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| | <div class="summarytext1"> | | <div class="summarytext1"> |
| | <div class="part"> | | <div class="part"> |
| − | <video height="100%" controls> | + | |
| − | <source src="https://static.igem.org/mediawiki/2015/8/81/Simulation1dCont.ogg" type="video/ogg">-->
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| − | Failed to load video.
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| − | </video>
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| − | | + | |
| − | <p>
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| − | <br/>
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| − | The video above shows how the proposed method for pattern formation works. Two cell types A and B are interacting. Type
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| − | A cells produce a repellent called leucine which causes the cells of type B to move away. At the same time type A cells
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| − | also produce OO-AHL, which is required by the cells of type B to move. Initially colonies of the two cell types are placed
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| − | at the center of the dish. As molecule production within the type A cells kicks in, the repellent and AHL concentrations
| + | |
| − | start to increase. This triggers the type B cells to move away from the center. Movement will contiue until the concentration
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| − | of AHL is insuficcent for the type B cells to move further.
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| − |
| + | |
| − | </br>
| + | |
| | The Keller segel model type model we used is given by the following equation system: | | The Keller segel model type model we used is given by the following equation system: |
| | $$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + \gamma A(1 - \frac{A}{k_{p}}),$$ | | $$\frac{\partial A}{\partial t} = D_a \bigtriangledown^2 A + \gamma A(1 - \frac{A}{k_{p}}),$$ |
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| | $$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 B + k_r A - k_{lossH} R $$ | | $$ \frac{\partial R}{\partial t} = D_r \bigtriangledown^2 B + k_r A - k_{lossH} R $$ |
| | $$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 B + k_h A - k_{lossR} H . $$ | | $$\frac{\partial H}{\partial t} = D_h \bigtriangledown^2 B + k_h A - k_{lossR} H . $$ |
| − | With: </br>
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| − | $$ P(B,H,R) = \frac{-B K_{c} H}{R}. $$
| + | |
| − | The model has been derived while looking at <sup><a href="#ref1">[1] </a></sup> and <sup><a href="#ref2">[2] </a></sup>
| + | |
| − | The terms that appear can be grouped into four categories. Every equation has a diffusion term given by
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| − | $D_x \bigtriangledown^2 X$, diffusion smoothes peaks by spreading them out in space. The two equations related to cell
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| − | densities contain logistic growth terms of the form $\gamma X(1-\frac{X}{k_x})$, which model the cell growth during
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| − | simulation time. Finally the second equation describing the moving cells comes with a variable coefficient Poisson term
| + | |
| − | $\bigtriangledown (P \bigtriangledown X)$ which describes the cell movement. Last but not least: the two bottom equations.
| + | |
| − | They model concentrations, contain linear production and degradation terms, which look like $kX$. </br>
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| − | | + | |
| − | To generate the video file above the system above has been discretized using a finite element approach in conjunction,
| + | |
| − | with an explicit euler scheme. Finally simulation has been done using the parameters given in the table below: <br/><br/>
| + | |
| − | </p>
| + | |
| − | <table style="width:100%">
| + | |
| − | <tr> <th>Parameter</th> <th>Value</th> <th>Unit</th> <th>Source</th> </tr> | + | |
| − | <tr> <td>$D_a$</td> <td>$0.072*10^{-3}$</td> <td>$cm^2/h$</td> <td>following <sup><a href="#ref1">[1]
| + | |
| − | </a></sup> </td> </tr>
| + | |
| − | <tr> <td>$D_b$</td> <td>$2.376*10^{-3}$</td> <td>$cm^2/h$</td> <td>following <sup><a href="#ref1">[1]
| + | |
| − | </a></sup></td> </tr>
| + | |
| − | <tr> <td>$D_r$</td> <td>$26.46*10^{-3}$</td> <td>$cm^2/h$</td> <td> as found in <sup><a href="#ref6">[6]</a></sup>
| + | |
| − | </td> </tr>
| + | |
| − | <tr> <td>$D_h$</td> <td>$50*10^{-3}$</td> <td>$cm^2/h$</td> <td>from <sup><a href="#ref3">[3]
| + | |
| − | </a></sup> </td> </tr>
| + | |
| − | <tr> <td>$K_{c}$</td> <td>$8.5*10^{-3}$</td> <td>$cm^2/h$</td> <td>guessed</td> </tr>
| + | |
| − | <tr> <td>$\gamma$</td> <td>$10^{-5}$</td> <td>$h^{-1}$ </td> <td>from <sup><a href="#ref1">[1]
| + | |
| − | </a></sup></td> </tr>
| + | |
| − | <tr> <td>$k_p$</td> <td>$1.0 * 10^2$</td> <td>$cl^{-1}$</td> <td>from <sup><a href="#ref1">[1]
| + | |
| − | </a></sup></td> </tr>
| + | |
| − | <tr> <td>$k_r$</td> <td>$1.584*10^{-4}$</td> <td>$nmol/h$</td> <td>computed using <sup><a href="#ref4">[4]
| + | |
| − | </a></sup> and <sup><a href="#ref5">[5] </a></sup> </td> </tr>
| + | |
| − | <tr> <td>$k_h$</td> <td>$1.5*1.584*10^{-4}$</td> <td>$nmol/h$</td> <td>guessed</td> </tr>
| + | |
| − | <tr> <td>$k_{lossH}$</td> <td>$10^{-5}$</td> <td>$nmol/h$</td> <td>guessed</td> </tr>
| + | |
| − | <tr> <td>$k_{lossR}$</td> <td>$10^{-5}$</td> <td>$nmol/h$</td> <td>guessed</td> </tr>
| + | |
| − | | + | |
| − | | + | |
| − | </table>
| + | |
| − | </div>
| + | |
| − | </div>
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| − | | + | |
| − | | + | |
| − | <div class="summaryheader">
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| − | <div class="summaryimg">
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| − | <img src="https://static.igem.org/mediawiki/2015/e/eb/KU_Leuven_fossilBackground.png" width="100%">
| + | |
| − | <div class="head">
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| − | <h2> 2-D continuous model </h2>
| + | |
| − | </div>
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| − | </div>
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| − | </div>
| + | |
| − | | + | |
| − | <div class="summarytext1">
| + | |
| − | <div class="part">
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| − | | + | |
| − | <video width="100%" controls>
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| − | <!--<source src="https://static.igem.org/mediawiki/2015/d/d2/2dSim.ogg" type="video/ogg">-->
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| − | <source src="https://static.igem.org/mediawiki/2015/c/c3/FinalSim8.ogg" type="video/ogg">-->
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| − | Failed to load video.
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| − | </video> <br/> <br/>
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| − | | + | |
| − | <video width="100%" controls>
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| − | <source src="https://static.igem.org/mediawiki/2015/9/95/FinalSim7.ogg" type="video/ogg">-->
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| − | Failed to load video.
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| − | </video> <br/> <br/>
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| − | | + | |
| − | <p> Using the equation system and described above, the model may also be simulated in two dimensions. Once more a finite
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| − | volume approach has been taken in connection with an explicit Euler scheme. All parameters have been kept constant with the one
| + | |
| − | exception of the chemotatctic sensitivity $K_c$. Which has been inreased to $Kc = 1.5 * 10^{-1}$, which leads to earlier pattern formation.
| + | |
| − | </p>
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| − | </div>
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| − | </div>
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| | <div class="part"> | | <div class="part"> |
| | <p id="ref1">[1] <a href="http://www.sciencedirect.com/science/article/pii/S0006349595804005#" target="_blank"> Temporal Patterns Generated by Salmonella typhimurium, D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. 0.Budrene,l and H. C.Berg , Biophysical Journal Volume 68 May 1995 2181-2189 </a></p> | | <p id="ref1">[1] <a href="http://www.sciencedirect.com/science/article/pii/S0006349595804005#" target="_blank"> Temporal Patterns Generated by Salmonella typhimurium, D. E. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. 0.Budrene,l and H. C.Berg , Biophysical Journal Volume 68 May 1995 2181-2189 </a></p> |
| − | <p id="ref2">[2]<a href= " http://link.springer.com/book/10.1007%2F978-3-642-35497-7" target="_blank"> Hybrid modelling of individual movement and collective behaviour, B. Franz and R. Erban, DISPERSAL,
| |
| − | INDIVIDUAL MOVEMENT AND SPATIAL ECOLOGY: A MATHEMATICAL PERSPECTIVE Book Series: Lecture Notes in Mathematics
| |
| − | Volume: 2071 Pages: 129-157</a> </p>
| |
| − | <p id="ref3">[3] <a href="http://www.jbioleng.org/content/supplementary/1754-1611-6-16-s1.pdf" target="_blank" >Supplement 1 to Engineered cell-cell communication via DNA messaging, Monica E Ortiz and Drew Endy,
| |
| − | Journal of Biological Engineering</a> </p>
| |
| − | <p id="ref4">[4] <a href="http://www.biomedcentral.com/1471-2164/9/102" target="_blank">Protein abundance profiling of the Escherichia coli cytosol, Yasushi Ishihama1, Thorsten Schmidt, Juri
| |
| − | Rappsilber, Matthias Mann, F Ulrich Hartl, Michael J Kerner and Dmitrij Frishman38,
| |
| − | BMC Genomics 2008, 9:102</a> </p>
| |
| − | <p id="ref5">[5] <a href="http://onlinelibrary.wiley.com/doi/10.1111/febs.12609/epdf" target="_blank">The specificity and kinetic mechanism of branched-chain amino acid amino transferase from Escherichia coli
| |
| − | studied with a new improved coupled assay procedure and the enzyme’s potential for biocatalysis, Xuejing Yu, Xingguo Wang
| |
| − | and Paul C. Engel, the FEBS Journal</a> </p>
| |
| − | <p id="ref6">[6] <a href="http://pubs.acs.org/doi/pdf/10.1021/je060149b" target="_blank"> Infinite Dilution Binary Diffusion Coefficients of Several $\alpha $-Amino Acids in Water over a Temperature Range from (293.2 to 333.2) K with the Taylor Dispersion
| |
| − | Technique, Tatsuya Umecky, Tomoyuki Kuga, and Toshitaka Funazukuri, J. Chem. Eng. Data 2006,51,1705-1710 </a> </p>
| |
| − | </div>
| |
| − | </div>
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| | <!------------------------------------------------------Source code files---------------------------------------------------------> | | <!------------------------------------------------------Source code files---------------------------------------------------------> |
