Team:Paris Bettencourt/Modeling/T7 diffusion

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<html>
<h1>Model for T7 RNA polymerase diffusion</h1>
<h2>Summary</h2>
<div style="margin-left:50px; margin-right:50px; padding: 5px; border:2px solid black;"><b><p>The model in 4 bullet points:
<ul>
     <li>Very sensitive system requiring less than 10 T7 RNA polymerases to be activated</li>
     <li>Adequate response time between 30 min and 1 hour before having a visible fluorescence signal</li>
	 <li>Promoter strengths need to be experimentaly evaluated in <i>B.subtilis</i></li>
     <li>Leakage is not included in the model but might occur nonetheless</li>
</ul></p></b></div>
<br/>
<p>The T7 RNA polymerase diffusion construct is one of the central point of our project. In this design, the T7 RNA polymerase is used <em>both as an auto-amplifier and as the signal being transmitted</em>. Because the T7 promoter is supposed to be orthogonal to both <i>B.subtilis</i> and <i>E.coli</i>, we used it in several of our other construct as an amplifier. We hoped its orthogonality would reduce leakiness and that the T7 RNA polymerase would prove to be an excellent amplifier for very low signals. It would however give us only an ON/OFF response in an individual cell, not a quantitative one.</p>
<p>You will find below the results for a simple simulation. All of our components are in one cell. Between t=7500s and t=12500s, <i>IPTG</i> is added to the cell, lifting repression by <i>LacI</i>. <i>RFP</i> is the reporter for the emitting construct and <i>GFP</i> for the receiving construct.</p>
<img src="https://static.igem.org/mediawiki/2011/4/4b/T7_results_reporters_only.jpg" style="width:100%;">
<p><b><center>Matlab simulation for the T7 polymerase construct (all in one cell, only reporters molecules and IPTG input)</center></b></p>
<p>Our simulation <em>exhibits the behavior we expected</em>. The model shows that in our experimental conditions the cell should produce a significant response in a reasonable time (approximately 20 to 30 minutes). Once started, the auto-amplification loop can not be stopped, as we can see with the GFP staying at its maximum even when RFP levels decrease. The major limit of this model could come from leakiness of promoters. If the <i>pT7</i> is not as reliable as we thought, the auto-amplification loop could trigger itself all the time.</p>
<p>With our parameters, the model is extremely reactive to T7 RNA polymerase. Having only one T7 RNA polymerase in a cell is sufficient to start the autoamplification loop. This is not surprising with a Hill coefficient of 1 and a dissociation constant of 4.8 molecules per cell for <i>pT7</i>.</p>
<b><p>Our <em>MATLAB files</em> are available here: <a href="https://2011.igem.org/File:T7_model.zip">download</a>.</p></b>
<br/>
<br/>
<h2>Design</h2>

<p>The <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/T7_diffusion">T7 diffusion design</a> is our first original construct. In this design, the T7 RNA polymerase acts both as the signal transmitted and as the amplifier (auto-amplification). T7 RNA polymerase is produced in the emitter cell, then diffuses through the nanotubes and arrives in the receiver cell. In this cell, the T7 RNA polymerase activates on a <i>pT7</i> promoter. Behind this promoter, we put a gene coding for the same T7 RNA polymerase (amplifier) and for GFP (reporter).</p>

<p>This construction was put in two different settings. One is what we just described, where the emitting gene network is in one cell and the receiving gene network is in another. In the other construction, everything is in one cell. We use the second construct as a control to really see the impact of the cell-to-cell communication on the behavior of the cells. Because of our parameters, only the "all in one cell" was relevant from a modeling point of view since we do not have detailed data on the nanotubes and only one T7 RNA polymerase triggers the system (see <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/T7_diffusion#discussion">discussion</a> for more details).</p>

<p>We ran our models for those two configurations. We used a steady flow of signaling molecules in the receiver cell for the "one emitting cell - one receiving cell" construction. You can find our justifications about this assumption <a href='https://2011.igem.org/Team:Paris_Bettencourt/Modeling#Steady_state_flow_from_the_nanotubes'>here</a>.</p>


<br/>
<br/>
<br/>
<h2>Model</h2>

<h4>LacI</h4>
<p>We use <i>LacI</i> as a repressor for the emitter gene construct. <i>LacI</i> repression can be canceled by <i>IPTG</i>. This way we can induce production of RFP and <i>T7'</i> by putting <i>IPTG</i> on the cells.</p>
<img src='https://static.igem.org/mediawiki/2011/c/c1/LacI_equations.png' style='width:100%;'>
<img src='https://static.igem.org/mediawiki/2011/f/fa/LacI_IPTG_interaction.png' style='width:100%;'>
<p>Inactivated <i>LacI</i> can not repress the <i>pHs</i> promoter  anymore. Note that we consider that the reaction between <i>IPTG</i> and <i>LacI</i> fires without any delay. This assumption is justified by the fact that this reaction is much faster than any other  in our gene network.</p>
<br>


<h4>Emitter gene construct - T7'</h4>
<p>The emitter gene construct is modeled by the following equations:</p>
<img src='https://static.igem.org/mediawiki/2011/5/54/T7%27_equations.png' style='width:100%;'>
<br>
<br>
<p>The reporter for the emitter gene construct (<i>RFP</i>) is modeled by the following equations:</p>
<img src='https://static.igem.org/mediawiki/2011/9/95/RFP_equations.png' style='width:100%;'>
<br>
<br>


<h4>Receiver and amplification gene construct - T7''</h4>
<p>The receiver and amplification gene construct is modeled by the following equations:</p>
<img src='https://static.igem.org/mediawiki/2011/f/f2/T7%27%27_equations.png' style='width:100%;'>
<br>
<br>
<p>The reporter for the receiver and amplification gene construct (<i>GFP</i>) is modeled by the following equations:</p>
<img src='https://static.igem.org/mediawiki/2011/2/2f/GFP_equations.png' style='width:100%;'>
<br>
<br>


<br/>
<br/>
<br/>
<h2>Parameters</h2>

<p>This design relies on the T7 RNA polymerase (which is noted <i>T7</i>) both as the signaling molecule going through the nanotubes and as the auto-amplification system when it acts on the <i>pT7</i> promoter. In our equations however, we chose to distinguish these functions.</p>
<ul>
<li><i>T7'</i> represents the signaling T7 RNA polymerase</li>
<li><i>T7''</i> represents is the auto-amplifying molecule</li>
</ul>
<!--<a href='https://2011.igem.org/File:Emitting_construct_schema.png'><img src='https://static.igem.org/mediawiki/2011/5/54/Emitting_construct_schema.png' style='width:10%;'></a> INSERT JAVASCRIPT TO HIDE/SHOW
<a href='https://2011.igem.org/File:Receiving_construct_schema.png'><img src='https://static.igem.org/mediawiki/2011/2/23/Receiving_construct_schema.png' style='width:10%;'></a> INSERT JAVASCRIPT TO HIDE/SHOW-->

<p>The parameters used in this model are:</p>


<table class="parameter">
<tr>
<th>Parameter</th>
<th>Description</th>
<th style="width:80px;">Value</th>
<th style="width:90px;">Unit </th>
<th>Reference</th>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/f/f2/LacI_concentration.png' style='height:22px;'></td>
<td>Active LacI concentration (LacI which is not inactivated by IPTG)</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/f/f7/IPTG_concentration.png' style='height:22px;'></td>
<td>IPTG concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/5/5e/LacI_inact_concentration.png' style='height:22px;'></td>
<td>Inactived LacI concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/e/e3/LacI_tot_concentration.png' style='height:22px;'></td>
<td>Total LacI concentration</td>
<td>10000</td>
<td>molecules <br>per cell</td>
<td>Steady state for equation, assumed realistic</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/c/cd/T7%27_concentration.png' style='height:22px;'></td>
<td>T7 RNA polymerase (emitter, T7') concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/8/89/MRNA_T7%27_concentration.png' style='height:22px;'></td>
<td>mRNA associated with T7' concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/3/35/T7%27%27_concentration.png' style='height:22px;'></td>
<td>T7 RNA polymerase (auto-amplification, T7'') concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/3/37/MRNA_T7%27%27_concentration.png' style='height:22px;'></td>
<td>mRNA associated with T7'' concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/f/f5/GFP_concentration.png' style='height:22px;'></td>
<td>GFP concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/5/5f/MRNA_GFP_concentration.png' style='height:22px;'></td>
<td>mRNA associated with GFP concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/4/47/RFP_concentration.png' style='height:22px;'></td>
<td>RFP concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/1/17/MRNA_RFP_concentration.png' style='height:22px;'></td>
<td>mRNA associated with RFP concentration</td>
<td>NA</td>
<td>molecules <br>per cell</td>
<td>Notation convention</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/6/63/Beta_const.png' style='height:22px;' /></td>
<td>Maximal production rate of pVeg promoter (constitutive)</td>
<td>0.02</td>
<td>molecules.s<sup>-1</sup> <br>or pops</td>
<td>Estimated, see the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Promoter_strengths">justification</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/d/d9/BetapLac.png' style='height:22px;' /></td>
<td>Maximal production rate of pHs promoter</td>
<td>0.02</td>
<td>molecules.s<sup>-1</sup> <br>or pops</td>
<td>Estimated, see the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Promoter_strengths">justification</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/5/5d/BetapT7.png' style='height:22px;' /></td>
<td>Maximal production rate of pT7 promoter</td>
<td>0.02</td>
<td>molecules.s<sup>-1</sup> <br>or pops</td>
<td>Estimated, see the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Promoter_strengths">justification</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/7/72/KIPTGLacI.png' style='height:22px;' /></td>
<td>Dissociation constant for IPTG to LacI</td>
<td>1200</td>
<td>molecules <br>per cell</td>
<td><a href='https://2009.igem.org/Team:Aberdeen_Scotland/parameters/invest_1'>Aberdeen 2009 wiki</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/0/0c/KLacI.png' style='height:22px;' /></td>
<td>Dissociation constant for LacI to LacO (pHs)</td>
<td>700</td>
<td>molecules <br>per cell</td>
<td><a href='https://2009.igem.org/Team:Aberdeen_Scotland/parameters/invest_1'>Aberdeen 2009 wiki</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/5/5d/KT7.png' style='height:22px;' /></td>
<td>Dissociation constant for T7 RNA polymerase to pT7</td>
<td>10</td>
<td>molecules <br>per cell</td>
<td>We used the classic assumption 1nM=1 molecule per cell and <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/T7_diffusion#references">[1]</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/6/64/NIPTG.png' style='height:22px;' /></td>
<td>Hill coefficient for LacI/IPTG interaction</td>
<td>2</td>
<td>NA</td>
<td><a href='https://2009.igem.org/Team:Aberdeen_Scotland/parameters/invest_1'>Aberdeen 2009 wiki</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/e/e0/NLacI.png' style='height:22px;' /></td>
<td>Hill coefficient for LacI/pHyperSpank interaction</td>
<td>2</td>
<td>NA</td>
<td><a href='https://2009.igem.org/Team:Aberdeen_Scotland/parameters/invest_1'>Aberdeen 2009 wiki</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/a/a3/Gamma_protein.png' style='height:22px;'  /></td>
<td>Translation rate of proteins</td>
<td>0.9</td>
<td>s<sup>-1</sup></td>
<td>Estimated, see the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Protein_translation_rate_justification">justification</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/0/05/Delta_dil.png' style='height:22px;' /></td>
<td>Dilution rate in exponential phase</td>
<td>2.88x10<sup>-4</sup></td>
<td>s<sup>-1</sup></td>
<td>Calculated with a 40 min generation time. See explanation</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/7/7d/Delta_mrna.png' style='height:22px;' /></td>
<td>Degradation rate of mRNA</td>
<td>2.88x10<sup>-3</sup></td>
<td>s<sup>-1</sup></td>
<td><a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/T7_diffusion#references">[3]</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/9/95/Delta_GFP.png' style='height:22px;' /></td>
<td>Degradation rate of GFP</td>
<td>10<sup>-4</sup></td>
<td>s<sup>-1</sup></td>
<td><a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=105188&ver=2&hlid=56398">BioNumbers</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/1/1c/Delta_RFP.png' style='height:22px;' /></td>
<td>Degradation rate of RFP</td>
<td>10<sup>-4</sup></td>
<td>s<sup>-1</sup></td>
<td>Estimated equal to GFP degradation rate</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/a/a8/TT7.png' style='height:22px;'/></td>
<td>Delay due to T7 RNA polymerase production and maturation</td>
<td>300</td>
<td>s</td>
<td><a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/tRNA_diffusion#references">[2]</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/f/f0/TGFP.png' style='height:22px;'/></td>
<td>Delay due to GFP production and maturation</td>
<td>360</td>
<td>s</td>
<td><a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=102972&ver=8">BioNumbers</a></td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/b/bc/TRFP.png' style='height:22px;'/></td>
<td>Delay due to RFP production and maturation</td>
<td>360</td>
<td>s</td>
<td>Estimated equal to GFP delay (similar molecules)</td>
</tr>
<tr>
<td><img src='https://static.igem.org/mediawiki/2011/9/9c/TmRNA.png' style='height:22px;' /></td>
<td>Delay due to mRNA production</td>
<td>30</td>
<td>s</td>
<td><a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=104902&ver=5&hlid=58815 2kb">BioNumbers</a> with an approximation: all our contructs are around 1-2kb</td>
</tr>
</table>
<div id="citation_box">
<p id="references">References</p>
<ol>
<li><i>Cytoplasmic expression of a reporter gene by co-delivery of T7 RNA polymerase and T7 promoter sequence with cationic liposomes</i>,
X Gao and L Huang, accessible <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC309671/pdf/nar00061-0090.pdf">here</a></li>
<li><i>Molecular Biology for Masters</i> by Dr. G. R. Kantharaj, accessible<a href="http://mol-biol4masters.masters.grkraj.org/html/Prokaryotic_DNA_Replication13-T7_Phage_DNA_Replication.htm"> here</a></li>
<li><i>An Introduction to Systems Biology: Design Principles of Biological Circuits</i>, Uri Alon</li>
<ol>
</div>
  
<br/>
<br/>
<br/>
<h2 id="discussion"> Results & discussions </h2>

<p>We launched this simulation in matlab and obtained the following results:</p>
<a href="https://static.igem.org/mediawiki/2011/0/01/Results_T7.jpg"><img src="https://static.igem.org/mediawiki/2011/0/01/Results_T7.jpg" style="width:100%;"></a>
<p><b><center>Matlab simulation for the T7 polymerase construct (all in one cell). Click to enlarge the picture.</center></b></p>
<br>
<p>The behaviour of the cell is as expected. The IPTG input removes the repression on the <i>mRNA T7'</i> production, which then is translated in <i>T7'</i>. This <i>T7'</i> polymerase activates the <i>mRNA T7''</i> production. Finally, this last mRNA is translated into <i>T7''</i>.</p>
<p>This is <em>exactly the results we wanted</em>. The T7 RNA polymerase acts both as a transmission molecule and an amplifier. Once the <i>pT7</i> is activated it auto-amplifies itself and gives us a clear result.</p>
<p>The IPTG imput is here theoritical. We can not in an experiment remove the IPTG from the medium. However, this input signal is an excellent way to understand the way the system behaves. After IPTG disappears, we can see the levels of <i>mRNA T7'</i> and <i>T7'</i> decreasing as expected since they are regulated by <i>pHyperSpank</i>. On the other hand, <i>mRNA T7''</i> and <i>T7''</i> regulated by <i>pT7</i> are not affected.</p>
<p>Because of the time scale it is hard to realize, but <em>all delays have been correctly implemented</em> and the order of appearance/disparitions of each product is what we expected. For instance, mRNA appear 30 seconds after the conditions for initiation of transcription are met.</p>
<p>The reporter proteins <i>RFP</i> and <i>GFP</i> follow closely the behaviour of the T7 RNA polymerase they are working on. Note that the slight difference between T7 RNA polymerase and reporter protein maximum levels of expression is due to <i>RFP</i> and <i>GFP</i> degradation rate. This does not however pose a problem for following the behaviour of cells, as expected.</p>
<p>With our parameters, the model is extremely reactive to T7 RNA polymerase. Having only one T7 RNA polymerase in a cell is sufficient to start the autoamplification loop. This is not surprising with a Hill coefficient of 1 and a dissociation constant of 10 molecules per cell for <i>pT7</i>. Since we know that a single molecule takes only a few second to find the <i>pT7</i> promoter(see our <a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis">hypotheses</a>), it is of no consequence if our parameters are correct.</p>

<h4>Limits</h4>
<p>The most obvious limit is that we supposed the <i>pT7</i> promoter to be not leaky at all, since it needs very little T7 RNA polymerase to be activated. If the leak is too important in pratice the model and the design might need some adjustments.</p>
<p>Most parameters are well defined, but promoter strengths tend to be quite difficult to find or to evaluate (see our <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Promoter_strengths">justification</a> for our choices of promoter strengths). In this model, changing moderately these strengths does not impact much on the overall behaviour of the system. It could be troubling however in our experiments with two cells if very few T7 RNA polymerase pass through the nanotubes.</p>


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<div id="scroll_left"><a href="https://2011.igem.org/Team:Paris_Bettencourt/T7_diffusion"><img src="https://static.igem.org/mediawiki/2011/0/0a/Arrow-left-big.png" style="width:100%;"></a><a href="https://2011.igem.org/Team:Paris_Bettencourt/T7_diffusion">Design for T7 RNA polymerase diffusion</a></div>
<div id="scroll_right"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Experiments/T7_diffusion"><img src="https://static.igem.org/mediawiki/2011/e/e0/Arrow-right-big.png" style="width:100%;"></a><a href="https://2011.igem.org/Team:Paris_Bettencourt/Experiments/T7_diffusion">Characterizing the T7 RNA polymerase system</a></div>
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