Revision as of 01:19, 22 September 2011

Modelling : Introduction

Introduction

The pINDEL plasmid can be divided into $2$ functional units:

the IN function which is composed of the gam, exo and bet genes coding for the $\lambda$ Red recombinase system \cite{dat,yu}; and
the DEL function which is based on the flp gene encoding the FLP site-specific recombinase \cite{dat,yu}.

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. ThearaC 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 flp gene is repressed at 90\% at $30^\circ$C, while at $42^\circ$C the flp 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}.

In addition, pINDEL contains the repA101ts gene encoding the RepA101Ts protein and the origin of replication (ori) \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.

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 flp 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})).

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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 <div id="sousm">
 <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>
 </div>
@@ Line 611: / Line 615: @@
 	<div id="maintext">
 		<div id="hmaint">
-			Modelling : Phase at 30°C		</div>
+			Modelling : Introduction			</div>
 			<div id="maint">
-<h1>Phase at $30^\circ$C on arabinose</h1>
-\label{Ph30}
-<h2>Transcriptional interference: computer simulation</h2>
-\label{IntTranscr}
-<p>
-In this section, we study the interference in the transcription provoked by the simultaneous expression of the gene coding for the flippase (which is performed only for $\ldots\%$ at $30^\circ$C), and of the three genes $i$ for the Red recombinase (which is performed for $100\%$).
-</p>
-<p>
-\ldots[Jo\&Pierre]
-</p>
-<h2>Preparation: electroporation and night culture</h2>
-<p>
-We electropore <em>E. coli</em> with Pindel plasmids. Given that the plasmids include a resistance gene to ampiciline, we can see, by testing that resistance, which bacteria actually received a Pindel plasmid. One colony of those bacteria is then cultivated at $30^\circ$C in $10$ml, where it attains saturation (between $2\cdot10^9$ and $5\cdot10^9$ bacteria per ml). The solution is diluted $100$ to $1000$ times, then cultivated again, until we reach an optic density (OD) (at $600$nm) of $0.2$, which corresponds to approximatively $10^8$ bacteria. Those bacteria are then put in touch with arabinose at $30^\circ$C.
-</p>
-<h1>Model</h1>
-<h2>Getting the equations</h2>
-\label{Mod30}
-<p>
-At the initial time ($t=0$), the concentration of bacteria ($N(t)$) is $N_0:=N(0)\approx10^8\mbox{bact}/\mbox{ml}$. It seems natural to use 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 maximum concentration of bacteria that is possible in the culture environment and where $k_N$ corresponds to the growth rate one would observe in the limit where the saturation would be inexistent. In our case, the saturation density slightly exceeds $1$ OD (at $600$nm), that is approximatively $N_{max}\approx 2\cdot10^9\mbox{bact}/\mbox{ml}$. On the other hand, since our <em>E. coli</em> ideally duplicate 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{\footnotesize{min}}} \quad\Rightarrow k_N\approx \frac{\log{2}}{20\cdot60}\mbox{s}^{-1}.
-\label{k_N}\end{equation}
-</p>
-<p>
-At this point, all the bacteria contain Pindel. At $30^\circ$C, RepA101 becomes active and is present in sufficient quantities to allow the plasmid's replication; the evolution of $E_{tot}$ and $E$ thus do not matter. However, we shall note that the amount of plasmids per bacterium cannot exceed a certain number $P_{max}\approx20$ (because the origin of replication of the plasmid is <em>low copy</em>). At initial time, the amount of Pindel plasmids per bacterium is $P_0:=P(0)\approx19$. 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>
-By the same reasoning we used for $k_N$ (eq(\ref{k_N})), we compute $k_P\approx\frac{\log{2}}{11}\mbox{s}^{-1}$, since our plasmid replicates itself every $11$s (reference?). Moreover, we have to consider the dilution of those plasmids through the population, due to its increase. In that purpose, let us suppose for a moment that the plasmids don't 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>
-Remark 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 amount of Pindel plasmids, 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 amount of plasmids has to be of the form
-\begin{equation}
-\frac d{dt}(NP)=NP\cdot(b(N,P)-d(N,P))
-\end{equation}
-in term of a birth rate of new plasmids $b(N,P)$ 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 birth rate, it has to diminish when $P$ increases, but is obviously unlinked with the amount of bacteria $N$; the easiest is then to postulate an affine function $b(N,P)=\alpha-\beta P$, so that we find
-\begin{equation}
-\frac d{dt}(NP)=NP(\alpha-\beta P)
-\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>
-Arabinose activates Pbad (the promotor of the three-gene sequence $i$ on Pindel), in order that those $3$ genes are expressed. Keeping in mind that the expressed proteins naturally deteriorate, the easiest way to modelise the evolution of their quantity ($G_i$) is
-\begin{equation}
-\dot{G_i}=C_iP-D_iG_i \quad (i=1,2,3)
-\label{Gi30}\end{equation}
-where $C_i$ is the production rate of the protein $i$ by the Pindel plasmid and $D_i$ the deterioration rate of that same protein. We estimated $C_i\approx\ldots$ and $D_i\approx\ldots$.
-</p>
-<p>
-The promotor of flippase is repressed by a thermo-sensible repressor, and, at $30^\circ$C, is only partially activated ($\ldots\%$); in addition, the transcription is hindered by a possible interference with the transcription of the genes $i$. By a computer simulation, we have been able to estimate $p_{simul}$, the probability that the flippase sequence gets entirely transcribed (see section \ref{IntTranscr}). Remark that at $30^\circ$C flippase is entirely active. If we take furthermore the natural deterioration of flippase in account, we can write
-\begin{equation}
-\dot{F}=C_Fp_{simul}P-D_FF
-\label{F30}\end{equation}
-where $C_F$ is the production rate of flippase by Pindel (in ideal conditions, at $100\%$ of its activity) and $D_F$ the natural deterioration rate of flippase. We estimated that $C_F\approx\ldots$ et $D_F\approx0.01$.
-</p>
-<p>
-We thereby obtain the following system (see eqs (\ref{N30}), (\ref{P30}), (\ref{Gi30}) and (\ref{F30})):
-\begin{numcases}{}
-\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 \qquad (i=1,2,3)\label{Gi30f}\\
-\dot{F}=C_Fp_{simul}P-D_FF\label{F30f}
-\end{numcases}
-</p>
-<h2>Solving the equations</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{equation}
-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)}\overset{\star}{\approx} N_0e^{k_Nt}
-\end{equation}
-where the approximation ($\star$) 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}
-</p>
+<h1>Introduction</h1>
 <p>
-Saturation is reached when $t\approx9000\mbox{s}=2\mbox{h}30\mbox{min}$, as we can see on the following graph (obtained for realistic values of the parameters)
+The pINDEL plasmid can be divided into $2$ functional units:
-\[\mbox{[insert graphic 1]}\]
+<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>
-The equation for $P$ (eq(\ref{P30f})) then becomes
+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}.
-\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,
-\[\mbox{[insert graphic 2]}\]
-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})), can also be solved  via <em>Mathematica</em>: for realistic constants,
+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.
-\begin{description}
-\item{for $F$:}
-\[\mbox{[insert graphic 3]}\]
-\item{for $G_i$:}
-\[\mbox{[insert graphic 4]}\]
-\end{description}
 </p>
 <p>
-As soon as $t\approx\ldots$, $G_i$ reaches its asymptotic maximum $C_iP_{max}/D_i\approx\ldots$ and $F$ reaches its asymptotic maximum $C_Fp_{simul}P_{max}/D_F\approx\ldots$.
+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})).
 </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. [mettre Žventuellement un lien vers une page avec tous les fichiers cdf]
+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.
 </p>