  Background: In Escherichia coli, overlapping rounds of DNA replication allow
the bacteria to double in faster times 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
ATP-bound DnaA proteins at the origin of replication. The synthesis of DnaA in
the cell is controlled by a growth-rate dependent, negatively autoregulated
gene found near the origin of replication. Both the regulatory and initiation
activity of DnaA depend on its nucleotide bound state and its availability.
  Results: In order to investigate the contributions of the different
regulatory processes to the timing of initiation of DNA replication at varying
growth rates, 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 describes the average-cell oscillations in
DnaA-ATP/DNA during the cell cycle, for varying growth rates. We evaluate the
conditions under which this ratio attains the same threshold value at the time
of initiation, independently of the growth rate.
  Conclusions: We find that a quantitative description of replication
initiation by DnaA must rely on the dependency of the basic parameters on
growth rate, in order to account for the timing of initiation of DNA
replication at different cell doubling times. We isolate two main possible
scenarios for this. One possibility is that the basal rate of regulatory
inactivation by ATP hydrolysis must vary with growth rate. Alternatively, some
parameters defining promoter activity need to be a function of the growth rate.
In either case, the basal rate of gene expression needs to increase with the
growth rate, in accordance with the known characteristics of the dnaA promoter.
