Team:Paris Bettencourt/Modeling/Diffusion

From 2011.igem.org

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<h1>Brownian diffusion model</h1>
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<h1>Passive diffusion model</h1>
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<h2>Summary</h2>
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<p>Our passive diffusion model uses a random walker approach to create a simulation of a population of particles randomly moving from one cell to another via a nanotube. We ran this simulation for different size of nanotubes, different molecules, and two different simple collisions models. The results of those simulations shows that passive diffusion alone can give us a very efficient transfer in only a few seconds.</p>
 +
<div style="margin-left:50px; margin-right:50px; padding: 5px; border:2px solid black;"><b><p>What we learned from our passive diffusion simulations:
 +
<ul>
 +
    <li>Passive diffusion is much faster (a few seconds) than gene network characteristic response times (a few hours)</li>
 +
    <li>Passive diffusion depends more on nanotube size than on the molecules size</li>
 +
    <li>We need to focus on molecule natures, sizes and numbers in our experiment, not diffusion time</li>
 +
</ul></p></b></div>
<h2>Introduction</h2>
<h2>Introduction</h2>
 +
<p>While working on the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Assisted_diffusion">assisted diffusion model</a>, we quickly realized that it may not fully explain the GFP diffusion observed in the original paper. We therefore investigated <em>passive diffusion</em>.</p>
 +
<p>We want to know if passive diffusion can, in theory, explain the transfer behaviour of the Dubey/Ben-Yehuda paper. We created a simulation to have an <em>estimation of the order of magnitude of passive diffusion time</em> through the nanotubes.</p>
 +
<p>We used a model similar to the stochastic diffusion model presented in our <a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis">assumptions</a>. Our methodology and our results are presented below.</p>
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<table>
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<h2>Model description</h2>
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<tr>
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<h3>Dividing the cell</h3>
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<td>
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<p>The core of the model is to <em>divide the cell</em> in occupation sites. Each of these occupation sites is roughly the size of the particle that we want to study. For instance if the particle has a length of 10nm, we will divide the cell in occupation sites of 10nm by 10 nm by 10 nm.</p>
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<img style="width:150px; margin-top:20px;" src="https://static.igem.org/mediawiki/2011/1/1a/Passive-diff-button.png">
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<p>When we want to study one particle we therefore model the cell as a <em>3D matrix(M*M*M)</em>. Its size depends of the particle characteristic size. We use the following parameters:</p>
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</td>
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<ul>
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<td>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
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<li><i>V</i> volume of the cell (for <i>B.subtilis</i> 10<sup>-18</sup> m<sup>3</sup>)</li>
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<td>
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<li><img src='https://static.igem.org/mediawiki/2011/d/d2/Lambda.png' style='width:20px;'/> characteristic size of the particle (m)</li>
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<b><p>In the original paper, <em>the authors claim</em> that the diffusion that is happening through the nanotube <em>is an active process</em>, because the obderved diffusion goes too fast regarding the speed that could be achieved with only passive brownian motion, they said. This statement <em>intrigated us much</em>, and we wanted to investigate further. Indeed, if the diffusion is active, it means that there is some <em>specificity</em> of the transporter for the transported object, and the transported coponent that had been observed so far have noting to do with the regular coponent of the cell.</p></b>
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</ul>
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</td>
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<p>We therefore divided our cell in <img src='https://static.igem.org/mediawiki/2011/7/72/Occupation_sites_general.png' style='width:80px;'/> occupation sites, giving us the size of the 3D matrix we used in out simulation: <img src='https://static.igem.org/mediawiki/2011/7/7b/Paris2011Frm1.png' style='width:80px;'/></p>
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</tr>
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<center><img src="https://static.igem.org/mediawiki/2011/b/be/Diff_model.png" style='width:350px; margin-right:10px;'/><img src="https://static.igem.org/mediawiki/2011/d/d8/Paris2011Sch1.png" style='margin-left:10px;'/></center>
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</table>
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</br>
 +
<h3>Random walker movements</h3>
 +
<p>We now have the layout for modeling our cells and the nanotube connecting between them. How does a particle moves in such a matrix? We were able to evaluate the <em>characteristic transition time</em> from one occupation site to one of its direct neighbours. Using <i>D</i>, diffusion coefficient of the particle (m<sup>2</sup>.s<sup>-1</sup>), we have the characteristic time associated with the transition:<img src='https://static.igem.org/mediawiki/2011/8/81/Characteristic_time_general.png' style='width:90px;'/> <a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis#references">[3]</a></p>
 +
<p>The simulation is then simple. We put a certain number of those particles in an emitter cell connected to an empty receiver cell via a nanotube. At <em>each time step</em> (corresponding to the characteristic transition time of the particle), the <em>particles move from the occupation site where they are to one of its neighbour</em>. This is a <em>synchronous model</em> where all the particles move at the same time. We repeat this process and count the particles going through the nanotubes and arriving in the receiver cell.</p>
 +
<p>We also had to take into account collisions between particles, with the cell membrane or with the nanotube membrane. We used <em>two configurations for dealing with collision problems</em>.</p>
 +
<h4>Wait and see</h4>
 +
<p>In this collision model, when a particle collides with another object, it <em>stays in its place</em> and waits for the next time step. This model resulted in massive simulation problems. It took several hours to run our java programme for a few hundred time steps. This problem can be explained by the fact that particles kept colliding in the nanotube and prevented other molecules from entering the nanotube by colliding with them. We therefore had to do a lot of debugging before <em>successfully implementing this idea</em>.</p>
 +
<h4>Restart</h4>
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<p>In this collision model, when a particle collides with another object, it is <em>reset to its starting position</em> in the first cell. This model is stastistically correct because of the random definition of the particle movement. We can compare the "teleportation" of one particle to its <em>degradation and the synthesis</em> of a new one. This is the model we used in our simulation.</p>
 +
</br>
 +
<p>The behaviour of the particles regarding collisions is probably somewhere between these two assumptions. The restart for instance probably gives us a longer diffusion time.</p>
 +
<p>Our simulation is particle-specific because it relies on parameters specific to a particle both for the spatial grid and the characteristic time.</p>
 +
</br>
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<p>Ours fisrt calculations of the diffusion of the molecule inside a cell shows that  <em>the diffusion inside a cell</em> is a very <em>fast process</em> (see the bottom of <a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis">this page</a>). It takes from 10 seconds to one minute for a coponent of the cell to have the average movement in the order of magnitude of the size of the cell. We wanted to explore further about the speed the passive diffusion can be achieved for a molecule to pass with passive diffusion, to see if <em>the author is wright or not in his statement</em>.</p>
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<h3>"Ping-pong" particles</h3>
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<p>We first assumed that particles, once they enter the receiver cell, would not go back in the nanotubes. This proved to be false. Particles passing in the receiver cell exhibit a "ping-pong" behaviour. They tend to go back and forth from one cell to the tube entrance. That would not be a problem except our simulation counts particle that pass through th exit of the nanotube. This phenomenon could account for up to half of the particles passing the entrance (or exit). We adapted our software so that the count would be correct even with those "ping-pong" particles.</p>
 +
<center><img style="width:850px" src="https://static.igem.org/mediawiki/2011/d/db/Pingpongparticles.png"/></center>
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<p>On the other hand, we <em>try to propose several ideas</em> about the origin of the motion if this motion <em>is indeed faster</em> than the diffusion (see <a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Assisted_diffusion">toward assisted diffusion</a>). We also propose some experiments to verify these models.</p>
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<br>
 +
<br>
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<p>Our <em>java software</em> to simulate this model is downloadable <a href="https://static.igem.org/mediawiki/2011/a/a7/Paris2011Model2.zip">here</a>.
 +
Each simulation is done with a certain  base amount of molecules (1000 in most of the next example).
 +
All the molecules of the simulation will start in one cell and <em>move randomly</em> until at least <em>a certain amount decided by the user (50 in most of the example) molecules pass through the nanotube</em> into the other cell. At that point our "counter" annonces that a significant number of molecules has passed the tube.
 +
</p>
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<p>The designs had been devised so that we can <em>try to measure the speed</em> of the diffusion through the tubes (see the <a href="https://2011.igem.org/Team:Paris_Bettencourt/Designs">Design & concepts page</a>). We aimed to show if the diffusion is indeed a passive diffusion, and the author is wrong, or, if we have to think about <em>more sophiticated models</em>.</p>
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<h2>Results</h2>
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<p>We ran simulations changing three major parameters:<ul>
 +
<li>Nanotube size</li>
 +
<li>Type of collision (see above)</li>
 +
<li>Nature of the particle</li>
 +
</ul>
 +
</p>
 +
<p>This last parameter allowed us to compare theoritical diffusion times for <em>molecules of different sizes and natures</em>. The characteristic sizes and diffusion coefficients are summed up below:</p>
 +
<center>
 +
<img src="https://static.igem.org/mediawiki/2011/3/30/Tabl2.png">
 +
</center>
 +
<p>We ran a quick 2D simplifed version of our model to give you an idea of how it works (the timescale might not be coherent with results from our 3D model). Here, we modelize diffusion for GFP with the "wait and see" collision model.</p>
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<p>Our 3D simulation, the actual model, gave us the following results:</p>
 +
<center>
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<a href="https://static.igem.org/mediawiki/2011/b/bd/Diff_per_nanotube_sizePARIS2011.png"><img src="https://static.igem.org/mediawiki/2011/b/bd/Diff_per_nanotube_sizePARIS2011.png" style='width:800px;'></a>
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</center>
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<p>This first graph represents the evolution of diffusion speed for differents size of nanotubes.
 +
We ran this simulation for 3 size of molecules</p>
 +
<center>
 +
<a href="https://static.igem.org/mediawiki/2011/6/69/Comparaison_collisionPARIS2011.png"><img src="https://static.igem.org/mediawiki/2011/6/69/Comparaison_collisionPARIS2011.png" style='width:470px;'></a><a href="https://static.igem.org/mediawiki/2011/a/ab/Relative_errorPARIS2011.png"><img src="https://static.igem.org/mediawiki/2011/a/ab/Relative_errorPARIS2011.png" style='width:470px;'></a>
 +
</center>
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<p>Two other interesting informations are the comparison between the two models of collision used ("wait and see" and "restart"), and the average relative error, which helped us to choose the right number of molecules for simulating valid models.</p>
-
<h2>About the brownian diffusion model</h2>
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<h2>Conclusions</h2>
 +
<p>What we obtain for passive diffusion is several orders of magnitude slower than for assisted diffusion, but it remains <em>really fast compared to cell division</em>. This means that <em>we can not measure diffusion time directly through genetic network response</em>. The time for significant GFP diffusion for instance is under a minute when response time for a genetic network is around an hour. We therefore need to focus on <em>molecule sizes, natures and numbers</em> in our experiments, not on diffusion time.</p>
 +
<p>This <em>software can be extended for other molecules</em>, but <em>diffusion coefficent</em> is an ambiguous data <em>subject do debate</em> in papers, so simulating for a molecule is subject to error due to different definitions for the same parameter.
 +
</p>
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<p>Earlier, we did some maths to calculate the <em>speed of the diffusion inside a cell</em>. But we were in shperical coordinates, that is to say a mathematics friendly conditions.Here, we have to deal with a more complex geometry, so we had to be helped by a computer.</p>
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<!--<h2>Extras</h2>
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<p>We also design a three dimensional graphics representation of the model in Maya. </p>
 +
<center><iframe width="425" height="349" src="http://www.youtube.com/embed/jrX1Wa-CgCc?hl=fr&fs=1" frameborder="0" allowfullscreen></iframe></center>-->
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<p>The essence of the model remains the same than the stochastical motion approach, but in this space we introduce <em>boundary limits</em> with the shape of two cells linked by a tube. We observe the passage of the molecule through the tube and we calculate the time taken for a molecule (like a transcription factor) to leave the emittor cell, diffuse though the tube and <em>meet a target</em> in the <em>receiver cell.</em></p>
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<h2>Data</h2>
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<p>You can find the full dataset for this page here:</p>
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<h2>Model description</h2>
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<table>
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<tr>
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<td>we wanted to know if nanotubes are active or passive medium, so we define a model for passive diffusion.
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</td>
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</tr>
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<tr>
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<td>
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<!--image du modèle d'une cellule : normal, maya -->
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</td>
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<td>We modelise each bacteria as a 3D M*M*L matrix.</td>
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</tr>
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<tr>
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<td>
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<!--image du modèle cellule+nanotube : normal, maya -->
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</td><td>
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The nanotube are also modelised as a 3D M'*M'*L' matrix.
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</td>
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</tr>
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</table>
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<br/>
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We adapte each cell to the molecule we want to modelise, so the molecul have a size of 1*1*1.
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so the matrix is divided in site of the size of a molecule.
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<br/>
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For example : for the glucose, if we want to have a size of 1*1*1 the matrix representing the cell need to be 1000*1000*4000 and the nanotube 100*100*600.
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<br/>
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All the 100 molecules of the simulation will start in one cell and move randomly until at least 10 molecule pass throw the nanotube to the other cell.
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<br/>
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We use a synchronous modelisation so all the 100 molecules move simultaneously. at each row (step of execution), all the molecule move randomly from there site to one of the 26 other possible position.
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<br/>
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We take acount of :
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<ul>
<ul>
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<li>Brownian movment of particle : each particle have a random movment at each row.</li>
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<li><a href="https://static.igem.org/mediawiki/2011/6/69/DataDIFFPARIS2011.zip">Full data</a></li>
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<li>colision : we have 2 model of colision :
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<li><a href="https://static.igem.org/mediawiki/2011/a/ab/ErrorDIFFPARIS2011.zip">Data for the error calculus</a></li>
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<ul>
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<li><a href="https://static.igem.org/mediawiki/2011/0/06/Collision_typeDIFFPARIS2011.zip">Data for collision models comparison</a></li>
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<li>if a molecule colide another object, it will reset to is start point.</li>
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<li><a href="https://static.igem.org/mediawiki/2011/9/9f/Nano_size_variationDIFFPARIS2011.zip">Data for nanotube size comparison</a></li>
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<li>if a molecule colide another object, it will stay at it's position for this row.</li>
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</ul>
</ul>
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</li>
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<div id="citation_box">
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<li>colision with cell membrane, nanotube and self colision</li>
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<p id="references">References</p>
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</ul>
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<ol>
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for each type of molecule, we do 10 simulation and we take the average time of those 10.
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<li><i>Diffusion-based Channel Characterization in Molecular Nanonetworks.</i> Llatser, I., Alarcón, E. and Pierobon, M., to appear in Proc. of the 1st IEEE International Workshop on Molecular and Nano Scale Communication (MoNaCom), held in conjunction with IEEE INFOCOM, Shanghai (China), April 2011</li><!--[http://www.ece.gatech.edu/research/labs/bwn/papers/2011/c6.pdf]-->
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</ol>
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</div>
<br/>
<br/>
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<h3>How do we calculate the real time ?</h3>
 
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<br/>
 
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A row correspond to the movment frome one site to an other, so in reality it take <a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis#tau"><img src='https://static.igem.org/mediawiki/2011/8/84/Tau.png' style='width:20px;'/></a>
 
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So we optain the realtime of diffusion ofr ten molecules throw the nanotube with : Rtime=<img src='https://static.igem.org/mediawiki/2011/8/84/Tau.png' style='width:20px;'/>*(number of row).
 
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<br/>
 
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finaly we obtain th time for diffusion from a cell to an other with : Rtime/10.
 
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<br/>
 
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this model is mapped in Maya for a user friendly aspect.
 
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<iframe width="425" height="349" src="http://www.youtube.com/embed/jrX1Wa-CgCc?hl=fr&fs=1" frameborder="0" allowfullscreen></iframe>
 
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<h2>Results</h2>
 
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<!--tab of result -->
 
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<h2>Conclusion</h2>
 
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Latest revision as of 15:35, 28 October 2011

Team IGEM Paris 2011

Passive diffusion model

Summary

Our passive diffusion model uses a random walker approach to create a simulation of a population of particles randomly moving from one cell to another via a nanotube. We ran this simulation for different size of nanotubes, different molecules, and two different simple collisions models. The results of those simulations shows that passive diffusion alone can give us a very efficient transfer in only a few seconds.

What we learned from our passive diffusion simulations:

  • Passive diffusion is much faster (a few seconds) than gene network characteristic response times (a few hours)
  • Passive diffusion depends more on nanotube size than on the molecules size
  • We need to focus on molecule natures, sizes and numbers in our experiment, not diffusion time

Introduction

While working on the assisted diffusion model, we quickly realized that it may not fully explain the GFP diffusion observed in the original paper. We therefore investigated passive diffusion.

We want to know if passive diffusion can, in theory, explain the transfer behaviour of the Dubey/Ben-Yehuda paper. We created a simulation to have an estimation of the order of magnitude of passive diffusion time through the nanotubes.

We used a model similar to the stochastic diffusion model presented in our assumptions. Our methodology and our results are presented below.

Model description

Dividing the cell

The core of the model is to divide the cell in occupation sites. Each of these occupation sites is roughly the size of the particle that we want to study. For instance if the particle has a length of 10nm, we will divide the cell in occupation sites of 10nm by 10 nm by 10 nm.

When we want to study one particle we therefore model the cell as a 3D matrix(M*M*M). Its size depends of the particle characteristic size. We use the following parameters:

  • V volume of the cell (for B.subtilis 10-18 m3)
  • characteristic size of the particle (m)

We therefore divided our cell in occupation sites, giving us the size of the 3D matrix we used in out simulation:


Random walker movements

We now have the layout for modeling our cells and the nanotube connecting between them. How does a particle moves in such a matrix? We were able to evaluate the characteristic transition time from one occupation site to one of its direct neighbours. Using D, diffusion coefficient of the particle (m2.s-1), we have the characteristic time associated with the transition: [3]

The simulation is then simple. We put a certain number of those particles in an emitter cell connected to an empty receiver cell via a nanotube. At each time step (corresponding to the characteristic transition time of the particle), the particles move from the occupation site where they are to one of its neighbour. This is a synchronous model where all the particles move at the same time. We repeat this process and count the particles going through the nanotubes and arriving in the receiver cell.

We also had to take into account collisions between particles, with the cell membrane or with the nanotube membrane. We used two configurations for dealing with collision problems.

Wait and see

In this collision model, when a particle collides with another object, it stays in its place and waits for the next time step. This model resulted in massive simulation problems. It took several hours to run our java programme for a few hundred time steps. This problem can be explained by the fact that particles kept colliding in the nanotube and prevented other molecules from entering the nanotube by colliding with them. We therefore had to do a lot of debugging before successfully implementing this idea.

Restart

In this collision model, when a particle collides with another object, it is reset to its starting position in the first cell. This model is stastistically correct because of the random definition of the particle movement. We can compare the "teleportation" of one particle to its degradation and the synthesis of a new one. This is the model we used in our simulation.


The behaviour of the particles regarding collisions is probably somewhere between these two assumptions. The restart for instance probably gives us a longer diffusion time.

Our simulation is particle-specific because it relies on parameters specific to a particle both for the spatial grid and the characteristic time.


"Ping-pong" particles

We first assumed that particles, once they enter the receiver cell, would not go back in the nanotubes. This proved to be false. Particles passing in the receiver cell exhibit a "ping-pong" behaviour. They tend to go back and forth from one cell to the tube entrance. That would not be a problem except our simulation counts particle that pass through th exit of the nanotube. This phenomenon could account for up to half of the particles passing the entrance (or exit). We adapted our software so that the count would be correct even with those "ping-pong" particles.



Our java software to simulate this model is downloadable here. Each simulation is done with a certain base amount of molecules (1000 in most of the next example). All the molecules of the simulation will start in one cell and move randomly until at least a certain amount decided by the user (50 in most of the example) molecules pass through the nanotube into the other cell. At that point our "counter" annonces that a significant number of molecules has passed the tube.

Results

We ran simulations changing three major parameters:

  • Nanotube size
  • Type of collision (see above)
  • Nature of the particle

This last parameter allowed us to compare theoritical diffusion times for molecules of different sizes and natures. The characteristic sizes and diffusion coefficients are summed up below:

We ran a quick 2D simplifed version of our model to give you an idea of how it works (the timescale might not be coherent with results from our 3D model). Here, we modelize diffusion for GFP with the "wait and see" collision model.

Our 3D simulation, the actual model, gave us the following results:

This first graph represents the evolution of diffusion speed for differents size of nanotubes. We ran this simulation for 3 size of molecules

Two other interesting informations are the comparison between the two models of collision used ("wait and see" and "restart"), and the average relative error, which helped us to choose the right number of molecules for simulating valid models.

Conclusions

What we obtain for passive diffusion is several orders of magnitude slower than for assisted diffusion, but it remains really fast compared to cell division. This means that we can not measure diffusion time directly through genetic network response. The time for significant GFP diffusion for instance is under a minute when response time for a genetic network is around an hour. We therefore need to focus on molecule sizes, natures and numbers in our experiments, not on diffusion time.

This software can be extended for other molecules, but diffusion coefficent is an ambiguous data subject do debate in papers, so simulating for a molecule is subject to error due to different definitions for the same parameter.

Data

You can find the full dataset for this page here:

References

  1. Diffusion-based Channel Characterization in Molecular Nanonetworks. Llatser, I., Alarcón, E. and Pierobon, M., to appear in Proc. of the 1st IEEE International Workshop on Molecular and Nano Scale Communication (MoNaCom), held in conjunction with IEEE INFOCOM, Shanghai (China), April 2011