A Roadblock-and-Kill Mechanism of Action Model for the DNA-Targeting Antibiotic Ciprofloxacin

To understand the response of E. coli to ciprofloxacin (CIP), we first measured the long-term (steady-state) growth rate at different CIP concentrations: the growth inhibition curve. Previous work (25) indicated that the inhibition curve of E. coli can be modeled by use of a Hill function with a plateau at low concentrations. However, these experiments might not have been conducted while the bacteria were in a state of balanced growth, as the bacteria were exposed to CIP for only 1 h.

To determine the steady-state growth rate for different CIP concentrations, we used two different methods (Fig. 1; see also Fig. S1 in the supplemental material). We first measured E. coli growth curves for a series of CIP concentrations by incubating bacteria in microplates (200 μl/well) in a plate reader and sampling the optical density (OD) every few minutes over 1 to 2 days (see Materials and Methods). We used two strains: the K-12 strain MG1655 and a mutant derivative, AD30. AD30 does not produce functional fimbriae and therefore sticks less to surfaces (Fig. 1B; see Materials and Methods), preventing biofilm growth during the experiment. To minimize potential problems, such as the dependence of the optical density on cell shape (26), which changes during CIP-induced filamentation (14, 27), we extracted the growth rates from the time shifts between the growth curves for cultures with different initial cell densities (see Materials and Methods). Both strains produced very similar growth inhibition curves with a characteristic inverted parabola-like shape (Fig. 1A and B). This shape is consistent with previous results for ciprofloxacin (25) but differs from that produced by many other antibiotics (5, 25).

In parallel, we measured exponential growth rates for a range of CIP concentrations using steady-state cells grown in a turbidostat, a continuous culture device that dilutes cells once they reach a threshold density, maintaining exponential growth over long times (see Materials and Methods and Fig. S1C and D). This could be done only for strain AD30, because wild-type strain MG1655 rapidly formed a biofilm in the turbidostat. The growth rates in the turbidostat agreed with those obtained from plate reader growth curves (Fig. 1B).

If a culture is in a state of balanced exponential growth, all components of the bacterial cell must replicate at the same rate (28). Therefore, the measured exponential growth rate should be the same as the rate of DNA synthesis. To confirm this, we measured the total amount of DNA at multiple time points in an exponentially growing culture for different CIP concentrations and extracted the DNA production rate (see Materials and Methods). Figure 1C shows that, indeed, the rate of DNA production matched the exponential growth rate, as measured in our plate reader and turbidostat experiments.

Taken together, these results show that the long-term, steady-state rate of DNA production is a nonlinear, inverted parabola-like function of the CIP concentration, with only a small slope at zero CIP. If each DSB caused by CIP contributed (with probability p) independently to the probability of cell death and the number of DSBs was n, the per-cell death rate would be proportional to 1 − (1 – p)ⁿ ≈ 1 − e^−pn. Assuming that n increases proportionally to the CIP concentration (c), we would then expect a concave relationship between the net growth rate (birth minus death) and c, with a negative slope at low c. As this is not the case, a cooperative effect may be at play, which causes the number of DSBs to increase faster than linearly with c. Alternatively, one might imagine a mechanism in which the number of DSBs is proportional to c but must exceed a certain threshold before its effects on the growth rate become visible. We show that the first hypothesis (a nonlinear increase in number of DSBs) is strongly supported by the data (see below), whereas the alternative hypothesis (a threshold number of DSBs is needed for growth inhibition) is not (see below).

To understand how the rate of DNA synthesis is affected by ciprofloxacin, we developed a quantitative model (Fig. 2). The model includes reversible replication fork stalling by CIP-poisoned gyrases and both replication-dependent and replication-independent double-strand breakage.

In our model, a bacterial culture is represented by an ensemble of replicating circular chromosomes. New chromosomes are synthesized on the template of the parent chromosomes and remain attached to them via replication forks. The forks start from the origin of replication (oriC) and end at the terminus (ter). Initiation occurs at time intervals drawn from a normal distribution, with a mean new fork initiation time (τ_fork) of 24 min being chosen to reproduce the CIP-free growth rate from Fig. 1B and a standard deviation σ(τ_fork) of 5 min (an arbitrary value) being chosen. Once initiated, replication forks progress at a constant rate (v_f) of 30 kb/min (29). When a chromosome successfully completes replication, it separates from the parent chromosome.

Poisoned gyrases can appear anywhere along the chromosome at a rate of k₊L/L₀, where k₊ is the DNA-poisoned gyrase binding rate, L is the current chromosome length, and L₀ is the birth length of a fully replicated chromosome. We assume that the rate k₊ is proportional to the extracellular CIP concentration (c) with an unknown proportionality constant, q [with units of 1/(time × concentration)]: k₊ = qc. Poisoned gyrases can also dissociate from the chromosome at a rate of 1/τ_gyr, where τ_gyr is the poisoned gyrase turnover time. The number of poisoned gyrases per chromosome (N_gyr) fluctuates, with the average value (<N_gyr>) being determined by the balance between the binding and removal rates: <N_gyr> = k₊ τ_gyr L/L₀.

If a replication fork encounters a poisoned gyrase, it stops and remains stalled until the poisoned gyrase is removed. The poisoned gyrase can also damage the entire chromosome irreversibly at the rate p_kill (Fig. 2C). Damaged chromosome conglomerates (i.e., chromosomes plus any connected DNA loops) are removed from the simulation. The exact nature of the DNA damage is not important for the model, but a biologically plausible mechanism would be the creation of a DSB that does not get repaired (15). The process of repair is not modeled explicitly, but its effectiveness is implicitly included in the value of p_kill (e.g., a large value of p_kill corresponds to impaired DNA repair, since a poisoned gyrase is more likely to cause irreversible damage).

Our model has three unknown parameters: τ_gyr, p_kill, and the proportionality constant q that relates the extracellular concentration of CIP to the rate k₊ at which poisoned gyrases appear on the chromosome.

We first checked if the model could reproduce the growth inhibition curve from Fig. 1. To do this, we calculated the rate of exponential increase in the total amount of DNA predicted by the model as a function of the CIP-proportional poisoned gyrase binding rate, k₊ (Fig. 3A and B). Figure 3B shows predicted growth inhibition curves for a fixed τ_gyr value of 15 min (arbitrary value) and a range of values of p_kill. The simulated curves resemble the experimental curve (Fig. 1A). As expected, the rate of DNA synthesis decreases as the parameter k₊ increases, mimicking the increasing CIP concentration.

We next systematically explored the parameter space (p_kill, τ_gyr, q) to find a range of parameter combinations that quantitatively reproduce our experimental data. Figure 3C shows that such a range indeed exists (the dark blue region in Fig. 3C); the best-fit parameters are a p_kill value of (7 ± 2) × 10⁻⁵min⁻¹, a τ_gyr value of 25 ± 5 min, and a q value of 0.030 ± 0.005 ml ng⁻¹min⁻¹. This combination produces an excellent fit to the experimental data (Fig. 3D). Our fitted value for τ_gyr is about half the turnover time (∼55 min) that has been estimated from in vitro reconstitution assays (30); this discrepancy is perhaps not surprising since the in vitro assay lacks DNA repair systems (23) that may actively remove poisoned gyrases.

One can also extract from the model the average number of poisoned gyrases per chromosome (N_gyr) for a given CIP concentration (Fig. S2). For a CIP concentration of 10 ng/ml, which corresponds to a 2-fold reduction in the growth rate, we obtained an N_gyr of ≈4. The model thus suggests that a small number of poisoned gyrases is enough to inhibit growth (a typical gyrase copy number in the absence of CIP is ∼500).

Our model explains why the growth inhibition curve assumes a parabolic shape. At low concentrations of CIP, there are very few poisoned gyrases present; DNA replication proceeds at almost normal speed and the chromosome topology is almost normal (since there are few blocked replication forks). Since the rate at which a chromosome conglomerate is damaged by CIP is proportional to the total amount of DNA in the conglomerate and p_kill is small, chromosome death is negligible at low CIP concentrations. However, as the CIP concentration (c) increases, replication forks become blocked more often. As a consequence, new replication forks are initiated before the parent and daughter chromosomes separate, producing large interconnected DNA conglomerates. Because the total amount of DNA per conglomerate increases, the number of poisoned gyrases that are bound to the DNA also increases. This produces a faster-than-linear increase in the degree of growth inhibition as c increases.

To confirm this interpretation of our model, we considered a modified model in which the damage caused by a poisoned gyrase does not kill the entire chromosome conglomerate but kills only the chromosome segment to which it is attached. There is some evidence that this might be the case for an E. coli strain that is deficient in DSB repair (31). This modified model predicts a very different growth inhibition curve (Fig. S3), which lacks the plateau at low CIP concentrations.

Our model has been parameterized to reproduce the inhibition curve for steady-state growth in the presence of ciprofloxacin. To check if the model is able to predict the dynamical response of E. coli to ciprofloxacin (for which it has not been parameterized), we exposed the ΔfimA strain AD30 to a step-up in ciprofloxacin concentration and measured the dynamical changes in the growth rate over many generations in the turbidostat while maintaining cells in the exponential growth phase. Interestingly, we observed that for low concentrations of ciprofloxacin, the growth rate did not decrease immediately on antibiotic addition. Both the time until the growth rate began to decrease and the time to achieve a new steady-state growth rate depended on the CIP concentration (Fig. 4A).

Our model cannot predict the bacterial growth rate directly, as it focuses on the rate of DNA synthesis, which does not have to be the same as the population-level growth rate during periods of unbalanced growth. However, the model can be used to predict the time to the new steady state (Fig. 4B; see Materials and Methods). The predicted values agree well with the results of our experiments.

We next checked if the model also correctly predicts the dynamical response of DNA synthesis to ciprofloxacin exposure in single cells. We treated E. coli (MG1655) cells with ciprofloxacin for 1 h, stained the cells with 4′,6-diamidino-2-phenylindole (DAPI) to visualize the DNA, and imaged the cells in the bright-field and fluorescent channels (Fig. 5). To prevent cell division and thus enable a direct comparison with the model, we used cephalexin (8 μg/ml), which inhibits PBP3, a component of the E. coli septation machinery (32). As expected, all the cells grew as filaments, without dividing (Fig. 5A).

The bacterial elongation rate was extracted from our measured filament length distributions by assuming exponential elongation at a constant rate (α) starting from the initial length distribution of untreated cells (see Materials and Methods). For cells treated with cephalexin only, the experimental length distribution was best fit by an elongation rate (α) of 1.85 ± 0.28 h⁻¹, similar to the growth rate obtained in plate reader experiments without any antibiotic (1.70 ± 0.10 h⁻¹) (Fig. 1B and 5B). Therefore, cephalexin prevented cell division without visibly decreasing the biomass growth rate.

Remarkably, the cell length distribution (and, hence, the biomass growth rate) remained unchanged when cells were exposed to both ciprofloxacin (up to 15 ng/ml) and cephalexin (Fig. 5B). This observation is consistent with previous microscopy data (14). Even at the highest CIP concentration used (50 ng/ml, ∼2.5× MIC for this strain), the elongation rate was only slightly reduced (Fig. 5B, right).

We next characterized the DNA organization in single cells following exposure to CIP and cephalexin. Figure 5C shows that cells treated solely with cephalexin had clearly defined, evenly spaced chromosomes. The overall chromosome density was consistent with that of antibiotic-free growth; for example, for a cephalexin-treated filament with a length of 24 μm, we observed ∼16 chromosomes, while E. coli with a length of 3 μm grown on LB antibiotic-free medium typically has ∼2 chromosomes (Fig. S6A). However, in the presence of CIP, DNA becomes less ordered and, as the CIP concentration increases, fewer distinct chromosomes can be identified. This suggests the presence of large entangled DNA structures and the failure of chromosome separation.

Our model makes a very specific prediction for how the total amount of DNA in a filamentous cell should depend on the CIP concentration after 1 h of exposure (Fig. 6A). To test this prediction, we quantified the total amount of DNA per cell by measuring the DAPI fluorescence in microscopic images of cells for different concentrations of CIP. We obtained excellent quantitative agreement between our simulations and the results of the experiments (Fig. 6B), without any additional fitting. Thus, our model, once fitted to the steady-state data, correctly predicts the early-time dynamical response to ciprofloxacin in single cells.

Ciprofloxacin-bound DNA gyrase has been hypothesized to cause both replication-dependent and replication-independent DNA double-strand breaks (18, 19, 21). To test the role of replication-dependent versus replication-independent killing, we simulated a version of the model in which chromosome damage occurs only via fork-associated poisoned gyrase (see Materials and Methods). This model turned out to reproduce the growth inhibition curve equally well (Fig. S7). Thus, models with only replication-dependent DNA breaks or with both replication-dependent and replication-independent DNA breaks produce the same growth inhibition dynamics.

Our model does not explicitly include repair of DNA double-strand breaks, which happens in E. coli via the RecBCD machinery, triggered by the SOS response (15, 33). We tested the role of DNA repair using a recA deletion mutant that cannot trigger the SOS response (see Materials and Methods). We first investigated the growth of the ΔrecA strain in the absence of ciprofloxacin. ΔrecA cells were similar in length and width to WT cells but had less organized chromosomes (Fig. S6B). In microplate cultures, the ΔrecA strain showed a reduced growth rate compared to that of the WT MG1655 strain (∼1 h⁻¹ versus 1.7 h⁻¹ for the WT). However, upon treating ΔrecA cells with cephalexin and measuring the cell-length distribution after 1 h, we found that individual ΔrecA cells elongated at the same rate as WT cells, although in the majority of the cells, the DNA looked less organized (Fig. 7A and B). To resolve this apparent contradiction, we imaged microcolonies of the ΔrecA and WT strains growing on agar pads. Interestingly, the ΔrecA colonies were significantly smaller and many colonies (∼30%) did not grow at all (Fig. S9). This suggests that the reduced population-level growth rate of ΔrecA cultures is due to an increased fraction of nongrowing cells rather than a decreased single-cell growth rate. This is consistent with previous observations that cultures of bacteria deleted for recA can contain a significant subpopulation of nongrowing cells (34, 35).

We also wondered if our model could predict the shape of the growth inhibition curve for the ΔrecA strain. We measured the growth inhibition curve of the ΔrecA strain in the plate reader (Fig. 7C). The MIC of this strain (∼1.5 ng/ml) was an order of magnitude lower than that of the WT. Moreover, the shape of the growth inhibition curve was significantly different from the parabola-like shape of the curve for the WT (Fig. 1): for the ΔrecA strain, the growth rate decreased approximately linearly with increasing CIP concentration, without a plateau at low CIP concentrations. We hypothesized that these features could be reproduced in our model by an elevated rate of DNA damage associated with CIP-poisoned gyrases, combined with a basal DNA damage rate in the absence of CIP, with both being due to the lack of the DSB repair mechanism. A modified model, in which the basal DNA damage rate (p_kill0) of 0.0033 ± 0.0002 min⁻¹ was fixed by fitting to the population growth rate in the absence of CIP, reproduced the experimental growth inhibition curve very well (Fig. 7C; p_kill and q were fitted to the inhibition curve). The same model also reproduced the growth inhibition curve for the ΔrecA ΔfimA double mutant (Fig. S8). One can intuitively understand the origin of the negative slope at a zero drug concentration: the basal damage rate acts as if a nonzero CIP concentration was present even when the actual concentration of the antibiotic was zero. This causes the parabolic shape of the curve for the WT to shift to the left, leaving only the part that is almost linear in the CIP concentration.

To investigate if our model could also predict the dynamical response, we repeated the turbidostat experiment for which the results are shown in Fig. 4 with the ΔrecA ΔfimA double mutant. Figure 8 shows that the time to reach the new steady-state growth rate after a CIP upshift (T_ss) is very well predicted by the model. However, we note that the agreement depends on how T_ss is calculated (see Materials and Methods). A change in the threshold growth rate used to obtain T_ss can change the predicted times by up to 1 h. Together, the results show that even though our model does not explicitly include DNA repair, an implicit modeling of DNA repair via the parameters p_kill0 and p_kill is sufficient to quantitatively reproduce our experimental data.

Our model reproduces all our experimental observations, but could an alternative model based on a different microscopic mechanism explain them equally well? To investigate this, we considered a biologically plausible model in which the parabolic shape of the inhibition curve arises due to a nonlinear response of the DNA repair mechanism to the CIP concentration, rather than from a nonlinearity in the amount of DNA damage, as in the previous model.

In this alternative model, for CIP concentrations above the MIC, DSB repair mechanisms become saturated, which causes the accumulation of DSBs. Below the MIC, however, we assume that RecBCD-mediated DSB repair (36) is very effective. Specifically, we assume that the number of DSBs [n(t)] evolves as (dn/dt) = b − min(r_max, rn^γ). Here, DSBs are created at a rate b that is proportional to the CIP concentration and are removed via repair at a rate rn^γ, which cannot exceed the maximum rate (r_max). The exponent γ characterizes the strength of the feedback between the number of DSBs and the rate of repair; a γ value of 1 corresponds to a linear response, whereas a γ value of <1 means that repair mechanisms are strongly triggered even by a small number of DSBs. We further assume that each DSB has an equal probability p of killing the cell; hence, the net growth rate is proportional to exp(−pn).

This model, which does not consider the dynamics of DNA replication, reproduces the steady-state growth inhibition curve quite well (Fig. S10) for a γ value of ≈0.5. However, the model predicts that the time to reach a new steady-state growth rate following an upshift of CIP should be proportional to b^{(1/γ) − 1} ≈ b. The time to the new steady state is thus predicted to increase with the CIP concentration (since b increases with c), which disagrees with what we observed experimentally (Fig. 4). Therefore, this model fails to reproduce the dynamics of CIP inhibition.

The growth inhibition curve for CIP is parabola-like (Fig. 1). Inhibition curves for many antibiotics, including CIP, have traditionally been approximated using the Hill function (25). This choice is often based on a qualitative description of the shape rather than on mechanistic insight. The Hill function is also a popular choice in population-level models of antibiotic treatment (37–39). However, some antibiotics can have very different inhibition curves that are not well approximated by a Hill function (5). This is potentially important for modeling the evolution of resistance to antibiotics, because differently shaped inhibition curves are expected to produce different fitness landscapes (40, 41), leading to different levels of selection for resistant mutants and, hence, different trajectories to resistance.

We checked how well our measured growth inhibition curve can be reproduced using a Hill function (see Fig. S11 in the supplemental material). The fit was slightly less good than that produced by our model. The Hill exponent (κ; which is equal to 4.4 ± 0.5) also differs significantly from the κ value of 1.1 ± 0.1 that has been reported before (25). We conclude that careful measurements of the steady-state growth inhibition curve, combined with physiological models of the antibiotic response, not only can shed light on the mechanism of inhibition but also are required for quantitative models of the evolution of antibiotic resistance. It may also be interesting to see whether, for models valid for other antibiotics, the models’ parameters are sufficiently restricted by fitting to the inhibition curve (the long-term response) to enable short-term predictions or whether perhaps the parameters must be measured independently in another way.

The cellular response to DNA damage was not explicitly included in our model but rather entered through the parameter values. In others’ work, the SOS response has been modeled in the context of the UV response (42–45). To check how realistic it was to omit details of the SOS response in our model, we adapted the model from reference 42 to our scenario. We set the initial number of DSBs (parameter N_G from reference 42) to 0 and added a term proportional to the CIP concentration to the equation dN_G/dt, which describes the rate of change of the number of DSBs. We calculated the time that it takes for LexA (the protein whose inactivation triggers the response) to reach a new steady state after a step-up in stimulus (10% above the infinite-time-limit concentration). Figure S12 shows that this time is less than 10 min for a broad range of DSB creation rates, indicating that the SOS response occurs much faster than the growth rate response that we report in Fig. 4. When we fit this alternative model to the data from Fig. 4B (the fitting parameter is the proportionality factor between the CIP concentration and the rate of production of DSBs), the reduced χ² value of ≈200 for the best-fit curve is many times larger than the value reported in the legend to Fig. 4B for our main model. Based on this and the excellent agreement between our main model and the results of the experiments, we conclude that key features of the growth inhibition in response to a sub-MIC of ciprofloxacin (the shape of the inhibition curve and the dynamics of inhibition) can be understood without modeling the SOS response explicitly. This does not mean that the SOS response is not important; on the contrary, SOS response-induced changes in bacterial physiology (e.g., expression of low-fidelity polymerases) are very important for the evolution of resistance (14, 46), and the role of the SOS response in mediating growth inhibition is also implicit in our model through the parameters p_kill and p_kill0.

In this work, we did not model individual cells; rather, we considered a collection of replicating chromosomes. While this seemed to be enough to reproduce the population-level growth rate response to ciprofloxacin and the DNA dynamics in single cells, it could not account for some aspects of behavior at the cellular level, such as the cell length distribution (in our experiments, we avoided this issue by treating the cells with cephalexin). More work will be required to create a model that is able to, for example, predict the cell length distribution (Fig. 5), cell division and budding (14), or antibiotic-induced fluctuations in the number of cells in small populations (47).

Based on the proposed mechanism, we expect the results to be generalizable to other fluoroquinolones, as long as gyrase is the primary target. This seems to be the case for E. coli (48–50). Topoisomerase IV, the other potential target, becomes important only in combination with resistance mutations in gyrA (51). Topoisomerase IV has a stronger affinity to fluoroquinolones in other bacterial species than topoisomerase IV does in E. coli (50); we do not expect the model to quantitatively reproduce the short- and long-time response for such cases. We note, however, that parabolic inhibition curves have been reported for the bacterium Mycobacterium smegmatis treated with nalidixic acid and novobiocin (see Fig. S2 in reference 52). This may suggest that the long-term response (and perhaps also the mechanism behind it) may be similar in other bacterial species.

A predictive understanding of how antibiotics inhibit bacteria could help in the design of better treatment strategies. Traditionally, models for antibiotic treatment have assumed an instantaneous response of bacteria to the antibiotic (53, 54); models that take intracellular dynamics into account are still rare (55, 56). Our research shows that ignoring the transient behavior (here, the short-term bacterial response delay) can be problematic because bacteria with this transient behavior can last for many generations at sub-MICs of the antibiotic, and the probability of developing resistance is the highest for these bacteria (55, 57, 58). Our physiological model could be integrated into population-level evolutionary models, allowing better prediction of the chances of resistance emergence by taking account of the cell-level dynamical response. Such effects are almost universally neglected in current evolutionary models. We postulate that, rather than using ordinary differential equation-based models (38) or stochastic models, such as birth-death processes (47, 59), one could use individual-based simulations with bacterial physiology modeled explicitly, similar to what has been done in biofilm modeling (60). In such a model, individual chromosomes, simulated according to our (or a similar) model, would also mutate; this would be represented by changing the model parameters to account for, e.g., an increased MIC for resistant mutants (a decreased number of poisoned gyrases). Since our model is computationally expensive, it can be used only for small populations of cells (up to a few million). This may be still very useful for modeling the laboratory evolution of resistance in microfluidic devices, which is gaining popularity (61, 62). For large population sizes, such as those required to model human infections (tens or hundreds of millions of cells), a hybrid model in which only a small number of cells (e.g., new mutants) have explicit internal dynamics and the bulk of the population is described using coarse-grained models could be used. Such hybrid models are used in cancer modeling (63, 64) but have not yet been applied in evolutionary microbiology.

In conclusion, we have proposed and tested a model that predicts the bacterial response to fluoroquinolone antibiotics. Our model complements those that have recently been proposed for other classes of antibiotics; taken together, such models may eventually be useful in providing an understanding of and predicting the bacterial response to clinically relevant treatment strategies, such as the effect of combination therapies (65–67).

We used MG1655, a K-12 strain of the bacterium E. coli, and two mutant derivatives, AD30 (MG1655 ΔfimA mutant), MG1655 ΔrecA, and EEL01 (MG1655 ΔrecA ΔfimA double mutant). The ΔfimA strain was constructed by P1 transduction from JW4277 (the fimA deletion strain in the strain BW25113 background from the Keio Collection) into MG1655 (68). The kanamycin resistance cassette was removed using Flp recombinase expressed in pCP20. Strain construction was confirmed by PCR using a combination of kanamycin resistance cassette-specific primers and gene-specific primers.

The ΔrecA mutant was donated by the Meriem El Karoui lab. This mutant is MG1655 in which ΔrecA::Cm^r was introduced by P1 transduction from DL0654 (David Leach, laboratory collection). The ΔrecA ΔfimA strain was created by P1 transduction of the recA deletion with a chloramphenicol resistance selection marker from the MG1655 ΔrecA strain. Briefly, the donor strain MG1655 ΔrecA was incubated overnight and inoculated at 37°C for 25 min with different dilutions of the P1vir phage in the presence of MgSO₄ and CaCl₂, before being mixed with molten top agar and spread onto an LB plate, left to set, and incubated at 37°C overnight. Donor phage was harvested from the top agar by mixing with phage buffer and a few drops of chloroform, the debris spun was down, and the supernatant containing the donor phage was used for transduction into the recipient strain (the ΔfimA strain). For the transduction, the recipient strain was incubated overnight, harvested, and resuspended in LB with MgSO₄ and CaCl₂, and the suspension was mixed with the P1 donor phage and incubated at 37°C for 30 min before the addition of sodium citrate. The cells were then incubated (37°C, 200 rpm), to allow for the expression of chloramphenicol resistance; spun down; and plated onto LB plates with chloramphenicol for selection of the ΔrecA::Cm^r construct. Following an overnight incubation at 37°C, colonies were purified twice on chloramphenicol plates with sodium citrate.

All our experiments were performed in LB medium at 37°C. LB liquid medium was prepared according to Miller's formulation (10 g tryptone, 5 g yeast extract, 10 g NaCl per liter). The pH was adjusted to 7.2 with NaOH before autoclaving at 121°C for 20 min. To create LB in 1.5% agar, agar (agar bacteriological, no. 1; Oxoid) was added before autoclaving.

Ciprofloxacin solutions were prepared from a frozen stock (10-mg/ml CIP hydrochloride in double-distilled H₂O) by diluting into LB to achieve the desired concentrations.

A stock solution of cephalexin (10 mg/ml) was prepared by dissolving 100 mg of cephalexin monohydrate in 10 ml of dimethyl sulfoxide.

To determine the growth rate at a given concentration of CIP, we used two different methods.

We incubated bacteria in a microplate inside a plate reader (BMG Labtech FLUOstar Optima with a stacker) starting from two different initial cell densities and measured the optical density (OD) of each culture every 2 to 5 min to obtain growth curves.

The plates were prepared automatically using a BMG Labtech CLARIOstar plate reader equipped with two injectors to create different concentrations of CIP in each column of a 96-well plate (total injected volume, 195 μl per well). Bacteria were diluted from a thawed frozen stock 10³ and 10⁴ times in phosphate-buffered saline (PBS), and 5 μl of the suspension was added to each well (a 10³ dilution was added to rows A to D, a 10⁴ dilution was added to rows E to H). After adding the bacteria, the plates were sealed with a transparent film to prevent evaporation and put into a stacker (temperature, 37°C; no shaking), from which they were periodically fed into the FLUOstar Optima plate reader (37°C, orbital shaking at 200 rpm for 10 s prior to OD measurement).

Assuming that all cultures grow at the same rate when the cell density is low (OD < 0.1), the time shift (ΔT) between the curves from rows A to D and E to H (see Fig. S1A in the supplemental material) is related to the exponential growth rate as follows: α = ln10/ΔT.

We used this relationship to calculate α time shifts between 4 pairs of replicate experiments (rows A and E, B and F, C and G, and D and H) for 12 concentrations of ciprofloxacin (range, 0 to 30 ng/ml). To validate the method, we also calculated growth rates by fitting an exponential curve (A + Be^αt, where t is time) to the low-OD (OD < 0.1) part of the growth curve. The time-shift method gives results more accurate than but overall similar to the results obtained by the exponential curve fitting (Fig. S1B) or maximum growth rate measurement methods (69). Our fitting method is not sensitive to the relationship between the OD and the true cell density (which depends on the cell shape and size), and it gives the average growth rate over many more generations (growth from approximately 10⁴ to 10⁸ cells, ≈13 generations) than curve-fitting based methods (OD = 0.01 to 0.1, 3 generations) (Fig. S1B).

To confirm that our measurements correspond to steady-state growth, we also measured the growth rate in a turbidostat (Fig. S1C), in which bacteria are kept at an approximately constant optical density (OD = 0.075 to 0.1) for many generations by diluting the culture with fresh medium (with the concomitant removal of the spent medium and bacteria) whenever the OD reaches a threshold value. The growth rate is obtained by fitting an exponential function to the background-corrected OD data between consecutive dilutions.

We found that strains MG1655 and AD30 have similar but not identical susceptibilities to ciprofloxacin: while the MG1655 wild type showed an MIC of 19 ± 3 ng/ml, in agreement with previous measurements (16), AD30 was slightly less susceptible, with an MIC of 24 ± 3 ng/ml. The MIC values were determined from the zero-growth point of the growth inhibition curves (3 to 6 replicate experiments).

To obtain the data presented in Fig. 1C, cells were grown in LB medium with or without CIP in shaken flasks (3 replicates) and diluted periodically with fresh medium to maintain steady-state exponential growth. Cells were sampled every ∼20 to 30 min and fixed (1 ml of culture was fixed with 250 μl of 1.2% formaldehyde), and their OD was measured using both a stand-alone spectrophotometer (Cary 100 UV-visible) and a plate reader (CLARIOstar) for cross-validation. DAPI was added to the fixed samples to a concentration of 2 μg/ml (27). After 30 min of incubation with DAPI, the cells were washed 3 times with PBS, and the DAPI fluorescence intensity was measured in the plate reader (CLARIOstar). Growth rates were extracted from the fluorescence and the OD-versus-time curves by least-squares fitting of an exponential function.

To obtain the data presented in Fig. 5 and 7, exponentially growing cells (LB flasks) were treated with ciprofloxacin and/or cephalexin. The samples were fixed with formaldehyde and incubated for 30 min with DAPI (2 μg/ml) (27) and 0.1% Triton X-100 to increase cell permeability. The fixed cells were put on agarose pads (2% agar in water) and imaged on a Nikon Eclipse Ti epifluorescence microscope using a 100× oil objective (excitation, 380 to 420 nm; emission, >430 nm; exposure time, 100 ms). Cell lengths and widths and the fluorescence intensity were extracted using the Fiji plug-in MicrobeJ (70). For measuring the area of the microcolonies (Fig. S9), we used the semiautomated ImageJ plug-in JFilament (71). After extracting the coordinates of the microcolony contours from the phase-contrast images, the colony area was calculated as the area of the corresponding polygon (72, 73).

The computer code used to simulate our model was written in Java. Each chromosome is represented as a one-dimensional lattice of L₀ equal to 1,000 sites. The ends of the lattice are linked either to each other (to represent a circular chromosome) or to another chromosome lattice at points corresponding to the current positions of the replication forks. Poisoned gyrases are identified by the index of the chromosome on which they sit and their position (lattice site) within that chromosome. The simulation proceeds in discrete time steps [dt =N_bp/(L₀ v_f)], where N_bp is the number of base pairs in the E. coli chromosome and is equal to 4,639,675, and v_f is the fork speed and is equal to 30,000 bp/min. At each time step, the position of each fork that can move (i.e., each fork that is not blocked by a gyrase) is advanced by one lattice unit. Gyrases bind and detach with probabilities proportional to the corresponding rate times (dt). Chromosomes are killed with probability p_kill dt times the number of poisoned gyrases and removed from the simulation. Chromosomes are separated when two forks reach the endpoints of the mother chromosome. A pair of new forks is added every τ_fork time units, where τ_fork is drawn from a normal distribution with a mean of 24 min and a standard deviation of 5 min. In simulations of the model with DNA damage occurring only at the forks, only stalled forks kill chromosomes (probability p_kill dt per stalled fork).

All simulations were initiated with a single chromosome at time zero and stopped at a time (t) of 6 h (Fig. 3 and 7; Fig. S5) or 5 h (Fig. 6). Between 1,000 and 5,000 independent runs were performed to obtain averaged curves. The step of CIP in Fig. 6 was simulated by running the simulation with k₊ equal to 0 at a t of <100 min and switching to a k₊ value of >0, corresponding to the desired CIP concentration at a t of >100 min.

To fit the model to the experimental growth inhibition curves, we systematically explored the space of the parameters p_kill and τ_gyr (Fig. 3). The parameter p_kill was varied over the range of 5 × 10⁻⁵ to 10⁻³min⁻¹ for 11 data points, and τ_gyr was varied over the range of 0 to 80 min in 5-min steps. For a given pair of values for p_kill and τ_gyr, we simulated the model with different values of k₊ and varied the scaling factor q to fit the experimentally obtained growth inhibition curve by minimizing the sum of squares between the experimental and simulated inhibition curves. The best fit was obtained for p_kill equal to (7 ± 2) × 10⁻⁵ min⁻¹, τ_gyr equal to 25 ± 5 min, and q equal to 0.030 ± 0.005 ml ng⁻¹min⁻¹ for the model with replication-independent killing and for p_kill equal to (2 ± 1.5) × 10⁻⁵min⁻¹, τ_gyr equal to 30 ± 5 min, and q equal to 0.040 ± 0.005 ml ng⁻¹min⁻¹ for the model with replication-dependent killing (at the forks).

To extract growth rates from the filament length distributions in Fig. 5 and 7, each cell was assigned an initial length (l₀) from the experimentally observed distribution (Fig. S5B) and a random growth rate (α) taken from a Gaussian distribution characterized by its mean and standard deviation [α, σ(α)]. The new cell length after a time (t) of 1 h was calculated as l = l₀ exp(αt). A histogram of 642,000 predicted cell lengths was compared with the experimentally obtained cell length distribution for cephalexin-treated cells. The best match was obtained for an α value of 1.86 h⁻¹ and a σ(α) value of 0.22 h⁻¹, using the P value from the Kolmogorov-Smirnov test as the goodness-of-fit measure. The best-fit mean growth rate was similar to the growth rate measured in the plate reader (1.7 h⁻¹) (Fig. 1A), indicating that cephalexin-treated cells continued to elongate at the same rate for at least 1 h in the presence of CIP. The spread of elongation rates given σ(α) is similar to that observed for untreated cells (74, 75).

The time to a new steady state (T_ss) was calculated from the experimental data (growth rates versus time) as the time from the step-up of CIP to the point at which the growth rate decreased to the threshold value 0.1k₀ + 0.9k_ss, where k₀ is the growth rate before CIP addition and k_ss is the steady-state growth rate (Fig. 4 and 8). In the case of experiments in which the CIP concentration was greater than the MIC, k_ss was assumed to be 0 h⁻¹. To calculate T_ss in the simulations, we used the same approach with the threshold growth rate 0.01k₀ + 0.99k_ss. Different thresholds for experimental/simulated data were used to balance systematic errors: difficulty in detecting the true steady state in the experiments and growth rates representing two different quantities (the OD-based growth rate in the experiments, the DNA-concentration-based growth rate in the simulations).

Our turbidostat device (Fig. S1C) encompasses 4 replicate cultures with a culture volume of approximately 26 ml. The growth medium used in all experiments was LB broth (Miller), and the E. coli strain used was AD30, to avoid biofilm formation. In the turbidostat, all cultures are connected to a bottle of LB medium and a bottle of LB and CIP (where CIP is present at a concentration 10 times the desired CIP concentration in the culture) through a system of computer-controlled syringe pumps and valves. The optical density is measured every 20 s using custom-made photometers (a separate photometer for each bottle) to which 3 to 4 ml of each culture is aspirated and dispensed back into the culture using a syringe pump. When the optical density reaches 0.1 or after 30 min since the last dilution (whichever happens first), 25% of the culture is replaced with fresh medium to maintain exponential growth. An appropriate volume of CIP-containing LB medium is injected 2 h after an OD of 0.1 is reached for the first time to achieve the required concentration (5 to 100 ng/ml) in the culture. Smaller volumes are injected in all subsequent dilution steps to maintain the prescribed concentration of CIP for the rest of the experiment. All cultures were kept in an incubator set to 37°C, continuously stirred using magnetic stirrers, and aerated with an air pump to keep the dissolved oxygen (measured using a PyroScience FireStingO2 meter) well above 50% of the saturation concentration at 37°C (aerobic conditions).