RNA polymerase sliding on DNA can couple the transcription of nearby bacterial operons

Abstract

DNA transcription initiates after an RNA polymerase (RNAP) molecule binds to the promoter of a gene. In bacteria, the canonical picture is that RNAP comes from the cytoplasmic pool of freely diffusing RNAP molecules. Recent experiments suggest the possible existence of a separate pool of polymerases, competent for initiation, which freely slide on the DNA after having terminated one round of transcription. Promoter-dependent transcription reinitiation from this pool of post-termination RNAP may lead to coupled initiation at nearby operons, but it is unclear whether this can occur over the distance- and time-scales needed for it to function widely on a bacterial genome in vivo. Here, we mathematically model the hypothesized reinitiation mechanism as a diffusion-to-capture process and compute the distances over which significant inter-operon coupling can occur and the time required. These quantities depend on previously uncharacterized molecular association and dissociation rate constants between DNA, RNAP and the transcription initiation factor σ⁷⁰; we measure these rate constants using single-molecule experiments in vitro. Our combined theory/experimental results demonstrate that efficient coupling can occur at physiologically relevant σ⁷⁰ concentrations and on timescales appropriate for transcript synthesis. Coupling is efficient over terminator-promoter distances up to ∼ 1, 000 bp, which includes the majority of terminator-promoter nearest neighbor pairs in the E. coli genome. The results suggest a generalized mechanism that couples the transcription of nearby operons and breaks the paradigm that each binding of RNAP to DNA can produce at most one messenger RNA.

INTRODUCTION

The core RNA polymerase (RNAP) in bacteria, which is composed of five subunits (β, β′, α^I, α^II , and ω), can catalyze the synthesis of RNA, but cannot recognize specific promoter sequences. To recognize promoters, RNAP must first bind an initiation factor such as the E. coli housekeeping σ⁷⁰, forming an RNAP holoenzyme (σ⁷⁰RNAP) ^1–3. DNA transcription initiates when σ⁷⁰RNAP is recruited from cytoplasm to bind to the promoter region of a gene. Controlling this process is thought to be the principal means through which transcription repressors and activators modulate gene transcription.

Bacterial metabolism requires the coordinated expression of multiple genes ⁴. A basic way in which this coordination is achieved in bacterial cells is by the organization of functionally related genes into operons, which are groups of consecutive genes that can be transcribed from the same promoter ^5,6. Functionally related operons often reside in contiguous regions of the bacterial genome ⁷. Proximally located operons show higher levels of correlated expression than distant operons in E. coli ^8–10. The same is true of closely spaced genes in eukaryotes ^11,12.

While specific groups of bacterial operons may have correlated activities simply because they have common regulatory proteins (e.g., alternative sigma factors), there are also proximity-based mechanisms that can couple transcription of adjacent operons. Terminator readthrough, in which the RNAP fails to read a terminator signal and keeps elongating the mRNA molecule, can generate the joint transcription of co-directional neighboring operons. Transcription-coupled DNA supercoiling ¹³ can induce coupled transcription of divergently transcribed genes ^14,15.

Recently, a new mechanism of proximity-based transcription coupling was observed. Using single-molecule microscopy, Harden et al. ¹⁶ and Kang et al. ^17,18 observed that RNAP can remain bound to DNA after termination for at least hundreds of seconds in vitro. This post-termination RNAP-DNA complex may retain a partially open bubble in the DNA ¹⁹. The retained RNAP exhibits one-dimensional diffusive sliding over hundreds or thousands of base pairs along the DNA. In the presence of σ⁷⁰ in solution, the sliding RNAP can re-initiate transcription at a nearby promoter. This post-termination behavior of bacterial RNAP may couple transcription of nearby operons in a way that is dependent on both the distance between the two transcription units and the available concentration of σ⁷⁰. Genome-wide transcription measurements are consistent with this mechanism, but do not prove that it operates in vivo in both E. coli and B. subtilis ¹⁶.

In this work, we test whether RNAP post-termination sliding followed by σ⁷⁰ rebinding can efficiently couple the transcription of nearby operons. First, we mathematically model the mechanism as a diffusion-to-capture process, in which the association of a σ⁷⁰ molecule with the sliding RNAP is required for reinitiation at a nearby promoter sequence. Next, we use single-molecule microscopy experiments under conditions designed to mimic the ionic composition of bacterial cytoplasm to measure the values of the model kinetic parameters. Finally, we input the measured values into the model to predict the distances and times over which post-termination sliding of RNAP could couple expression of neighboring genes.

RESULTS

Model of operon coupling by sliding RNAP

We model transcriptional coupling between proximal operons using the sliding RNAP mechanism depicted in Fig. 1. The distance between the terminator of the first operon and the promoter of the second operon is d. Upon reaching the terminator sequence T of the primary operon (at time t = 0), the RNAP releases an RNA transcript but remains non-specifically bound, enabling it to diffuse along the DNA with a diffusion coefficient D. During this time interval of sliding, the RNAP can either dissociate from the DNA with rate k_off, or bind a σ⁷⁰ molecule from solution with a rate k_b[σ⁷⁰], where [σ⁷⁰] denotes the free σ⁷⁰ solution concentration. In the latter case, the RNAP-σ⁷⁰ complex continues diffusing along the DNA molecule, and can dissociate with a rate k_off,s, or can encounter and be captured by the promoter for the secondary operon. We define the time it takes the captured RNAP-σ⁷⁰ complexes to find the secondary promoter as the search time t_f. In this mechanism, σ⁷⁰ can have conflicting effects because it can stimulate RNAP dissociation from DNA via the k_off,s step and yet is also required for secondary promoter capture.

Figure 1: Model of operon coupling by sliding RNAP.

(i) An RNAP molecule (green) terminates transcription of the primary operon (blue), and (ii) starts sliding along the DNA molecule with a diffusion constant D. (iii) While sliding, the RNAP can either dissociate from the DNA with rate k_off, or bind σ⁷⁰ (gray) with rate k_b[σ⁷⁰]. (iv) After binding of σ⁷⁰, the RNAP-σ⁷⁰ complex can either dissociate with a rate k_off,s, or find the promoter (bent arrow) for the secondary operon (pink), which is located is at a distance d along the DNA from the primary operon terminator (T).

For simplicity, we assume that the binding of σ⁷⁰ to sliding RNAP is irreversible. This is equivalent to assuming that the unbinding of σ⁷⁰ from the sliding RNAP is significantly slower than the dissociation of the RNAP-σ⁷⁰ complex from the DNA. This assumption is reasonable, given previous measurements of σ⁷⁰ dissociation from free ^20,21 and DNA-bound RNAP ¹⁷. Also for simplicity, we assume that the diffusion coefficient on DNA of the RNAP-σ⁷⁰ complex and RNAP are the same.

In this work we will refer to the complex that is formed by binding of σ⁷⁰ to DNA-bound RNAP as RNAP-σ⁷⁰-DNA and to the complex formed by binding σ⁷⁰RNAP holoenzyme to DNA as holoenzyme-DNA. It is not currently known whether these different orders of assembly produce complexes with the same structure (see Discussion).

Calculation of coupling efficiency

To quantify how transcriptional coupling by sliding RNAP changes with varying distance between operons and with [σ⁷⁰], we define the coupling efficiency (E) as the probability that an RNAP molecule, which terminates transcription of the primary operon, reaches the promoter of the secondary promoter by the sliding RNAP mechanism. To reach the secondary promoter, (i) the sliding RNAP has to bind a σ⁷⁰ molecule from solution before falling off the DNA, and (ii) the RNAP-σ⁷⁰ complex then has to reach the secondary promoter before falling off the DNA. The probability of (i), P_bind, is given by the partition ratio

$$P_{\text{bind}}=\frac{k_{\text{b}}\left[{\sigma }^{70}\right]}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}},$$ (1)

while the probability of (ii), P_find, can be computed as the probability that a RNAP-σ⁷⁰ complex remains bound to DNA for at least the time needed to find the secondary promoter before dissociating.

To determine P_find, we combine the distribution of times it takes RNAP-σ⁷⁰ complexes to encounter the secondary promoter for the first time, p_{first passage}(t), and the probability that the complex will stay bound on the DNA long enough for the encounter to happen. Thus, P_find is given by

$$P_{\text{find }}={\displaystyle {\int }_0^{\infty }{p}_{\text{first passage }}\left(t^{\prime }\right)\exp \left(-k_{\text{off,s }}{t}^{\prime }\right)d{t}^{\prime }}.$$ (2)

Here we use uppercase P to refer to probabilities, and lowercase p to represent probability density functions (PDFs).

RNAP terminates transcription of the primary operon at x = 0 and starts performing a one-dimensional random walk along the DNA with diffusion coefficient D. How long it takes for a RNAP-σ⁷⁰ complex to first encounter the secondary promoter will depend on its position on the DNA (x_b) when it starts the search, i.e., when σ⁷⁰ binds the sliding RNAP molecule. x_b in turn depends on how long after termination at the primary terminator σ⁷⁰ binds (t_b). At a time t_b drawn at random from the exponential distribution

$$p_{\text{bind },\text{t}}\left(t_{\text{b}}\right)=\left(k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}\right)\exp \left(-\left(k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}\right)t_{\text{b}}\right),$$ (3)

the sliding RNAP will either bind a σ⁷⁰ molecule or dissociate from the DNA. If it binds a σ⁷⁰ molecule, the binding position will be a random value drawn from

$$p_{\text{bind },\text{x}}^{\text{cond }}\left(x_{\text{b}}\mid t_{\text{b}}\right)=𝒩\left(0,2D{t}_{\text{b}}\right),$$ (4)

where $𝒩(\mu ,var)$ describes a normal distribution with mean μ and variance var. Eq. 4 represents the conditional probability distribution for x_b given the binding time t_b. Now we can calculate the unconditional distribution of binding positions

$$\begin{array}{l}{p}_{\text{bind },\text{x}}\left(x_{\text{b}}\right)={\displaystyle {\int }_0^{\infty }{p}_{\text{bind },\text{x}}^{\text{cond }}\left(x_{\text{b}}|t^{\prime }\right)p_{\text{bind },\text{t}}\left(t^{\prime }\right)d{t}^{\prime }}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}=\sqrt{\frac{k_{\text{b}}[\sigma ]+k_{\text{off}}}{4D}}\exp \left(-\sqrt{\frac{k_{\text{b}}[\sigma ]+k_{\text{off}}}{D}}\left|x_{\text{b}}\right|\right).\end{array}$$ (5)

The distribution of times t_fp it would take the RNAP-σ⁷⁰ complex to reach the secondary promoter at x = d for the first time is then given by the first-passage-time density for a one-dimensional random walk ²²

$$p_{\text{first passage }}^{\text{cond }}\left(t_{\text{fp}}\mid x_{\text{b}}\right)=\frac{\left|d-x_{\text{b}}\right|}{\sqrt{4\pi D{t}_{\text{fp}}^3}}\exp \left(-\frac{{\left(d-x_{\text{b}}\right)}^2}{4D{t}_{\text{fp}}}\right),$$ (6)

which is conditional on x_b. The unconditional distribution of first passage times is then calculated as

$$p_{\text{first passage }}\left(t_{\text{fp}}\right)={\displaystyle {\int }_{-\infty }^{+\infty }{p}_{\text{first passage }}^{\text{cond }}\left(t_{\text{fp}}\mid x^{\prime }\right)p_{\text{bind},\text{x}}\left(x^{\prime }\right)d{x}^{\prime }}.$$ (7)

Finally, substituting Eq. 7 into Eq. 2 to get P_find, and multiplying by P_bind (Eq. 1), we get the expressions in Eq. 8.

$$E\left(d,\left[{\sigma }^{70}\right]\right)=\left\{\begin{array}{ll}\frac{k_{\text{b}}\left[{\sigma }^{70}\right]}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}-k_{\text{off},\text{s}}}\left(\exp \left(-\sqrt{\frac{k_{\text{off},\text{s}}}{D}d}\right)-\sqrt{\frac{k_{\text{off},\text{s }}}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}}\exp \left(-\sqrt{\frac{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}{D}}d\right)\right),\hfill & \text{if}\hspace{0.33em}{k}_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}\ne k_{\text{off},\text{s }}\hfill \\ \frac{k_{\text{b}}\left[{\sigma }^{70}\right]}{2}\exp \left(-\sqrt{\frac{k_{\text{off},\text{s}}}{D}}d\right)\left(\frac{d}{\sqrt{k_{\text{off},\text{s}}D}}+\frac{1}{k_{\text{off},\text{s}}}\right),\hfill & \text{ if }{k}_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}=k_{\text{off,s }}\hfill \end{array}\right.$$ (8)

Coupling regimes and calculation of coupling distance

We can distinguish three different coupling regimes depending on the availability of free σ⁷⁰ molecules. For this, we define a critical σ⁷⁰ concentration at which the diffusion time intervals available before and after σ⁷⁰ binding to RNAP are equal, ${\left[{\sigma }^{70}\right]}_c=\frac{k_{\text{off},\text{s }}-k_{\text{off }}}{k_{\text{b}}}\approx \frac{k_{\text{off},\text{s }}}{k_{\text{b}}}$. Here we used k_off << k_off,s based on prior studies ²³and our experimental results (see below).

1. When [σ⁷⁰] ≫ [σ⁷⁰]_c, the coupling efficiency at small d is given by P_bind (Eq. 1) since any RNAP that binds σ⁷⁰ will subsequently encounter the promoter. For larger d the efficiency decays exponentially,

$$E\approx P_{\text{bind }}\exp \left(-\sqrt{\frac{{\text{k}}_{\text{off,s }}}{D}}d\right).$$ (9)

In this regime, we can define the coupling distance as the characteristic decay distance of the coupling, $d_c^{\gg }\approx \sqrt{\frac{D}{k_{\text{off},\text{s}}}}$.

2. When [σ⁷⁰] ≪ [σ⁷⁰]_c, the decay is also exponential but in this case,

$$E\approx P_{\text{bind}}\sqrt{\frac{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}{k_{\text{off},\text{s}}}}\exp \left(-\sqrt{\frac{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}{D}}d\right),$$ (10)

and the characteristic distance is $d_c^{\ll }\approx \sqrt{\frac{D}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}}$. However, in this case $\sqrt{\frac{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}{k_{\text{off},\text{s}}}}\ll 1$, which means that there is no significant coupling between adjacent transcription units no matter the distance between them.

3. For [σ⁷⁰] ≈ [σ⁷⁰]_c, we get the bottom expression in Eq. 8. Even though it is not exponential, we can still define a coupling distance $d_c^{\approx }$ over which the coupling efficiency decays by a factor of e. Following the calculations in Appendix S1, we get

$$\begin{array}{l}{d}_c^{\approx }\approx 2.15\sqrt{\frac{D}{k_{\text{off, s }}}}=2.15\sqrt{\frac{D}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=1.08\left(\sqrt{\frac{D}{k_{\text{off},\text{s}}}}+\sqrt{\frac{D}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}}\right).\end{array}$$ (11)

In simple terms, the regimes differ by whether most of the diffusional search for the secondary promoter takes place after (regime 1) or before (regime 2) the binding of σ⁷⁰. In addition, given that $\sqrt{\frac{D}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}}\ll \sqrt{\frac{D}{k_{\text{off, },s}}}$ in regime 1, and $\sqrt{\frac{D}{k_{\text{off},\text{s}}}}\ll \sqrt{\frac{D}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}}$ in regime 2, for all three regimes we can then approximate the coupling distance as $d_c\approx \sqrt{\frac{D}{k_{\text{b}}[\sigma ]+k_{\text{off}}}}+\sqrt{\frac{D}{k_{\text{off},\text{s}}}}$, which is roughly the sum of the root mean squared displacements of the RNAP before and after binding a σ⁷⁰ molecule (Fig. S1). Efficiency curves as a function of distance scaled by the critical distance are shown for all three regimes in Fig. 2. As expected, when the concentration of σ⁷⁰ is well below its critical concentration, the efficiency is small for any distance between the two promoters, while the efficiency can be of order one when [σ⁷⁰] is well above [σ⁷⁰]_c.

Figure 2: Predicted relationship of Coupling efficiency E to the distance between operons.

The coupling efficiency is the probability of an RNAP that terminated transcription at the end of the primary operon reaching the secondary promoter. Efficiency curves are shown for the three regimes of σ⁷⁰ concentration described in the text, for a chosen set of parameter values: k_b = 10⁷ M⁻¹s⁻¹, k_off = 10⁻³ s⁻¹, k_off,s = 1 s⁻¹, D = 4 × 10⁴ bp²s⁻¹. Curves were calculated using Eq. 8. Distance is represented in units of the coupling distance $d_c\approx \sqrt{D{\left(k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}\right)}^{-1}}+\sqrt{D{\left(k_{\text{off},\text{s}}\right)}^{-1}}$. Inset: enlarged plot of the curve for [σ⁷⁰] ≪ [σ⁷⁰]_c.

Single-molecule microscopy experiments to measure model parameters

To estimate the coupling efficiency E, coupling distance d_c, and search times t_f that can be achieved via the proposed mechanism (Fig. 1), we need values for the model parameters D, k_b, k_off, and k_off,s.

The diffusion coefficient of RNAP post termination, D = (3.9 ± 0.5) × 10⁴ bp²s⁻¹, was experimentally measured in ref. ¹⁶. Those investigators also sometimes observed a non-diffusing post termination RNAP-DNA complex in their experiments, but attributed this to RNAP binding to the ends of the linear DNA molecules they used.

We performed single-molecule experiments to measure k_b, k_off and k_off,s. Specifically, we quantified the dwell times of RNAP on promoterless DNA templates in the presence of different concentrations of σ⁷⁰ (Fig. 3A). These experiments allow us to measure all three rate constants. This is because at low σ⁷⁰ concentrations the measured dwell times are limited by the rate of RNAP dissociation from DNA, at intermediate σ⁷⁰ concentrations they are limited by the rate of σ⁷⁰ binding to the RNAP-DNA complex, and at high σ⁷⁰ concentrations they are limited by the rate of RNAP-σ⁷⁰ complex dissociation from DNA.

Figure 3: Single-molecule experiments to measure k_b, k_off, and k_off,s.

a. Experiment schematic. Fluorescently labeled, promoterless circular DNA templates (${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$; black circles) were tethered to the surface of a glass flow chamber (blue) through polyethylene glycol linkers (dotted black curves). The chamber was then incubated with fluorescently labeled core RNAP (RNAP⁵⁴⁹) to form RNAP-DNA nonspecific complexes, which correspond to the post-termination complex. In each of six experiments, a different concentration of σ⁷⁰ (gray) was introduced at t = 0, and the lifetime of each RNAP⁵⁴⁹ that colocalized with a surface-tethered DNA molecule was monitored by single-molecule fluorescence microscopy. B. Example of experiment record. Left: ${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$ and RNAP⁵⁴⁹ fluorescence images of the same field of view (65 µm diameter) upon introducing 1.19 µM σ⁷⁰ at t = 0. Right: Time record excerpt of RNAP⁵⁴⁹ fluorescence at the location of a single DNA molecule. Gallery shows 5 × 5 pixel images centered on the DNA molecule; graph shows the summed, background-corrected intensity of the 3 × 3 pixels centered on the DNA. t_dwell represents the duration of the fluorescent spot. C. t_dwell survival probability distributions in the presence of 0, 0.59, and 1.19 µM σ⁷⁰. D. Rates (with 68% C.I.s) of σ⁷⁰-dependent dissociation of RNAP⁵⁴⁹ from ${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$ (black) and ${\text{DNA }}_{\text{short}}^{\text{Cy}5}$ (gray) as a function of σ⁷⁰ concentration, and global fit (red; see eq. (12) and accompanying text).

For experimental convenience, we did not use core RNAP-DNA complexes that were formed after termination of transcription. Instead, we directly formed sequence non-specific RNAP-DNA complexes by adding core RNAP to a DNA that lacks known promoter sequences. The two types of complexes have the same protein composition and have similar properties: both are long-lived, in both RNAP slides on DNA, both are sensitive to the polyanion heparin ¹⁶, and both are rapidly disassembled by the bacterial SNF2 ATPase RapA ²⁴.

To implement these experiments we designed and synthesized a biotinylated, 3,033 bp circular DNA lacking known promoter sequences that was labeled with the red-excited dye Cy5 (we refer to this construct as ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$). Circular DNAs were used to avoid possible binding of RNAP to DNA ends ²⁵, which are largely non-physiological since the E. coli chromosome is circular.

We immobilized ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ molecules on the surface of a glass flow chamber via a biotin-streptavidin linkage (Fig. 3A). We then incubated the chamber with a solution containing E. coli core RNAP labeled with a green-excited dye (RNAP⁵⁴⁹) for ∼ 10 min, and washed it out at time t = 0 with a solution containing σ⁷⁰ in the 0 to 1.2 μM range. Single-molecule total internal reflection microscopy was performed with alternating red and green excitation for observation of ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ and RNAP⁵⁴⁹, respectively. An example of the fluorescence records used for extracting the dwell times of the RNAP⁵⁴⁹ molecules on the ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ template for each experiment is shown in Fig. 3B.

Given that these experiments study sequence-nonspecific interactions between RNAP⁵⁴⁹ and ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$, it was expected that multiple RNAP⁵⁴⁹ molecules could be bound to the same ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ template simultaneously. To reduce complications in the dwell time measurements arising from multiple RNAP⁵⁴⁹ molecules bound to the same template, we restricted the analysis to only those ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ locations with a single colocalized RNAP⁵⁴⁹. The number of RNAP⁵⁴⁹ molecules bound to each ${\text{DNA}}_{\text{Long }}^{\text{Cy}5}$ template was quantified by counting the number of decreasing steps present in the RNAP⁵⁴⁹ fluorescence intensity records (Fig. S2 and Appendix S3).

Distributions of RNAP dwell times on DNA

Example dwell time probability distributions of RNAP⁵⁴⁹ on promoterless circular DNA templates for different concentrations of σ⁷⁰ are shown in Fig. 3C. Consistent with the results in ²³, σ⁷⁰ accelerates the dissociation of RNAP from DNA.

In the absence of promoter sequences in the DNA, the model in Fig. 1 predicts that the dwell time distributions for RNAP obtained in the limits of low and high [σ⁷⁰] are exponential. Theoretically, at intermediate [σ⁷⁰] the dwell time distributions are non-exponential, due to the presence of two sequential steps (k_b and k_off,s). Still, for reasonable values of the rate constants the distribution is well approximated by an exponential and the effective rate constant has a hyperbolic dependence on [σ⁷⁰],

$$k_{\text{eff}}\approx k_{\text{off}}+\frac{k_{\text{off},\text{s}}{k}_{\text{b}}\left[{\sigma }^{70}\right]}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off},\text{s}}}.$$ (12)

However, the experimental distributions are in fact described not by a single exponential, but by the sum of multiple exponential components with very different rate constants. Specifically, for the experiments in which [σ⁷⁰] > 0 the distributions are well fit by a sum of three exponentials with characteristic rates k_slow, k_inter, and k_fast (Fig. S3). This suggests that in these experiments there are at least three types of RNAP-σ⁷⁰-DNA complexes. The resulting fits, obtained using a maximum likelihood method, are shown in Fig. S4, and the fit parameters are summarized in Table 1. In all cases, non-specific binding of RNAP⁵⁴⁹ to the chamber surface was minimal (Table S1), and therefore was not considered when fitting the data.

Table 1:

Parameters for fits to dwell time distributions of RNAP⁵⁴⁹-promoterless DNA complexes at different σ⁷⁰ concentrations.

[σ70] (μM)	N	Nd	afast (×10−1)	ainter (×10−1)	kfast (×10−2 s−1)	kinter (×10−3 s−1)	kslow (×10−4 s−1)	DNA preparation
0	65	60	8.6(6.8 − 9.4)	-	0.16(0.13 − 0.22)	-	1.59(0 − 3.99)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 1
0.15	60	42	1.2(0.4 − 2.0)	1.6(0.4 − 2.9)	1.63(1.24 − 2.26)	3.11(2.27 − 4.52)	1.90(1.44 − 2.33)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 2
0.30	139	85	1.8(1.3 − 2.1)	1.3(0.7 − 3.8)	2.22(1.80 − 2.90)	1.23(0.53 − 2.04)	1.27(0.48 − 1.52)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 2
0.59	106	100	3.4(2.5 − 4.8)	5.4(4.1 − 6.2)	2.59(1.52 − 3.77)	2.61(1.87 − 3.37)	2.18(0.26 − 3.63)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 1
0.74	106	60	1.8(1.2 − 2.2)	2.0(1.0 − 5.2)	2.57(1.77 − 3.96)	0.81(0.34 − 1.72)	1.05(0 − 1.54)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 3
1.19	74	71	3.4(2.3 − 4.9)	5.8(4.4 − 6.8)	3.90(2.47 − 5.47)	3.88(2.45 − 5.26)	1.61(0 − 3.24)	${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$, preparation 1
1.19	76	68	5.8(4.0 − 6.7)	3.0(2.0 − 4.7)	4.91(3.63 − 8.32)	5.77(4.03 − 10.09)	0.72(0 − 1.56)	${\text{DNA }}_{\text{Short}}^{\text{Cy}5}$

The models used for the fit are described in Methods (Eqs. 13 and 14). N is the number of DNA sites with co-localized RNAP that were used in the analysis. N_d is the number of DNA sites for which the co-localized RNAPs disappeared before the end of the experiment. The values are presented with 68% CI. In some cases, the lower confidence limit on k_slow is poorly defined because $k_{\text{slow}}^{-1}$ exceeds the duration of the experiment. ${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$ preparations 1, 2, and 3 were made by the same method on different occasions.

For the experiments with [σ⁷⁰] > 0, k_fast showed a hyperbolic dependence of [σ⁷⁰] (Fig. 3D); k_inter and k_slow did not (Table 1). Therefore, we hypothesize that the fastest component corresponds to σ⁷⁰-induced dissociation of RNAP⁵⁴⁹ bound to DNA, k_eff = k_fast. This suggests that at low σ⁷⁰ concentrations binding of σ⁷⁰ to the sliding RNAP is rate-limiting so that the dissociation rate of RNAP from DNA increases linearly with σ⁷⁰ concentration, while at high concentrations the dissociation of the RNAP-σ⁷⁰ complex from DNA becomes limiting, and the dissociation rate saturates. Possible origins of the longer-lived RNAP-DNA complexes with [σ⁷⁰]-independent dissociation rates k_inter and k_slow are discussed in the Appendix S4.

To confirm that k_fast depends on [σ⁷⁰] and not on the DNA template used, we repeated the 1.19 μM σ⁷⁰ experiment using a different DNA template, the promoterless 586 bp circular ${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$. Similar values were obtained for k_fast for both templates (Table 1, Fig. 3D), supporting the idea that k_fast depends on σ⁷⁰ concentration, and not on the length or sequence of the DNA template used.

Two characteristic rates were observed for the experiment with [σ⁷⁰] = 0. The slower one is similar to the values of k_slow observed in the presence of σ⁷⁰ (Table 1). The faster one is similar to the mean dissociation rate observed for the post-termination RNAP-DNA complex in the absence of σ⁷⁰, as well as to the mean dissociation rate observed for core RNAP sequence-nonspecifically bound to DNA ¹⁶. Therefore, we assume that the faster rate corresponds to the dissociation rate of RNAP⁵⁴⁹ from DNA in the limit where [σ⁷⁰] = 0.

Extraction of model parameters k_b, k_off, and k_off,s

Having established the hyperbolic dependence of the σ⁷⁰-induced dissociation rate k_fast on σ⁷⁰ concentration, we can now determine the values for the Fig. 1 model parameters k_b, k_off, and k_off,s. For this, we jointly fit the data from experiments at different σ⁷⁰ concentrations to a global model that incorporates our conclusions about the origins of the different components of the dwell time distributions (see Appendix S4). The σ⁷⁰-independent rates k_inter and k_slow were globally fit for all six experiments performed with ${\text{DNA}}_{\text{Long}}^{\text{Cy5 }}$ (Table S3 and Fig. S8). A separate set of parameters $k_{\text{inter}}^{\text{'}}$ and $k_{\text{slow}}^{\text{'}}$ were obtained by fitting the dwell time distribution from the experiment with ${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$. The global model explicitly included the [σ⁷⁰] dependency of k_fast (Eq. 12, where k_eff = k_fast).

The model fit well to the data (Fig. S8) and gave well-constrained values for the rate constants (Table 2). The rate constants, together with the diffusion coefficient D measured in ref. ¹⁶ provide the information needed to calculate the extent and kinetics of operon coupling.

Table 2:

Global model parameters.

Parameter	Description	Value (68% C.I.)	Source
D	Diffusion coefficient of sliding RNAP	3.9 × 104 bp2s−1†	16
k b	Binding rate of σ70 to sliding RNAP	1.2(0.7 − 2.7) × 105 M−1s−1	Fit
k off	Dissociation rate of RNAP before binding σ70	1.6(1.3 − 1.9) × 10−3 s−1	Fit
k off,s	Dissociation rate of RNAP-σ70 complex	5.1(3.4 − 12.2) × 10−2 s−1	Fit

^†Kang et al ¹⁷ measured a somewhat lower value corresponding to 0.8 × 10⁴bp²s⁻¹ at similar ionic strength but in a different buffer.

Extent of operon coupling by sliding RNAP

Operon coupling cannot be biologically functional if it takes an infeasibly long time for RNAP to find the secondary promoter after terminating transcription at the terminator of the primary operon. To compute the distribution of search times, we used the experimental results for D, k_b, k_off, and k_off,s to simulate the mechanism for a realistic σ⁷⁰ concentration and terminator-promoter spacing (Fig. 4A). The distribution of the search times is roughly exponential with a mean ⟨t_f⟩ ∼ 7 s, which is comparable to the time for transcription initiation at well-studied promoters (a few seconds to a few minutes ^26,27). This indicates that transcription re-initiation by sliding RNAP is capable of effectively increasing expression of the secondary operon.

Figure 4: Extent of operon coupling predicted by the sliding RNAP model, using the kinetic parameter values from Table 2.

a. Distribution of search times that end in a promoter encounter t_f obtained by simulating the model in Figure 1 for d = 600 bp and [σ⁷⁰] = 5 μM. b. Coupling efficiency dependence on the distance d between primary operon terminator and secondary operon promoter, for two possible free σ⁷⁰ concentrations. Shaded areas show the 68% C.I.s. c. Distribution of distances between operon final terminators and the nearest operon initial promoter in the E. coli genome determined from data in ref. ²⁹.

Inputting the experimental results for D, k_b, k_off, and k_off,s into Eq. 8, we can predict the value of the efficiency as a function of the distance between operons and the σ⁷⁰ concentration. The total concentration of σ⁷⁰ in E. coli is on order 10 μM²⁸, but its availability is highly regulated through sequestration by anti-σ factors, whose activity is also tightly regulated. This means that at any time, the free σ⁷⁰ concentration could be anything below roughly 10 μM. Thus, the free σ⁷⁰ concentration in the cell could be either above or below [σ⁷⁰]_c, which we calculate to be 0.4 μM. To test whether the model predicts appreciable coupling at typical operon spacings, we calculated the predicted coupling efficiency as a function of the distance d between the primary terminator and the secondary promoter at high and low σ⁷⁰ concentrations (Figure 4B). At 5 μM σ⁷⁰, this calculation predicts efficient coupling at distances d up to 1, 000 bp. At a much lower free σ⁷⁰ concentration of 50 nM, the model predicts a smaller but still significant amount of coupling on this distance scale, with less dependence on operon spacing. Regulation of the free σ⁷⁰ concentration would therefore allow the cell to vary the amount of coupling in response to internal and environmental conditions.

The spacing between the final terminator of an operon and the nearest operon initial promoter has a broad distribution in the E. coli genome (Figure 4C). Nevertheless, based on our calculations, a large fraction of these pairs are capable of efficient coupling by sliding RNAP. For example, 52% of the terminator-promoter pairs are at distances where the coupling efficiency is at least 50% at [σ⁷⁰] = 5 μM. In other words, at this σ⁷⁰ concentration (and in general when [σ⁷⁰] >> [σ⁷⁰]_c ≈ 0.4 μM) the predicted critical distance d_c ≈ 1, 000 bp is of the same order of magnitude as the typical inter-operon distance (median 600 bp). This could allow many pairs of adjacent operons in the genome to be coupled, while at the same time enabling other operons to be transcribed independently of their neighbors, depending on the terminator-promoter spacing. Thus, the model predicts significant coupling under relevant cellular conditions and predicts that coupling can be regulated by tuning these conditions.

The model makes the simplifying assumption that every encounter of the RNAP-σ⁷⁰ complex with a promoter is productive and leads to synthesis of a transcript. To the extent that not all encounters are productive, the model will overestimate efficiency (Fig. 4B) and underestimate search time (Fig. 4A).

DISCUSSION

Using a combination of theory, stochastic simulations, and single-molecule microscopy experiments, we characterized the potential spatial and temporal reach of transcriptional coupling between adjacent operons mediated by diffusive sliding of RNAP that remains bound to DNA following transcription termination. We predict that σ⁷⁰ has both stimulatory and inhibitory effects on reinitiation. The stimulatory effect arises from the fact that only RNAP with bound σ⁷⁰ can recognize the secondary promoter. On the other hand, we show that RNAP-σ⁷⁰ has only a short lifetime on DNA, during which the sliding σ⁷⁰RNAP must find a promoter on the fly to reinitiate transcription. Despite the latter difficulty, we show that reinitiation is expected to be common and efficient for physiological ranges of terminator-promoter spacings and σ⁷⁰ concentrations. Thus, our results show that the proposed reinitiation mechanism is consistent with experiments that demonstrate reinitiation in vitro ^16,17 and in vivo ¹⁶.

To quantitatively define the reinitiation process, we measured three previously uncharacterized rate constants: the second-order rate constant for binding of σ⁷⁰ to the RNAP-DNA complex, k_b = 1.2 × 10⁵ M⁻¹s⁻¹, the rate constant for the dissociation of the RNAP-DNA complex, k_off = 1.6 × 10⁻³ s⁻¹, and the rate constant for the dissociation of the RNAP-σ⁷⁰-DNA complex, k_off,s = 5.1 × 10⁻² s⁻¹. The value obtained for k_b is an order of magnitude smaller than the rate constant of formation of a stable σ⁷⁰-RNAP complex in the absence of DNA, 1.5 × 10⁶ M⁻¹s^{−1 21}, which suggests that the presence of bound DNA significantly impedes σ⁷⁰ association with RNAP. The value obtained for k_off,s is an order of magnitude smaller than the value obtained for $k_1^{\text{D}}$, the dissociation rate of the RNAP holoenzyme-DNA complex (see Appendix S4 and Table S2). This suggests that the RNAP-σ⁷⁰-DNA complex (formed by RNAP-DNA binding σ⁷⁰ from solution) and the holoenzyme-DNA complex (formed by mixing σ⁷⁰RNAP with non-promoter DNA) have different conformations, despite them having the same protein and DNA constituents. It is possible that in the two complexes different subsets of σ⁷⁰ subregions interact with RNAP and/or DNA. More information, kinetic and structural, will be required to understand these differences.

The search for target sequences by proteins sliding on DNA has been demonstrated both in vitro and in vivo (e.g., ^30,31). Post-termination sliding of core RNAP on DNA is atypically slow compared to a sample of other DNA binding proteins ^16,32, possibly because RNAP maintains an open bubble of non-base-paired DNA in the post-termination RNAP-DNA complex ¹⁹. The presence in cells of sliding RNAP molecules that may take on order of 10 s after termination to reinitiate transcription (Fig. 4A) is consistent with demonstration of a substantial population in vivo of slowly diffusing RNAP molecules that are neither bound to a fixed site on DNA nor freely diffusing in solution ^33,34.

Rapid, efficient reinitiation of transcription through sliding of post-termination RNAP over relevant genomic distances may have significant implications for transcription homeostasis and regulation in both natural and engineered genomes. Under particular growth conditions, transcription activity is often concentrated in clusters of genes or operons in confined genomic regions ^4,6,7,35,36. Sliding-mediated reinitiation may help to maintain a localized pool of RNAP molecules that are efficiently reused in these transcriptionally active regions. Indeed, the efficiency of reinitiation by sliding core RNAP compared to conventional initiation by RNAP holoenzyme from solution may be one of the factors that confers a selective advantage to the clustering of functionally related operons. In the context of synthetic biology, reinitiation by sliding might cause problems by giving rise to non-intended connectivity between transcription units that are intended to act modularly, but conversely could be a used as a tool to introduce correlations in designed genetic circuits.

MATERIALS AND METHODS

Plasmids

Plasmid pDT4 is identical to pCDW116¹⁶ except for mutation of CTGGAGTGCG to CTGGAGACCG to introduce a second BsaI site.

DNA templates

Circular DNA templates ${\text{DNA}}_{\text{Long}}^{\text{Cy5 }}$ and ${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$ were built by Golden Gate Assembly ³⁷ using a plasmid or PCR product and a synthetic ‘ligator” duplex oligonucleotide containing both dye and biotin modifications. Ligator was made by annealing two complementary oligonucleotides: 5′-CGATTAGGTCTCGGGCTAGTAC TGGTTTCTAGAG/iCy5/GTTCCAAGCC/iBiodTCACGGCGGCCGCCCATCGAGACCGGTTAACC-3′ and 5′-GGTTA ACCGGTCTCGATGGGCGGCCGCCGTGAGGCTTGGAACCTCTAGAAACCAGTACTAGCCCGAGACCTAATCG-3′ (IDT).

For making template ${\text{DNA}}_{\text{Long}}^{\text{Cy5 }}$, two identical Golden Gate Assembly reactions were carried out by mixing 7 μl of each ∼ 20 nM DT4 plasmid and ∼ 20 nM ligator fragment with 1 μl of Golden Gate Mix (New England Biolabs) in T4 DNA Ligase Buffer (New England Biolabs), in total volumes of 20 μl. The mixtures were incubated for alternating cycles of 5 min at 37°C and 10 min at 16°C 35 times, followed by 5 min at 55°C. After the reaction, the ligase was inactivated for 10 min at 65°C. The resulting 40 μl of reaction product was mixed with 4 μl of T5 Exonuclease (New England Biolabs) in NEB Buffer 4, to a total of 50 μl and incubated at 37°C for 30 min. The digestion was stopped by adding 15 mM of EDTA, and a Qiagen PCR Cleanup Kit was used to remove the cleaved nucleotides and enzymes.

For making template ${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$, a linear DNA fragment was first amplified by PCR from plasmid pCDW114 (¹⁶, Addgene #70061), using primers 5′-GAAGGTCTCCAGCCGTACCAACCAGCGGCTTATC-3′ and 5′-CCGGGTCTCACCATACCCGCTGTCTGAGATTACG-3′. A Golden Gate Assembly reaction was carried out by mixing 2 μl of 343 nM PCR product, 0.6 μl of 1 μM ligator and 1 μl of Golden Gate Mix in T4 DNA Ligase Buffer, in a total volume of 20 μl. The mixture was incubated for alternating cycles of 5 min at 37°C and 10 min at 16°C 35 times, followed by 5 min 55°C. After the reaction, the ligase was inactivated for 10 min at 65°C. The resulting reaction product was mixed with 1 μl Exonuclease V (New England Biolabs) and 3 μl of 10 mM ATP in NEB Buffer 4, in a total volume of 30 μl, and incubated at 37°C for 30 min. Exonuclease V was then inactivated for 30 min at 70°C. Finally, a Qiagen PCR Cleanup Kit was used to remove the cleaved nucleotides and enzymes.

Proteins

Fluorescent labeling of core RNAP

E. coli core RNAP with a SNAP tag on the C-terminus of β′ ³⁸ (RNAP-SNAP, gift from the Robert Landick lab) was labeled with SNAP-Surface 549, yielding RNAP⁵⁴⁹, as follows: 13.65 μM SNAP-RNAP (core) and 45.5 μM of SNAP-Surface 549 were mixed in a buffer containing 9 mM Tris-Cl^- pH 7.9, 5 mM MgCl₂, 1 mM DTT, 20% glycerol, and 90 mM NaCl, and incubated for 30 min at room temperature. The sample was then mixed with an equivalent amount of Dilution buffer (11 mM Tris-Cl^- pH 8.0, 30% glycerol, 110 mM NaCl, 1 mM DTT), flash-frozen in liquid nitrogen and stored at −80°C.

Expression and purification of His-tagged σ⁷⁰

His₆-tagged E. coli σ⁷⁰ (σ⁷⁰) was overexpressed in T7 Express cells (New England Biolabs) as inclusion bodies from the pET-28a-σ⁷⁰ overexpression plasmid ³⁹ by growing the cells at 37°C to an OD₆₀₀ of ∼ 0.8, and then inducing by addition of IPTG to 0.4 mM. The temperature was decreased to 20°C and cells were left shaking at 200 rpm overnight. Cells were then harvested by centrifugation at 4°C, followed by sonication in Lysis buffer (50 mM Tris-Cl^-, pH 7.9, 5 mM imidazole, 5% [v/v] glycerol, 233 mM NaCl, 2 mM EDTA, 10 mM β-mercaptoethanol) plus 1× cOmplete^™protease inhibitor cocktail (Roche). The lysate was centrifuged at 22, 000 × g for 30 min at 4°C and the supernatant was discarded. To remove E. coli membrane and cell wall material, the pellet was resuspended in 10 ml of 2 M Urea Cleaning buffer (20 mM Tris-Cl^- pH 8.0, 500 mM NaCl, 2 M urea, 2% Triton X-100, 10 mM β-mercaptoethanol) and sonicated. The resulting sample was centrifuged again at 22, 000 × g for 30 min at 4°C and the supernatant was discarded. Four consecutive resuspension-centrifugation cycles were carried out, two of them in 2 M Urea Cleaning buffer, and the other two in Wash buffer (20 mM Tris-Cl^- pH 8.0, 500 mM NaCl, 7% glycerol, 20 mM imidazole, 10 mM β-mercaptoethanol) to remove Triton X-100 from the pellet. To solubilize and denature the protein, the washed pellet was resuspended in 6 M Guanidine Binding buffer (20 mM Tris-Cl^- pH 8.0, 500 mM NaCl, 5 mM imidazole, 6 M guanidine hydrochloride, 2 mM β-mercaptoethanol), stirred for 1 hour, and centrifuged at 22, 000 × g for 30 min at 4°C. The supernatant was passed through a 0.22 μm filter and injected into a 1 ml HisTrap column (Cytiva Life Sciences), followed by a wash with Wash buffer supplemented with 6 M urea. Refolding of the bound protein was performed using a linear 1-hour-long 6 M to 0 M urea gradient in Wash buffer. The refolded protein was eluted with 500 mM imidazole in Wash buffer. The purified protein was dialyzed overnight into σ⁷⁰ storage buffer (10 mM Tris-Cl^-, pH 8.0, 30% [v/v] glycerol, 0.1 mM EDTA, 100 mM NaCl, 20 μM ZnCl₂, 1 mM MgCl₂, 0.1 mM DTT) and aliquots were flash-frozen in liquid N₂ and stored at −80°C.

Fluorescent labeling of σ⁷⁰

An N-terminal His₆-tagged single-cysteine derivative of E. coli σ⁷⁰ (C132S C291S C295S S366C) ⁴⁰ (see Appendix S4) was overexpressed and purified following the same protocol used for σ⁷⁰. The purified protein was concentrated 5× using an Amicon Ultra-0.5ml 30K filter by centrifuging for 11 minutes at 14, 000 × g at 4°C. For fluorescent labeling, the concentrated protein was mixed with Cy5-maleimide dye (Cytiva) (1:15 protein:dye ratio), incubated first for 10 min at room temperature, and then left overnight at 4° C. The excess dye was then removed using a Centrispin 20 column (Princeton Separations). After addition of glycerol and BSA to 30% and 1 mg/ml respectively, the samples were flash-frozen in liquid N₂, and aliquots were stored at −80°C.

Reconstitution of doubly-labeled holoenzyme

Cy5-σ⁷⁰RNAP⁵⁴⁹ holoenzyme (see Appendix S4) was reconstituted by incubating 121 nM of RNAP⁵⁴⁹ and 280 nM of Cy5-σ⁷⁰ for 30 min at 37°C.

Colocalization Single-Molecule Spectroscopy (CoSMoS) Experiments

Single-molecule total internal reflection fluorescence microscopy was performed ⁴¹ at excitation wavelengths 532 and 633 nm, for observation of DNA^Cy5 template (and/or Cy5-σ⁷⁰) and RNAP⁵⁴⁹, respectively; focus was automatically maintained ⁴². A stage heating device was used to keep the samples at 30°C. Single-molecule observations were performed in glass flow chambers (volume ∼ 30 μl) passivated with a mPEG-SG2000:biotin-PEG-SVA5000 (Laysan Bio) 200 : 1 w/w mixture as described in ⁴³. Neutravidin (#21125; Life Technologies) was introduced at 220 nM in KO buffer (50 mM TrisOAc, 100 mM KOAc, 8 mM Mg(OAC)₂, 27 mM NH₄(OAc), 0.1 mg/ml bovine serum albumin (BSA) (#126615 EMB Chemicals), pH 8.0), incubated for 45 s, and washed out (this and all subsequent wash steps used two washes each of two chamber volumes of KO buffer). The chamber was then incubated with ∼1 nM Cy5-DNA (${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$ or ${\text{DNA}}_{\text{Long}}^{\text{Cy5 }}$) in KO buffer for ∼ 20 min and washed out with Imaging buffer (KO buffer supplemented with an O₂ scavenging system: 4.5 mg/ml glucose, 40 units/ml glucose oxidase, and 1, 500 units/ml catalase ⁴³).

For the experiments to measure the dwell time of RNAP⁵⁴⁹ on DNA at different concentrations of σ⁷⁰, ∼ 1 nM of RNAP⁵⁴⁹ was introduced into the chamber in Imaging buffer supplemented with 3.5% w/v PEG 8, 000 and 1 mg/ml BSA for ∼ 10 min. Image acquisition was performed by alternating 1 s exposures to 532 and 633 nm, at 450 and 200 μW respectively (all laser powers were measured incident to the micromirror optics), and the flow chamber was washed with Imaging buffer supplemented with 3.5% w/v PEG 8, 000 and containing 0, 148 nM, 297 nM, 593 nM, 741 nM or 1.19 μM of σ⁷⁰. In the cases where the concentration of σ⁷⁰ was lower than 1.19 μM, the appropriate amount of σ⁷⁰ storage buffer was added in replacement so that all experiments were performed at the same solute concentrations: 47.0 mM TrisOAc, pH 8.0, 93.0 mM KOAc, 7.4 mM Mg(OAc)₂, 25 mM NH₄(OAc), 3% w/v 8, 000 PEG, 0.2 mM Tris-Cl^-, 2.0 mM NaCl, 0.4 μM ZnCl₂, 20 μM MgCl₂, 4.5 mg/ml glucose, 40 units/ml gluclose oxidase, 1, 500 units/ml catalase, 0.6% glycerol, 2 μM EDTA, 0.1 mg/ml BSA, 2 mM DTT, 10 nM DTT-quenched Cy5.5 maleimide dye.

The experiments to measure dwell time of σ⁷⁰RNAP on DNA (see Appendix S4) were performed similarly to the ones described in the previous paragraph, with three differences. First, a photobleaching step was performed after DNA surface attachment. Cy5 photobleaching was induced by 633 nm excitation at ∼ 1 mW in the presence of Imaging buffer without DTT. Second, instead of σ⁷⁰, ∼ 1 nM of Cy5-σ⁷⁰RNAP⁵⁴⁹ (which also contained an additional 1.5 nM Cy5-σ⁷⁰) was introduced into the chamber in Imaging buffer supplemented with 3.5% w/v PEG 8, 000 and no subsequent wash was performed. The final composition of the solution was 47.5 mM of TrisOAc, pH 8.0, 95.1 mM KOAc, 7.4 mM Mg(OAc)₂, 25.7 mM NH₄(OAc), 3.3% w/v 8, 000 PEG, 0.1 mg/ml BSA, 1 mM DTT, 0.1 mM Tris-Cl^-, 0.3% glycerol, 1 mM NaCl, 8 μM MgCl₂, 36 nM ZnCl₂, 1.8 μM EDTA, 1.1 nM SNAP-RNAP, 2.5 nM unreacted SNAP-Surface 549, 2.5 nM Cy5-σ⁷⁰, 4.5 mg/ml glucose, 40 units/ml glucose oxidase, 1, 500 units/ml catalase. Third, image acquisition was performed by continuous exposure to 532 and 633 nm lasers, at 450 and 200 μW respectively, at an acquisition rate of 1 frame per second.

CoSMoS Data analysis

Analysis of CoSMoS video recordings was done using custom software and algorithms for mapping between wavelength channels, spatial drift correction, and detection of spot colocalization as described ⁴¹. In each recording, we selected ${\text{DNA}}_{\text{Long}}^{\text{Cy5 }}$ or ${\text{DNA}}_{\text{Short }}^{\text{Cy5 }}$ fluorescence spots that co-localized with RNAP⁵⁴⁹ spots at t = 0. For the selected DNA molecules, we computed RNAP⁵⁴⁹ fluorescence intensity time records by summing the intensity over 3 × 3 pixel squares centered at DNA molecule locations in each recorded frame. Fluorescence intensity values were corrected for background fluorescence and non-uniform illumination across the microscope field of view, yielding normalized values for spot intensities ⁴⁴. This allowed us to directly compare integrated intensity values for different spots located throughout the field of view. The numbers of decreasing intensity steps in the resulting time traces were counted to assess the initial number of RNAP⁵⁴⁹ molecules present at each DNA molecule location (Fig. S2A). Records that showed more than a single RNAP⁵⁴⁹ molecule bound at t = 0 were excluded from subsequent analysis. The times of the first image with no spot at each DNA location were taken to be the dwell times of the RNAP⁵⁴⁹ molecules present at the beginning of the recording. Spots that persisted until the end of the recording were separately counted as censored dwell times equal to the recording duration.

Fits to RNAP-DNA complex dwell time distributions

The probability distribution of RNAP-DNA complex dwell times measured in each individual experiment was modeled as the sum of two exponential terms

$$f_1=a_{\text{fast }}{k}_{\text{fast }}\exp \left(-k_{\text{fast }}t\right)+\left(1-a_{\text{fast }}\right)k_{\text{slow }}\exp \left(-k_{\text{slow }}t\right)$$ (13)

or the sum of three exponential terms

$$\begin{array}{l}{f}_2=a_{\text{fast }}{k}_{\text{fast }}\exp \left(-k_{\text{fast }}t\right)+a_{\text{inter }}{k}_{\text{inter }}\exp \left(-k_{\text{inter }}t\right)\\ \hspace{1em}\hspace{1em}+\left(1-a_{\text{inter }}-a_{\text{fast }}\right)k_{\text{slow }}\exp \left(-k_{\text{slow }}t\right).\end{array}$$ (14)

For each distribution, lifetimes of RNAP⁵⁴⁹ binding events that terminated by disappearance of the fluorescent spot, and those that were censored by the end of the experiment, were jointly fit using the maximum likelihood algorithm by an approach analogous to the one used in ⁴⁵. Confidence intervals were calculated by bootstrapping ⁴¹.

Extraction of model parameters k_b, k_off, and k_off,s

To get values for k_b, k_off, and k_off,s, we globally fit dwell times collected at all concentrations of σ⁷⁰ to the models in equation 13 (for [σ⁷⁰] = 0) or equation 14 (for [σ⁷⁰] > 0), where k_fast was explicitly constrained to depend on [σ⁷⁰]:

$$k_{\text{fast }}=k_{\text{off}}+\frac{k_{\text{off},\text{s}}{k}_{\text{b}}\left[{\sigma }^{70}\right]}{k_{\text{off},\text{s}}+k_{\text{b}}\left[{\sigma }^{70}\right].}$$ (15)

In this formulation, k_inter and k_slow are assumed to be independent of [σ⁷⁰], and therefore were globally fit across distributions obtained for template ${\text{DNA }}_{\text{Long }}^{\text{Cy}5}$. For the experiment performed with template ${\text{DNA }}_{\text{Short}}^{\text{Cy}5}$, we fit independent values $k_{\text{inter}}^{\text{'}}$ and $k_{\text{slow}}^{\text{'}}$ for these parameters. Censored data were treated using the same approach as described above for the individual experiment fits to dwell time distributions.

Fit to holoenzyme-DNA dwell time distribution

To account for non-specific binding of holoenzyme to the glass flow chamber (see Appendix S4), we first randomly selected locations on the chamber surface that did not contain DNA molecules. We then fit the distribution of dwell times for Cy5-σ⁷⁰RNAP⁵⁴⁹ holoenzyme molecules bound at these non-DNA locations to a biexponential model

$$f_{\text{nD}}(t)=\left[a_1^{\text{nD}}{k}_1^{\text{nD}}\exp \left(-k_1^{\text{nD}}t\right)+\left(1-a_1^{\text{nD}}\right)k_2^{\text{nD}}\exp \left(-k_2^{\text{nD}}t\right)\right]/f_{\text{nD}}^0,$$ (16)

with

$$f_{\text{nD}}^0=a_1^{\text{nD}}\exp \left(-k_1^{\text{nD}}{t}_{\min }\right)+\left(1-a_1^{\text{nD}}\right)\exp \left(-k_2^{\text{nD}}{t}_{\min }\right),$$ (17)

where f_nD(t) is normalized so that it integrates to 1 over all dwell times t greater than the minimum detectable dwell time t_min = 0.2 s.

By analogy to ⁴¹, the dwell time distribution for Cy5-σ⁷⁰RNAP⁵⁴⁹ at DNA locations was then fit to the background-corrected model

$$\begin{array}{l}{f}_{\text{D}}(t)=\left(F_{\text{D}}-F_{\text{nD}}\right)\left[a_1^{\text{D}}{k}_1^{\text{D}}\exp \left(-k_1^{\text{D}}t\right)+a_2^{\text{D}}{k}_2^{\text{D}}\exp \left(-k_2^{\text{D}}t\right)\right.\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\left.+\left(1-a_1^{\text{D}}-a_2^{\text{D}}\right)k_3^{\text{D}}\exp \left(-k_3^{\text{D}}t\right)\right]/f_{\text{D}}^0+F_{\text{nD}}{f}_{\text{nD}}^{\text{ML}}(t),\end{array}$$ (18)

with

$$\begin{array}{l}{f}_{\text{D}}^0=a_1^{\text{D}}\exp \left(-k_1^{\text{D}}{t}_{\text{min}}\right)+a_2^{\text{D}}\exp \left(-k_2^{\text{D}}{t}_{\text{min }}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+\left(1-a_1^{\text{D}}-a_2^{\text{D}}\right)\exp \left(-k_3^{\text{D}}{t}_{\min }\right),\end{array}$$ (19)

where F_D and F_nD represent the total binding frequency at DNA and non-DNA locations, respectively, and $f_{\text{nD}}^{\text{ML}}$ stands for f_nD evaluated on the maximum likelihood estimators obtained by fitting the non-DNA locations data. As before, we jointly fit the censored and uncensored data using the maximum likelihood method ⁴⁵.

Simulation of search times

In order to calculate the mean search time ⟨t_f⟩ required for the RNAP to find the secondary promoter, we used numerical simulations of the mechanism depicted in Fig. 1. In particular, we used the Gillespie algorithm ⁴⁶ to generate the stochastic trajectory of an RNAP molecule on a DNA molecule. In the simulation, the state of the system is characterized by the position of the RNAP on DNA and whether it is bound to σ⁷⁰ or not. Initially, the RNAP is at position x = 0 (which corresponds to the position of the terminator from the primary transcription unit), and it is not bound to σ⁷⁰. We then draw a time t₁ at random from an exponential distribution with p(t) = λ exp (−λt) with λ = k_b[σ⁷⁰] + k_off, and choose between two possible transitions: binding σ⁷⁰ or dissociating from the DNA. Which of the transitions takes place is chosen at random according to their relative weights $\frac{k_{\text{b}}\left[{\sigma }^{70}\right]}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off}}}$ and $\frac{k_{\text{off }}}{k_{\text{b}}\left[{\sigma }^{70}\right]+k_{\text{off }}}$. If the RNAP dissociates from the DNA, its attempt to find the secondary promoter is considered unsuccessful, and the simulation starts over with a new trial. If the RNAP binds σ⁷⁰, the position on the DNA x₁ at which binding occurs is dependent on the amount of diffusion away from the primary terminator. x₁ is determined by drawing at random from a normal distribution with mean μ = 0 and variance var = 2Dt₁. The time t₂ that is required to diffuse from x₁ to the secondary promoter located at x_p is then drawn at random from the first passage time density

$$p\left(t_2\right)=\frac{\left|x_p-x_1\right|}{\sqrt{4\pi D{t}_2}}\exp \left(-\frac{{\left(x_p-x_1\right)}^2}{4D{t}_2}\right).$$

An RNAP-σ⁷⁰ complex dissociation time t₃ is drawn at random from an exponential distribution with λ = k_off,s. If t₂ < t₃, the RNAP-σ complex is considered to have found the secondary promoter at time t_f = t₁ + t₂. If t₂ > t₃, the attempt to find the secondary promoter was unsuccessful. The whole process is repeated multiple times, generating a distribution of search times.

Genome-wide analysis of terminator-promoter distances

Using the promoter and terminator annotations reported by Conway et al ²⁹, we measured the distance of each operon-ending terminator to the nearest operon initial promoter in the E. coli genome.

SIGNIFICANCE STATEMENT.

After transcribing an operon, a bacterial RNA polymerase can stay bound to DNA, slide along it, and reinitiate transcription of the same or a different operon. Quantitative single-molecule biophysics experiments combined with mathematical theory demonstrate that this reinitiation process can be quick and efficient over gene spacings typical of a bacterial genome. Reinitiation may provide a mechanism to orchestrate the transcriptional activities of groups of nearby operons.