Matthew A. A. Grant¹, Chiara Saggioro², Ulisse Ferrari³, Bruno Bassetti⁴, Bianca Sclavi²,Marco Cosentino Lagomarsino⁵ and Pietro Cicuta<sup>1

1</sup>Department of Physics, JJ Thomson Avenue, Cambridge, CB3 OHE, UK
²LBPA, UMR 8113 du CNRS, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 CACHAN, France
³Dip. Fisica, Università Sapienza, Piazzale A. Moro 2, I-00185, Rome, Italy
⁴Università degli Studi di Milano, Dip. Fiscia. Via Celoria16, 20133, Milano, Italy
⁵Genomic Physics Group, FRE 3214 CNRS “Microorganism Genomics” and University Pierre et Marie Curie, 15 rue de L’École de Médecine, 75006 Paris, France

Background:
In Escherichia coli overlapping rounds of DNA replication allow the bacteria to achieve doubling times that are faster than the time required to copy the genome. The precise timing of initiation of DNA replication is determined by a regulatory circuit that depends on the binding of a critical number of DnaA-ATP proteins at the origin of replication resulting in the melting of the DNA and the assembly of the replication complex. The synthesis of DnaA in the cell is controlled by a growth-rate dependent, negatively autoregulated gene found near the origin of replication. The activity of the protein depends on its nucleotide bound state and its availability compared to the number of nonspecific binding sites on the genome, both of which are dependent on the rate of DNA replication.

Results:
In order to obtain an improved understanding of the contribution of the different regulatory processes in the timing of initiation of DNA replication as a function of growth rate we formulate a minimal quantitative model of the initiator circuit that includes the key ingredients known to regulate the activity of the DnaA protein. This model can describe the oscillations in DnaA-ATP/DNA as a function of the cell cycle, achieving the same maximum value at the time of initiation independently of the growth rate by a continuous change in the value of the parameters controlling the rate of DnaA synthesis. We compare different possible scenarios to characterize the roles of DnaA autoregulation and of the DnaA-bound ATP-hydrolysis regulatory process under different growth conditions.

Conclusions:
A continuous increase in the rate of DnaA gene expression dependent on the growth rate of the cell can account for the timing of initiation of DNA replication at different cell doubling times. Auto-repression of the DnaA gene is not required for this model to work and the rate of RIDA can be varied 10 fold without significantly affecting the outcome. Both RIDA and auto-repression however can contribute to dampen the amplitude of the oscillations of DnaA-ATP/DNA during the cell cycle and result in a smaller range in the change of the average amount of DnaA-ATP per genome equivalent as a function of growth rate. This is required if DnaA's activity as both the initiator of DNA replication and as a transcription factor is to reflect the status of the DNA replication process in response to perturbations of the replication forks independently of the growth rate.