Modeling method overview

We simulate a one-dimensional topological domain containing a single gene of length L, with two flanking regions of distance D to the domain barriers, giving an overall domain size of 2D + L (Fig 1). Our model is based on a modeling approach conceptually similar to several previous works, including ours [16,21,28–30], with differences and improvements summarized in S1 Table and in the dedicated paragraph below. The main elements of the model are described in Table 1 and are described in detail in the following paragraphs. At each timestep (of 0.04 s), transcription initiation may occur following a one-step SC-dependent stochastic initiation rate. Elongating RNAPs move by one bp, with a stalling threshold on upstream and downstream SC, upon reaching the terminator. Elongating RNAPs affect the SC levels upstream and downstream, which diffuse instantaneously (at the timescale of transcription and over kb-scale distances) and are modulated by the stochastic activities of topoisomerases.

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Table 1. Summary of deterministic and stochastic model processes.

RNAP elongation, termination, and SC diffusion are simulated deterministically. Elongation occurs at a rate of v_RNAP when the upstream and downstream SC levels are within an acceptable range, and stalls otherwise. Termination occurs instantaneously when an RNAP reaches the terminator. SC diffusion occurs instantaneously whenever topoisomerases act or RNAPs bind, unbind, or elongate. Initiation, topoisomerase binding, and RNA decay are simulated stochastically. Initiation occurs at a rate with a negatively sloped sigmoidal dependence on the local SC density, σ. This sigmoidal curve is defined by its midpoints, σ_i, and width, β_i. Gyrase binding has a positively sloped sigmoidal SC dependence defined by similar parameters σ_G and β_G, and TopoI binding has a negatively sloped sigmoidal SC dependence defined by σ_T and β_G. RNA decay occurs according to a Poisson process with rate t_RNA^-1.

https://doi.org/10.1371/journal.pcbi.1012764.t001

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Fig 1. Model of supercoiling-coupled transcription.

(A) Model setup. For transcription, we consider the following fives aspects: (1) initiation, a single step process dependent on SC; (2) elongation, which introduces positive and negative supercoils downstream and upstream respectively; it occurs at a constant rate except when stalled under extreme supercoiling levels; (3) termination, which occurs instantaneously and produces a single RNA transcript; (4) topoisomerase activities, including gyrase and TopoI; and (5) RNA degradation, which occurs via a Poisson process. The gene length, L, is the distance between the transcription start site (TSS) and terminator; the barrier distance, D, is the sequence length between the TSS and upstream topological barrier and between the terminator and downstream topological barrier. (B) Modeled relative activity (normalized to the maximum possible rate) of gyrase binding (green), transcription initiation (orange), and TopoI binding (blue) at varying SC densities (σ). (C) Gyrase preferentially binds positively supercoiled DNA and decreases the linking number by 2 per catalytic cycle. Gyrase acts processively, performing four catalytic cycles per binding event on average. TopoI preferentially binds negatively supercoiled DNA and increases the linking number by 1 per binding event. (D) Example simulation time courses of supercoiling density (σ) upstream (blue) and downstream (red) of the gene under three different conditions: normal, insufficient Topo I activity, and insufficient gyrase activity. (E) Corresponding time courses of produced RNA copy number and number of bound RNAP molecules under the three conditions. The stalling SC thresholds, ± σ_s, are shown as black dashed lines. Under the normal condition (left, gyrase activity 0.24 Lk/kb/s, TopoI activity 0.24 Lk/kb/s), transcription maintains continuous elongation with infrequent stalling. Under the insufficient TopoI activity condition (middle, gyrase activity 0.24 Lk/kb/s, TopoI activity 0.095 Lk/kb/s), transcription frequently stalls due to accumulated negative SC. Under the insufficient gyrase activity condition (right, gyrase activity 0.095 Lk/kb/s, TopoI activity 0.24 Lk/kb/s), transcription also stalls frequently due to accumulated positive SC. Under normal conditions, the RNA copy number consistently exceeds the RNAP copy number, indicating that RNAPs quickly transcribe the entire gene. In contrast, when either topoisomerase is insufficient, the RNA copy number decreases while the RNAP copy number increases, indicating that the RNAPs are stalled during elongation and spend more time on the gene. All examples are for strong genes (k_i = 0.2 s^-1) with intermediately distant barriers (D = 10 kb). All cartoons were created with BioRender.com.

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Computation of SC values

On a stretch of DNA, SC is quantified by the difference between the linking number—the number of times the DNA strands wind around each other—of the DNA stretch and that of an equivalent length of relaxed B-form dsDNA. The linking number consolidates both the twist (the wrapping of DNA strands around their helical axis) and writhe (the wrapping of the DNA duplex along itself) into a single number; as both configurations present similar topological constraints to RNAP, and supercoiling density is only related to the change in the linking number, we do not consider the distinction between these two forms of SC.

We consider a list of DNA-bound RNAP molecules (RNAP₁, …, RNAP_k), with RNAP₁ being furthest upstream (or the newest RNAP molecule on the DNA). Each RNAP is considered to occupy a DNA stretch of length w = 30 bp to which other RNAP molecules cannot bind. For each RNAP_i, we track the location of its center, x_i, and the linking number of the DNA segment separating it from the nearest upstream RNAP_i-1 or barrier, Lk_i^(up), and the nearest downstream RNAP_i+1 or barrier, Lk_i^(down). From these linking numbers, we calculate a supercoiling density upstream (σ_i^(up)) and downstream (σ_i^(down)) of each RNAP molecule. The supercoiling density on a stretch of DNA is then defined by the formula

where Lk is the linking number of a DNA stretch and Lk₀ is the linking number of the most relaxed state of the same length of DNA (10.5 bp^-1). To manage the case where no RNAPs are bound, we also track Lk_dom, the linking number of the domain. We initialize Lk_dom based on the initial supercoiling density, σ_start, an arbitrary value from which the simulation equilibrates. In the absence of topoisomerases, Lk_dom is a constant value. Our simulation utilizes time steps of length Δt = 1/v, where v = 25 bp/s is the RNAP velocity [3,42].

Initiation

We consider transcription initiation to occur in a single step facilitated by negative SC values. We assume that the concentration of RNAP is in great excess of DNA, so the concentration of the unbound RNAP pool stays constant. When initiation occurs, a new RNAP molecule is created with its center at the promoter position , where w is the footprint of an RNAP molecule, so that the downstream end of the RNAP is at the transcription start site (TSS). This newly added RNAP is indexed as RNAP₁, and the indices of the existing RNAPs are increased by one (RNAP₁ becomes RNAP₂, RNAP₂ becomes RNAP₃, etc.). Linking numbers are calculated and updated for the newly bound RNAP molecule and, if present, the adjacent downstream RNAP molecule. Initiation cannot occur if the promoter is occluded by an already-bound RNAP molecule, i.e., when RNAP(s) are bound and

When the promoter is free, the initiation rate is dependent on the supercoiling density, σ, (Fig 1B) according to the following sigmoidal function, where k_i′ is the initiation rate; k_i is the maximal initiation rate (under infinitely negative SC); σ_i is the SC density at which k_i′ is half of k_i; and β_i is the crossover width, which characterizes the sensitivity of k_i′ to changes in σ:

Based on in vitro data on transcription, we note that initiation reaches a maximum around σ = -0.05 and declines to a very low level around σ = -0.03 to σ = -0.01 [38]. To match this curve, we therefore use σ_i = -0.04 and β_i = 0.005 except where specified otherwise.

Elongation

At each timestep, the position x_i of each RNAP molecule (RNAP_i) is increased by one bp, commensurate with a basal elongation speed of 25 bp/s [3,42], in the absence of stalling. Stalling occurs for RNAP_i at position x_i if σ_i^(up) <-σ_s or σ_i^(down)> σ_s, where σ_s= 0.062 is the stalling threshold calculated from experimental observations [3,21]. We assume that the polymerase is sufficiently drag-limited that only the DNA, and not the RNAP, rotates during elongation, meaning we hold Lk_i^(up) and Lk_i^(down) constant during elongation. After each translocation, supercoiling densities for the translocated and neighboring RNAP molecules are then recalculated based on their updated distances. The average elongation speed of each RNAP molecule is calculated by dividing the gene length by the time the RNAP molecule spends from initiation to termination. Thus, a frequently stalled RNAP molecule will have a slow elongation speed.

We note that several approaches have been taken to model RNAP stalling. While our work, as in [21], considers the upstream and downstream torques to act independently, others base stalling on the net torque experienced by RNAP, defined by a function of the difference between upstream and downstream superhelical densities [25,30]. Still others use a combination of both approaches [28]. Considering the upstream and downstream torques independently and deterministically has the advantage that the upstream and downstream torques required to stall RNAP have been individually determined by experiment [3,28], but no similar data exists for torques exerted simultaneously upstream and downstream. Still, future work should isolate the consequences of these varied models of RNAP elongation, including varied deterministic and stochastic stalling criteria.

Termination and RNA

Termination occurs deterministically and instantaneously when an RNAP molecule reaches the termination site, i.e., x_k = L + D. Each termination event increases the RNA copy number by one. Upon termination, RNAP_k is removed. If there are additional RNAP molecules bound to the gene, the linking number of the new downstream RNAP molecule after removal is updated to reflect the merged domains.

We assume that RNA molecules are degraded by a Poisson process, where each RNA molecule is assigned a lifetime drawn from an exponential distribution with expected lifetime t_RNA, after which it is removed from the simulation. We use t_RNA = 120 s, on par with past experimental measurements [42].

We quantify transcriptional noise by calculating the Fano factor, calculated from the distribution of RNA copy numbers at each time point of the simulation by dividing the variance by the mean. This statistic is the natural choice for discriminating between Poisson and bursty transcription because a Poisson process has a Fano factor of one regardless of the mean level. Transcriptional bursting (bunched RNA production during “on” periods separated by inactive “off” periods) is characterized by a Fano factor greater than one.

Topoisomerases

Topoisomerase activities are simulated stochastically. Gyrase and TopoI bind nonspecifically upstream and downstream of the gene. TopoI binds to the DNA at a rate determined by a decreasing sigmoidal function of SC defined by parameters k_T, σ_T, and β_T, where k_T is the TopoI binding rate under optimal conditions, σ_T is the SC density where gyrase binds at half its maximum rate, and β_T is the crossover width of the SC dependency. The dependence of TopoI binding on SC density has not been fully determined experimentally. Therefore, except when noted otherwise, we take σ_T = σ_i = -0.04 and β_T = β_i = 0.005 on the grounds that both promoter opening and TopoI activity require DNA melting [10], as depicted in Fig 1B. These values approximately match existing indirect estimates [43]. However, a dependence on the DNA sequence is also expected and might be characterized in future studies (with possibly significant impact on the interplay of topoisomerase I with transcription initiation). We did not model any direct interaction between TopoI and RNAP. Gyrase binds at a rate determined by an increasing sigmoid function of SC with analogous parameters k_G, σ_G, and β_G. In line with available experimental data [44,45], we take σ_G = 0 and β_G = 0.015, also shown in Fig 1B.

When a TopoI molecule binds to a DNA stretch, it increases the linking number by one and unbinds instantaneously. This linking number is tracked by both adjacent RNAP molecules and the domain linking number (Lk_dom).

When a gyrase molecule binds to a DNA stretch with a threshold of σ > σ_⍴ = -0.11, it can perform multiple catalytic cycles before unbinding, as previously shown experimentally [46]. Gyrase processivity is simulated by drawing the number of catalytic cycles from a geometric distribution with mean ρ_G = 4 (obtained by fitting the distribution from [46]), where ρ_G describes the average number of catalytic cycles per binding event. We then cap the number of cycles such that gyrase never acts on DNA with σ < σ_ρ = -0.11; this value has been shown experimentally to be the lower bound on the SC level that gyrase can bind [44]. This procedure is equivalent to simulating successive catalytic cycles following the experimentally observed distribution of linking number changes introduced by gyrase [44,46]. Each catalytic cycle decreases the linking number by two, which is handled the same way as with TopoI. When reporting “topoisomerase activities” for plotting purposes, we report 2k_Gρ_G as the gyrase activity and k_T as the TopoI activity to ensure comparable rates in units of Lk/kb/s.

Simulation

Our model is implemented in Python 3.9.18 using NumPy 2.1.3. Analysis and plotting were performed in R 4.3.1. Initiation, topoisomerase activities, and RNA decay are simulated stochastically (Table 1) by randomly selecting the waiting time before each upcoming event from the appropriate time distribution. Within each 0.04 s timestep (the time for an RNAP molecule to elongate by one bp), we resolve any stochastic events scheduled to occur before simulating elongation and termination deterministically. Supercoiling levels are updated instantaneously whenever any SC-altering event occurs. Sections of simulation code were taken from [21], and new sections were introduced to account for our treatment of initiation, topoisomerase activities, and RNA degradation. Parameter choices are indicated in Table 2. All code for our simulations and analysis is available on GitHub (DOI: https://doi.org/10.5281/zenodo.14586118).

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Table 2. Model parameter definitions and values.

https://doi.org/10.1371/journal.pcbi.1012764.t002

Comparison to previous models

The main additions over existing works are the description of (nonspecific) topoisomerase activity, for which we employ a gradual response function to local SC, and the introduction of gyrase processivity, based on experimental evidence [44,46,47]. Compared to [21], we do not add a sequence-specific term in topoisomerase activity. Compared to [16,28], we do not consider the possibility for RNAP to rotate during elongation. While several of these modeling ingredients remain subject to improvements, we verified that the main conclusions of this study are robust to minor changes in choices of functions and parameters regarding initiation and topoisomerase activities. For example, we tested both logistic and step functions for initiation and topoisomerase binding rates and did not observe qualitatively different conclusions. We chose to adopt the logistic functions because the width of the curve did not modify the overall results (within a reasonable range of values). Therefore, the use of logistic functions enhances the stability of our simulations in response to small changes in SC levels without significantly increasing the model’s parameterization.

Transcription of a freely rotating plasmid without topoisomerases

We simulated the case of a gene transcribed on a circular plasmid without topoisomerases (as in a usual in vitro setup), where positive and negative SC introduced by elongation diffuse around the plasmid and annihilate one another. This annihilation leads to a constant global SC level. We simulated short genes (L = 500 bp) with a strong promoter (k_i = 0.4 s^-1) that opens at different SC densities (σ_i = -0.06, -0.05, and -0.04). Since our simulation scheme was designed for linear DNAs, we approximated a circular plasmid by using extremely distant barriers (D = 10⁹ bp), resulting in negligible SC accumulations, which mimic the annihilation of positive and negative supercoiling around a plasmid. While this approximation differs from the desired “reflective boundaries” simulation method, it is computationally cost-effective, and we confirmed that the SC level fluctuations caused by RNAP elongation were less than 0.0003, insignificant in the context of our model to cause RNAP stalling.

For each condition, we ran 21 simulations, varying the SC density from σ_start = -0.1 to σ_start = 0 in steps of 0.005. Simulations were run for 10⁶ timesteps (11.11 hours simulated time), using approximately 1.7 CPU minutes each. To calculate the transcription rate (number of RNA/time), we divided the number of termination events (i.e., the number of produced RNA) by the total time of each simulation for individual plasmid topoisomers. To compare the simulations with transcription levels from a distribution of topoisomers, we assumed a normal distribution of linking numbers with a standard deviation of 0.004, a typical value for plasmids extracted from cells and measured by agarose-chloroquine gels [17]. We then calculated the transcription rates for distributions of plasmid topoisomers with mean SC densities ranging from -0.1 to 0.

Transcriptional regulation of a topologically isolated gene

We simulated transcription of a gene on linear DNA flanked by topological barriers in the presence of topoisomerases, mimicking a topological domain in vivo. In all simulations, we used a gene length of L = 1 kb. We varied four variables between these simulations: promoter strength (k_i), barrier distance (D), TopoI binding rate (k_T), and gyrase binding rate (k_G). Promoter strengths were chosen based on the expected number of RNAP molecules on the gene, assuming conditions permissive of initiation and elongation to simulate different transcription regimes: a weak promoter (k_i = 0.008 s^-1, average RNAP molecules per gene = 0.32) simulates intermittent transcription, a moderate promoter (k_i = 0.05 s^-1, avg. RNAP molecules per gene = 2) simulates mostly continuous transcription with occasional breaks, and a strong promoter (k_i = 0.2 s^-1, avg. RNAP molecules per gene = 8) simulates continuous transcription. As a control, we also consider the case of no transcription. We consider barrier distances of D = 1 kb (close barriers), 10 kb (intermediate barriers), and 100 kb (far barriers).

For each barrier distance, we calculate the steady-state binding rates of both TopoI and gyrase such that TopoI removes the upstream negative SC and gyrase removes the downstream positive SC at the same rate that SC is introduced by elongation, assuming TopoI and gyrase bind at their maximum rates. We term these values k_T^* and k_G*, and we use them to guide our parameter choices, as we would expect different behaviors when k_T < k_T* and k_T > k_T*, and likewise when k_G < k_G* and k_G> k_G*. We simulate each combination of k_T and k_G achieved by varying k_T from 0 to 2k_T^* in increments of 0.1k_T^* and k_G from 0 to 2k_G^* in increments of 0.1k_G^*. Note that k_T^* and k_G* are inversely proportional to the barrier distance D (since we assume non-specific binding of topoisomerases). We run additional sets of simulations with 10-fold increased and decreased topoisomerase activities compared to those described above to enable comparisons between the effects of altering barrier distance and topoisomerase activities. To simulate the effects of topoisomerase inhibition, we run simulations with a 5-fold decreased gyrase or TopoI activity separately in the conditions of 10 kb and 100 kb barrier distances. Finally, we run simulations to match the experimental setup of [21]: we consider an upstream barrier distance of 250 bp and a downstream barrier distance of 320 bp; we then incrementally increase the upstream barrier distance to 3200 bp or the downstream barrier distance to 3408 bp independently, calculating the ratio in expression strength (Far/Close).

We therefore generated 6237 total simulations (11 TopoI conditions, 11 gyrase conditions, 4 promoter conditions, 3 barrier conditions, and 3 topoisomerase conditions for the primarily analysis; 11 TopoI conditions, 11 gyrase conditions, 3 promoter conditions, and 2 barrier conditions for the individual topoisomerase inhibition simulations; 11 TopoI conditions, 11 gyrase conditions, 3 promoter conditions, and 3 barrier conditions for varying the upstream/downstream distances independently; and 3 promoter conditions and 22 barrier conditions for the fine-grained varying of upstream/downstream distances). For each simulation, we run timesteps (22.22 hours simulated time), utilizing approximately 4 CPU minutes per simulation. In most cases, we disregard the first 45,000 timesteps (30 minutes) to focus on the steady-state behavior of transcription (in cases with 100 kb barrier distances, we had to remove 60 minutes of data, as the simulations took longer to reach the steady state). To isolate the impact of topological variables on transcription from the strength of the promoter, we calculate a normalized transcription rate by dividing the observed average transcription rate by the maximal promoter initiation rate (k_i). We also calculate a “free promoter fraction”, representative of the proportion of time steps during which the promoter is not blocked from binding by an RNAP. Low free promoter fraction indicates that transcription is repressed by slow elongation, as RNAP does not clear the promoter in a timely manner. This value is calculated by dividing the observed transcription rate by the mean value of the effective initiation rate given the promoter SC density (k_i’). In our model, all RNAPs that initiate eventually produce an RNA transcript, and initiation is determined exclusively by the promoter SC density and the occupancy of the promoter by RNAP. Hence, this value isolates the effect of promoter occupancy and characterizes the fraction of time during which the promoter is free for RNAP binding.

Simulation of in vitro transcription recapitulates optimal expression at physiological negative superhelical levels

To validate our model, we first compared its predictions to in vitro transcription data obtained using circular plasmids of different superhelical densities incubated with RNAP. These data have been a major source for identifying the SC-sensitivity of bacterial promoters [36–38]. Because supercoils generated by transcription can diffuse and annihilate each other in a circular freely rotating plasmid, the SC level can be approximated as constant in such templates. Previous experiments have shown that a gene typically transcribes maximally at an optimal negative SC level and exhibits decreased transcription when the negative SC level is too low or too high (S1 Fig). However, no stochastic model has reproduced this non-monotonic behavior, a basic yet crucial observation of transcription in response to negative SC.

In our simulations, transcription ceased completely at very negative SC densities (σ < σ_s) due to RNAP stalling (Fig 2A, left side of the red vertical line), and decreased continuously when the SC density increases above this threshold (Fig 2A, right side of the red vertical line), due to a reduced initiation rate. This transcriptional response to the SC density of the plasmid holds true for promoters with different initiation dependencies on SC (σ_i, Methods) (Fig 2A, compare data points of different colors).

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Fig 2. Transcription on Freely Rotating Plasmids.

We simulate transcription on circular plasmids in the absence of topoisomerases, mimicking in vitro transcription assays. (A) Transcription rates of three promoters that open at different σ_i, simulated on plasmids with different SC densities, σ, but with only a single topoisomer. The RNAP stalling threshold σ_s is indicated in red. (B) Transcription rates of the same three promoters were simulated on plasmids with normally distributed SC densities with multiple topoisomers, which mimics typical topoisomer distributions of plasmids extracted from cells [36–38]. Inset plots show the distribution of SC densities for plasmids of a given mean SC level (a Dirac delta function for A, and a Gaussian distribution for B).

https://doi.org/10.1371/journal.pcbi.1012764.g002

To further compare with experiments performed on plasmid samples of different mean SC densities and with a distribution of topoisomers (with different linking numbers, as commonly extracted from cells), we computed the total expression level from the weighted averages of different templates (each corresponding to one plasmid topoisomer) and calculated the effective transcription rate of samples with different mean SC densities. Fig 2B shows the resulting continuous SC-dependency where transcription is maximized by a moderately negative SC density, and decreased in either direction from this optimum, recapitulating previous experimental results [36–38]. This result validates our model for different promoters, suggesting that it can faithfully recapitulate the non-monotonic dependence of transcription on negative SC levels. The very sharp decrease towards very negative SC levels suggests that in in vitro transcription systems, RNAP might experience weaker frictional drag due to the absence of translation and a less crowded environment than that simulated in our model.

Regulatory effects of topoisomerases’ activities depend on promoter strength

Next, we simulated transcription on linear DNAs with topological barriers in the presence of topoisomerase activities. This situation mimics typical transcription occurring in bacterial chromosomal topological domains. We simulated topological barriers at distances of 10 kb flanking a gene of a typical size, 1 kb, leading to a 21 kb topological domain, in line with experimentally reported sizes of chromosomal domains in E. coli [48]. TopoI and gyrase molecules can bind upstream or downstream of elongating RNAPs, depending on their binding rates and the local SC densities (see Methods). In the absence of transcription, the relative activities of the two topoisomerases maintain a domain-wide (homeostatic) SC level of about σ = -0.06 to -0.02, consistent with that typically measured in bacterial cells (S2A Fig). Meanwhile, transcription substantially alters the spatiotemporal distribution of SC in the domain (S2C Fig) while maintaining a similar average domain-wide SC density (S2B Fig).

In our simulations, we define the transcription rate as the number of RNA molecules generated per unit time, normalized to the maximal transcription rate that the promoter can achieve (Fig 3, column i). Note that, as we do not consider premature termination in our simulations, all RNAP molecules that initiate will eventually produce a transcript. Therefore, the transcription rate should be equivalent to the transcription initiation rate and independent of the elongation speed (Fig 3, column ii). However, an RNAP molecule may stay at the promoter and occlude the binding of the next RNAP molecule to the promoter if it is stalled at the promoter due to an unfavorable SC level or blocked from moving past the promoter due to piled, stalled RNAP molecules in the gene body. Therefore, a slow elongation speed, which reflects stalling during initiation and/or elongation, could impact the transcription rate through the availability of the promoter to the next RNAP molecule. To illustrate this point, we calculated the free promoter fraction, which is the proportion of time steps during which the promoter is free for RNAP binding (Fig 3, column iii). As such, the free promoter fraction is a direct indicator of the impact of RNAP’s stalling on the transcription rate, with lower free promoter fractions indicating transcription repression caused by stalled RNAP. Fig 3A-C shows the values of these statistics for all topoisomerase conditions we tested, while rows Fig 3D-E isolate the effects of either TopoI or gyrase activity.

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Fig 3. Regulatory effects of topoisomerases on transcription.

(column A) Mean normalized transcription rate, k_obs/k_i; i.e., the expression strength normalized by the promoter strength. (column B) Mean elongation speed. (column C) Free promoter fraction, i.e., the proportion of timesteps during which the promoter is not sterically occluded by a bound RNAP molecule. We consider a 1 kb gene flanked by 10 kb distances to topological barriers. We vary topoisomerase activities (k_G from 0 to 2k_G^* and k_T from 0 to 2k_T *, where the k_G* and k_T* and the topoisomerase activities calculated to remove the supercoiling introduced by continuous elongation) and plot heatmaps of each of these variables calculated for weak (row i, k_i = 0.008 s^-1), moderate (row ii, k_i = 0.05 s^-1), and strong (row iii, k_i = 0.02 s^-1) promoters. We also fix either k_T = k_T^* (row iv) or k_T = k_T^* (row v) and vary the other topoisomerase’s activity (i.e., we trace along the blue lines in rows i-iii) and plot each statistic described above.

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From these simulations, we observe that TopoI and gyrase impact promoters of varied strengths in a qualitatively different manner, with each promoter reaching its maximal transcription rate at a different TopoI activities (Fig 3D) and having different sensitivities to gyrase activities (Fig 3E). Specifically, for a weak promoter, high gyrase activity (leading to more negative SC densities) increases the transcription rate (Fig 3A and 3Ei), while a high TopoI activity (leading to less negative SC density) inhibits the transcription rate (Fig 3Ai and 3Di). Interestingly, the highest transcription rate of the weak promoter does not correspond to the fastest elongation rate (compare Fig 3Ai to 3Bi). In other words, slow elongation speed, which signals frequent RNAP stalling, does not appear to reduce the overall transcription rate for the weak promoter. However, as the free promoter fractions across different topoisomerase conditions remain high (Fig 3Ci), it indicates that the weak promoter is limited by transcription initiation but not RNAP stalling. The promoter is maximally transcribed under topoisomerase conditions that result in the most negative SC (low TopoI and high gyrase activities), which facilitate promoter opening and elongation to increase the transcription rate. Under this condition, RNAP stalling due to insufficient topoisomerases’ activities still occurs; however, it does not substantially affect the transcription rate because the initiation rate is limiting.

As we increase the promoter strength, we observe markedly different behaviors. Moderate and strong promoters reach their maximum transcription rate at moderate TopoI activities and moderate to high gyrase activities. Increasing TopoI activities increases the elongation speed until minimal stalling occurs, at which point the free promoter fraction is near one. Past this point, the addition of TopoI increases the SC density at the promoter, thus slowing initiation and, therefore, transcription. As gyrase activity increases, a maximum free promoter fraction is reached more quickly. However, additional gyrase continues to increase transcription by decreasing SC density at the promoter, facilitating initiation. The strong gene requires more TopoI activity to reach a free promoter fraction of one, suggesting that strong genes are more sensitive to RNAP-stalling-mediated transcription repression than weak to moderate genes.

Taken together, our simulations show that the transcription rate can be regulated by combined Topo I and Gyrase activities through the availability of the promoter. A high free promoter fraction, such as that in the case of the weak promoter, signals that the promoter SC level is limiting for RNAP to initiate transcription. Meanwhile, a low free promoter fraction, such as that in the case of the strong promoter, suggests that RNAP stalling is limiting by making the promoter unavailable for the next RNAP molecule. Specifically, in conditions where the SC-dependent initiation rate, k_i′, is low, the free promoter fraction is high, and transcription is regulated primarily by the promoter SC level for initiation. Therefore, transcription is enhanced by increasing gyrase activity and decreasing TopoI activity to increase negative SC for promoter opening. As k_i′ increases, the increased frequency of RNAP binding means that the free promoter fraction becomes sensitive to RNAP stalling. Thus, the transcription rate becomes highly dependent on the presence of adequate levels of both TopoI and gyrase to keep up with the SC introduced by elongation to avoid stalling. With high k_i′, the favorability of the promoter SC level for initiation becomes relevant only when comparing multiple conditions in which TopoI and gyrase are sufficient for avoiding stalling. In general, fast initiation leads to transcription being dependent on RNAP stalling, while slow initiation leads to transcription being dependent on the promoter SC level.

Effects of topological domain size are mediated by the binding of topoisomerases

We next investigated the dependency of transcription on topological domain size. Specifically, we reduced and lengthened barrier distances to investigate the effects of introducing and resolving topological constraints. We keep the same gene length (1 kb) and promoter strengths and show the log2-fold change in transcription rate in response to reducing the barrier distances from 10 kb to 1 kb as a function of TopoI and gyrase activities (Fig 4A). This results in changes to transcription that are more positive for weaker genes and high TopoI conditions. We note that these effects are similar to the effects of reducing the topoisomerase activities 10-fold (Fig 4B) and are ameliorated by simultaneously decreasing the barrier distance and increasing the topoisomerase activities 10-fold (Fig 4C). This relationship between changes to barrier distances and topoisomerase activities was expected as in our model, topoisomerases act nonspecifically, meaning that proportionally fewer binding events of topoisomerases occur on the smaller domain. This shortening of barrier distances mimics cells in which chromosomal domains form in the presence of the same cellular pool of topoisomerases.

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Fig 4. Effects of barrier distances mediated by topoisomerase binding.

We fix either gyrase (left) or TopoI (right) activity at the steady-state value (blue lines in Fig 3, computed as the value sufficient to remove the SC induced by elongation), and vary the other topoisomerase’s activity. We plot the log2 fold change in the expression rate in response to (A) reducing the barrier distances from 10 kb to 1 kb, (B) reducing nonspecific topoisomerase activities 10-fold, (C) reducing the barrier distances from 10 kb to 1 kb and increasing topoisomerase nonspecific activities 10-fold to compensate, (D) increasing the barrier distances from 10 kb to 100 kb, (E) increasing the nonspecific topoisomerase activities 10-fold, and (F) increasing the barrier distances from 10 kb to 100 kb and decreasing the nonspecific topoisomerase activities 10-fold to compensate. Cartoons were created with BioRender.com.

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When we increase the barrier distance from 10 kb to 100 kb, we note that at moderate to high TopoI levels, weaker genes are enhanced while stronger genes are repressed; this trend reverses at low TopoI activities (Fig 4D). This change mimics the opening of a topological domain. This effect is replicated by increasing topoisomerase activities 10-fold while holding the barrier distance constant (Fig 4E) and ameliorated by simultaneously increasing barrier distances and decreasing topoisomerase activities 10-fold (Fig 4F). The full data for this analysis, including all simulated topoisomerase conditions, are shown in S4 Fig [49–51].

Taken together, these observations suggest that the size of a topological domain imposes an additional layer of transcription regulation, primarily through the availability of topoisomerases in the domain. The formation of a smaller topological domain reduces the number of topoisomerase binding events, altering the local SC level relevant to the genes within the domain. Strong genes in smaller domains are hence repressed by the inability of topoisomerases to keep up with the SC introduced by elongation. Meanwhile, weak to moderate genes respond differentially depending on the combination of the two topoisomerase activities. On large domains, transcription is favored by low TopoI and high gyrase levels, as the effective topoisomerase activities increase, resulting in a high free promoter fraction. The interaction between promoter strength, topoisomerase binding rates, and topological context must therefore be considered to quantitatively understand a gene’s regulation, consistent with [21].

SC modulation of the off-state leads to transcriptional bursting

Transcriptional bursting is the phenomenon of a gene possessing multiple states in which transcription occurs at different rates (typically on or off) [52]. The bursting phenomenon results in transcriptional noise greater than that of a random birth-and-death Poisson process, typically inducing variable gene expression in an isogenic cell population, which enhances its resilience to environmental uncertainty [34]. Since SC modulates transcriptional initiation in our model, we wished to understand whether it could serve as a generator of transcriptional bursting in the absence of additional factors such as stochastic binding of regulators or other proteins affecting the promoter accessibility. We therefore analyzed the simulations to characterize the conditions that result in transcriptional bursting and identify underlying mechanisms.

The level of noise can be quantified using the normalized variance of expression (variance divided by the square of the mean). We computed this normalized variance at all promoter strengths, topoisomerase binding rates, and barrier distances, and observed that this parameter decreases with the expression level, making it challenging to compare simulations involving different promoter strengths (Fig 5A). Rather, we computed Fano factors (variance divided by the mean) for the same simulations (Fig 5B). We note that the Fano factor is not scale-free (i.e., the Fano factor depends on the transcription strength), however regardless of scale a Fano factor of one indicates Poissonian expression (i.e., constant initiation rate), whereas values greater than one indicate transcriptional bursting. We observed Fano factor values ranging from 0.7 to 4.8, with bursty transcription appearing in diverse promoter and topological conditions. The datapoints separate horizontally along the axis of mean RNA. Copy numbers belong to three clouds corresponding to simulations with weak, moderate, and strong promoters, respectively. For most conditions, the distance to barriers has little impact on transcription burstiness. However, promoter strength appears to play a role. Weak promoters’ transcription is close to Poissonian (Fano factors close to 1, squares, Fig 5B), with moderate promoters exhibiting consistently higher Fano factors (about 1–2, triangles, Fig 5B), indicative of moderate burstiness. We observed a large number of simulations exhibited Fano factors below 1, i.e., sub-Poissonian, suggesting that transcription becomes less random because initiation events are more regularly spaced by the time RNAP molecules take to clear the promoter.

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Fig 5. Mechanism of Transcriptional Bursting.

(A) Normalized variance (noise) and mean RNA copy numbers for simulations run using varied promoter strengths (squares, k_i = 0.008 s^-1, moderate: triangles, k_i = 0.05 s^-1, strong: circles, k_i = 0.2 s^-1), barrier distances (close: red, D = 1 kb, intermediate: green, D = 10 kb, far: blue, D = 100 kb), and topoisomerase binding rates (one point per simulation). (B) Fano factors and mean RNA copy numbers for the same simulations. (A-B) are shown on a log-log scale. (C) Fano factors for a moderate gene (k_i = 0.05 s^-1) flanked by intermediate barriers (D = 10 kb). Squares correspond to individual simulations with varied topoisomerase activities. Cyan lines denote the steady-state topoisomerase activities calculated to eliminate the supercoils generated by elongation. Scenarios (1), (2), and (3) are analyzed in the remainder of the figure. (D) Distributions of RNA copy numbers (black) throughout one simulation compared with the Poisson distribution with the same mean (blue). We indicate ±σ_s with black dashed lines. (E) Features relevant to transcriptional bursting are indicated, including an RNAP molecule binding to the gene and an RNAP molecule unbinding after completing transcription (middle) and gyrase and Topo I binding (right). Note that RNAP unbinding leads to the merging of the upstream (red) and downstream (cyan) SC domains. (F) The time course of copy numbers of RNA (green) and bound RNAP (purple) molecules.

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To further explore transcription bursting dynamics, we consider in-depth the case of a moderate promoter (k_i = 0.05 s^-1) with intermediate barriers (D = 10 kb). This promoter strength and domain size (green triangles in Fig 5B) might be representative of some bacterial genes and exhibit weakly bursty transcription independent of barrier distance (S5 Fig), matching experimental observations of Fano factors typically between 1 and 3 for native genes [34]. In Fig 5C, we show Fano factors for simulations of a moderate gene (k_i = 0.05 s^-1) flanked by intermediate barriers (D = 10 kb) for different combinations of TopoI and gyrase activities, and chose three scenarios for detailed analyses: scenario 1, with topoisomerase activities matched to the rate of SC introduced by elongation; scenario 2, with low gyrase activity; and scenario 3, with low TopoI activity. In scenario 1, the Fano Factor is 1.6 and the steady-state topoisomerase binding rates exactly cancel the SC introduced by elongation; in scenario 2, the Fano Factor is at 2.0 and the gyrase activity is lower than that in Scenario 1; in scenario 3, the Fano Factor is 0.88 and the TopoI activity is lower than that in Scenario 1. For each scenario, we consider the distribution of the RNA copy numbers (Fig 5D), the time course of the upstream and downstream SC density (Fig 5E), and the time course of the RNA copy number and number of bound RNAP molecules (Fig 5F).

Scenarios 1 and especially 2 describe a mechanism for transcriptional bursting in which the upstream negative SC induced by elongation facilitates the binding of additional RNAP molecules at the promoter, hence facilitating more frequent initiation events. In scenario 2, apparent “on-off” transitions are observed in accordance with a higher Fano factor at 2: the very low gyrase activity resulted in a relatively high SC density at the promoter that mostly, but not completely, suppresses initiation (the “off” state) when the gene is not bound by any RNAP molecules, whereas a single transcription event turns the gene into the “on” state by reducing the promoter SC level (Fig 5E). Whenever all RNAP molecules stochastically unbind, the positive SC accumulated downstream diffuses onto the promoter, suppressing initiation and returning the gene to the off state as the existing RNA fully decays. The off-state lasts on the order of 10 minutes, in line with experimental data [53]. Scenario 1 demonstrates on- and off-states by the same mechanism, but the off-state is shorter as the relatively high gyrase activity results in a steady-state SC level more favorable for initiation before previous RNA molecules are degraded in the off-state (Fano factor of 1.60). This mechanism implies a cooperative mode of multiple RNAP molecules as predicated previously and is interesting for its alignment with the experimental observation that transcriptional bursting is related to low gyrase expression [34]. Scenario 2 displays a greater Fano factor than scenario 1 (2.0 vs 1.6) because the greater steady-state SC level in the absence of bound RNAP molecules results in less frequent RNAP binding and a more prolonged off-state.

In scenario 3, we do not observe transcriptional bursting; in fact, we observe a sub-Poisson Fano factor (0.88). The SC level is strongly negative, almost always sustaining efficient initiation, hence a high transcription level, but so much so that elongation is frequently stalled due to insufficient TopoI activity. Therefore, several RNAPs are almost always bound to the gene, and the promoter never reaches an “off” state. Because these abundant RNAPs frequently block the promoter from binding additional RNAPs, initiation events are more evenly spaced than in a Poisson distribution, explaining the low Fano factor.

We conclude that transcriptional bursting is dependent on a steady-state SC level in the “off” state, which represses, although does not eliminate, initiation, while negative SC induced by elongation turns the promoter into a transitory “on” state (i.e., transcription is self-activating).

In vivo expression data suggest moderate topoisomerase regulation of transcription, dependent on species and growth conditions

Our modeling effort so far has demonstrated that changes in topoisomerase activities could lead to qualitatively different effects on transcription. It is yet difficult to predict what parameter values are relevant within live bacterial cells, since we ignore the quantitative topoisomerase activities affecting different topological domains, as well as the latter’s exact locations and boundaries. We therefore confronted our simulation results with in vivo expression data to identify if and what simulated conditions are compatible with the observed supercoiling regulation of transcription and to possibly infer some physiologically relevant information.

As a first source of data, we compiled previously published transcriptomic datasets [13–21,39] obtained in evolutionarily distant bacterial species (S2 Table), where cells were treated with a gyrase inhibitor (fluoroquinolones or coumarins) or TopoI inhibitor (seconeolitsine). These datasets provide the quantitative response of broad sets of genes, which differ by their topological context (nucleoid-associated proteins, neighboring genes, etc.), regulators, and surrounding sequences. Since the simulations above exhibited a strong influence of transcription strength on its sensitivity to topoisomerases, and the expression strength is experimentally available (in contrast, e.g., to the topological domain size of each gene), we looked to see if a similar relationship was detectable and robust across datasets. Because the latter are heterogeneous in the type of species, culture conditions, drug dosages, treatment time, and experimental methods used, we grouped genes into expression quartiles to improve readability. Fig 6A shows typical relations between gene expression strength and transcriptional response to gyrase inhibition, for Mycoplasma pneumoniae and Salmonella enterica. In both cases, we clearly observed that genes of higher expression were associated with transcriptional repression when gyrase is inhibited, albeit with a very different magnitude in the two species (the absolute expression ratio between the first and last quartiles is around 30-fold for M. pneumoniae vs around 2-fold for S. enterica). Note that in these analyses, the expression strengths are normalized to the total transcriptional output, so it cannot be concluded that weak genes are upregulated, but rather that their share in total mRNA numbers has increased after the treatment. We therefore show the relative ratio between weakly (1^st quartile) and strongly (4^th quartile) expressed genes in response to gyrase (Fig 6B) or TopoI (Fig 6D) inhibition for all datasets obtained in five species.

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Fig 6. Transcriptome Response to Topoisomerase Inhibition.

(A) Experimentally determined fold-change in expression of genes in Escherichia coli and Mycoplasma pneumonia in response to gyrase inhibition. Genes are split into quartiles based on expression strength. (B) Experimentally determined log2-ratio of fold-changes in strong (upper quartile) vs weak (lower quartile) expressed genes in response to TopoI inhibition. The symbol *** indicates P < 10^-3 for the associated t-test. The list of conditions is detailed in S2 Table. (C) Simulated log2-ratio of fold-changes in simulated transcription rate of strong vs weak genes (k_i = 0.02 s^-1 vs k_i = 0.008 s^-1) in response to a 5-fold reduction in gyrase activity (mimicking the action of gyrase inhibitors) for varied baseline topoisomerase activities (k_G ranging from 0 to 2k_G^* and k_T range from 0 to 2 k_T^*). (D) and (E) are the same as (B) and (C), respectively, but instead consider TopoI inhibition. Numbers in (C) and (E) indicate simulated topoisomerase conditions which produce log2-ratios compatible with the experimental results in (B) and (D) respectively (in several cases, we show one among several compatible conditions, see text).

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Two main conclusions can be drawn. First, all analyzed datasets exhibit a clear tendency towards repression of highly expressed genes when either gyrase or TopoI is inhibited, a notable robustness considering the strong heterogeneity of these datasets. Second, the magnitude of the effect is quite variable across datasets, and quite moderate for most of them (as illustrated for S. enterica in Fig 6A), with strong genes being repressed less than two-fold compared to weak genes. Together, these two observations suggest that topoisomerases contribute to the expression of strong bacterial genes, but only in a mild manner. Their effect also seems species- and growth condition-dependent, since D. dadantii is more affected by TopoI inhibition than S. pneumoniae (possibly due to a stronger drug dosage, both grown in minimal medium) while their sensitivity to gyrase inhibition is similar, and E. coli responded much stronger to novobiocin (dataset index 2) than to norfloxacin (dataset index 6) in rich medium. We obtained similar conclusions with an alternative calculation based on the correlation between expression strength (without arbitrary classification into quartiles) and expression fold-change.

We then tested if simulations could recapitulate the observed behavior, and in what parameter range. We note that while our model does not consider transcription factor binding and other complex regulatory elements in cells, it is meaningful to compare the simulation and experimental results qualitatively to identify the role and contribution of supercoiling in gene regulation. Furthermore, since the exact effect of drugs on topoisomerase activity in vivo is not precisely known (and likely variable in our dataset), we can only derive a qualitative comparison. We ran simulations mimicking an (arbitrary) 5-fold reduction in either enzyme, considering the log2 fold change in expression level induced in three representative genes of strong, intermediate, and weak promoter strengths (S6 Fig). Since we are interested in the relative effects of topoisomerase inhibition on genes of varied strengths, we show the ratio of these log2 fold changes between strong and weak genes in response to gyrase (Fig 6C) and TopoI (Fig 6E) inhibition (comparable to the experimental values in Fig 6B and 6D respectively) for simulations across all gyrase and topoisomerase activity values.

Contrary to naive expectations, the effect is quite variable, with gyrase inhibition resulting in a relative repression of strong genes only in a small part of the parameter space, at low gyrase activities and moderate TopoI activities (blue region of Fig 6C). In contrast, TopoI inhibition represses strong genes in almost all conditions (Fig 6E). When we consider 100 kb barriers (rather than 10 kb), we obtained essentially similar results (S6 Fig), showing some robustness to variations in simulation parameters. Although a direct comparison between our simulations and experimental values remains tentative due to the many confounding factors affecting in vivo transcription, we can therefore pinpoint a parameter range where the model is compatible with each dataset (see indices in Fig 6C and 6E). For two datasets, the combination of values observed with both gyrase and TopoI inhibition data (S. Pneumoniae, 4, and D. dadantii, 5) is satisfied only for specific combinations of topoisomerase activities (we indicated the most likely region for both cases with an index). The very strong repression observed in Mycoplasma pneumoniae (1) suggests a very low initial gyrase activity (or much stronger inhibition than in our simulation) coupled to moderate TopoI activity. For other species showing mild repression by gyrase inhibition, the parameter range is broader (light blue region of Fig 6C) but still well-defined to a regime of weak gyrase activity (along the whole left edge of the graph).

A second source of in vivo data is provided by recent expression measurements of a gene in a plasmid-borne topological domain (Boulas et al. [21], whose results we reproduce in S7A Fig), showing that it is sensitive to the upstream but not the downstream distance to topological barriers (S7 Fig). Using a similar approach to that used with transcriptomics data, we tested what range of topoisomerase activity values recapitulated this asymmetric sensitivity. We found a qualitative agreement with the data under low TopoI activity, and preferentially under high gyrase activity (S7B-C Fig), i.e., a different regime than that identified from transcriptomics data (this discrepancy is addressed in the Discussion). In this regime, our model recapitulates the general observed trend, although the quantitative effect of increasing the distance does not fully recapitulate the data (S7D Fig). This difference was expected since our model includes a minimal (nonspecific) description of topoisomerase binding to DNA, which consists of a description of gyrase processivity but lacks additional terms (such as the specific binding of topoisomerase I and gyrase at certain locations) that the authors of the original study included to explain their data. We note that a comprehensive modeling of topoisomerase binding will likely involve a complex combination of DNA sequence and 3D deformations.