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

## ABSTRACT

## INTRODUCTION

## RESULTS

### The parabolic shape of the growth inhibition curve suggests a cooperative inhibition mechanism.

*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*

^{−}

*. Assuming that*

^{pn}*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).

### Quantitative model for the action of ciprofloxacin.

*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*) of 30 kb/min (29). When a chromosome successfully completes replication, it separates from the parent chromosome.

_{f}*k*

_{+}

*L*/

*L*

_{0}, where

*k*

_{+}is the DNA-poisoned gyrase binding rate,

*L*is the current chromosome length, and

*L*

_{0}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*/

*L*

_{0}.

*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).

_{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.

### The model reproduces the growth inhibition curve.

*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.

*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

^{−5 }min

^{−1}, a τ

_{gyr}value of 25 ± 5 min, and a

*q*value of 0.030 ± 0.005 ml ng

^{−1 }min

^{−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.

*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).

*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.

### The model predicts the dynamical response of E. coli to ciprofloxacin.

*fimA*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).

^{−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.

### Replication-dependent and replication-independent DNA damage predicts the same shape of growth inhibition curve.

### Basal DNA damage is sufficient to model a DNA repair-deficient mutant.

*recA*deletion mutant that cannot trigger the SOS response (see Materials and Methods). We first investigated the growth of the Δ

*recA*strain in the absence of ciprofloxacin. Δ

*recA*cells were similar in length and width to WT cells but had less organized chromosomes (Fig. S6B). In microplate cultures, the Δ

*recA*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 Δ

*recA*cells with cephalexin and measuring the cell-length distribution after 1 h, we found that individual Δ

*recA*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 Δ

*recA*and WT strains growing on agar pads. Interestingly, the Δ

*recA*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 Δ

*recA*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).

*recA*strain. We measured the growth inhibition curve of the Δ

*recA*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 Δ

*recA*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 Δ

*recA*Δ

*fimA*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.

*recA*Δ

*fimA*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.

### An alternative hypothesis based on saturation of repair mechanisms does not explain the data.

*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*).

*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.

## DISCUSSION

*recA*mutant that cannot activate the DNA repair machinery and that is significantly more sensitive to ciprofloxacin (Fig. 8). Thus, the SOS system can significantly alter the parameters of the model but, importantly, does not control the dynamics of the response. Instead, the dynamics are controlled by the DNA replication rate and the binding/unbinding rates of gyrase from the DNA.

### Shape of the growth inhibition curve.

### Role of the SOS response.

*N*from reference 42) to 0 and added a term proportional to the CIP concentration to the equation

_{G}*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 χ

^{2}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}.

### Importance of chromosome segregation.

### Other fluoroquinolones and bacterial species.

*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.

### Relevance for bacterial infections.

## MATERIALS AND METHODS

### Bacterial strains.

*fimA*mutant), MG1655 Δ

*recA*, and EEL01 (MG1655 Δ

*recA*Δ

*fimA*double mutant). The Δ

*fimA*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.

*recA*mutant was donated by the Meriem El Karoui lab. This mutant is MG1655 in which Δ

*recA*::Cm

^{r}was introduced by P1 transduction from DL0654 (David Leach, laboratory collection). The Δ

*recA*Δ

*fimA*strain was created by P1 transduction of the

*recA*deletion with a chloramphenicol resistance selection marker from the MG1655 Δ

*recA*strain. Briefly, the donor strain MG1655 Δ

*recA*was incubated overnight and inoculated at 37°C for 25 min with different dilutions of the P1

*vir*phage in the presence of MgSO

_{4}and CaCl

_{2}, 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 Δ

*fimA*strain). For the transduction, the recipient strain was incubated overnight, harvested, and resuspended in LB with MgSO

_{4}and CaCl

_{2}, 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 Δ

*recA*::Cm

^{r}construct. Following an overnight incubation at 37°C, colonies were purified twice on chloramphenicol plates with sodium citrate.

### Growth media and antibiotics.

_{2}O) by diluting into LB to achieve the desired concentrations.

### Growth inhibition curves.

### (i) Method 1.

^{3}and 10

^{4}times in phosphate-buffered saline (PBS), and 5 μl of the suspension was added to each well (a 10

^{3}dilution was added to rows A to D, a 10

^{4}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).

*T*) 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/Δ

*T*.

*A*+

*Be*

^{α}

*, where*

^{t}*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 10

^{4}to 10

^{8}cells, ≈13 generations) than curve-fitting based methods (OD = 0.01 to 0.1, 3 generations) (Fig. S1B).

### (ii) Method 2.

### Measurements of DNA production.

### Microscopy.

### Computer simulations of the DNA replication model.

*L*

_{0}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}/(

*L*

_{0}

*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*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 (

_{f}*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).

*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.

*p*

_{kill}and τ

_{gyr}(Fig. 3). The parameter

*p*

_{kill}was varied over the range of 5 × 10

^{−5}to 10

^{−3 }min

^{−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

^{−1 }min

^{−1}for the model with replication-independent killing and for

*p*

_{kill}equal to (2 ± 1.5) × 10

^{−5 }min

^{−1}, τ

_{gyr}equal to 30 ± 5 min, and

*q*equal to 0.040 ± 0.005 ml ng

^{−1 }min

^{−1}for the model with replication-dependent killing (at the forks).

### Model for exponentially growing filaments (cephalexin).

*l*

_{0}) 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*=

*l*

_{0}exp(α

*t*). 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).

### Finding *T*_{ss}.

*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 0.1

*k*

_{0}+ 0.9

*k*

_{ss}, where

*k*

_{0}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 0.01

*k*

_{0}+ 0.99

*k*

_{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).

### Turbidostat.

## ACKNOWLEDGMENTS

*recA*mutant strain. We also thank the anonymous reviewers for helpful comments and suggestions.

## Supplemental Material

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