Team:ULB-Brussels/modeling/loss

From 2011.igem.org

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<h1>Introduction</h1>
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<h1>Loss of the pINDEL plasmid in LB liquid medium at $42^\circ$C</h1>
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\label{Ph3}
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<p>
<p>
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The pINDEL plasmid can be divided into $2$ functional units:
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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.
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<ol>
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  <li>the IN function which is composed of the <em>gam</em>, <em>exo</em> and <em>bet</em> genes coding for the $\lambda$ Red recombinase system \cite{dat,yu}; and</li>
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  <li> the DEL function which is based on the <em>flp</em> gene encoding the FLP site-specific recombinase \cite{dat,yu}.</li>
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</p>
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<h2>Model</h2>
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<h3>Definitions</h3>
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<p>
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The expression of $\lambda$ Red recombinase genes is under the control of the pBAD promoter.  This promoter is repressed by the AraC transcriptional regulator in absence of arabinose and activated by the same protein in the presence of arabinose.  The<em>araC</em> gene is also encoded in the pINDEL plasmid.  The expression of the FLP recombinase is under the control of the $\lambda$ pR promoter.  This promoter is repressed at  $30^\circ$C by the thermosensitive CI857 repressor which is also encoded in the pINDEL plasmid.  We will consider that expression of the <em>flp</em> gene is repressed at 90\% at $30^\circ$C, while at $42^\circ$C the <em>flp</em> gene is fully expressed. However it is reported that at this temperature, the activity of FLP is drastically reduced as compared to lower temperature \cite{buch}.
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Let us begin with a proper definition of the different biological functions that are considered in our model:
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<ul>
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  <li>$N$: total number of bacteria in the considered population;</li>
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  <li>$P$: average number of pINDEL plasmids per bacterium;</li>
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  <li>$E$: average amount of active RepA101 enzymes per bacterium;</li>
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  <li>$F$: average amount of active FLP per bacterium;</li>
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  <li>$G_i (i=1,2,3)$:} average amount of the Red recombinase protein $i$ (1 is Gam, 2 is Exo and 3 is Bet), per bacterium.</li>
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</ul>
</p>
</p>
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<h2>Getting the equations</h2>
<p>
<p>
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In addition, pINDEL contains the <em>repA101ts</em> gene encoding the RepA101Ts protein and the origin of replication (<em>ori</em>) \cite{dat,yu}. The RepA101Ts protein initiates replication at $30^\circ$C by specifically binding to the ori. The RepA101Ts protein becomes rapidly inactive when the culture is shifted at 42¡C and is therefore not able to mediate replication initiation at this temperature. The pINDEL plasmid also contains the Amp resistance gene for plasmid selection.
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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
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$$
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\left\{
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\begin{array}{c}
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\dot E=C_EP-A_EE-k_NE\label{E42f2}\\
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\dot P=k_P\frac E{E_0}P\left(1-\frac P{P_{max}}\right)-k_NP\label{P42f2}
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\end{array}
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\right.
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$$
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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, $E_0:=E(0)\approx5\cdot10^3$.
</p>
</p>
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<h2>Solving the equations</h2>
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<p>
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Those equations can be solved numerically via \textit{Mathematica}; for realistic values of the parameters, we obtain (fig(\ref{graph10})) and (fig(\ref{graph11})).
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<br>
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\includegraphics{figure101.pdf}\\\includegraphics{figure102.pdf}
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\caption{\label{graphe10}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 $P_0=19$.}
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<br>
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\includegraphics{figure11.pdf}
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\caption{\label{graphe11}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 $P_0=19$.}
<p>
<p>
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The Red recombinase promotes the insertion of a gene of interest (gene X) coupled to an antibiotic resistance gene flanked of FRT' sites (FRT'-Cm-FRT', our biobrick BBa\_K551000 for the selection of the insertion event in the bacterial chromosome.  FLP on the other hand is responsible for the site-specific excision of the antibiotic resistance gene, after insertion of the gene of interest, leaving a FRT' site. Thus, the IN and DEL functions are antagonist. Even under <em>flp</em> repression condition ($30^\circ$C), we cannot exclude that a small amount of FLP is produced due to the $\lambda$ pR promoter leakiness. This could drastically affect the frequency of insertion because excision of the Cm resistance gene could occur prior insertion of the X gene in the bacterial chromosome. To overcome this problem, we designed a particular configuration in which the IN and DEL functional units are encoded on the opposite strands and are facing each other. Our hypothesis is that the expression of the IN function (induced by arabinose) would inhibit the DEL function expression by a mechanism denoted as transcriptional interference. First, we will study by a computer simulation whether a potential transcriptional interference occurs between these 2 opposite-oriented functional units (see section (\ref{IntTranscr})).
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We can observe that $E$ decreases very rapidly and cancels out, in such a way that
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\begin{equation}
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\dot P\approx-k_NP\quad\Rightarrow P\approx P_0e^{-k_Nt}
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\end{equation}
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The average amount of Pindel plasmids per bacterium becomes insignificant as soon as $t\approx 1/k_N\approx 1730\mbox{s}=28\mbox{min}50\mbox{s}$.
</p>
</p>
<p>
<p>
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In our different models, we will consider a few parameters and we will estimate their values based on biological considerations. We will then analyze the coherence of our predictions together with the results of the experiments, and adapt the model if necessary.
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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.
</p>
</p>

Revision as of 03:40, 22 September 2011

Modelling : Loss of the pINDEL

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

\label{Ph3}

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})).
\includegraphics{figure101.pdf}\\\includegraphics{figure102.pdf} \caption{\label{graphe10}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 $P_0=19$.}
\includegraphics{figure11.pdf} \caption{\label{graphe11}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.

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