown).

Transcription of rRNA in Wild Type and AT Mutants.

Finally we extended our model to the structure of E. coli rrn operons as shown in Fig. 1A, where the AT complex is assembled in the leader region of the transcript, followed by a termination site, and reassembled in the spacer region between the 16S and 23S genes, again followed by a termination site (see Methods in SI Text for the modeling details). We used this full version of our model to study the effect of AT mutants. Quan et al. (25) have recently characterized viable NusA and NusB mutants. They showed that these strains exhibit polarity between the 16S and 23S genes and quantified this observation using electron microscopy of the rrn operons. To see whether our model can account for these observations, we simulated transcription of rRNA with the full model. We chose the initiation attempt rate to be α = 120 min¹ so that the RNAP density for perfect AT matches the measured density for the wild type (see Table S1, rows 1 and 2). For this value of α, RNAP traffic is dense, ≈73% saturated (data not shown). The transcription rate predicted by our model is 58 min¹ (Table S1, row 2). The transcription rate was not directly measured by Quan et al. (25). However, by using the known relation between growth rate and rrn transcription for wild-type E. coli cells (2, 3), we can predict a corresponding doubling time of ≈24 min, which agrees with the one observed (25). We then varied the degree of AT to mimic defective AT due to NusB or NusA mutations and determined the numbers of RNAPs on the 16S and 23S genes (Table S1), as well as histograms of RNAP–RNAP distances (Fig. 3 B and C). We obtain quantitative agreement with the experimental values for the number of RNAPs per gene, if we assume 60% of the RNAPs to be not antiterminated in the Nus mutants (Table S1, rows 3–5). We also determined the corresponding transcription rate (of full length transcripts) and estimated the corresponding doubling time to be ≈36–38 min for these mutants (Table S1, row 5). This estimate agrees well with the measured doubling time for the NusA mutant (38 min). The NusB mutant exhibits rrn transcription very similar to the NusA mutant but has a substantially larger doubling time of 58 min, possibly indicating an additional defect of this mutant beyond ribosomal AT.

For the wild-type cells and the NusB mutant, Quan et al. (25) also determined histograms of the RNAP–RNAP distances measured in electron microscopy images of the rrn operons. The corresponding histograms obtained from our model are shown in Fig. 3 B and C. For perfect AT (Fig. 3B), we find that the histograms are almost the same for the 16S and 23S genes, with a strong peak at distances of ≈90 nt and a very small fraction of RNAP–RNAP distances >250 nt, in good agreement with the histograms of ref. 25 for wild-type cells (Table S1). For defective AT with only 40% of the RNAPs antiterminated (Fig. 3C), we find that the distribution for the 23S gene is much broader than the one for the 16S gene and that the distributions for both genes are broadened compared with the case of perfect AT with an increased fraction of large distances (Table S1), in good agreement with the results from ref. 25 for the NusB mutant.