Team:Paris Bettencourt/Modeling

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<h1>Modeling introduction</h1>
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<h1>Modeling</h1>
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<h2>Modeling in our project</h2>
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<p>Our modeling was organized around two main questions:
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<h2>What is modeling in synthetic biology?</h2>
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<p>Synthetic biology relies heavily on genetic constructs designed to perform a specific task. In order to predict the behaviour of such systems, we need to be able to model it. Knowing in advance how well or how poorly a construct will work can help us during the genetic design phase or the preparation of our experiments. We need to describe the future reactions of our system by creating a mathematical model of it. The parameters of the model are then fitted to the experimental results.</p>
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<p>The bible for synthetic biology modeling is Uri Alon's book <em>An Introduction to Systems Biology: Design Principles of Biological Circuits</em> <a href="#references">[1]</a>. Most of our models are based on his approach to biological circuits. We are now going to explain quickly what is the basic structure behind our simulations.</p><br>
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<p>The gene ''geneX'' is responsible for the production of the corresponding protein <em>X</em>. The promoter <em>pX</em> controlling the expression <em>geneX</em> can be:</p>
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<li>A constitutive promoter, <em>geneX</em> is activated whatever the conditions</li>
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<li>Can we <em>explain the transfer</em> through nanotubes?</li>
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<li>Positively regulated, <em>geneX</em> is more active when an inducer is present</li>
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<li>What will be the <em>behaviour of our constructs</em> and how will it impact our experimental designs?</li>
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<li>Negatively regulated,  <em>geneX</em> is less active when an repressor is present</li>
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<p align="center"><img src='http://2011.igem.org/wiki/images/0/03/Constitutive_promoter_general.png' style='width:50%;'/></p>
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<p>Answering those two questions was essential for our project. We needed to know <em>what to expect in order to design our experiments</em> properly and to know what kind of restults we should obtain.</p>
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<p class="legende_schema" align="center">Constitutive promoter</p>
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<p align="center"><img src='http://2011.igem.org/wiki/images/1/16/Positive_promoter_general.png' style='width:50%;'/></p
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<p class="legende_schema" align="center">Positive autoregulation</p>
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<p align="center"><img src='http://2011.igem.org/wiki/images/0/0c/Negative_promoter_general.png' style='width:50%;'/> </p>
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<p class="legende_schema" align="center">Negative autoregulation</p>
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<p>The inducer or repressor can be another protein or even the product itself. In the latter case, the gene is auto-regulated, wether positively or negatively. To begin, let's assume <em>pX</em> is a constitutive promoter for now. We will make another simplification by taking into account the translation step and assuming that the gene "directly" produces protein X.</p>
 
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<p>We will know take a look at the parameters involved in modeling this network.</p>
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<h2>Investigating nanotube transfer</h2>
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<ul>
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<li><em>[X]</em> is the concentration of protein <em>X</em></li>
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<li><img src='http://2011.igem.org/wiki/images/a/a8/Alpha.png' style='width:20px;'/> is the expression rate of protein <em>X</em> (molecule.s<sup>-1</sup>). It mainly depends on the constitutive promoter.</li>
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<li><img src='http://2011.igem.org/wiki/images/0/05/Delta_dil.png' style='width:24px;'/> is the dilution rate, due to cell division (s<sup>-1</sup>)</li>
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<li><img src='http://2011.igem.org/wiki/images/6/62/Delta_deg.png' style='width:24px;'/> is the degradation rate of protein <em>X</em> (s<sup>-1</sup>)</li>
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<br>
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<p>The equation and solution modeling the behaviour of this system are the following:</p>
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<p>In order to answer the first question, we investigated the <em>physical properties of cell membrane</em> and <em>passive diffusion</em> to comprehend how the transfer could occur. We came up with two different ideas that could explain molecule transfer through the nanotubes, and based our original models on these assumptions, done in Java for passive diffusion and in Matlab for assisted diffusion.These two novel models show that transfer through the nanotubes, whether happening by passive diffusion or the so-called assisted diffusion, is happening too quickly to be accurately measured by fluorescent microscopy. As nanotube transfer is too fast compared to genetic response to allow us to measure its time span correctly, our conclusion was that our designs would not allow us to determine which one of these two processes (passive or assisted diffusion) is dominant during the transfer. Even though this makes it impossible to create a definitive model of molecule transfer through the nanotubes, the information provided by our two alternative models gave us an <em>insight on the time scale of the transfer</em>.</p>
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  <td style="width:200px; text-align:center"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Diffusion"><img style="width:150px; margin-top:20px;" src="http://2011.igem.org/wiki/images/1/1a/Passive-diff-button.png"></a>
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  <td><b><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Diffusion">Passive diffusion in nanotubes</a></b> We investigate here the hypothesis of passive diffusion through nanotubes.
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  <td style="width:200px; text-align:center"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Assisted_diffusion"><img style="width:150px; margin-top:20px;" src="http://2011.igem.org/wiki/images/b/b9/Active-diff-button.png"></a>
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  <td><b><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Assisted_diffusion">Assisted diffusion</a></b> We propose here a model explaining how diffusion through nanotubes could be "assisted" by the tension differential between cell walls.
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<p align="center"><a href='http://2011.igem.org/File:Constitutive.png'><img src='http://2011.igem.org/wiki/images/5/5f/Constitutive.png' style='width:60%;'/></a></p>
 
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<p class="legende_schema" align="center">Constitutive expression of protein X over time</p>
 
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<p>Now, let's assume that <em>pX</em> is auto-regulated, either positively or negatively. We need to introduce a few new parameters.</p>
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<h2>Predicting the behaviour of our designs</h2>
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<ul>
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<p>The second question was a crucial step in the design of our experiments. Taking into account the predicted time scale and other information provided by our two general models, we were able to build <em>models of each of our genetic networks</em>. These models are an improvement of Uri Alon's approach in <i>An Introduction to Systems Biology: Design Principles of Biological Circuits</i> and were done in Matlab. With these models we showed that some designs might work better that the others. For instance, we prioritized the T7 RNA polymerase diffusion and tRNA diffusion systems and decided to concentrate our wet lab experiments and characterizatons in these systems. The ComS system, on the other hand, was less developed because of some disadvantages that our model predicted (high background even without induction, very high activation threshold mainly). Moreover, our models let us <em>evaluate the response time of each of our our constructs</em>. With these estimations, were able to prepare protocols for our <a href ="http://2011.igem.org/Team:Paris_Bettencourt/Experiments/Microscopy">microscopic experiments</a> (by evaluating the characteristic response time of our system, the activation thresholds, etc.).</p>
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<li><em>K</em> is the dissociation constant, representing the binding of a inducer/repressor to the promoter</li>
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<li><em>n</em> is the Hill coefficient of the function</li>
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<li><img src='http://2011.igem.org/wiki/images/3/3f/Beta.png' style='width:20px;'/> is the maximum production rate of our gene (molecule.s<sup>-1</sup>)</li>
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<li><img src='http://2011.igem.org/wiki/images/a/a8/Alpha.png' style='width:20px;'/> now represents the basal expression ("leaking") of the gene (molecule.s<sup>-1</sup>)</li>
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<p>The constants used in our examples below are arbitrary and chosen to amplify the behaviours expected. The equations and solutions are now:</p>
 
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<p align="center"><img src='http://2011.igem.org/wiki/images/a/af/Positive_equation_general.png' style='width:40%;'/><a href='http://2011.igem.org/File:Positive.png'><img src='http://2011.igem.org/wiki/images/6/6b/Positive.png' style='width:60%;'/></a></p>
 
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<p class="legende_schema" align="center">Positive autoregulation equation and behaviour</p>
 
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<p align="center"><img src='http://2011.igem.org/wiki/images/a/ac/Negative_equation_general.png' style='width:40%;'/><a href='http://2011.igem.org/File:Negative.png'><img src='http://2011.igem.org/wiki/images/d/d2/Negative.png' style='width:60%;'/></a></p>
 
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<p class="legende_schema" align="center">Negative autoregulation equation and behaviour</p>
 
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<p>You will note that the regulation is modeled as a Hill function. This type of function is used to represent most of the regulation that takes place in a genetic network.</p>
 
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<p>Of course most genetic networks are more complex than simply auto-regulated nodes. The product of one gene can regulate the activation of another which in turn inhibits a third, etc. By coupling this kind of equations together, we achieved modeling most of our genetic networks pretty easily.</p><br>
 
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<h2>Our choices for modeling</h2>
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  <td style="width:200px; text-align:center"><a href="http://2011.igem.org/Team:Paris_Bettencourt/what_is_modeling"><img style="width:150px" src="http://2011.igem.org/wiki/images/2/2b/Question_mark_button.png"></a>
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  <td><b><a href="http://2011.igem.org/Team:Paris_Bettencourt/what_is_modeling">The basics about genetic networks modeling</a></b> You can find here an introduction to our methods and the general idea behind most gene network models.
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  <td style="width:200px; text-align:center;"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Hypothesis"><img style="width:150px; margin-top:20px;" src="http://2011.igem.org/wiki/images/2/21/Hypotheses_button.png"></a>
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  <td><b><a href="http://2011.igem.org/Team:Paris_Bettencourt/Hypothesis">Our assumptions</a></b> Because of the specificities of our project, we had to adapt the "classic" model to better represent our current situation. On top of that we made and justified a few other hypotheses detailed in this section.
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  <td style="width:200px; text-align:center"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Designs"><img style="width:150px; margin-top:20px;" src="http://2011.igem.org/wiki/images/a/ac/Graph-button.png"></a>
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  <td><b><a href="http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Designs">Modeling our designs</a></b> Models predicting the behaviour of our designs are detailled in this section.
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<h3>Steady state flow from the nanotubes</h3>
 
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<p>A lot of our designs are based on a simple <em>emitter cell - nanotubes - receiver cell</em> principle. The basic models however can not take into account the nanotubes transfer. We decided to make separate models for particles transmission through the nanotubes and to assume that the number of signaling protein going through them reaches a "steady state" in the receiver cell. It can be reached when the flow through nanotubes into the cell exactly compensate protein degradation and dilution. The reason for such a choice is to simplify the interpretation of data. If we were using a non-steady flow of signaling protein, understanding the behaviour of our reporters would have been much more difficult.</p>
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<p>The way we present our models is the following. We first model both emitting and receiving genes in the same cell to give us a control. We then see what would be the response of the receiver genes alone in a cell for different signaling protein steady states. We can then compare our models to reality and evaluate the influence of nanotubes on the response of the system.</p>
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<h3>Translation</h3>
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<p>We introduced the translation step in our models. This means each equation described above is transformed into two equations: one describing mRNA production and one describing the translation of this mRNA into a protein. We then need to introduce the protein production rate <img src='http://2011.igem.org/wiki/images/4/4d/Gamma.png' style='width:20px;' /> (s<sup>-1</sup>). We also need to introduce a different degradation rate for each product. The first one will be for the degradation of <em>mRNA</em> associated with <em>geneX</em> and the second one with protein <em>X</em> itself. The equations for the auto-repression (other regulation systems are similar) are now:</p>
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<p>This is justified notably because of our tRNA construction. This design relies directly on the translation process, we therefore need to model it if we want to compare models for different designs.</p>
 
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<h3>Delays for maturation</h3>
 
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<p>We also introduced a delay for protein production and maturation. Most models can ignore this but our experiments rely heavily on time measurements which means that for proteins with maturation time around 5 min (to compare with the cell division time 40 min) we need to take this into account. We chose to model this simply by adding a delay to the response time as it is shown in the following equations:</p>
 
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<p>The delay for mRNA production is significantly lower than the one for protein production: <img src='http://2011.igem.org/wiki/images/2/24/T1.png' style='width:20px;'/> << <img src='http://2011.igem.org/wiki/images/4/49/T2.png' style='width:20px;'/>.</p>
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<h3>No delays for diffusion in receptor cell</h3>
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<p>Most of our designs rely on the following scheme: an emitter cell creates a signaling protein which is received through the nanotubes by a receiver cell which is then activated. However, one could argue that after entering the receiver cell, the signaling proteins spend a significant time diffusing in it before reaching the receptor gene construct. We discussed this at length before finally agreeing not to consider this as a delay.</p>
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<p>Several models showed us that diffusion, even of a single signaling molecule, happen too fast to add any delay to the response time of the system. We will discuss two of these models here.</p>
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<h4>Using diffusion equations</h4>
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<p>The first step is to study the general principles of diffusion and to apply them to a single molecule. We expected to estimate the order of magnitude for diffusion time of molecules with this model, not to have a precise understanding of the movement of molecules in a cell. Most of our experimental designs rely on time measurement to characterize the nanotubes, it was therefore crucial to see if diffusion time could add a significant delay to the response of receiver cells.</p>
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<p>The principle of this model is quite simple. We use the statistical diffusion equation with a new normalization constant so that it describes the behaviour of one molecule. Rather than obtaining a concentration field, we end up with a distribution of the density of probability to find the molecule at a certain position and a certain time. We did not use any kind of boundary conditions, we therefore only model the "movements" of one molecule floating in an infinite water medium.<br>
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The equation of diffusion is the following:</p>
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<p align="center"><img src='http://2011.igem.org/wiki/images/4/4a/Diffusion_equation.png' style='width:230px;'/></p>
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<p>Where <em>c</em> is the concentration of particles in the cell, function of <img src='http://2011.igem.org/wiki/images/0/06/X_vector.png' style='width:20px;'/> (position) and <em>t</em> (time). <em>D</em> is the diffusion coefficient.</p>
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<p>We decided to use as initial conditions a Dirac function centered on the origin of space. The solution for such an equation is:</p>
 
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<p>This solution shows that through time, the density of probability "diffuses" in all directions equally. So we have take advantage of the spherical symetry of the problem.</p>
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<a href='http://2011.igem.org/wiki/images/3/3b/Diffusion_3D.png'><img src='http://2011.igem.org/wiki/images/3/3b/Diffusion_3D.png' style='width:100%;'/></a>
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<a href='http://2011.igem.org/File:Diffusion_1D.png'><img src='http://2011.igem.org/wiki/images/0/0d/Diffusion_1D.png' style='width:100%;'/></a>
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<p class="legende_schema">(Note that this serie of graph is here only as an example. The units are arbitrary.)</p>
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<p>What is of interest for us is the instant when we can consider that the probability of presence of the molecule is "the same" for every point in the cell. The cell here is an arbitrary boundary which has no direct impact on the model. In our approach, the cell is a sphere centered on the origin and with a diameter of 1 micrometer (roughly Subtilis length).  We chose to consider that the probability is "the same" when the lowest probability within the cell is 95% of the highest probability in the cell (i.e. in the center).</p>
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<p align="center"><a href='http://2011.igem.org/File:95_percent_1D.png'><img src='http://2011.igem.org/wiki/images/f/f7/95_percent_1D.png' style='width:60%;'/></a></p>
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<p>We can then solve analytically the equation. Let us call <img src='http://2011.igem.org/wiki/images/8/8d/T0.png' style='width:20px;' /> the time when the probability is the same (at 95%) within the cell boundaries. This means that for concentration for <em>x=cell radius</em> is 0.95 times concentration for <em>x=0</em>.</p>
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<p align="center"><a href='http://2011.igem.org/File:95_percent_diffusion_time_equation.png'><img src='http://2011.igem.org/wiki/images/2/2b/95_percent_diffusion_time_equation.png' style='width:100%;'/></a></p>
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<p>You will find in the table below a list of such characteristic times for different molecules.</p>
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<p align="center"><img src='http://2011.igem.org/wiki/images/9/9c/Time_table_difusion.png' style='width/100%;'/></p>
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<h4>Using a stochastic model</h4>
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<p>This model is very similar to the one used in [Ribosome kinetics and aa-tRNA competition determine rate and fidelity of peptide synthesis. Fluitt A, Pienaar E, Viljoen H. Comput Biol Chem. 2007 Oct;31(5-6):335-46. Epub 2007 Aug 15.]. We consider that the particle diffusing in the cell is a random walker.</p>
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<p>We want to see how long it takes for a particle of a given size to diffuse to any point of a cell. We use the following parameters:</p>
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<li><em>V</em> volume of the cell (10<sup>-18</sup> m<sup>3</sup>)</li>
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<p>We divide the cytoplasm volume <em>V</em> into <img src='http://2011.igem.org/wiki/images/7/72/Occupation_sites_general.png' style='width:80px;'/> occupation sites for the walker. The characteristic time <img src='http://2011.igem.org/wiki/images/8/84/Tau.png' style='width:20px;'/>  associated with the transition from one site to another is:<img src='http://2011.igem.org/wiki/images/8/81/Characteristic_time_general.png' style='width:90px;'/> [http://www.ece.gatech.edu/research/labs/bwn/papers/2011/c6.pdf]</p>
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<p>If we have <em>R</em> walkers of this type, the probability that a molecule arrives at a given occupation site during the time interval <img src='http://2011.igem.org/wiki/images/8/84/Tau.png' style='width:20px;'/> is: <img src='http://2011.igem.org/wiki/images/b/bb/Probability_general.png' style='width:80px;'/>. In our case we study only one molecule (worst case scenario) so let us assume R=1.</p>
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<p>The average time that elapses before the arrival of a particle is: <img src='http://2011.igem.org/wiki/images/9/9d/Characteristic_time_dev_general.png' style='width:190px;'/>.</p>
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<p>You will find below a table of time before arrival for different molecules.</p>
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<h3>Parameters</h3>
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<p>Finally, we made some assumptions for certain parameters that are used in most of our models. These assumptions are discussed at length in the <a href='http://2011.igem.org/Team:Paris_Bettencourt/Modeling/Parameters'>parameters</a> section.</p>
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<p id="references">References</p>
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<li>Test</li>
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<li>Re-Test</li>
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<div id="scroll_left"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Designs"><img src="http://2011.igem.org/wiki/images/0/0a/Arrow-left-big.png" style="width:100%;"></a><a href="http://2011.igem.org/Team:Paris_Bettencourt/Designs">Design overview</a></div>
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<div id="scroll_right"><a href="http://2011.igem.org/Team:Paris_Bettencourt/Experiments/List"><img src="http://2011.igem.org/wiki/images/e/e0/Arrow-right-big.png" style="width:100%;"></a><a href="http://2011.igem.org/Team:Paris_Bettencourt/Experiments/list">Experiments overview</a></div>
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== Direct observation ==
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== Characterization ==
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[[File:Parameters.png|thumb|center|upright=3.0|Relevant parameters for modeling]]
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[[File:T7_equations.png|thumb|center|upright=3.0|Allosteric equations for modeling]]
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=== T7 system ===
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[[Team:Paris_Bettencourt/Modeling/T7_diffusion|The T7 diffusion model.]]
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=== tRNA_amber system ===
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[[Team:Paris_Bettencourt/Modeling/tRNA_diffusion|The tRNA diffusion model.]]
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== Distribution of tRNA_amber in mRNA populations ==
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[[Team:Paris_Bettencourt/Modeling/tRNA_distribution|The tRNA distribution model.]]
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Latest revision as of 03:14, 29 October 2011

Team IGEM Paris 2011

Modeling

Modeling in our project

Our modeling was organized around two main questions:

  • Can we explain the transfer through nanotubes?
  • What will be the behaviour of our constructs and how will it impact our experimental designs?

Answering those two questions was essential for our project. We needed to know what to expect in order to design our experiments properly and to know what kind of restults we should obtain.

Investigating nanotube transfer

In order to answer the first question, we investigated the physical properties of cell membrane and passive diffusion to comprehend how the transfer could occur. We came up with two different ideas that could explain molecule transfer through the nanotubes, and based our original models on these assumptions, done in Java for passive diffusion and in Matlab for assisted diffusion.These two novel models show that transfer through the nanotubes, whether happening by passive diffusion or the so-called assisted diffusion, is happening too quickly to be accurately measured by fluorescent microscopy. As nanotube transfer is too fast compared to genetic response to allow us to measure its time span correctly, our conclusion was that our designs would not allow us to determine which one of these two processes (passive or assisted diffusion) is dominant during the transfer. Even though this makes it impossible to create a definitive model of molecule transfer through the nanotubes, the information provided by our two alternative models gave us an insight on the time scale of the transfer.

Passive diffusion in nanotubes We investigate here the hypothesis of passive diffusion through nanotubes.
Assisted diffusion We propose here a model explaining how diffusion through nanotubes could be "assisted" by the tension differential between cell walls.

Predicting the behaviour of our designs

The second question was a crucial step in the design of our experiments. Taking into account the predicted time scale and other information provided by our two general models, we were able to build models of each of our genetic networks. These models are an improvement of Uri Alon's approach in An Introduction to Systems Biology: Design Principles of Biological Circuits and were done in Matlab. With these models we showed that some designs might work better that the others. For instance, we prioritized the T7 RNA polymerase diffusion and tRNA diffusion systems and decided to concentrate our wet lab experiments and characterizatons in these systems. The ComS system, on the other hand, was less developed because of some disadvantages that our model predicted (high background even without induction, very high activation threshold mainly). Moreover, our models let us evaluate the response time of each of our our constructs. With these estimations, were able to prepare protocols for our microscopic experiments (by evaluating the characteristic response time of our system, the activation thresholds, etc.).

The basics about genetic networks modeling You can find here an introduction to our methods and the general idea behind most gene network models.
Our assumptions Because of the specificities of our project, we had to adapt the "classic" model to better represent our current situation. On top of that we made and justified a few other hypotheses detailed in this section.
Modeling our designs Models predicting the behaviour of our designs are detailled in this section.