Team:ULB-Brussels/modeling/42

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<h1>Introduction</h1>
+
 
 +
<h1>Insertion model</h1>
 +
 
 +
<h2>Definitions</h2>
 +
 
 +
<p>
 +
Let us begin with a proper definition of the different biological functions that are considered in our model:
 +
<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>
-
The pINDEL plasmid can be divided into $2$ functional units:
+
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.
-
<ol>
+
-
  <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>
+
-
  <li> the DEL function which is based on the <em>flp</em> gene encoding the FLP site-specific recombinase \cite{dat,yu}.</li>
+
-
</ol>
+
</p>
</p>
 +
 +
<h2>Getting the equations of the insertion step</h2>
 +
<p>
<p>
-
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}.
+
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}
 +
\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, <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>
-
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.
+
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}
</p>
</p>
<p>
<p>
-
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})).
+
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}
 +
</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>
<p>
<p>
-
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.
+
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>
</p>

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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