The Standard Promoter Model Does Not Produce Bursting

Golding et al. showed that their results were not consistent with transcription initiation being a single Poisson process. By considering the McClure model of transcription initiation (Figure 1A) we show that initiation as a single Poisson process is a special case where only one step is rate limiting, and that while the more general case is not a single Poisson process it is still unable to fit the results of Golding et al.

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Figure 1. The Hawley-McClure 3-step model of transcription initiation.

(A) First an RNAP forms a closed complex at the promoter with some on (k_b) and off rates (k_u), and subsequently forms an open complex with rate O. This process is a directed non-equilibrium transition. Finally, the open complex escapes the promoter into productive transcription with the one way rate E. (B–C) The distribution of intervals between transcription events for the standard 3-step model. For binding and unbinding rates of the closed complex we use k_b = k_u = 60 [1/min]. The average total strength is F = 1/(20 [min]), whereas l = 35 [bp] and v = 25 [bp/sec]. (B) Class I: Probability distribution for time between transcripts Δt for a promoter with a single rate limiting step. Here, it is isomerisation with O = 1/(9.7 [min]). The escape rate from the open complex is E = 1/(0.19[min]). Red dots: stochastic simulation results. Solid line: predicted distribution, (1/τ)exp[−Δt/τ] with τ = 19.4 [min]. (C) Class II; Δt probability distribution for a promoter with two rate limiting steps because the isomerisation and escape rates are similar (O = 1/(4.9[min]), E = 1/(9.7 [min])). Solid line shows the predicted distribution, (1/τ)² Δt exp[−Δt/τ] with τ = 9.7 [min].

https://doi.org/10.1371/journal.pcbi.1000109.g001

In prokaryotes, the initiation of transcription requires the binding of an RNAP to the promoter, the isomerisation of the RNAP through several intermediate forms, rounds of abortive initiation and then finally release from the promoter. Here we consider the McClure model of transcription [11]–[13] (Figure 1A), where transcription initiation requires three steps: RNA polymerase (RNAP) binding to the promoter to form a closed complex, followed by isomerisation of the closed complex to an open complex in which the DNA at the promoter is melted, and the escape of the open complex to form an RNAP complex engaged in elongation of the transcript. The closed complex is assumed to be in rapid equilibrium with free RNAP, while isomerisation and escape are treated as being slower and irreversible. This model is a simplified but useful version of the full kinetics of initiation.

The kinetics of each elementary reaction in initiation determines the final distribution of transcription initiation. Transcription is often treated as a Poisson process, i.e. the probability of initiation at a given moment is a constant, which results in an exponential distribution of times between transcripts. Golding et al. were able to show through several methods that the distribution of transcription initiation was non-Poisson. However, the exponential distribution is a special case where there is only one rate limiting step in the initiation of transcription.

For the analytical analysis of the McClure model, we make the assumption that the rates of binding k_b and unbinding k_u of the closed complex are relatively fast, and therefore that there are only two kinetically significant steps, isomerisation of the closed complex to an open complex, and promoter escape by the open complex. We assume that each step is elementary, i.e. that it can be approximated as a single chemical reaction. We also ignore the effect of self-occlusion, where an RNAP prevents further initiation at the promoter until it has transcribed far enough to no longer occlude the promoter (50 bp), as the time needed to transcribe this distance (1–4 seconds) is negligible compared to the time between initiations in the Golding et al. experiments. The average time needed to complete the first step, t_o, is therefore t_o = (1+K)/O, where K = k_u/k_b is the equilibrium constant of dissociation for the closed complex and O is the rate of transition from closed to open complex. The inverse of the rate of the open to elongating transition (E) gives the average time needed for the second kinetically important step, t_E (Figure 1A). The average time taken for initiation (and therefore the time gap between initiations, 〈Δt〉, with 〈…〉 indicating the average) is the sum of two exponentially distributed random variables, 〈Δt〉 = 〈τ_O+τ_E〉. The probability distribution of time gaps between initiations is given by(1)for t_O≠t_E. For t_O = t_E = t, we get(2)

In the case where one step is much slower than the other (Class I), there is only one rate-limiting step in initiation and the distribution of Dt approaches a single exponential with mean t_L = max(t_O,t_E) (Equation 1; Figure 1B), i.e. it approaches a single Poisson process. Here, the data points in Figure 1B) have been obtained by simulating the model of the promoter in Figure 1A) using the Gillespie algorithm [14], which stochastically determines the next reaction to occur and the time interval between reactions based on the given rates. The other extreme, where t_O = t_E (Class II), is shown in Figure 1C. In Class II, the chance of rapid successive firings faster than the average (Dt<<t_O+t_E) is smaller than for a Class I promoter, as for a Class II promoter a low Dt requires both the isomerisation and the escape to productive transcription to occur in rapid succession, whereas for a Class I promoter a low Dt requires the rapid occurrence of the rate limiting step only. As a consequence the distribution in Class II shows a peak at non-zero Dt. Promoter models that specify more kinetically significant reaction intermediates produce more extreme versions of the Class II distribution, with a larger peak centered around 〈Δt〉, resulting in more regular firing intervals.

The Class I type promoter shows the most fluctuation in Dt, and the effect of adding more kinetically significant intermediate steps is to reduce the amount of variability in Dt. Therefore neither the standard model nor models that take into account more intermediates can reproduce the bunched activity observed by Golding et al. [10], which show greater fluctuations in Dt than a Poisson process. In order to reproduce the bunched activity, it is necessary to consider a model with a branched pathway, where the system can go into either an active state or an inactive state with a switching mechanism between them.

Previously Proposed Mechanisms for Bunched Activity

Here we consider several hypotheses for the mechanism of transcriptional bursting and argue that they are unlikely to be correct. The promoter used by Golding et al. [10], P_lac/ara, can be repressed about 70-fold by the lac repressor and activated about 30 fold by AraC [15]. Therefore, a simple hypothesis put forward by Golding et al. is that the silent periods are periods where the lac repressor is bound to the promoter, and the bursts are periods of activity when the promoter is free. However, the mean duration of off-periods is 37 min while on periods are only 6 min in duration, despite the fact that the promoter has been fully induced by 1 mM IPTG. It seems impossible for the lac repressor to remain bound to the DNA for 37 minutes under these conditions; especially considering that 1 mM IPTG derepresses the lac promoter in less than 5 sec [16].

A similar idea is that the off-periods represent periods where AraC is not bound to the promoter [10]. To make this feasible the on rate for AraC in an E.coli cell would have to be exceedingly small given the large off periods. This is unreasonable in view of the high association rate for AraC to other operators [17]. Presumably association rate is diffusion limited, meaning that it would take one AraC molecule less than a minute to bind to the operator [18]. In conclusion we find it unlikely that binding AraC is sufficient to produce bunched activity.

Another hypothesis put forward by Golding et al. is that RNAP might be able to re-initiate after termination, aided by the retention of sigma factor during transcription [19]. Presumably the RNAP would have to be positioned to rebind to the same promoter after termination for re-initiation to occur with any reliability, and it is not clear how this would be caused. One possibility is that a transcription factor might remain in contact with both the RNAP and the promoter via a DNA loop. This would render the promoter unavailable during transcription, which has some support from the data in that the lengths of the observed on-periods were approximately equal to the number of initiations multiplied by the time taken to transcribe the reporter mRNA for both P_lac/ara and P_RM (Golding, private communication), which would be expected if transcription does not occur simultaneously. However, this data is somewhat anecdotal, and stands in contradiction to the simultaneous transcription observed with electron microscopy [20]. Also, this mechanism requires binding of a closed complex to the DNA to be the rate limiting step that causes the 37 minute long off-period, and we consider it unlikely that simple recognition of the promoter by RNAP would take this long, especially given that closed complex formation is often thought to be a rapid equilibrium process.

Multiple RNAP can cooperate to overcome pause sites [21]. It might therefore be possible that the burst is due to multiple RNAP building up at a pause site and overcoming it together. However, this would require the RNAP to pause for a length of time on the same scale as the off-period; such an extreme pause is unlikely given that even the strongest pauses measured in vitro only last for around one minute.

Bursting could also result if there were distinct regions of high and low transcriptional activity within bacteria, akin to the idea of transcription factories in eukaryotes, and the promoter moved in and out of these regions on a slow time scale [22],[23]. Although this is an interesting possibility, not enough is known to evaluate such a mechanism in bacteria in much detail.

Fluctuations in the availability of free RNAP within the cell could contribute to variable initiation rates but it is difficult to see how such severe and long-lasting fluctuations capable of producing extended periods of complete inactivity could occur in cells where ∼3000 RNAPs [24] produce >10⁵ RNAs per generation.

Supercoiling-Mediated Recruitment

There is both theoretical [25] and experimental evidence [26],[27] that an elongating RNAP can increase the negative supercoiling of the DNA behind it.

Promoters can be very sensitive to supercoiling; for example, in vitro the activity of the LacP promoter increases by more than a factor of 10 when the super-coilings is changed from zero to −0.065 (which is the average supercoiling of DNA in E. coli) [26]. We therefore consider it a possibility that the bursts of transcription might be caused by a transcribing RNAP assisting the recruitment of further RNAP via the wake of supercoiling left behind it. In principle one could argue that perturbed supercoil states could relax quickly in a plasmid [25] like the one used by Golding et al., but it has been demonstrated that a promoter can induce huge changes in supercoiling of a plasmid [28].

Consider a promoter where open complex formation is a rate limiting step that is assisted by negative supercoiling. To model this, we assume that the negative supercoiling assists this step to the extent that it is no longer rate limiting. We parameterize this effect of supercoiling into a single number q, the probability that supercoiling left in the wake of a prior RNAP allows a subsequent RNAP to rapidly form an open complex before the supercoiling is relaxed (Figure 2A). This then creates two possible behaviors at the promoter. If the promoter is in the supercoiled state, open complex formation is enhanced to the point where it is not rate limiting, and transcription events occur at rate E and are exponentially distributed. If the promoter is not in the supercoiled state, then open complex formation is very much slower and now rate limiting; transcriptional events are still exponentially distributed but now with the much lower rate O. This creates the long periods of inactivity associated with off periods (Figure 3A) and holds when O≪E, and gives a distribution(3)(shown in Figure 3B).

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Figure 2. Revisiting mechanisms for bunched firing.

We propose (A) supercoiling assisted open complex formation, and (B) possible stalling of an RNAP into a “moribund” complex. The yellow background indicates the state where most of the time is spent in the off period. (A) An elongating RNAP might recruit a subsequent RNAP into an open complex with probability q, thus by-passing the time needed to recruit an RNAP and form open complex. In the limit of large k_b, the firing rate is given by with . (B) Two alternative open complexes, of which one is productive and the other is a dead end complex that is removed with rate d. In this case Q denotes the probability that a closed complex enters into the productive open complex. In the limit of large k_b, the firing rate is given by with . The detailed calculations and equations for limiting k_b are given in Text S1. Both the dead-end and the recruitment model can be simulated on-line using the java applet on http://www.cmol.nbi.dk/models/transcription/RNAPInitiation.html.

https://doi.org/10.1371/journal.pcbi.1000109.g002

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Figure 3. Transcriptional bursting from three step model with supercoiling assisted recruitment and the dead-end complex model.

(A–D) show the results from the recruitment model with the probability of recruitment q = 0.545. The average firing rate is 1/20 [1/min]. k_b = k_u = 60 [1/min], E = 1/2.9 [1/min], O = 1/18.3 [1/min], l = 35 [bp], v = 25 [bp/sec]. (A) Accumulated number of mRNAs, showing on periods and off-periods. Transcriptional bursting can be seen around 62 [min] to 70 [min], where 5 firings occur in rapid succession. See Materials and Methods for the choice of parameters and definitions of t_on and t_off. (B) The distribution of the durations between firing, P(Δt). The solid line shows Equation 3. (C) The distribution of the number of firings per on-period, P(Δn). The filled circles show experimental data from Golding et al [10], and the open circles are Δn from the recruitment model. (D) The distribution of the “on-times” t_on (open circles) and the “off-times” (open boxes) t_off. The experimental data from Golding et al. for the on-time (filled circles) and off-time (filled boxes) are also shown. (E) The distribution of intervals between initiations for the dead-end complex model (Figure 2B). The probability of productive elongation is Q = 0.545, and the rate of removal of dead-end complexes is d = 1/τ_dead = 1/(20 [min]). The average firing rate is 1/22 [1/min]. k_b = k_u = 60[1/min], E = 1/1.5[1/min], O = 1/0.7 [1/min], l = 35 [bp], v = 25 [bp/sec]. This parameter choice corresponds to a Class 2 promoter in the on periods, which gives a round curve for short timescales (around 3 [min]). The solid line shows QΔt exp[−Δt/τ]/τ²+(1−Q) exp[−Δt/(τ_dead/Q)]/(τ_dead/Q) with τ = τ_O = τ_E.

https://doi.org/10.1371/journal.pcbi.1000109.g003

The supercoiling need not persist for the full length of the on-period, or for the length of time between two initiations. In the scheme we present here, it is only required that the supercoiling persists long enough to allow an open complex to form rapidly. The final escape step is assumed to be neutral with respect to supercoiling and hence as soon as an open complex has formed at the promoter the supercoiling can be relaxed without interrupting the on-period. This assumption can be varied without changing the general behavior of the model.

If the supercoiling is relaxed before an open complex is formed, the promoter has switched to an off-period where initiation occurs at a much slower rate. The parameter q determines the size of the on-periods, as after each initiation there is a probability q that another open complex will be recruited and the on-period will continue, or a probability 1-q that an off-period will start. Therefore, the probability of getting a burst of 〈Δn〉 initiations is proportional to q^Dn−1. In this model a promoter is in the on-state when it is in the supercoiled state or when it has an open complex. Table 1 gives equations relating model parameters to the average 〈Δn〉, 〈t_on〉, and 〈t_off〉 (Derivations are given in Text S1).

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Table 1. Relations between model parameters and the average 〈Δn〉, 〈t_on〉, and 〈t_off〉.

https://doi.org/10.1371/journal.pcbi.1000109.t001

This mechanism can reproduce the observations of Golding et al. [10] with the parameters t_O = 37 [min], t_E = 29 [min] and q = 0.545. We simulated the recruitment model using the Gillespie algorithm [14]. It gives the expected shape for the P(Dt) distribution (Figure 3B) and matches the distribution of Dn measured by Golding et al. (3C) and also the distributions of on and off-periods measured by Golding et al. (3D). In these plots the on-periods are defined as being the time intervals when there is rapid successive initiation (Figure 3A), following the procedure in Golding et al. [10]; the detailed definition is given in the Materials and Methods section.

Formation of a Dead-End Complex

Another possibility is that the off periods are due to the formation of long-lived non-productive initiation complexes at the promoter [29]–[31]. These non-productive complexes have been observed in vitro and may be arrested backtracked complexes or complexes that cannot exit the abortive initiation state into productive elongation. In both cases initiation can be made more efficient by the GreA/B RNAP-binding factors [29],[30]. The random formation of such ‘dead-end’ complexes could block the promoter for extended periods of time, causing productive transcription to be confined to those times when the promoter is free. For the promoter lPR the lifetime of these complexes was found to be in the order of 10–20 minutes under in-vitro conditions, thus dead-end complexes can last long enough to cause the observed off-periods [31].

For the analytical treatment of this model we call the probability that a promoter bound complex will undergo a productive initiation Q, and the probability that the promoter bound complex enters a moribund state is therefore 1- Q. We assume that removal of the moribund complexes is a Poisson process with a rate d, which gives 〈t_off〉 = τ_dead/Q with t_dead = 1/d, which allows for the fact that a single off-period can be caused by multiple subsequent moribund complexes (Table 1). Here we consider a promoter to be in the off-period if it is occupied by dead-end complexes; otherwise it is on. The derivations of on- and off-times are given in Text S1. The dead-end complex mechanism is also capable of causing the behavior observed by Golding et al. The data of Golding et al. are reproduced with Q = 0.545, t_dead = 20 [min], and t_O+t_E = 2.9 [min]. Figure 3E shows the distribution P(Dt) with these parameters obtained by the simulation using the Gillespie algorithm [14]. It has been confirmed that the distributions of Dn, t_on, and t_off are reproduced as well as the recruitment model (data not shown).

The formation of dead-end complexes is favored by low temperatures at the lac UV5 promoter [32]. If this were also the case for the P_lac/ara promoter, it could be part of the explanation for why the P_lac/ara promoter is so weak in the conditions used by Golding et al. (22°C) when it is reported to be a strong promoter elsewhere [15]. However, the activity of the promoter observed by Golding et al. at 37°C is still rather low compared the previously reported estimate [15]. This could be associated with the fact that there is almost no activation of the promoter caused by AraC/arabinose under their experimental conditions (see Figure 1E in Golding et al.). Another possibility could be the presence of an unknown terminator, which would imply that the number of complete transcripts represents only a fraction of the transcription initiation events.

Control of Transcriptional Noise

One of observations made by Golding et al. that was used as evidence for transcriptional bursting was that the Fano factor for the distribution of number of transcripts N, ν = 〈(N−〈N〉)²〉/〈N〉, was approximately 4 for the P_lac/ara promoter at 37°C, rather than 1 predicted for Poisson transcription. The Fano factor is a measure of noise; higher values indicating a more noisy process. When the on-periods are much shorter than the off-periods, the Fano factor n is linked to the burst size Dn as ν≈〈Δn〉. If the on-time is sizable, on the other hand, 〈Δn〉 needs to be much larger to give the same n.

By considering a population of cells where transcripts are degraded with rate g, we can relate n to model parameters. Figure 4 shows how n varies with model parameters for each model while keeping 〈N〉 = 10 obtained by analytical calculations (The detailed calculations are in the Text S1.). In the recruitment case the Fano factor is larger for smaller a and larger q, i.e., when the open complex formation is the rate limiting step and once a firing has occurred further recruitment occurs successively. In the dead-end model the Fano factor is larger for smaller b = (t_O+t_E)/t_dead and larger Q, which occurs when moribund persist for long periods of time, but transcription during the on periods is rapid and occurs many times before another off period occurs. One should note that the Fano factor can be changed depending on parameters for a given 〈N〉; This means that the noise can be tuned for a given promoter strength under either model, which can allow the promoter noise to evolve to reflect a level that provides the best fitness for the cell.

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Figure 4. Parameter dependence of Fano factor.

(A) Fano factor in the recruitment model for 〈N〉 = F/γ = 10. The horizontal axis shows the aspect ratio α = O′/E, and the vertical axis shows the recruitment probability q. The fluctuations are larger for smaller α, where the formation of the closed complex is the rate-limiting step. (B) Fano factor for the dead-end model, with , for 〈N〉 = 10 and α = 1. The horizontal axis is β = (τ_O+τ_E)/τ_dead, the ratio of the average time required for successful firing to the average time taken to remove a dead-end complex, and the vertical axis is the probability of successful firing, Q. Small β and large Q gives large fluctuations, which enables bust like firing through successive normal firings (from large Q) and long silent periods until the dead-end complex is removed (from small β). The detailed calculations are given in Text S1.

https://doi.org/10.1371/journal.pcbi.1000109.g004

Testing the Recruitment Model

The recruitment model implies a number of predictions that can be tested. In particular, promoters with bunched transcription initiation will be highly sensitive to negative supercoiling of the DNA. And conversely, promoters that are insensitive to supercoiling will have transcription events which are separated by more regular time intervals.

For promoters that are sensitive to supercoiling, one could selectively shorten the long off periods by introducing a second nearby promoter. One option is to add a divergent promoter that might be able to donate its negative supercoil wake. Such a construct was investigated by Opel et al. [27], who reported that a second promoter could indeed increase the activity of a supercoiling sensitive promoter in the ilvYC operon. This predicts that if a similar experiment was done with the P_lac/ara promoter, then reduced off periods would be observed.

Another prediction is that for promoters with bunched activity the isomerisation step is rate limiting. Thus the fraction of time spent in open complex is small compared to the time between transcription initiations. One might be able to show an inverse correlation between the noisiness of a promoter and the occupancy of the promoter by open complexes using potassium permanganate DNA footprinting [35].

Testing the Dead-End Model

The dead-end mechanism implies that the promoter is mostly occluded by an RNAP with an open transcription bubble. This could be identified permanganate footprinting [35].

The availability of GreA/B could affect the rate of removal of the dead-end complex, d [29],[30]. Overexpression of GreA/B could increase d and reduce off-periods, while longer off-periods, due to lower d, could be observed in greA/B mutants.

It is possible that the dead-end complexes could be removed by a collision with an RNAP transcribing from a second promoter in a fashion similar to the removal of an open complex by transcriptional interference [36]. The off-times of a promoter could therefore in principle be shortened by using other RNAP's initiated from another promoter that transcribes across the promoter in question. If a promoter spent a substantial fraction of the time occupied by a dead-end complex, it could be strongly activated by tandem or even convergent promoters, which would be a novel twist on the usually repressive effect of transcriptional interference. If d is reduced in Table 1, the “off-times” could be reduced by a factor set by the ratio of the strength of the two promoters, and the promoter activity could increase. Thus, if P_lac/ara activity is affected by dead-end complex formation, then placing a weak divergent promoter upstream should not increase P_lac/ara activity but placing this promoter in a convergent orientation may activate P_lac/ara.

Perspectives for the Regulation of Transcriptional Noise

The sensitivity of a promoter to supercoiling mediated recruitment or dead-end complex formation provides additional avenues for control of overall promoter strength, either by evolution or by regulatory factors.

DNA supercoiling can increase or decrease promoter activity both in vitro [26] and in vivo [37] in a promoter specific manner. Supercoiling can affect RNAP binding to the promoter and open complex formation in vitro and presumably can affect other steps as well. RNAP recruitment induced by the supercoiling created by an elongating transcription complex may contribute significantly to the activity of certain promoters. We expect that, except for very active promoters, rapid dissipation of the supercoil wake would make inhibition of a supercoiling-repressed promoter by this mechanism unlikely. Stimulation by the departing elongating complex should similarly only apply to the early steps in initiation. Thus only promoters whose early steps are rate-limiting and can be enhanced by supercoiling should be stimulated by this mechanism.

The reduction of promoter activity by the formation of dead-end complexes is potentially very strong. The effect increases with the probability of forming such a complex (1-Q) and with the lifetime of the complex (1/d), parameters which could be determined both by the promoter sequence and by the availability of factors such as GreA/B that may remove the complex [29],[30]. This mechanism would seem to be an inefficient way to set the strength of a promoter, as it would sequester an RNAP. However, it would allow regulation by transcription factors that change the fraction of RNAPs that enter into dead-end complexes or that stabilized the dead-end complex. As a consequence, genes which are silenced through this mechanism will have relatively high fluctuations in expression level, and thereby some cells can explore advantages afforded by relatively high expressions, even when most cells are kept at near zero expression. Bunched activity for a near silenced promoter could, for example, be important in the pathway for the spontaneous induction of lysogeny for some temperate phages, like P2.

High noise in protein levels can also be obtained at the translation level. If a single mRNA molecule is rapidly translated many times the result is a burst of protein production. Therefore transcriptional bursting is not strictly required for protein production to occur in bursts. However, transcriptional bursting might allow for additional modes of regulation by transcription factors or other proteins that influence the state of the DNA around the promoter site. It may also complement bursts of protein production produced by rapid translation by removing constraints placed on burst size by the upper limits of mRNA translation rate.

Dynamics and the interplay between timescales presents an open, and until recently, quite unexplored part of molecular biology. The present analysis suggests a new mechanism for in vivo regulation, where long silent timescales emerge as the result of some particularly large rate limiting step in the promoter. These steps are open for new levels of regulation by transcription factors, which naturally will be most effective when they influence the rate limiting step of transcription initiation [38].

Calculation Methods

To calculate the activity of a promoter we first calculate the probability that the promoter will be occupied by closed (h) and open (q) complexes using steady state conditions. The total activity of the promoter is given by F = Eq for the standard model and the recruitment model, and F = QEq for the dead-end model. Details of the calculation are found in the Text S1.

The time between subsequent initiations is calculated by considering the time needed for each step as described in the Text S1. For class I there is only one step and the distribution is a simple exponential. For class II there is two steps. If these steps take an average time of t_o and t_E , the total waiting time between events is distributed with(4)giving eq. (1) in the main text for t_O≠t_E. For one t much greater than the other, this distribution degenerates into a simple exponential. For t_O = t_E, eq. (4) gives eq. (2) in the main text.

For the recruitment model, the intervals between initiations are partitioned between the supercoiling assisted or unassisted outcomes, with a partitioning ratio given by q. Details are in the Text S1. For the dead-end model the distribution is similarly partitioned between the two distributions with a partition ratio given by Q. Details are in the Text S1. In the Text S1 we also show how to calculate the distribution of “on” and “off” times from q or Q. Finally, we calculate the Fano factor ν = 〈(N−〈N〉)²〉/〈N〉 by using generating functions as described in the Text S1.

Protocol to Determine On-Periods and Off-Periods

We distinguish “on-periods” and “off-periods” in the simulation data following the procedure used by Golding et al. [10]. They analyzed the experimentally obtained time series of fluorescent signal manually. The system is considered to be in “off-period” when the signal does not change for a while, and otherwise it is in “on-period”. The specific time resolution to detect an “off-period” was not given, but the shortest off-time measured was around 6 [min] (Golding, private communication); in other words, transcription events separated by less than 6 [min] were considered to be in the same “on-period”.

During an on-period, the number of messages transcribed, Dn≥1, and the duration t_on were recorded; the time to transcribe one message D was 2.5 [min] [10], which corresponds to the on-time for Dn = 1 case.

Considering this protocol used by Golding et al. [10], we defined Δn, t_on, and the duration of the off-time t_off out of the time series of firings from our model (Figure 3A) as follows: (i) When firings are separated by more than τ_c = 6 [min]+Δ = 8.5[min], the promoter is in an off period. (iii) Otherwise, if successive firings are separated by an interval less than τ_c, the gene is considered to be on until we observe an interval greater than τ_c. This defines the on-time t_on, and we count the number of transcripts per on-time Δn.