Team:Paris Bettencourt/Modeling

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<h1>Modeling</h1>
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<br>
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<h2>Modeling in our project</h2>
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= Modeling =
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<p>Our modeling was organized around two main questions:
 +
<ul>
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<li>Can we <em>explain the transfer</em> through nanotubes?</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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</ul></p>
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== What is modeling in synthetic biology? ==
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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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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 when we need to prepare our experiments. We need to describe the future reactions of our system is to create a mathematical model of it.<br>
 
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The bible for synthetic biology modeling is Uri Alon's book ''An Introduction to Systems Biology: Design Principles of Biological Circuits''. 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.<br>
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<h2>Investigating nanotube transfer</h2>
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The gene ''geneX'' is responsible for the production of the corresponding protein ''X''. The promoter ''pX'' controlling the expression ''geneX'' can be:
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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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* A constitutive promoter, ''geneX'' is activated whatever the conditions
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<table>
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* Positively regulated, ''geneX'' is more active when an inducer is present
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<tr>
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* Negatively regulated, ''geneX'' is less active when an repressor is present
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  <td style="width:200px; text-align:center"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Diffusion"><img style="width:150px; margin-top:20px;" src="https://static.igem.org/mediawiki/2011/1/1a/Passive-diff-button.png"></a>
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  </td>
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  <td><b><a href="https://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>
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</tr>
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<tr>
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  <td style="width:200px; text-align:center"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Assisted_diffusion"><img style="width:150px; margin-top:20px;" src="https://static.igem.org/mediawiki/2011/b/b9/Active-diff-button.png"></a>
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  </td>
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  <td><b><a href="https://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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  </td>
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</tr>
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</table>
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INSERT IMAGE HERE
 
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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 ''pX'' 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.<br>
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<h2>Predicting the behaviour of our designs</h2>
 +
<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 ="https://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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We will know take a look at the parameters involved in modeling this network.
 
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''X'' is the concentration of protein ''X''
 
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''alpha'' is the expression rate of protein ''X'' (molecule.s^-1). It mainly depends on the constitutive promoter.
 
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''delta_dil'' is the dilution rate, due to cell division (s^-1)
 
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''delta_deg'' is the degradation rate of protein ''X''
 
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The equation and solution modeling the behaviour of this system are the following:
 
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INSERT IMAGE HERE (initial conditions zero)
 
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Now, let's assume that ''pX'' is auto-regulated, either positively or negatively. We need to introduce a few new parameters.
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<table>
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<tr>
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  <td style="width:200px; text-align:center"><a href="https://2011.igem.org/Team:Paris_Bettencourt/what_is_modeling"><img style="width:150px" src="https://static.igem.org/mediawiki/2011/2/2b/Question_mark_button.png"></a>
 +
  </td>
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  <td><b><a href="https://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>
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</tr>
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<tr>
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  <td style="width:200px; text-align:center;"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Hypothesis"><img style="width:150px; margin-top:20px;" src="https://static.igem.org/mediawiki/2011/2/21/Hypotheses_button.png"></a>
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  </td>
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  <td><b><a href="https://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>
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</tr>
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<tr>
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  <td style="width:200px; text-align:center"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Modeling/Designs"><img style="width:150px; margin-top:20px;" src="https://static.igem.org/mediawiki/2011/a/ac/Graph-button.png"></a>
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  </td>
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  <td><b><a href="https://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.
 +
  </td>
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</tr>
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</table>
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''K'' is the dissociation constant, representing the binding of a inducer/repressor to the promoter
 
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''n'' is the Hill coefficient of the function
 
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''beta'' is the maximum production rate of our gene (molecule.s^-1)
 
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''alpha'' now represents the basal expression ("leaking") of the gene (molecule.s^-)
 
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The equations and solutions are now:
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INSERT IMAGE HERE (initial conditions zero, parameters noted)
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You will note that the regulation is modeled as a Hill function. This type of functioncan help us model most of the regulation that takes place in a genetic network.<br>
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      title='Recent changes'>Recent changes</a></li>
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Of course most genetic networks are more complex than simply auto-regulated nodes. The product of one gene can regulated 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.<br>
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== Our choices for modeling ==
 
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We introduced the translation step in our models. This means each equation described aboved 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 ''gamma''. The equations are now:
 
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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.
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We also introduced a delay for protein production and maturation. Most models can ignore this but our experiments rely heavily on time measurement 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:
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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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<div id="scroll_left"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Designs"><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/Designs">Design overview</a></div>
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=== T7 system ===
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<div id="scroll_right"><a href="https://2011.igem.org/Team:Paris_Bettencourt/Experiments/List"><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/list">Experiments overview</a></div>
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[[Team:Paris_Bettencourt/Modeling/T7_diffusion|The T7 diffusion model.]]
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[[File:T7_scheme.png|thumb|center|upright=3.0|T7 genetic design]]
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[[File:0711_Modelling_T7_without_delay.jpg|thumb|center|upright=3.0|First T7 model without delay between receptor and amplifier]]
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[[File:0711_Modelling_T7_with_delay.jpg|thumb|center|upright=3.0|T7 model with delay between receptor and amplifier]]
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=== tRNA_amber system ===
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[[File:TRNA_scheme.png|thumb|center|upright=3.0|tRNA Amber genetic design]]
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[[Team:Paris_Bettencourt/tRNA_diffusion|The amber suppressor tRNA diffusion.]] The idea of the system is to pass tRNA amber molecules through the nanotubes. At every moment of time in the receiver cell there is a certain amount of transcribed mRNA-T7 among the others mRNA. The behavior of tRNA amber that arrived in a receiver cell is random, so in order to describe its interaction with mRNA-T7 and its further translation we can reason in terms of probability.
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We can reason in two steps : first a tRNA amber molecule gets close to a mRNA molecule. Then, it binds it's anti-codon with a codon of the mRNA. This reasoning is similar to the problem of boxes and balls. There are two types of boxes: 'a' of the first type and 'b' of the second (which corresponds to the set of mRNA-T7 and mRNA-non-T7), and there are 't' balls(tRNA amber). All the balls are randomly distributed in the boxes. If there are two or more balls in some box of the first type (two or more tRNA amber per mRNA-T7) then a T7 molecule will be produced with a chance P_0.
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We have defined two models for this system which both rely on the following assumptions :
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* Each mRNA is defined as a 'box'
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* All the tRNA molecules are uniformly distributed in the boxes.
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* The number of tRNA_amber diffused through the nanotubes is much more smaller than the one of the mRNA. Thus the chance that three or more tRNA amber will "find" one mRNA-T7 is negligible comparing to the one of two tRNA amber (finding a mRNA-T7). In our model we will consider that at one moment of time each mRNA interacts with 0, 1 or 2 tRNA ambers.
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* The tRNA_amber placed in a correct box are always used
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== Distribution of tRNA_amber in mRNA populations ==
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We have defined two models for this system which both rely on the following assumptions :
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* Each mRNA is defined as a 'box'
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* All the tRNA molecules are uniformly distributed in the boxes.
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* The number of tRNA_amber diffused through the nanotubes is much more smaller than the one of the mRNA. Thus the chance that three or more tRNA amber will "find" one mRNA-T7 is negligible comparing to the one of two tRNA amber (finding a mRNA-T7). In our model we will consider that at one moment of time each mRNA interacts with 0, 1 or 2 tRNA ambers.
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* The tRNA_amber placed in a correct box are always used
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Let's define two types of boxes: mRNA_amber ('A') and other type of mRNA ('B'). We note the mRNA_amber producing T7 as mRNA*_amber. The latter appears if we have two tRNA_amber in one box.
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This model treats the repartition of tRNA_amber in the different boxes.
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[[File:Boxes_scheme.png|thumb|center|upright=3.0|A-box and b-box]]
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We note P(x) the probability of having x A-boxes containing two tRNA_amber. Thus P(x=1) corresponds to the probability of finding a couple of tRNA_amber in an A-box, thus to produce x T7 molecules. 't' is the number of tRNA_amber in the cell.
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We note:<br>
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* P(x=1)= (probability that 2 balls choose A-boxes) * (probability that these 2 balls choose the same A-box) + (probability that 3 balls choose A boxes) * (probability that 2 balls out of 3 choose the same A-box and the third doesn't) + ... + (probability that t balls choose an A-box) * (probability that 2 of these t balls choose the same A-box and the rest don't).
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* P(x=2)= (probability that 4 balls choose A-boxes) * (probability that these 4 balls choose 2 A-boxes, one A-box per pair of balls) + (probability that 5 balls choose A-boxes) * (probability that 4 balls out of 5 choose 2 A-boxes, one A-box per pair of balls and the fifth doesn't) + ... + (probability that t balls choose A-boxes) * (probability that 4 out of these t balls choose [t/2] A-boxes, one A-box per pair of balls and the rest don't).
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* ...
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* P(x=i)= (probability that 2i balls choose A-boxes) * (probability that these 2i balls choose i A-boxes, one A-box per pair of balls) + (probability that (2i + 1)  balls choose A-boxes) * (probability that 2i balls out of (2i + 1) choose i A-boxes, one A-box per pair of balls and the rest don't) + ... + (probability that t balls choose A-boxes) * (probability that 2i out of these t balls choose [t/2] A-boxes, one A-box per pair of balls and the rest don't).
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<br> Hence:
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[[File:P(1).png|thumb|center|upright=3.0|A-box and b-box]]
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== Master/Slave ==
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== Bi-directional communication ==
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== Brownian motion and diffusion ==
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A significant part of our designs relies on diffusion of very few molecules activating the amplification and reporter systems. For instance, we know that in the T7 design, we need less than ten T7 RNA polymerases to trigger the amplification. Knowing how a molecule moves in a cell just after its transport through nanotubes was necessary. To model this kind of behaviour we had to look into the mechanisms of diffusion for single molecules in a cell. This meant studying the motions of diffusion.<br>
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Please note that we are ''not'' talking here about the movement of molecules inside the nanotubes, which requires other types of modeling.<br>
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=== Diffusion without boundary conditions ===
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The first step was to study the general principle 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.<br>
+
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The principle of this first model is quite simple. We use the statistical diffusion equation with a new normalisation 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:<br>
+
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[[File:diffusion_equation.png|thumb|center]]
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-
 
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Where ''c'' is the concentration of particles in the cell, function of <math>\vec{x}</math> (position) and ''t'' (time). ''D'' is the diffusion coefficient.<br>
+
-
 
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The solution for such an equation is:<br>
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[[File:Diffusion_solution_form.png‎ |thumb|center|upright=3.0]]
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The star * represents the convolution of the two functions. The first part represents the initial conditions which are in our case a Dirac function centered on the origin of space (there is only one molecule at (0,0,0) at t=0). the second part of this solution is the so-called fundamental solution.
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<br><br><br><br><br><br><br><br>
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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.

Retrieved from "http://2011.igem.org/Team:Paris_Bettencourt/Modeling"