Phase at $42^\circ$C

Preparation: electroporation and night culture

Once the E. Coli population obtained after the phase at $30^\circ$C on arabinose (see section (\ref{Ph30})) reaches $1$ OD (at $600$nm), that is (almost) saturation, we move the bacteria in liquid (to have roughly $10^3$ per ml), and electropore them with the DNA fragment to insert (i.e. the gene X followed by the chloramphenicol resistance). The bacteria are then put in a culture with chloramphenicol, eliminating the ones that eventually would not have integrated the DNA fragment. That way, we obtain colony's of E. Coli having the fragment..

Such a colony is then moved to a $30^\circ$C culture in $10$ml, where she grows until saturation (between $2\cdot10^9$ and $5\cdot10^9$ bacteria per ml). The solution is then diluted $100$ to $1000$ times, then cultivated, until it reaches an optic density (OD) (at $600$nm) of $0.2$ (which corresponds approximatively to $10^8$ bacteria). The bacteria are then put in presence of glucose at $42^\circ$C. The absence of arabinose desactivates Pbad (the promotor of the three-genes sequence $i$), and the presence of glucose reinforces this desactivation.

Modelisation of the $42^\circ$C phase on glucose

At initial time ($t=0$), the amount of bacteria is $N_0:=N(0)\approx10^8$. Like in section (\ref{Mod30}), we use a logistic model: \begin{equation} \dot N=k_NN\left(1-\frac N{N_{max}}\right) \label{N42}\end{equation} where $N_{max}$ is the maximum amount of bacteria that the culture environment is able to contain and where $k_N$ corresponds to the growth rate one would observe in the limit where the saturation would be inexistent. Like in section (\ref{Mod30}), we have that the density of bacteria at dew point corresponds to a little more then $1$OD (at $600$nm), that is approximatively $N_{max}\approx2\cdot10^9$, and $k_N\approx \frac{\log{2}}{20\cdot60}\mbox{s}$.

Let us observe how $P$ evolves in time. At initial time in this phase, the average amount of Pindel plasmids per bacterium is $P_0:=P(0)\approx19$. The replication of those plasmids is activated by RepA101, non-stop produced by Pindel. The total amount of those enzymes thus follows the equation \begin{equation} \dot E_{tot}=C_EP-D_EE_{tot} \label{Etot42}\end{equation} in terms of the production rate $C_E$ of the enzyme per plasmid and of the natural deterioration rate $D_E$. However, at $42^\circ$C, the enzyme becomes quickly inactive. We obtain for the average active enzymes $E$ the relation: \begin{equation} \dot E=\dot E_{tot}-A_EE \label{E42}\end{equation} where $A_E$ is the natural deterioration rate of RepA101. We estimated $C_E\approx\ldots$, $A_E\approx\ldots$ and $D_E\approx\ldots$ (note that the value of $A_E$ is huge, relatively to the other parameters). At initial time, we can estimate that $E_0:=E(0)=E_{tot}(0)\approx\ldots$.