Standard models of bacterial regulatory circuits were adapted to situations where the growth rate is fixed [42, 53]. The notion that these quantitative descriptions must account for bacterial physiology through the growth-rate dependent basic partitioning of the cell physico-chemical components is now entering the field of systems biology through a combination of new work [41, 48, 49, 54] and reconsideration of the classics [8, 40, 55].
The dependency of the basic parameters on growth rate can produce notable effects on a genetic circuit, and complicates the standard descriptions . In our case, the task is more difficult, as the circuit under examination is active in determining some features of the bacterial physiology and not only affected by them. Furthermore, on the technical level, one must produce a time-dependent description the expression of DnaA over cell-cycles of a range of durations. Perhaps also for this reason, despite the fact that the regulation of DNA replication has been a subject of intense study for over 50 years [24, 57], many questions remain open. Given these obstacles, we have shown that, under a series of simplifying hypotheses, a consistent mean-field description for the DnaA/replication initation circuit is possible with varying growth rate.
Our description includes the processes that are believed to be most important for initiation of replication . In these respects, it is broadly compatible with previous modelling approaches [4–7]. Its originality lies in the minimality and in the attention given to growth-rate dependency. We focused on the minimal ingredients necessary in order for the basic tenet that the ratio DnaA-ATP/DNA attains a constant threshold at initiation to hold [58, 59]. The validity of this tenet is confirmed by the recent observations that initiation time is not affected by adding an extra origin on the chromosome  and on the compensatory mutations emerging in Hda mutants .
We have defined the DNA replication initiation potential, determining the (synchronous) timing of DNA replication, as the DnaA-ATP to DNA ratio, r. Molecular titration has been shown to result in ultrasensitive "all or none" responses , which further justifies using r as the threshold and could explain the synchrony of initiation in cells containing oriC minichromosomes . We assume that its value at the time of initiation, r(X), is independent of the specific growth rate. The amount of DnaA-ATP at the time of initiation thus needs to increase as a function of growth rate in order for r(X) to remain constant as a function of doubling time, and we found that consequently, some of the model's parameter values must be allowed to vary. This assumption has not been verified directly. On the other hand, we feel that our point of view would still be useful in case of a growth-rate dependent r(X), as it is unlikely that this dependence would automatically match the dependence of all the other parameters.
We have defined two main scenarios in which different subsets of the parameters are allowed to change. In Scenario 1 the RIDA rate (per replication fork), k
, is held constant as a function of growth rate, but the binding affinities of RNAP and DnaA-ATP to the DNA need to vary with growth rate (note that in addition, there are two technical sub-scenarios to Scenario 1 due to the possibility of either fixing the growth rate dependence of P, the number of available RNA polymerase molecules a priori to the trend of ref. (Scenario 1a) or allowing it to be free (Scenario 1b)). In Scenario 2 the binding constants (c.f. c
1 and c
2) are independent of growth rate but the RIDA rate, k
, must vary. We have verified that both scenarios are consistent with the eperimentally tested predictions of RNAP availability with growth rate  and with previous measurements  and our own experimental evaluation of total DnaA expression (Chiara Saggioro, Anne Olliver, Bianca Sclavi: Multiple levels of regulation in the growth rate dependence of DnaA expression, submitted), and also with a number of "in silico mutations" inspired by the available literature [24, 37]. Thus, the scenarios appear as possibilities that are testable, but for the moment remain open. Note that the property that the initiation threshold holds constant with respect to growth rate changes is not related to the specific set of parameters we used, or any set of parameters. Our analysis shows that in general, for any fixed parameter set at a given growth rate, a transformation is necessary in order to keep the threshold constant while moving to another growth rate. In order to provide specific examples, we have produced plots in the style of those in Figure 5, with different curves corresponding to choosing different values of the initial input parameters. These demonstrate that the qualitative behaviour of the transformation is independent of these parameters (Additional File 1, Figures A8, A9 and A10). This exercise is also important to show that the parameter changes with growth rate are not numerically negligible for empirically plausible parameters, so that the question of keeping the initiation threshold constant is not purely academic.
It is then interesting to ask which of these scenarios is more reasonable considering the known biological processes. We speculate that scenario 2 is less likely, since until now there is no evidence pointing to a possible change in the intrinsic RIDA rate as a function of growth rate. The DnaA-related protein Hda (Homologous to DnaA) mediates this process . Experiments with mutants over-and underexpressing Hda , with corresponding increases and decreases in the RIDA rate, suggest a possible mechanism by which the k
term in the equations could vary by a growth rate-dependent expression of the Hda gene. There may also be other, as yet unknown, factors that affect the growth rate dependence of the RIDA rate. Alternatively, we can speculate that the decrease in the rate of RIDA with growth rate could be caused effectively by the action of the reverse process of DnaA-ATP recycling by the recently discovered recycling regions . Figure 5b shows that the RIDA rate should increase with cell cycle time and thus decrease with growth rate. This growth rate increase causes overlapping replication rounds, and thus higher chromosome copy number. Since more recycling regions are present there is more recycling, i.e. a decrease in the effective RIDA rate, compatibility with the requirement imposed by our results. However, considering explicitly this model variant, we find that the balancing recycling cannot by itself impose a constant threshold.
Conversely in scenario 1, the RIDA rate per replication fork is constant, and one has to rationalize the variation of the binding affinities. It seems possible that the binding affinities could change with growth rate through changes in supercoiling, in similar ways to those seen in Figure 5 and Additional File 1, Figure A5 [62, 63]. The levels of average negative supercoiling are known to increase as the growth rate increases . However, it makes sense to challenge the validity of the basic assumption that the ratio of DnaA-ATP to DNA at the time of initiation is constant. This model assumes that the affinity for DnaA-ATP binding to its own promoter can change with growth rate but its affinity for the origin does not. The first assumption mainly allows the model to change the magnitude of negative autoregulation as a function of growth rate, and it may indeed be explained by the changes in global cellular parameters such as negative supercoiling. We have considered how the activation threshold in the model (estimated in Additional File 1, Figure A11 and corresponding caption) would be affected if the binding affinity for DnaA to the origin would vary in the same way as its value at the dnaA promoter, required by Scenario 1, for a set of realistic parameters. We found that these changes in r(X) are less than 10% over a wide range of growth rates, suggesting that this scenario might be robust. Indeed, the observed threshold is certainly approximately constant when compared to the untransformed case i.e. the different values of the ratio at t = X shown in Figure 3A. More generally, the initiation of DNA replication has been significantly simplified in this model; all it requires is a specific amount of available DnaA-ATP molecules. However we know that other factors, such as the binding of nucleoid proteins FIS, IHF, H-NS and HU, may contribute to the formation of an open complex at the origin. On the other hand, other recent results have shown that at slower growth (slower than the range considered here) the cell contains a greater average amount of DnaA-ATP per origin that results in initiation events that are independent of the novel synthesis of DnaA-ATP . These results suggest that the regulation of the initiation process at the origin might indeed be dependent on the growth rate and that these changes still remain to be characterized quantitatively before they can be included in a theoretical model.
Interestingly, the basal rate of transcription of the dnaA gene, k
must vary in both scenarios. Figure 5 shows that k
decreases as the cells grow more slowly. This is what is expected from a promoter like the one of the dnaA gene that closely resembles ribosomal RNA promoters. This family of promoters have a GC-rich sequence at the transcription initiation site called a discriminator region. This region renders the activity of the promoter sensitive to the degree of negative supercoiling, which activates transcription by enhancing DNA melting, and leads to its inhibition by the accumulation of ppGpp at slower growth rates .