DNA Replication and Cell Growth for Escherichia coli
Joseph M. Mahaffy
Department of Mathematical Sciences
San Diego State University
Collaborators:
Prof. Judith W. Zyskind
Wai H. Lee
Abstract:
Escherichia coli can divide every 20 min in a rich medium, yet its DNA replication requires about 45 min. So how does this organism regulate its cell cycle? Initiation of DNA replication in E. coli appears to be the primary controlling element in determining the length of the cell cycle. DNA replication begins at oriC with a large complex of proteins. We developed a model, which supports the hypothesis that the protein DnaA regulates the rate of initiation of DNA replication. Our model depends on several growth rate dependent parameters. A model for the growth of E. coli is being developed to include the stringent response or negative feedback by tetraphosphoguanine (ppGpp) as nutrient pools are depleted. The model tracks the major pools of proteins, RNAs, and mass of the growing cell. This talk includes a discussion of the key steps that are known in growth rate regulation and DNA replication along with the models developed to date.


Initiation of DNA Replication and DnaA
DnaA
Elongation and the C Period
Mathematical Model
Below is a schematic representing one model for the Control of Initiation of DNA Replication by the protein DnaA.
1. Probabilistic Part of the Model
A Monte Carlo simulation is used to determine if the active DnaA, DnaA^{.}ATP, binds to oriC at any of the 4 binding sites with the consensus sequence. The probability that one molecule of DnaA^{.}ATP binds to oriC is approximately
It is assumed that there is at most one binding event during the time interval, Dt/J. The probability that one molecule dissociates is
where N is the number of exposed molecules. The simulation follows how many DnaA molecules are bound to the 4 binding sites of oriC for each time step in the program.
2. Deterministic Part of the Model
To determine the random binding of DnaA to the consensus sequence elsewhere along the replicating DNA forks (GS), we evaluate
where k_{gs} is the rate of formation of consensus sequence sites per elongating chromosome, AC.
The differential equation for the dynamics of the concentration of the various forms of DnaA protein (bound to the chromosome, inactive, and free with ATP are given by the following:
where the k's are the kinetic constants, m = ln(2)/t for t the generation time, and g a fraction representing both different available quantities of [DnaA_{i}] and differing binding rates for these forms.
3. Program Controls  Initiation and Division
Whenever 30 or more molecules of DnaAATP bind to oriC, then initiation is assumed to occur and a new oriC is created. The new oriC and the one that initiated are not allowed to bind DnaA for 8 min.
The length of the C and D periods for replication are fixed, so it is assumed that 70 min after initiation, division occurs. Note that this initiation event will have occurred in a past generation, so the program tracks specific oriCs. At division, the volume and quantities of the chemical species are halved, and the generation time t is reset to reflect the length of the last generation.
4. Simulations of the Model
Below are graphs showing typical simulations of the model for a given set of growth parameters.
t = 45
t = 58
Summary of the Results
Model of Growth of E. coli
Scenario for Model
Assume a mutant strain of E. coli limited by some essential Amino Acid. Track a culture as it goes through the early lag phase into exponential growth, then when nutrient becomes depleted, it becomes stressed and enters a stationary growth phase.
Transcription
Translation
Production of ppGpp
Mathematical Model
Results and Future Directions