Loss of the pINDEL plasmid in LB liquid medium at $42^\circ$C

This experiment is performed in order to determine the kinetic of pINDEL loss at $42^\circ$C in a growing \textit{E. coli} population. A culture of \textit{E. coli} containing pINDEL is grown at $42^\circ$C and diluted at appropriate times to maintain the bacteria in logarithmic growth phase.

Model

Definitions

Let us begin with a proper definition of the different biological functions that are considered in our model:

• $N$: total number of bacteria in the considered population;
• $P$: average number of pINDEL plasmids per bacterium;
• $E$: average amount of active RepA101 enzymes per bacterium;
• $F$: average amount of active FLP per bacterium;
• $G_i (i=1,2,3)$:} average amount of the Red recombinase protein $i$ (1 is Gam, 2 is Exo and 3 is Bet), per bacterium.

Getting the equations

Our equation for $E$ and $P$ are the same as in the previous section (section (\ref{Mod42})), but we have $\frac{\dot N}N=k_N$ so that our equations read $$ \left\{ \begin{array}{c} \dot E=C_EP-A_EE-k_NE\label{E42f2}\\ \dot P=k_P\frac E{E_0}P\left(1-\frac P{P_{max}}\right)-k_NP\label{P42f2} \end{array} \right. $$ At initial time in this experiment ($t=0$), the average amount of pINDEL plasmids per bacterium is $P_0:=P(0)\approx19$, that is slightly less than the maximum: immediately after the night culture we must have theoretically $P=P_{max}$, but we have to take into account the possible accidents during the manipulations before the beginning of the excision step. Moreover, $E_0:=E(0)\approx5\cdot10^3$.

Solving the equations

Those equations can be solved numerically via \textit{Mathematica}; for realistic values of the parameters, we obtain (fig(\ref{graph10})) and (fig(\ref{graph11})).

This is obtained for $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $C_E=\frac1{18}\mbox{s}^{-1}$, $A_E=\frac{\log{2}}{30}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $E_0=5\cdot10^3$, $P_{max}=20$ and $P_0=19$.}
This is obtained for $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $C_E=\frac1{18}\mbox{s}^{-1}$, $A_E=\frac{\log{2}}{30}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $E_0=5\cdot10^3$, $P_{max}=20$ and $P_0=19$.}

We can observe that $E$ decreases very rapidly and cancels out, in such a way that \begin{equation} \dot P\approx-k_NP\quad\Rightarrow P\approx P_0e^{-k_Nt} \end{equation} The average amount of Pindel plasmids per bacterium becomes insignificant as soon as $t\approx 1/k_N\approx 1730\mbox{s}=28\mbox{min}50\mbox{s}$.

It is important to point out that, again, the solution of our model only shows a small sensitivity to the parameters, around the estimated values: a small error on the parameters only leads to a small error in the solution, as we can easily notice by moving the parameters in a confidence interval around the estimated values.