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Fluoroquinolones - antibiotics that cause DNA damage by inhibiting DNA topoisomerases - are clinically important, but their mechanism of action is not yet fully understood. In particular, the dynamical response of bacterial cells to fluoroquinolone exposure has hardly been investigated, although the SOS response, triggered by DNA damage, is often thought to play a key role. Here we investigate growth inhibition of the bacterium Escherichia coli by the fluoroquinolone ciprofloxacin at low concentrations. We measure the long-term and short-term dynamical response of the growth rate and DNA production rate to ciprofloxacin, at both population- and single-cell level. We show that despite the molecular complexity of DNA metabolism, a simple ‘roadblock-and-kill’ model focusing on replication fork blockage and DNA damage by ciprofloxacin-poisoned DNA topoisomerase II (gyrase) quantitatively reproduces long-term growth rates in the presence of ciprofloxacin. The model also predicts dynamical changes in DNA production rate in wild type E. coli and in a recombination deficient mutant, following a step-up of ciprofloxacin. Our work highlights that bacterial cells show a delayed growth rate response following fluoroquinolone exposure. Most importantly, our model explains why the response is delayed: it takes many doubling times to fragment the DNA sufficiently to inhibit gene expression. We also show that the dynamical response is controlled by the timescale of DNA replication and gyrase binding/unbinding to the DNA, rather than by the SOS response, challenging the accepted view. Our work highlights the importance of including detailed biophysical processes in biochemical-systems models to quantitatively predict the bacterial response to antibiotics.