Team:ULB-Brussels/modeling/introduction

We will consider a population of Escherichia Coli, that possesses the Pindel plasmid, inserted beforehand by electroporation. The replication origin (ORI) of this plasmid is activated by RepA101, a constitutive enzyme, produced by the plasmid itself; at $42^\circ$C, this enzyme (RepA101) becomes rapidly inactive. Also, the plasmid includes a three-gene sequence coding proteins (Red recombinase) inducing the insertion, in the DNA, of a gene "X" (at which we added, as a control, a resistance gene to chloramphenicol). The promotor of this three-genes sequence, Pbad, is activated by arabinose. Pindel includes in addition a coding sequence for flippase, whose promotor is blocked by a thermo-sensible repressor (at $42^\circ$C, the promotor isn't repressed at all, whereas at $30^\circ$C it is almost completely repressed, to $\ldots\%$). Those two sequences follow each other, but are encrypted in opposite directions, which eventually leads to an interference phenomenal. Furthermore, flippase, originally coming from yeast, is entirely active at $30^\circ$C, but is disabled at $42^\circ$C. Red recombinase recognizes the gene sequence to insert (the "X" gene followed by the resistance gene) and insert them in the bacteria's genome, while flippase excise the resistance to chloramphenicol package.

At start, we place bacteria possessing Pindel at $30^\circ$C on arabinose. The Red recombinase genes are then expressed because of the presence of arabinose. In parallel, a small amount of flippase is produced: in section (2.1), we will study, by a computer simulation, the interference that occurs during the transcription of those two opposite-oriented genes, which will give us an idea of the probability that the flippase sequence eventually is transcribed. Nevertheless, we do expect that the amount of flippase produced stays low. After a while, those bacteria are electroporated in order to insert the X gene.

By testing the resistance to chloramphenicol, we can determine which colony's integrated the X gene, coupled with the resistance gene. We take some of those bacteria and place them at $42^\circ$C on glucose. Flippase is then fully produced, and is able to excise the chloramphenicol-resistance package. At the same time, the RepA101-enzyme, allowing the Pindel plasmid to reproduce is strongly suppressed, and becomes fast completely inactive. In consequence, the total number of plasmids remains constant; and the increase in population will dilute that amount of plasmids among the bacteria, leaving most of them without any plasmid. In the same way, flippase stops being produced, and therefore, by natural degradation, disappears. We thus obtain a population possessing the X gene, but devoid of the chloramphenicol resistance nor the Pindel plasmid.

It is this whole process that we will try to modelise here. In our model, we considered a few different parameters, of which we will estimate the values, based on biological considerations. Afterwards, we will analyze the coherence of our predictions together with the results of the experiments, or if necessary, adapt the model itself.

Let us begin with a proper definition of the different biological functions we are to study:

• $N$ : total amount of bacteria in the considered population;
• $P$ : average amount of Pindel plasmids per bacterium;
• $E_{tot}$ : average amount of RepA101 enzymes per bacterium;
• $E$ : average amount of RepA101 enzymes active per bacterium;
• $F$ : average amount of flippase per bacterium;
• $G_i (i=1,2,3)$ : average amount of the protein $i$ for the Red recombinase, per bacterium.