Team:ULB-Brussels/modeling/42

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<a href="https://2011.igem.org/Team:ULB-Brussels">Home</a>
<a href="https://2011.igem.org/Team:ULB-Brussels">Home</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/project">Project</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/project">Project</a>
-
<a id="couleur"  href="https://2011.igem.org/Team:ULB-Brussels/modeling">Modeling</a>
+
<a id="couleur"  href="https://2011.igem.org/Team:ULB-Brussels/modeling">Modelling</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/human">Human practice</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/human">Human practice</a>
-
<a href="https://2011.igem.org/Team:ULB-Brussels/Results">Results</a>
+
<a href="https://2011.igem.org/Team:ULB-Brussels/Discussion">Discussion</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/parts">Parts</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/parts">Parts</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/safety">Safety</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/safety">Safety</a>
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<div id="sousm">
<div id="sousm">
-
<a  href="https://2011.igem.org/Team:ULB-Brussels/modeling/introduction">Introduction</a>
+
<a  href="https://2011.igem.org/Team:ULB-Brussels/modeling">Introduction</a>
-
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/30">Phase at 30°C</a>
+
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/30">Transcriptional interference</a>
-
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/42">Phase at 42°C</a>
+
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/42">Insertion model</a>
-
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/comparison">Comparison with the Wet Lab work </a>
+
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/excision">Excision model</a>
 +
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/loss">Loss of the pINDEL plasmid at 42°C</a>
 +
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/comparison">Comparison with data</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/conclusion">Conclusion</a>
<a href="https://2011.igem.org/Team:ULB-Brussels/modeling/conclusion">Conclusion</a>
</div>
</div>
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<div id="maintext">
<div id="maintext">
<div id="hmaint">
<div id="hmaint">
-
Modeling : Phase at 30°C </div>
+
Modelling : Insertion model </div>
<div id="maint">
<div id="maint">
-
<h1>Phase at $42^\circ$C</h1>
 
-
<h2>Preparation: electroporation and night culture</h2>
+
 
 +
<h1>Insertion model</h1>
 +
 
 +
<h2>Definitions</h2>
<p>
<p>
-
Once the <em>E. Coli</em> 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 <em>E. Coli</em> having the fragment..
+
Let us begin with a proper definition of the different biological functions that are considered in our model:
-
</p>
+
<ul>
 +
  <li>$N$: total number of bacteria per ml in the considered culture;</li>
 +
  <li>$P$: average number of pINDEL plasmid copies per bacterium;</li>
 +
  <li>$F$: average amount of active FLP per bacterium;</li>
 +
  <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>
 +
</ul>
 +
 
 +
 
 +
<h2>Experimental design of the insertion step</h2>
<p>
<p>
-
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.
+
A colony of <em>E. coli</em> containing the pINDEL plasmid is grown at $30^\circ$C in $10$ml of LB medium containing Amp to select for the presence of pINDEL. This culture reaches saturation after ON culture at $30^\circ$C (titer of the culture between $2\cdot10^9$ and $5\cdot10^9$ bacteria per ml of culture). This ON culture is then diluted $100$- to $1000$-fold and grown in logarithmic phase at $30^\circ$C in LB medium ($\mbox{OD}_{600\mbox{nm}}$ around $0.2$, corresponding to $10^8$ bacteria/ml of culture). Arabinose ($0.2$ to $1\%$) is then added to the culture to induce the expression of the IN function. These cells are electroporated with a linear PCR fragment containing the gene X of interest and the FRT'-Cm-FRT'. Transformants are selected on LB plates containing Cm without arabinose at $30^\circ$C.
</p>
</p>
-
<h2>Modelisation of the $42^\circ$C phase on glucose</h2>
+
<h2>Getting the equations of the insertion step</h2>
 +
 
<p>
<p>
-
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:
+
At the initial time ($t=0$), <em>i.e.</em> immediately after the dilution, the number of bacteria ($N(t)$) is $N_0:=N(0)\approx10^8 \ \mbox{bact}/\mbox{ml}$. We used the Verhulst's logistic model.
\begin{equation}
\begin{equation}
\dot N=k_NN\left(1-\frac N{N_{max}}\right)
\dot N=k_NN\left(1-\frac N{N_{max}}\right)
-
\label{N42}\end{equation}
+
\label{N30}
-
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 $k_N\approx \frac{\log{2}}{20\cdot60}\mbox{s}$.
+
\end{equation}
 +
where $N_{max}$ is the maximal number of bacteria in the culture and where $k_N$ corresponds to the growth rate one would observe in the limit where the saturation would be inexistent. In our case, at saturation, the $\mbox{OD}_{600\mbox{nm}}$ slightly exceeds $1$, which corresponds to approximately $N_{max}\approx 2\cdot10^9\mbox{bact}/\mbox{ml}$. On the other hand, since in our conditions, <em>E. coli</em> ideally divides every $20$min, if we are far from the saturation ($N_{max}=\infty$), we obtain
 +
\begin{equation}
 +
\dot N=k_NN \Rightarrow N_0e^{k_Nt}=N(t)=N_02^{t/20\mbox{\min}} \quad\Rightarrow k_N\approx \frac{\log{2}}{20\cdot60}\mbox{s}^{-1}.
 +
\label{k_N}\end{equation}
</p>
</p>
<p>
<p>
-
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
+
At this point, all the bacteria in the culture contain the pINDEL plasmid. Note that RepA101Ts is fully active for pINDEL replication at $30^\circ$C. However, we shall note that the number of plasmid copy per bacterium cannot exceed a certain number $P_{max}\approx20$ (as the origin of replication of pINDEL is low copy).  At initial time, the number of pINDEL plasmid copy 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 insertion step. Again, we naturally postulate a logistic model:
\begin{equation}
\begin{equation}
-
\dot E_{tot}=C_EP-D_EE_{tot}
+
\dot P=k_PP\left(1-\frac{P}{P_{max}}\right).
-
\label{Etot42}\end{equation}
+
\label{production}\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:
+
</p>
 +
 
 +
<p>
 +
As pINDEL is composed of $10800$ nt and as the replication rate is of about $750$ nt/s, we can estimate that the replication of pINDEL takes $\frac{10800}{750}\mbox{s}=14.4 \mbox{s}$. Using then the same reasoning we used for $k_N$ (eq(\ref{k_N})), we compute $k_P\approx\frac{\log{2}}{14.4}\mbox{s}^{-1}$. Moreover, we have to consider the contribution of the increase in population, which produce a dilution effect. In that purpose, let us suppose for a moment that the plasmids do not replicate any more; we then have $PN=\mbox{cst}$, thus
\begin{equation}
\begin{equation}
-
\dot E=\dot E_{tot}-A_EE
+
P=\frac{\mbox{cst}}N\quad\Rightarrow \dot P=-\mbox{cst}\frac{\dot N}{N^2}=-\frac{\dot N}NP.
-
\label{E42}\end{equation}
+
\label{dilution}\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 $E_0:=E(0)=E_{tot}(0)\approx\ldots$.
+
</p>
</p>
 +
<p>
 +
Combining both production (eq (\ref{production})) and dilution (eq(\ref{dilution})) effects, we get the evolution equation for $P$:
 +
\begin{equation}
 +
\dot P=k_PP\left(1-\frac P{P_{max}}\right)-\frac{\dot N}NP.
 +
\label{P30}\end{equation}
 +
</p>
 +
 +
<p>
 +
Note that this equation can be written as follow:
 +
\begin{equation}
 +
\frac d{dt}(NP)=k_PNP\left(1-\frac P{P_{max}}\right),
 +
\label{equNP}\end{equation}
 +
which allows a convenient interpretation: $NP$, the total number of pINDEL plasmid copy, follows a logistic model but where the saturation is only due to $P$. This seems quite natural, as we will see. The evolution of the total number of plasmid copy (per ml) has to be of the form
 +
\begin{equation}
 +
\frac d{dt}(NP)=NP\cdot(g(N,P)-d(N,P))
 +
\end{equation}
 +
in term of a generation rate of new plasmids $g(N,P)$ and a death rate $d(N,P)$. The death rate is <em>a priori</em> constant and even zero in our case: $d(N,P)=d=0$. Regarding the generation rate, it has to diminish when $P$ increases, but is obviously not correlated to the number of bacteria per ml ($N$); the easiest is then to postulate an affine function $g(N,P)=\alpha-\beta P$, so that we find
 +
\begin{equation}
 +
\frac d{dt}(NP)=NP(\alpha-\beta P)t
 +
\end{equation}
 +
which is equivalent to (\ref{equNP}). This observation thus justifies our equation for $P$ (eq(\ref{P30})), initially obtained by heuristic reasoning.
 +
</p>
 +
 +
<p>
 +
As explained above, arabinose induces expression from the pBAD promoter (the promoter controlling the expression of the three genes $i$ on pINDEL). Keeping in mind the three Red recombinase proteins natural decay, the easiest way to model the evolution of the total amount (per ml) of these proteins (<em>i.e.</em> $G_i\cdot N$) is
 +
\begin{align}
 +
&\frac d{dt}(G_iN)=C_iPN-D_iG_iN \quad (i=1,2,3)\\
 +
\Leftrightarrow\quad&\dot G_i=C_iP-D_iG_i-\frac{\dot N}NG_i \quad (i=1,2,3)
 +
\label{Gi30}\end{align}
 +
where $C_i$ is the production rate of the protein $i$ and $D_i$ the decay rate of the same protein. We can estimate that a pINDEL plasmid produces one protein $i$ every $40$s: in good approximation, we only have to consider the three genes transcription time and we may suppose the transcriptions are performed one by one; as <em>gam</em>, <em>bet</em> and <em>exo</em> consist of $417$nt, $786$nt and $681$nt respectively and as the transcription speed is about $51.5$nt/s (between $24$ and $79$ nt/s), we find a transcription time of about $40$s, so that $C_i\approx\frac1{40}\mbox{s}^{-1}$. Furthermore, as these three proteins are stable, we can estimate their half-life time to be around $60$min; we then obtain (by a similar reasoning as in (eq(\ref{k_N}))) $D_i\approx\frac{\log2}{60\cdot60}\mbox{s}^{-1}$.
 +
</p>
 +
 +
<p>
 +
The promoter of the <em>flp</em> gene is repressed by the CI857 thermosensitive repressor at $30^\circ$C. However, repression is not complete and we postulate that the pR leakiness is around $10\%$.  In addition, the pR transcription is inhibited by interference with the transcription of the IN genes $i$. By computer simulation, we have been able to estimate $p_{simul}$, the probability that the <em>flp</em> gene is entirely transcribed (see section (\ref{IntTranscr})). Note that at $30^\circ$C FLP is active. Keeping in mind FLP natural decay, the easiest way to model the evolution of the total amount (per ml) of FLP (<em>i.e.</em> $F\cdot N$) is
 +
\begin{align}
 +
&\frac d{dt}(FN)=10\%p_{simul}C_FPN-D_FFN\\
 +
\Leftrightarrow\quad&\dot{F}=10\%p_{simul}C_FP-D_FF-\frac{\dot N}NF
 +
\label{F30}\end{align}
 +
where $C_F$ is the production rate of FLP by pINDEL (in ideal conditions, at $100\%$ of its activity, without transcriptional interference nor repression) and where $D_F$ is the decay rate of FLP. We can estimate that a pINDEL plasmid produces one FLP every $24$s: in good approximation, we only have to consider the three genes transcription time and we may suppose the transcriptions are performed one by one; as <em>flp</em> consists of $1272$nt and as the transcription speed is about $51.5$nt/s (between $24$ and $79$ nt/s ), we find a transcription time of about $24$s, so that $C_F\approx\frac1{24}\mbox{s}^{-1}$. Furthermore, as FLP at $30^\circ$C is stable, we can estimate its half-life time to be around $60$min; we then obtain (by a similar reasoning as in (eq(\ref{k_N}))) $D_F\approx\frac{\log2}{60\cdot60}\mbox{s}^{-1}$.
 +
</p>
 +
 +
<p>
 +
We thereby obtain the following system (see eqs (\ref{N30}), (\ref{P30}), (\ref{Gi30}) and (\ref{F30})):
 +
\[
 +
\left\{
 +
\begin{array}{c}
 +
\dot{N}=k_NN\left(1-\frac N{N_{max}}\right)\label{N30f}\\
 +
\dot{P}=k_PP\left(1-\frac{P}{P_{max}}\right)-\frac{\dot N}NP\label{P30f}\\
 +
\dot{G_i}=C_iP-D_iG_i-\frac{\dot N}NG_i \qquad (i=1,2,3)\label{Gi30f}\\
 +
\dot{F}=10\%p_{simul}C_FP-D_FF-\frac{\dot N}NF\label{F30f}
 +
\end{array}
 +
\right.
 +
\]
 +
</p>
 +
 +
<h2>Solving the equations of the insertion step</h2>
 +
 +
<p>
 +
In order to solve the first equation (eq(\ref{N30f})), we pose $M=1/N$; the equation then reads
 +
\begin{equation}
 +
\dot M=-\frac{\dot N}{N^2}=-k_N\left(\frac1N-\frac1{N_{max}}\right)=-k_NM+\frac {k_N}{N_{max}}
 +
\end{equation}
 +
and easily get solved to give
 +
\begin{equation}
 +
M(t)=\frac1{N_{max}}+(\frac1{N_0}-\frac1{N_{max}})e^{-k_Nt}
 +
\end{equation}
 +
thus
 +
\begin{align}
 +
N(t)&=\frac{N_{max}N_0e^{k_Nt}}{N_0e^{k_Nt}+(N_{max}-N_0)}=N_0e^{k_Nt}\frac1{1+\frac{N_0}{N_{max}}\left(e^{k_Nt}-1\right)}\label{Nsol30}\\
 +
&\approx N_0e^{k_Nt}\label{approx30}
 +
\end{align}
 +
where the approximation (eq(\ref{approx30})) remains valid for short times, that is
 +
\begin{equation}
 +
t\ll\frac1{k_N}\log{(\frac{N_{max}}{N_0}+1)}\approx5271\mbox{s}=1\mbox{h}27\mbox{min}51\mbox{s}.
 +
\end{equation}
 +
Saturation is reached when $t\approx9000\mbox{s}=2\mbox{h}30\mbox{min}$, as we can see on the graph (fig(\ref{graph1})) (obtained for realistic values of the parameters).
 +
<br>
 +
<img src="https://static.igem.org/mediawiki/2011/9/99/Figure1%27.PNG" alt="">
 +
In blue is plot the exact solution for $N$, while in red is the exponential approximation (eq(\ref{approx30})). This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml and $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$.
 +
 +
</p>
 +
 +
<p>
 +
The equation for $P$ (eq(\ref{P30f})) then becomes, using eq(\ref{Nsol30}):
 +
\begin{equation}
 +
\dot P=k_PP\left(1-\frac{P}{P_{max}}\right)-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}P
 +
\end{equation}
 +
which cannot be solved analytically.  However, we can solve it numerically using <em>Mathematica</em>: for realistic values of the parameters, we obtain the graph (fig(\ref{graph2})).
 +
<br>
 +
<img src="https://static.igem.org/mediawiki/2011/3/37/Figure2%27.PNG" alt="">
 +
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$ and $P_{max}=20$.
 +
 +
We observe that $P(t)\approx P_{max}$ as soon as $t\gtrsim50\mbox{s}$.
 +
</p>
 +
 +
<p>
 +
The two last equations, for $F$ and $G_i$ (eqs (\ref{F30f}) and (\ref{Gi30f})), rewrite, using the solution for $N$ (eq(\ref{Nsol30})):
 +
 +
$$
 +
\left\{
 +
\begin{array}{c}
 +
\dot{G_i}=C_iP-D_iG_i-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}G_i \qquad (i=1,2,3)\label{tre}\\
 +
\dot{F}=10\%p_{simul}C_FP-D_FF-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}F\label{ert}
 +
\end{array}
 +
\right.
 +
$$
 +
 +
which can also be solved  via <em>Mathematica</em>; for realistic constants, we get the graphs (fig(\ref{graph3})) and (fig(\ref{graph4})) for $F$ and $G_i$ respectively.
 +
<br>
 +
<img src="https://static.igem.org/mediawiki/2011/c/cc/Figure3%27.PNG" alt="">
 +
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$, $P_{max}=20$, $D_F=\frac{\log2}{60\cdot60}\mbox{s}^{-1}$, $C_F=\frac1{24}\mbox{s}^{-1}$ and $p_{simul}=0.01$.
 +
 +
<br>
 +
<img src="https://static.igem.org/mediawiki/2011/5/5f/Figure4%27.PNG" alt="">
 +
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$, $P_{max}=20$, $D_i=\frac{\log2}{60\cdot60}\mbox{s}^{-1}$ and $C_i=\frac1{40}\mbox{s}^{-1}$.
 +
 +
Note that $G_i$ and $F$ increase to a stable asymptotic equilibrium:
 +
\begin{equation}
 +
\lim\limits_{t\rightarrow\infty}{G_i(t)}=\frac{C_iP_{max}}{D_i}\approx 2.59 \cdot 10^3
 +
\end{equation}
 +
and
 +
\begin{equation}
 +
\lim\limits_{t\rightarrow\infty}{F(t)}=\frac{10\%p_{simul}C_FP_{max}}{D_F}\approx\ 4.33
 +
\end{equation}
 +
as can be seen immediately from the equations (eq(\ref{tre})) and (eq(\ref{ert})).
 +
</p>
 +
 +
<p>
 +
It is important to point out that here, the solution of our model only presents a small sensitivity to the parameters around the estimated values: a small error on the parameters will only result in a small change in the solution, as we can observe if we vary the values of the parameters a little around their estimation.
 +
</p>
</div>
</div>

Latest revision as of 04:28, 22 September 2011

Modelling : Insertion model

Insertion 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 per ml in the considered culture;
  • $P$: average number of pINDEL plasmid copies 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.

Experimental design of the insertion step

A colony of E. coli containing the pINDEL plasmid is grown at $30^\circ$C in $10$ml of LB medium containing Amp to select for the presence of pINDEL. This culture reaches saturation after ON culture at $30^\circ$C (titer of the culture between $2\cdot10^9$ and $5\cdot10^9$ bacteria per ml of culture). This ON culture is then diluted $100$- to $1000$-fold and grown in logarithmic phase at $30^\circ$C in LB medium ($\mbox{OD}_{600\mbox{nm}}$ around $0.2$, corresponding to $10^8$ bacteria/ml of culture). Arabinose ($0.2$ to $1\%$) is then added to the culture to induce the expression of the IN function. These cells are electroporated with a linear PCR fragment containing the gene X of interest and the FRT'-Cm-FRT'. Transformants are selected on LB plates containing Cm without arabinose at $30^\circ$C.

Getting the equations of the insertion step

At the initial time ($t=0$), i.e. immediately after the dilution, the number of bacteria ($N(t)$) is $N_0:=N(0)\approx10^8 \ \mbox{bact}/\mbox{ml}$. We used the Verhulst's logistic model. \begin{equation} \dot N=k_NN\left(1-\frac N{N_{max}}\right) \label{N30} \end{equation} where $N_{max}$ is the maximal number of bacteria in the culture and where $k_N$ corresponds to the growth rate one would observe in the limit where the saturation would be inexistent. In our case, at saturation, the $\mbox{OD}_{600\mbox{nm}}$ slightly exceeds $1$, which corresponds to approximately $N_{max}\approx 2\cdot10^9\mbox{bact}/\mbox{ml}$. On the other hand, since in our conditions, E. coli ideally divides every $20$min, if we are far from the saturation ($N_{max}=\infty$), we obtain \begin{equation} \dot N=k_NN \Rightarrow N_0e^{k_Nt}=N(t)=N_02^{t/20\mbox{\min}} \quad\Rightarrow k_N\approx \frac{\log{2}}{20\cdot60}\mbox{s}^{-1}. \label{k_N}\end{equation}

At this point, all the bacteria in the culture contain the pINDEL plasmid. Note that RepA101Ts is fully active for pINDEL replication at $30^\circ$C. However, we shall note that the number of plasmid copy per bacterium cannot exceed a certain number $P_{max}\approx20$ (as the origin of replication of pINDEL is low copy). At initial time, the number of pINDEL plasmid copy 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 insertion step. Again, we naturally postulate a logistic model: \begin{equation} \dot P=k_PP\left(1-\frac{P}{P_{max}}\right). \label{production}\end{equation}

As pINDEL is composed of $10800$ nt and as the replication rate is of about $750$ nt/s, we can estimate that the replication of pINDEL takes $\frac{10800}{750}\mbox{s}=14.4 \mbox{s}$. Using then the same reasoning we used for $k_N$ (eq(\ref{k_N})), we compute $k_P\approx\frac{\log{2}}{14.4}\mbox{s}^{-1}$. Moreover, we have to consider the contribution of the increase in population, which produce a dilution effect. In that purpose, let us suppose for a moment that the plasmids do not replicate any more; we then have $PN=\mbox{cst}$, thus \begin{equation} P=\frac{\mbox{cst}}N\quad\Rightarrow \dot P=-\mbox{cst}\frac{\dot N}{N^2}=-\frac{\dot N}NP. \label{dilution}\end{equation}

Combining both production (eq (\ref{production})) and dilution (eq(\ref{dilution})) effects, we get the evolution equation for $P$: \begin{equation} \dot P=k_PP\left(1-\frac P{P_{max}}\right)-\frac{\dot N}NP. \label{P30}\end{equation}

Note that this equation can be written as follow: \begin{equation} \frac d{dt}(NP)=k_PNP\left(1-\frac P{P_{max}}\right), \label{equNP}\end{equation} which allows a convenient interpretation: $NP$, the total number of pINDEL plasmid copy, follows a logistic model but where the saturation is only due to $P$. This seems quite natural, as we will see. The evolution of the total number of plasmid copy (per ml) has to be of the form \begin{equation} \frac d{dt}(NP)=NP\cdot(g(N,P)-d(N,P)) \end{equation} in term of a generation rate of new plasmids $g(N,P)$ and a death rate $d(N,P)$. The death rate is a priori constant and even zero in our case: $d(N,P)=d=0$. Regarding the generation rate, it has to diminish when $P$ increases, but is obviously not correlated to the number of bacteria per ml ($N$); the easiest is then to postulate an affine function $g(N,P)=\alpha-\beta P$, so that we find \begin{equation} \frac d{dt}(NP)=NP(\alpha-\beta P)t \end{equation} which is equivalent to (\ref{equNP}). This observation thus justifies our equation for $P$ (eq(\ref{P30})), initially obtained by heuristic reasoning.

As explained above, arabinose induces expression from the pBAD promoter (the promoter controlling the expression of the three genes $i$ on pINDEL). Keeping in mind the three Red recombinase proteins natural decay, the easiest way to model the evolution of the total amount (per ml) of these proteins (i.e. $G_i\cdot N$) is \begin{align} &\frac d{dt}(G_iN)=C_iPN-D_iG_iN \quad (i=1,2,3)\\ \Leftrightarrow\quad&\dot G_i=C_iP-D_iG_i-\frac{\dot N}NG_i \quad (i=1,2,3) \label{Gi30}\end{align} where $C_i$ is the production rate of the protein $i$ and $D_i$ the decay rate of the same protein. We can estimate that a pINDEL plasmid produces one protein $i$ every $40$s: in good approximation, we only have to consider the three genes transcription time and we may suppose the transcriptions are performed one by one; as gam, bet and exo consist of $417$nt, $786$nt and $681$nt respectively and as the transcription speed is about $51.5$nt/s (between $24$ and $79$ nt/s), we find a transcription time of about $40$s, so that $C_i\approx\frac1{40}\mbox{s}^{-1}$. Furthermore, as these three proteins are stable, we can estimate their half-life time to be around $60$min; we then obtain (by a similar reasoning as in (eq(\ref{k_N}))) $D_i\approx\frac{\log2}{60\cdot60}\mbox{s}^{-1}$.

The promoter of the flp gene is repressed by the CI857 thermosensitive repressor at $30^\circ$C. However, repression is not complete and we postulate that the pR leakiness is around $10\%$. In addition, the pR transcription is inhibited by interference with the transcription of the IN genes $i$. By computer simulation, we have been able to estimate $p_{simul}$, the probability that the flp gene is entirely transcribed (see section (\ref{IntTranscr})). Note that at $30^\circ$C FLP is active. Keeping in mind FLP natural decay, the easiest way to model the evolution of the total amount (per ml) of FLP (i.e. $F\cdot N$) is \begin{align} &\frac d{dt}(FN)=10\%p_{simul}C_FPN-D_FFN\\ \Leftrightarrow\quad&\dot{F}=10\%p_{simul}C_FP-D_FF-\frac{\dot N}NF \label{F30}\end{align} where $C_F$ is the production rate of FLP by pINDEL (in ideal conditions, at $100\%$ of its activity, without transcriptional interference nor repression) and where $D_F$ is the decay rate of FLP. We can estimate that a pINDEL plasmid produces one FLP every $24$s: in good approximation, we only have to consider the three genes transcription time and we may suppose the transcriptions are performed one by one; as flp consists of $1272$nt and as the transcription speed is about $51.5$nt/s (between $24$ and $79$ nt/s ), we find a transcription time of about $24$s, so that $C_F\approx\frac1{24}\mbox{s}^{-1}$. Furthermore, as FLP at $30^\circ$C is stable, we can estimate its half-life time to be around $60$min; we then obtain (by a similar reasoning as in (eq(\ref{k_N}))) $D_F\approx\frac{\log2}{60\cdot60}\mbox{s}^{-1}$.

We thereby obtain the following system (see eqs (\ref{N30}), (\ref{P30}), (\ref{Gi30}) and (\ref{F30})): \[ \left\{ \begin{array}{c} \dot{N}=k_NN\left(1-\frac N{N_{max}}\right)\label{N30f}\\ \dot{P}=k_PP\left(1-\frac{P}{P_{max}}\right)-\frac{\dot N}NP\label{P30f}\\ \dot{G_i}=C_iP-D_iG_i-\frac{\dot N}NG_i \qquad (i=1,2,3)\label{Gi30f}\\ \dot{F}=10\%p_{simul}C_FP-D_FF-\frac{\dot N}NF\label{F30f} \end{array} \right. \]

Solving the equations of the insertion step

In order to solve the first equation (eq(\ref{N30f})), we pose $M=1/N$; the equation then reads \begin{equation} \dot M=-\frac{\dot N}{N^2}=-k_N\left(\frac1N-\frac1{N_{max}}\right)=-k_NM+\frac {k_N}{N_{max}} \end{equation} and easily get solved to give \begin{equation} M(t)=\frac1{N_{max}}+(\frac1{N_0}-\frac1{N_{max}})e^{-k_Nt} \end{equation} thus \begin{align} N(t)&=\frac{N_{max}N_0e^{k_Nt}}{N_0e^{k_Nt}+(N_{max}-N_0)}=N_0e^{k_Nt}\frac1{1+\frac{N_0}{N_{max}}\left(e^{k_Nt}-1\right)}\label{Nsol30}\\ &\approx N_0e^{k_Nt}\label{approx30} \end{align} where the approximation (eq(\ref{approx30})) remains valid for short times, that is \begin{equation} t\ll\frac1{k_N}\log{(\frac{N_{max}}{N_0}+1)}\approx5271\mbox{s}=1\mbox{h}27\mbox{min}51\mbox{s}. \end{equation} Saturation is reached when $t\approx9000\mbox{s}=2\mbox{h}30\mbox{min}$, as we can see on the graph (fig(\ref{graph1})) (obtained for realistic values of the parameters).
In blue is plot the exact solution for $N$, while in red is the exponential approximation (eq(\ref{approx30})). This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml and $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$.

The equation for $P$ (eq(\ref{P30f})) then becomes, using eq(\ref{Nsol30}): \begin{equation} \dot P=k_PP\left(1-\frac{P}{P_{max}}\right)-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}P \end{equation} which cannot be solved analytically. However, we can solve it numerically using Mathematica: for realistic values of the parameters, we obtain the graph (fig(\ref{graph2})).
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$ and $P_{max}=20$. We observe that $P(t)\approx P_{max}$ as soon as $t\gtrsim50\mbox{s}$.

The two last equations, for $F$ and $G_i$ (eqs (\ref{F30f}) and (\ref{Gi30f})), rewrite, using the solution for $N$ (eq(\ref{Nsol30})): $$ \left\{ \begin{array}{c} \dot{G_i}=C_iP-D_iG_i-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}G_i \qquad (i=1,2,3)\label{tre}\\ \dot{F}=10\%p_{simul}C_FP-D_FF-k_N\frac{N_{max}-N_0}{N_0e^{k_Nt}+(N_{max}-N_0)}F\label{ert} \end{array} \right. $$ which can also be solved via Mathematica; for realistic constants, we get the graphs (fig(\ref{graph3})) and (fig(\ref{graph4})) for $F$ and $G_i$ respectively.
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$, $P_{max}=20$, $D_F=\frac{\log2}{60\cdot60}\mbox{s}^{-1}$, $C_F=\frac1{24}\mbox{s}^{-1}$ and $p_{simul}=0.01$.
This is obtained for $N_{max}=2\cdot10^9$bact/ml, $N_0=10^8$bact/ml, $k_N=\frac{\log2}{20\cdot60}\mbox{s}^{-1}$, $k_P=\frac{\log2}{14.4}\mbox{s}^{-1}$, $P_ 0=19$, $P_{max}=20$, $D_i=\frac{\log2}{60\cdot60}\mbox{s}^{-1}$ and $C_i=\frac1{40}\mbox{s}^{-1}$. Note that $G_i$ and $F$ increase to a stable asymptotic equilibrium: \begin{equation} \lim\limits_{t\rightarrow\infty}{G_i(t)}=\frac{C_iP_{max}}{D_i}\approx 2.59 \cdot 10^3 \end{equation} and \begin{equation} \lim\limits_{t\rightarrow\infty}{F(t)}=\frac{10\%p_{simul}C_FP_{max}}{D_F}\approx\ 4.33 \end{equation} as can be seen immediately from the equations (eq(\ref{tre})) and (eq(\ref{ert})).

It is important to point out that here, the solution of our model only presents a small sensitivity to the parameters around the estimated values: a small error on the parameters will only result in a small change in the solution, as we can observe if we vary the values of the parameters a little around their estimation.

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