Construction of DDE equations

The ara operon is regulated by the AraC protein. If arabinose is absent, the dimer AraC protein represses the structural gene by binding to araI 1 and araO 2 and the DNA forms a loop. The loop prevents RNA polymerase from binding to the promoter of the ara operon, thereby blocking transcription .When arabinose is present, arabinose binds AraC and prevents AraC from interacting. This breaks the DNA loop. The two AraC-arabinose complexes bind to the araI site which promotes transcription. When arabinose is present, AraC acts as an activator and promotes the translation of Cre. Then we use Cre-Loxp recombination, a site-specific recombinase technology to carry out inversions in the DNA of cells. The description of the system contains the above mass actions as well as some hill kinetics, Henri-Michaelis-Menten. Since concentration of protein is the integrals of its mRNA, we assume that it is proportional to concentration of mRNA in a previous time. Thus we came up a set of DDEs with 12 equations:


The parameters are inherent factors determining the behaviors, properties of a system. Some of our parameters are derived from original ones, some of them are created to describe the new equations, and others are set for further testing.


We coded the system in MATLAB. The result following shows that the concentration of Cre and the concentration of Cre-Loxp which we use all the paper to denote the Cre protein which binds to Loxp sites grow to a peak and then decline to zero. So we can expect that the inversion of the DNA which is proportional to the concentration of Cre-Loxp happens only in a period when Cre-Loxp exists.

The concentration of Cre:

The concentration of Cre-Loxp :