 the rate of termination, combined with small decreases in k SD and k MD ( Figure 5A and B, Supplementary Figures S2 and S8C). The best fit was with k T = 0.0045 s −1, a 14-fold reduction compared with mfd +, which implies that 90% of RNAP termination at the roadblock in our system is due to Mfd. The 2-4-fold decreases in fitted k SD and k MD values in Δmfd cells (Supplementary Figure S2) suggest that forward translocation of the stalled RNAP by Mfd may sometimes lead to dislodgement of the roadblock without termination.
The simulations indicate that the decreasing readthrough at high promoter strengths in the Δmfd cells is due to promoter clogging, where the queue of RNAPs builds up at the roadblock and extends backward along the DNA, eventually preventing RNAP binding to the promoter. Once the RNAP flux is sufficient to clog the promoter, the rate of RNAP passage past the roadblock cannot increase (Figure 5D). In our system, promoter clogging limits the effect of cooperation, such that increased promoter strength can never completely overcome a strong roadblock. Clogging occurs at our highest k F values even in mfd + cells but is exacerbated when termination is reduced ( Figure 5C and D).
Clogging should become more significant as the roadblock is moved closer to the promoter. The predicted probability of clogging for the 102 bp spacer (able to accommodate four RNAPs) is quite low, even for the highest promoter strengths, but increases substantially with shorter spacers ( Figure 5E). However, for weak promoters, clogging is infrequent, even with very short spacers, except in Δmfd cells ( Figure 5E  ate promoters ( Figure 5D, Supplementary Figures S8A and  B).

The effect of reduced distance between the promoter and the roadblock
Our modeling (Figures 5E and 6A) predicts that reducing the distance between the promoter and the roadblock from a 4-RNAP spacing to a 2-RNAP spacing should have little effect on the queue length in wild-type cells and should thus not substantially alter the amount of cooperation by multiple RNAPs or the amount of promoter clogging. In contrast, a 1-RNAP spacing should eliminate cooperation, preventing any increase in Rf with increasing promoter strength. A 1-RNAP spacing should also lead to promoter clogging and reduced Rf at high promoter strengths.
To test this, we shortened the spacer between the transcription start site and the edge of Oid to either 60 bp (2-RNAP) or 30 bp (1-RNAP) and measured readthrough using 10 promoters and 250 nM LacI ( Figure 6B). As predicted, the effect of shortening to the 60 bp spacer was small, though readthrough was increased for the weaker promoters. However, the results for the 1-RNAP spacer were more complex ( Figure 6B). The cooperation effect was indeed lost as predicted, that is, there was no increase in readthrough with increasing promoter strength. Unexpectedly, the readthrough values for the weaker promoters were Error bars are 95% confidence intervals (n = 9). The solid curves show the optimal fits from stochastic simulation, obtained from fitting the data with the same termination rate (k T ) as for 102 bp promoter-operator spacer and increased dislodgement rates (Supplementary Figure S2).
systematically much higher than for the longer spacers. In addition, there was no observable clogging effect. A possible explanation of the increased readthrough is that the dissociation of the RNAP 70 subunit from the elongating complex takes some time (45,46), and the presence of 70 is known to interfere with Mfd action in vitro (30). In the 30 bp and 60 bp constructs, the RNAP can travel at most 25 bp or 55 bp before meeting the roadblock. Thus, at least some of the stalled RNAPs may retain 70 for some time and be resistant to Mfd termination. We found that reduced termination could allow the simulations to reproduce the 60 bp spacer data reasonably well. But the reductions in termination required to reproduce the high readthroughs for the weaker promoters at the 30 bp spacer produced strong clogging and reduced readthrough at high promoter strengths. Instead, reasonable fits to the data for both the 60 bp and 30 bp spacers could be obtained if dislodgement rates by single RNAP (k SD ) were increased some 2.1-fold and 5.3-fold, respectively ( Figure 6B and Supplementary Figure S2).

A simple model for transcriptional roadblocking
The effects on roadblocking of four of the five parameters we examined--RNAP flux (k F ), roadblocker concentration (k B ), roadblocker unbinding (k U ) and termination (k T )--can be well explained by remarkably straightforward mechanisms of RNAP-roadblock interaction (Figure 1C). The effect of the fifth parameter--promoterroadblock spacing--did not fully conform to the expectations of the model in the case of the shortest spacing, where readthrough was higher than expected. The modeling suggested that this result is not consistent with decreased termination of an RNAP stalled close to the promoter, but could be explained if a single promoter-proximal RNAP is more likely to dislodge the roadblock than a promoterdistal RNAP. We speculate that this may be due to sequence differences at the different stall sites (e.g. differential backtracking), which we have not controlled for. Alternatively, the RNAP may be different. The 70 subunit is lost with a half-life of ∼7 s (46) and various elongation regulating factors (e.g. NusA, NusG, rho) bind to the RNAP after it leaves the promoter (47). Such changes could decrease the ability of the RNAP to dislodge the roadblock. However, the 72 bp difference between the short and long spacers would provide only a short time (∼2 s) to establish a 5-fold difference in dislodgement capability. If the promoter proximity effect can be confirmed, experiments with a larger range of distances, better controlled stall-site sequences and elongation factor mutations could be used to examine this further.
The wild-type system has two main behavioral regimes. At low RNAP flux, where k F k T +k U +k SD, each RNAP almost always acts alone, either terminating or passing through the roadblock site without cooperation from a trailing RNAP. At higher RNAP flux levels, the wild-type system enters a more complex regime, where interactions between multiple RNAPs at the roadblock site provide significant cooperation. In our LacI/Oid-O1-O2 system, the low-flux regime applies for promoter firing rates of less than 1 every 30 s and the high-flux regime applies for firing rates >1 every 20 s, with an intermediate transition zone. For weaker binding roadblocker proteins, increased k U and likely increased k SD means that k F k T +k U +k SD can hold for considerably stronger promoters, extending the low-flux regime.
The critical variables in the low RNAP flux regime are k U and k SD, since k T is likely to be reasonably constant in wild-type cells ( Figure 1D). Although we have examined a 23-fold range of k U, associated with an estimated 4-fold range of k SD (Supplementary Figure S2), k U and k SD are likely to fall outside these ranges for some DNA-binding protein/operator combinations. Can we predict the level of roadblocking in these cases? Higher unbinding rates should lead to increased readthrough, and we can calculate using the analytical model ( Figure 1D; at 250 nM LacI without any increase in k SD ) that increasing k U 10-fold beyond that for LacI-O2 (to k U ∼0.1 s −1 ) should increase Rf from 26% to 68%, while a 100-fold increase in k U (to k U = 1.0 s −1 ) would almost eliminate roadblocking (Rf = 98%). A 100fold increase in k U would give an affinity comparable to the weak lacO3 operator (43). Allowing k SD to increase with increasing k U does not affect the calculated roadblocking substantially. Using a power law to extrapolate how k SD changes with k U (Supplementary Figure S9A) gave a ∼3fold increase in k SD for a 10-fold increase in k U and a cal-culated Rf = 72% (at 250 nM LacI), only slightly above the 68% without an increase in k SD. Thus, once unbinding is fast, dislodgement becomes unimportant.
In the high RNAP flux regime, most readthrough relies on roadblock dislodgement by multiple RNAPs, making k MD the critical parameter. We estimate k MD to be ∼4to 10-fold higher than k SD if all stalled RNAPs are subject to the same termination rate, and 2-to 4-fold higher than k SD if RNAPs can protect each other from termination (this difference depends on operator strength; Supplementary Figure S2). Frequent dislodgement and high flux can make rebinding of the roadblock protein limiting, so that k B also becomes significant in this regime. For weaker DNA-binding proteins, spontaneous unbinding can also be significant. Increasing k U to 0.1 s −1 would increase pL readthrough from Rf = 45% (for O2) to Rf = 73% (at 250 nM LacI). If k MD values continue to rise with increasing k U (Supplementary Figure S9A), then readthrough increases further; to Rf = 91% for pL (if k MD = 0.1 s −1 ).

Mechanisms of dislodgement
Our data and literature estimates of the LacI in vivo on