All living cells need to coordinate DNA replication with growth and division to generate cell cycles that are stable in time. The bacterium Escherichia coli initiates replication at a volume per origin that on average is independent of the growth rate. It also adds an on average constant volume per origin between successive initiation events, independent of the initiation size. Yet, a molecular model that can explain these observations has been lacking. Here, we develop a mathematical model of DNA replication initiation in E. coli that is consistent with a wealth of experimental data. We first show that the previously proposed initiator titration model, which is based on the accumulation of the initiator protein DnaA on chromosomal titration sites, is not consistent with the experimental data. We then present a model that is based on an ultra-sensitive switch between an inactive form of DnaA and an active form that induces replication initiation. Our model shows that at low growth rates the switch is predominantly controlled by activation of DnaA via lipids and deactivation via the chromosomal site datA, while at high growth rates DARS2 and RIDA become essential. Crucially, in our mean-field model DNA replication is initiated at a constant volume per origin, qualifying our model as a sizer. Yet, we show that in a stochastic version of the same model the inevitable fluctuations in the components that control the DnaA activation switch naturally give rise to the experimentally observed adder correlations.
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