BSS Group, Department of Physics, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, UK

LBPA, UMR 8113 du CNRS, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 CACHAN, France

Dip. Fisica, Università "Sapienza", and IPCF-CNR, UOS Roma Piazzale A. Moro 2, I-00185, Rome, Italy

Università degli Studi di Milano, Dip. Fisica. Via Celoria 16, 20133 Milano, Italy

I.N.F.N. Milano, Italy

Génophysique/Genomic Physics Group, UMR7238 CNRS "Microorganism Genomics

University Pierre et Marie Curie, 15 rue de l'École de Médecine, 75006 Paris, France

Abstract

Background

In

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, depending on the roles of DnaA autoregulation and DnaA ATP-hydrolysis regulatory process. 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

Background

The coordination of DNA replication with cell division in

Timing of DNA replication initiation as a function of the length of the cell cycle according to the Cooper and Helmstetter model

**Timing of DNA replication initiation as a function of the length of the cell cycle according to the Cooper and Helmstetter model**. A: Plots of the values of

In 1968, Donachie calculated that the correct timing would be guaranteed by a constant ratio of the cell size at the moment of initiation (termed the 'initiation mass') and the number of

In addition, DnaA exists under two forms, ATP or ADP bound. The first is required for activation of the origin, thus it is usually called the active form

- R

- I

- D

- A

Other processes can contribute to prevent reinitiation within the same cell cycle, such as the binding of the SeqA protein to the newly replicated, hemimethylated DNA

DnaA-ATP binding to the origin must determine the timing of initiation for a range of growth rates and thus in the presence of increasing genome amounts (providing non-specific binding sites). Thus, the amount of DnaA-ATP per cell needs to increase with the decrease in doubling time. The

It has previously been proposed that the presence of both autoregulation and RIDA contributes to increased robustness of the initiation regulatory network upon perturbations

The resulting equations describe, via a continuous change in parameter values with growth rate, the oscillations in DnaA-ATP per non-specific site and the attainment of a constant threshold as a function of growth rate. This shows that the circuit performing the timing of replication initiation must encode subtle information on the bacterial physiological state through the growth rate dependence of the parameters. This analysis also allows us to define a few scenarios consistent with the available experimental knowledge and to make testable predictions on the relative roles of DnaA autorepression and of the RIDA process at different growth rates. We use this model to elucidate the reciprocal roles of the known factors affecting DnaA activity in

Methods

Assumptions of the model

The model consists of a set of Ordinary Differential Equations (ODEs) describing DnaA-ATP production by the expression of the

We ask how the parameters of this model must vary in order for this assumption to hold in the range of doubling times between 20 and 60 minutes. In the absence of autoregulation, the only factor that contributes to a decrease in the ratio of DnaA-ATP to non-specific binding sites is the increase in DNA after DNA replication has begun. The complete model also includes the autoregulation of DnaA expression by DnaA binding to its own promoter

Initial values for the parameters in the model

**Parameter**

**Untransformed Value (at τ **=

**Units**

**Reference**

Basal transcription rate _{A}

75

molecules/min

RNAP binding

12/10000

Dimensionless

Dna-ATP binding

1/10000

Dimensionless

RNAP amount _{0}

5050

molecules

RIDA rate _{R}

10

molecules/min

Replication rate _{Λ}

1/40

genome equivalents/min

non-specific binding sites _{NS}

5 ^{6}

(genome equivalent)^{-1}

The parameters are fixed with the values in the table for

Formulation of the model

Timing of replication

We take into consideration the situation where the cell cycle repeats itself identically i.e. balanced, exponential growth. Following Cooper and Helmstetter, at a time

where ^{n }

Promoter term

The activity of the _{NS }
_{-}. We then use the assumption that the number of non-specific binding sites is proportional to the length of DNA in the cell (which we write as Λ), i.e. Λ = _{NS}

Thus the expression for the rate of transcription at the

**Additional Text and Figures**. Single pdf file containing the additional text and figures mentioned in the main text.

Click here for file

where _{A }
_{1 }and _{2 }depend on the binding energies Δ_{pd }
_{ad }
_{- }respectively to their promoter binding sites. The binding energies are determined from the ratio of specific vs non-specific binding affinities.

where the exponential terms are Boltzmann weights. _{2 }= 0 if the promoter is not autorepressed. A version of the promoter where DnaA binding to its sites is cooperative is described in Additional File

RIDA term

This term reflects the number of DnaA-ATP molecules that are converted to DnaA-ADP molecules per unit time by the RIDA process. As discussed in the introduction, RIDA is a process that takes place at the replication forks during DNA synthesis. We assume that the rate of conversion _{R }

For

and for

(note that this equation is intrinsically discrete since it relates to the physical number of replication forks) and so the conversion from DnaA-ATP to DnaA-ADP takes place at a rate

This leads to the following differential equations for DnaA-ATP (denoted _{-}) and DnaA-ADP (denoted _{+})

Term for the growth of the chromosome

The growth of the chromosome is controlled by the replication forks. Defining the rate of DNA synthesis of each pair of replication forks as _{Λ}, we can write

Assuming that _{Λ }is constant, and normalizing so that Λ = 1 is the length of one full chromosome, we have _{Λ }= 1

Main equation

Figure

Ingredients of the model

**Ingredients of the model**. A: illustration of 1) the autorepression of the _{- }(number of DnaA-ATP molecules), _{+ }(number of DnaA-ADP molecules), Λ (total genome length), _{A }_{1}, _{2 }(binding constants), _{R }_{Λ }(growth rate of the chromosome per replication fork). The first equation represents the change in the number of DnaA-ATP molecules, with a source term due to the

This equation describes the dynamics of the variable

Main Assumptions

The model relies on the following further assumptions

Numerical integration

The non-linearity of the main equation (11) necessitates the use of a numerical method of integration. We used a custom C++ implementation of the fourth-order Runge-Kutta method. The equation was integrated for values of the cell doubling time,

In order to test for the constant threshold condition, a transformation was performed by integrating the equation for

The dependence of the parameters in the different scenarios

**Scenario**

**Floating Parameters**

**Fixed parameters**

1a

_{A}_{1}, _{2}

_{R}

1b

_{A}_{2},

_{1}, _{R}

2

_{R}_{A}

_{2}, _{1}

In all the scenarios, _{A }

Results

A fixed set of parameters gives a varying initiation threshold with increasing growth rate

We first describe the behaviour of the model with a fixed parameter set. The ratio

The model imposes a specific DnaA-ATP threshold at the moment of initiation (

**The model imposes a specific DnaA-ATP threshold at the moment of initiation ( t = X)**. A: The model with fixed parameters cannot explain an 'initiation threshold' since a different value of the ratio DnaA-ATP:genome length (

This fact naturally leads to us consider a model in which the parameters are able to vary with growth rate. Biologically, this is a natural requirement, as one may well expect from previous observations of the change in cellular components as a function of growth rate

A constant threshold condition implies alternative scenarios of growth-rate dependency in the circuit architecture

The condition of a constant DnaA-ATP/DNA threshold at the time of initiation can be imposed by performing a mathematical transformation on the model and verifying the implications of this for the values of the parameters. The mathematical details of this transformation can be found in Additional File

All scenarios of the model are compatible with previous measurements and predictions

**All scenarios of the model are compatible with previous measurements and predictions**. A: Variation in the average number of RNAP molecules per cell with doubling time. Simulations from the three scenarios (connected triangles, squares and crosses) are compared to the (validated) predictions of ref.

Predictions of the model can distinguish between different scenarios

**Predictions of the model can distinguish between different scenarios**. Two possible scenarios can result in a constant initiation threshold for the model. In the first the binding affinity of DnaA-ATP to its repressor sites decreases with increasing cell doubling time, and in the second the RIDA rate increases with cell doubling time. In both scenarios _{A }

1. In the first scenario, the RIDA rate (per replication fork), _{R}
_{A }
_{1 }and _{1 }and

(a) In the first of these sub-scenarios, 1a, the variation in the number of non-specifically bound RNAP in the cell,

(b) In sub-scenario 1b, the binding affinity of RNAP to the

2. In scenario 2, the binding affinity of both RNAP to the _{A}
_{R }

In brief, two possible scenarios can result in a constant initiation threshold for the model. In the first, the binding affinity of DnaA-ATP to its repressor sites decreases with increasing cell doubling time and in the second the RIDA rate increases with cell doubling time. In both scenarios _{A }
_{A}
_{1 }and _{2 }terms). It describes how quickly RNAP moves through a gene when transcribing. The variation of this characteristic time with growth rate can be associated with variations in DNA supercoling (see Discussion).

In absence of RIDA (_{R }
_{2 }= 0), the transformation can still be performed. However, it implies that the ratio _{1}
_{1 }(which, for example, could also vary through changes in supercoiling) would have to compensate exactly for the changes in

The resulting scenarios are compatible with available knowledge on RNAP availability and total DnaA expression

Given these scenarios, we have asked whether the predicted parameter variation with growth rate and the observables quantities produced by the model were compatible with the measurements and observations available in the literature. Starting from the dependency of available RNAP with growth rate, this is predicted and matched with available experimental data in ref.

We now turn to the changes in measured expression of DnaA (averaged over a population) with growth rate. This can be measured by a reporter gene technique. Figure

Finally, at fixed growth rate, in order to determine whether this model reflects the main features of the regulatory network in the cell, we reproduced some of the experimental perturbations described in the literature. One of these experiments changed the rate of RIDA by changing the level of expression of the gene encoding for the Hda protein

Finally, once we obtained a set of parameters that satisfies the constant threshold constraint, we modified the RIDA rate while leaving the other parameters unchanged (Additional File

Model Variants

In order to gain confidence that the conclusion (that the parameters need to vary with growth rate) is not a consequence of the restricted set of biological ingredients included in this model, we considered some additional model variants, including some of the known factors that may influence the timing of replication initiation.

1. Delay in the synthesis of DnaA-ATP

We introduced a delay, representing the time necessary to obtain an active DnaA molecule from the binding of the RNA polymerase to the

2. Cooperativity of autorepression

Cooperativity of autorepression affects the growth rate dependence of gene expression

where the parameter _{1 }and _{2 }are the binding affinities of the two DnaA binding sites, multiplied by the proportionality constant between Λ and _{NS}

3. The

We considered the effect that the presence of the

4. The eclipse

No constraint for the eclipse period

5. DnaA-ATP Recycling regions

Genomic recycling regions catalyzing the reconversion of DnaA-ADP into DnaA-ATP _{R }

Since none of these model variants qualitatively changes the behaviour of the model with respect to attaining a constant initiation threshold, they were not included in the minimal model formulation, in order to avoid confusion and proliferation of parameters. However, as shown by the variants explored above, the qualitative behaviour that the parameters of the model must vary with growth rate does not hold strictly for the minimal model only, but might be more general.

Discussion

Standard models of bacterial regulatory circuits were adapted to situations where the growth rate is fixed

The dependency of the basic parameters on growth rate can produce notable effects on a genetic circuit, and complicates the standard descriptions

Our description includes the processes that are believed to be most important for initiation of replication

We have defined the DNA replication initiation potential, determining the (synchronous) timing of DNA replication, as the DnaA-ATP to DNA ratio,

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), _{R}
_{1 }and _{2}) are independent of growth rate but the RIDA rate, _{R}

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 (

- H

- D

- A

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

Interestingly, the basal rate of transcription of the _{A }
_{A }

Conclusions

All things considered, we can say that perhaps our main result is that the determination of the timing of initiation by DnaA, besides relying on the known "architecture" comprising autorepression, RIDA and a number of other "dedicated" processes, can be understood only in its complex interplay with bacterial physiology (comprising DNA supercoiling, ppGpp, growth-rate dependent partitioning of molecular machinery, etc.)

Nevertheless, it makes sense to ask whether this model allows us to elucidate some features of the reciprocal role of RIDA and DnaA autorepression, its two main ingredients. Biologically, RIDA renders the control of DnaA-ATP dependent upon ongoing DNA replication, and thus results in an increase in DnaA-ATP when replication forks are blocked. Autorepression however probably plays a larger role in the absence of RIDA at slow growth, or in bacteria that do not have RIDA at all (such as _{1}/

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MCL, BS, and BB designed research. MG, BB, MCL, UF, and CS performed research. MG, BS and MCL wrote the paper. All authors read and approved the final manuscript.

Acknowledgements

We are grateful to Matteo Osella, Rosalind Allen, Pietro Cicuta, Antonio Celani, Andrea Sportiello, Kunihiko Kaneko and Massimo Vergassola for useful discussions and feedback. The authors acknowledge support from the Human Frontier Science Program Organization (Grant RGY0069/2009-C) and from EPSRC.