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Supporting Information Supporting Information Klumpp and Hwa 10.1073/pnas.0806084105 SI Text similar effects. In particular, sequence-dependent stepping rates Methods. The model. The transcription by RNAPs in dense traffic have been investigated extensively and also lead to the build-up is modeled by a stochastic cellular automaton model (1), in which of traffic jams behind a slow site (8). We note however that in the RNAPs are represented as extended objects of size L such a model, all particles move slowly at a site with a low confined to move along a 1-dimensional lattice; sites of this stepping rate, while in our model only a fraction of RNAPs lattice represent the individual bases of a DNA template. Each pauses at each pause site (see below for a description of our RNAP can be in either the active, paused, or backtracked state simulations of sequence-dependent pauses). Premature termi- independently of the state of other RNAPs. An active RNAP nation as included in our model is related to unbinding of transcribes RNA by making a single nucleotide step forward at particles in models used to describe the traffic of cytoskeletal a stepping or elongation attempt rate ␧, provided that the access motors (9, 10); however, in these models unbinding typically to the next base is not blocked by the presence of another RNAP. occurs together with binding of particles to the track, which does During transcription, the active RNAP may switch stochastically not apply to RNAPs that initiate transcription from the pro- to the paused state with a rate f (see Parameter estimation below moter rather than from random positions along the DNA for a discussion of the stochasticity and sequence-dependence of template. pauses). A paused RNAP remains at the same site. It can switch Model simulations and analytical results. The model is simulated by Ϫ3 back to the active state with rate 1/␶, so that the average duration using discrete Monte Carlo steps that correspond to 5 ϫ 10 s. of a pause is ␶. In an extended version of our model, we also Each step consists of N moves, where N is given by the length of included backtracking of paused RNAPs, which has, however, the lattice or the number of nucleotides in the operon, for which Ϸ little effect on transcription in dense traffic (see Effect of we used the length of the rrnC operon, N 5,450 (4). At each backtracking below). Finally, for simulations of rho-dependent of these moves, a site is chosen randomly and updated according termination, termination is implemented by removing nonanti- to the described rates (11). All steps that would move an RNAP terminated, paused RNAPs with rate kt. This implementation to a site occupied by another RNAP are rejected. Simulations ϫ 6 mimics the effect that Rho can only displace the nonantitermi- were run for 5 10 steps, and data for steady-state average were ϫ 5 nated RNAPs when it catches up with those RNAPs in the taken starting after 5 10 steps. paused state (2). Localized termination sites were implemented In the limit where pauses are absent, our model reduces to the by allowing this termination process only within a template well-studied asymmetric simple exclusion process with extended stretch of 1,000 nt, so that RNAPs pause there, on average, once. particles for which a number of analytical results are known In simulations with partial AT, each initiating RNAP was (5–7), shown as thick solid lines in Fig. 2 A–C and Fig. S1. The randomly chosen to be either antiterminated (with probability transcription rate corresponds to the current in that model and 1 Ϫ p) or not antiterminated (with probability p). In the is given by simulations that included reloading of the AT complex in the J͑␣͒ ϭ ␣͑␧Ϫ␣͒/͓␧ϩ␣͑L Ϫ 1͔͒ and J ϭ␧/͑1 ϩ L1/2͒2, spacer region, this assignment was renewed with the same max probabilities at site 2000. In this case, termination was allowed [1] at the first 1,000 sites and between sites 2000 and 3000 to mimic the 2 termination sites after the leader and spacer boxA se- in the initiation-limited and in the elongation-limited situation, quences, respectively (indicated by T in Fig. 1A and discussed in respectively. These 2 regimes are separated by a continuous 1/2 Location of Rho-dependent termination sites within the rrn operons phase transition that occurs for ␣c ϭ ␧/(1 ϩ L ). The RNAP below). density is given by The initiation of transcription is described by the initiation ␳͑␣͒ ϭ ␣ ͓␧ϩ␣͑ Ϫ ͔͒ ␳ ϭ ͓ ϩ Ϫ1/2͔ attempt rate ␣ with which RNAPs are inserted at the first L ϭ L / L 1 and max 1/ 1 L , [2] 50 sites provided that these sites are not occupied. This rate is respectively. From these 2 known results, we obtain the elonga- taken to summarize all processes of the initiation stage such as tion velocity u using the relation u ϭ Js L/␳, which leads to RNAP binding to the promoter, open complex formation, and initiation of transcript elongation (3). Actively transcribing ͑␣͒ ϭ␧ ͑ Ϫ ␣ ␧͒ ϭ␧ ͑ ϩ Ϫ1/2͒ u s 1 / and umin s/ 1 L [3] RNAPs that reach the end of the operon are terminated, i.e., removed, when making a forward step from the last site. Fast in the initiation-limited regime and at maximal transcription, termination is implied by the absence of RNAP jamming at the respectively. end of the operon in a recent electron microscopy study of ribosomal RNA operons (4). We estimated the values for all Parameter Estimation. In this section, we describe in detail how we model parameters from in vitro and in vivo data (see Parameter estimated values for the parameters of our model using in vivo estimation below). The estimated values are summarized in and in vitro data from the literature. The parameter values are Table 1. summarized in Table 1. Our model is a variant of stochastic cellular automata or Pauses. The frequency f and duration ␶ of pauses have been driven diffusive systems and incorporates several features that directly measured in vitro in single-molecule experiments (12– have been extensively studied in the nonequilibrium statistical 16) Experiments with high resolution show that RNAPs exhibit physics literature. In particular, models with extended particles, both short and prolonged pauses (discussed below). The majority i.e., particles that occupy more than a single lattice site, have of pauses are short (12, 13). These pauses occur with a frequency been studied in detail, mostly motivated by the traffic of ribo- of 0.07–0.15 sϪ1 and have durations of Ϸ1 s (12–14). In vivo somes on mRNA (5–7), and we make use of some known values for these parameters have not been measured. However, analytical results for the case without pausing (see below). we expect these in vitro values also to be representative for the Pauses of the type observed for RNAPs have not been studied situation in vivo, despite the fact that transcription in vivo within these models, but some well-studied models exhibit quite proceeds faster than under the conditions of the single-molecule Klumpp and Hwa www.pnas.org/cgi/content/short/0806084105 1of13 experiments. The rationale for this assumption is given by the these systems. Our analysis of the effect of ribosomal AT also following 2 arguments: (i) The pause frequency has been ob- implies an upper bound on the stepping rate (described in served to be constant over the range of elongation rates acces- Constraints on the stepping rate ␧ below) at Ϸ110–150 sϪ1.We sible in vitro (14), so it may be extrapolated to the higher therefore expect the stepping rate in vivo to be Ϸ100 sϪ1. The elongation rate in vivo.* (ii) Assuming a stepping rate of 100 sϪ1 latter value has been used in all simulations described in the main (see below), the in vitro pause parameters are consistent with the text, but different values in the estimate range lead to similar in vivo elongation speeds of mRNA (17–19) and rRNA in the predictions for the effect of ribosomal AT (see Constraints on the absence of antitermination (18). In fact, when we determined the stepping rate ␧ below and Fig. S6). pause duration from the elongation speeed of rRNA without Size of the RNAP footprint. The number of nucleotides L occupied antitermination, we obtained values for the pause duration ␶ by a single RNAP is estimated from DNase footprinting exper- between1and2swithin the expected range of stepping rates, iments (23–27). These experiments exhibit footprints of 77 and see Table 2 and Constraints on the stepping rate ␧ below. 50 nt for promoter-bound RNAPs in open and closed complexes, In single-molecule experiments, pauses occur in a stochastic respectively. These footprints reach from positions Ϫ57 to ϩ20 fashion. The frequency of pauses exhibits rather small variation for the open complex (23, 24) and from Ϫ55 to Ϫ5 for the closed along the DNA template when measured with Ϸ100-nt resolu- complex (25, 26). An elongating RNAP has a footprint of 32 nt tion (13), and both the distance and time between subsequent (23). The footprint of an elongating RNAP that is stalled at pauses follow single-exponential distributions (12). Recent ex- position ϩ32 (i.e., after synthesizing a transcript of 32 nt) reaches periments with single-base resolution revealed a weak sequence to ϩ45; in this situation a second RNAP can bind to the dependence of the short pauses (14): Pauses occur preferentially promoter (27).
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