Team:Uppsala/Modeling PopulationLevel

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document.getElementById("tab1").innerHTML = '<h2>Introduction</h2><p>We wanted to evaluate the effectiveness of our system in the real intestine of a human. Since it requires a lot of ethical consideration and animal trials to test on real humans we decided to construct a model instead. The model will also give us insight into what parameters are most important when improving our system. Since the thought out medicine is thought of as a pill, see Policy & Practise, this is the premiss we will evaluate our modeling around.<br><br>Our final model is able to produce both a “density map” showing the small intestine as a two dimensional landscape and graphs of the total amount of cells and molecules over time. The first model is more heavy to run and can therefore only run at limited time intervals, while the later model is able to run at much longer intervals. The density map is good for analysing movement of molecules and cells in the domain, while the graphs are nice to show total effect of the system.</p><img src="https://static.igem.org/mediawiki/2014/7/73/Populationsummary1_Uppsala14.PNG"></img><p><i>Figure 1: Mathematical model of our system.</i></p><h2>Design</h2><p>To create our model we created a set of PDE:s to represent our system, see figure 1. The change of state is determined via a set of heaviside step functions(noted as theta) that are controlled by threshold concentrations noted as K_i, where i indicates which density is related to the threshold. Y.enterocolitica have been found to be immobile at 37 degrees celsius and do therefore not have a diffusion term. All parameters are defined in figure 2. You can read more about our design under the design page.</p><br><table id="partsT"><tr><th>Parameter</th><th>Value</th><th>Source</th></tr><tr><td>D_b</td><td>3*10^-4 mm^2/s</td><td>5</td></tr><tr><td>D_c</td><td>4.2*10^-5 mm^2/s</td><td>1</td></tr><tr><td>D_o</td><td>4.9*10^-6 mm^2/s</td><td>2</td></tr><tr><td>K_c</td><td>2.25*10^-7 molecules/mL</td><td>3</td></tr><tr><td>K_a</td><td>10^-8 molecules/mL</td><td>est.</td></tr><tr><td>beta_c</td><td>15.625 s^-1</td><td>1</td></tr><tr><td>beta_o</td><td>1 s^-1</td><td>est.</td></tr><tr><td>etac</td><td>1/1200</td><td>ets.</td></tr><tr><td>eta_o</td><td>1/1200</td><td>4</td></tr></table><p><i>Table 1. Table of parameter values</i></p><h2>Results</h2><p>The model was split into two scenarios depending if the Bactissile could coexist on the Y.enterocolitica or not. Being able to coexist with Y.enterocolitica was 6 times as effective as the case where the Bactissile could only surround the Y.enterocolitica, figure 1 and 2. Both models showed that the production of colicin is the rate limiting step, figure 3 and 4, while OHHL threshold and diffusion coefficients importance differed. In the coexist model it was only needed for the OHHL threshold to be less than the OHHL initial concentration. However in the non coexist model both OHHL and diffusion coefficients changes had a significant impact on the time to kill all Y.enterocolitica.</p><table><tr><td><img class="main_pic_left" src="https://static.igem.org/mediawiki/2014/4/48/Populationtest2_Uppsala14.jpg"></td><td><p><i>Figure 2: Bactissile and Y.enterocolitica coexists. 100s until elimination</i></p></td></tr></table><table><tr><td><img class="main_pic_left" src="https://static.igem.org/mediawiki/2014/8/83/Populationtest3_Uppsala14.jpg"></td><td><p><i>Figure 3: Bactissile and Y.enterocolitica does not coexist. 10% remains after 600s.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/a/a0/Populationtest6_Uppsala14.png"></td><td><p><i>Figure 4:  Coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/0/0f/Populationtest9_Uppsala14.png"></td><td><p><i>Figure 5:  Non coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p></td></tr></table><br><p>Based upon this, efforts should continue to form an adhesion system that ensures that the Bactissile can coexist. Further on measurements of the amount of colicin production rate should be taken to make better predictions, and work should be focused on increasing this production.<br><br>If no effective adhesion system can be built, improved effectivity could be achieved by taking another dose of Bactissiles within a short interval, e.g. every 20 minutes. This would act as a “reset” for the graph. Another alternative is to take the Bactissiles prior to sickness so that no Y.enterocolitica can infect in the first place. However unless the Bactissile can stay in the intestine for a long period of time you would frequently have to retake the bacteria. Having a large amount of cells in a living system is unstable since the amount of mutations increase, that could result in the system not working as intended.</p>';
document.getElementById("tab1").innerHTML = '<h2>Introduction</h2><p>We wanted to evaluate the effectiveness of our system in the real intestine of a human. Since it requires a lot of ethical consideration and animal trials to test on real humans we decided to construct a model instead. The model will also give us insight into what parameters are most important when improving our system. Since the thought out medicine is thought of as a pill, see Policy & Practise, this is the premiss we will evaluate our modeling around.<br><br>Our final model is able to produce both a “density map” showing the small intestine as a two dimensional landscape and graphs of the total amount of cells and molecules over time. The first model is more heavy to run and can therefore only run at limited time intervals, while the later model is able to run at much longer intervals. The density map is good for analysing movement of molecules and cells in the domain, while the graphs are nice to show total effect of the system.</p><img src="https://static.igem.org/mediawiki/2014/7/73/Populationsummary1_Uppsala14.PNG"></img><p><i>Figure 1: Mathematical model of our system.</i></p><h2>Design</h2><p>To create our model we created a set of PDE:s to represent our system, see figure 1. The change of state is determined via a set of heaviside step functions(noted as theta) that are controlled by threshold concentrations noted as K_i, where i indicates which density is related to the threshold. Y.enterocolitica have been found to be immobile at 37 degrees celsius and do therefore not have a diffusion term. All parameters are defined in figure 2. You can read more about our design under the design page.</p><br><table id="partsT"><tr><th>Parameter</th><th>Value</th><th>Source</th></tr><tr><td>D_b</td><td>3*10^-4 mm^2/s</td><td>5</td></tr><tr><td>D_c</td><td>4.2*10^-5 mm^2/s</td><td>1</td></tr><tr><td>D_o</td><td>4.9*10^-6 mm^2/s</td><td>2</td></tr><tr><td>K_c</td><td>2.25*10^-7 molecules/mL</td><td>3</td></tr><tr><td>K_a</td><td>10^-8 molecules/mL</td><td>est.</td></tr><tr><td>beta_c</td><td>15.625 s^-1</td><td>1</td></tr><tr><td>beta_o</td><td>1 s^-1</td><td>est.</td></tr><tr><td>etac</td><td>1/1200</td><td>ets.</td></tr><tr><td>eta_o</td><td>1/1200</td><td>4</td></tr></table><p><i>Table 1. Table of parameter values</i></p><h2>Results</h2><p>The model was split into two scenarios depending if the Bactissile could coexist on the Y.enterocolitica or not. Being able to coexist with Y.enterocolitica was 6 times as effective as the case where the Bactissile could only surround the Y.enterocolitica, figure 1 and 2. Both models showed that the production of colicin is the rate limiting step, figure 3 and 4, while OHHL threshold and diffusion coefficients importance differed. In the coexist model it was only needed for the OHHL threshold to be less than the OHHL initial concentration. However in the non coexist model both OHHL and diffusion coefficients changes had a significant impact on the time to kill all Y.enterocolitica.</p><table><tr><td><img class="main_pic_left" src="https://static.igem.org/mediawiki/2014/4/48/Populationtest2_Uppsala14.jpg"></td><td><p><i>Figure 2: Bactissile and Y.enterocolitica coexists. 100s until elimination</i></p></td></tr></table><table><tr><td><img class="main_pic_left" src="https://static.igem.org/mediawiki/2014/8/83/Populationtest3_Uppsala14.jpg"></td><td><p><i>Figure 3: Bactissile and Y.enterocolitica does not coexist. 10% remains after 600s.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/a/a0/Populationtest6_Uppsala14.png"></td><td><p><i>Figure 4:  Coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/0/0f/Populationtest9_Uppsala14.png"></td><td><p><i>Figure 5:  Non coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p></td></tr></table><br><p>Based upon this, efforts should continue to form an adhesion system that ensures that the Bactissile can coexist. Further on measurements of the amount of colicin production rate should be taken to make better predictions, and work should be focused on increasing this production.<br><br>If no effective adhesion system can be built, improved effectivity could be achieved by taking another dose of Bactissiles within a short interval, e.g. every 20 minutes. This would act as a “reset” for the graph. Another alternative is to take the Bactissiles prior to sickness so that no Y.enterocolitica can infect in the first place. However unless the Bactissile can stay in the intestine for a long period of time you would frequently have to retake the bacteria. Having a large amount of cells in a living system is unstable since the amount of mutations increase, that could result in the system not working as intended.</p>';
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document.getElementById("tab2").innerHTML = '<br><p>In this section we explain how we created our system of PDE:s, the parameters, the domain, the boundaries and the initial values. We also discuss how exact our model is and what its major flaws are. </p><h2>Inspiration</h2><p>A similar system that was controlled by switching on the CheZ-gene expression when the bacteria detected a specific chemical signal have been studied before (6). However no mathematical model was found. Although there are well established mathematical models for chemotaxis in <i>E.coli</i>, called Keller Segel(KS) models(7).The Keller Segel models are known to be intuitive and to capture the key-elements of chemotaxis, and many alterations are possible to improve upon the model(7).The KS model controls the movement via sensing a change in concentration of the sensed molecule, thereby following a gradient.A genetically controlled system can only return a expression result depending upon the rate of the chemical signal, or activates/deactivates once a threshold concentration has been reached. Our model can therefore not be based upon the KS model.</p><h2>Movement and target module</h2><p>Since we could not use the KS model we had to start from scratch, and the first step was the modeling of a random walk. Random walk was modeled as a diffusion via Ficks law of diffusion, as is the convention when modeling large number of cells.<br><br>To model the activation of our system when a threshold concentration of chemical signal( K_o) is reached, an additional bacteria density (b_a) for attacking Bactissiles was introduced. This enables a visual representation of the amount of attacking Bactissiles. This also solves the problem to turn off the random walk when the Bactissiles are attacking. The activation and de-activation of bacteria is controlled via a heavieside-step function, notated as theta.</p><img src="https://static.igem.org/mediawiki/2014/c/c8/Populationdesig1_Uppsala14.PNG"></img><h2>The <i>Y.enterocolitica</i></h2><p>The <i>Y.enterocolitica</i>, noted p, has been shown to be immobile at 37 degrees Celsius (8), and does therefore not need a term describing its random walk. Drift will be neglected since most bacteria attach to the intestinal wall and might therefore not be influenced by drift nor intestinal movement. Neither will growth of the <i>Y.enterocolitica</i> be included since the generation time is about 30min in optimal conditions for most bacteria. Considering the rough competitive climate in the intestine, it would be hard for the <i>Y.enterocolitica</i> to grow. It is therefore not worth to add a growth term due to its complexity.<br><br>The death of <i>Y.enterocolitica</i> due to sufficient concentrations of the colicin will be included in the model via a heaviside-step function. We can control when the <i>Y.enterocolitica</i> should be eliminated by setting up a threshold, K_c. When the colicin is above K_c the prey will be eliminated via the -p term, as seen in equation below.</p>';
+
document.getElementById("tab2").innerHTML = '<br><p>In this section we explain how we created our system of PDE:s, the parameters, the domain, the boundaries and the initial values. We also discuss how exact our model is and what its major flaws are. </p><h2>Inspiration</h2><p>A similar system that was controlled by switching on the CheZ-gene expression when the bacteria detected a specific chemical signal have been studied before (6). However no mathematical model was found. Although there are well established mathematical models for chemotaxis in <i>E.coli</i>, called Keller Segel(KS) models(7).The Keller Segel models are known to be intuitive and to capture the key-elements of chemotaxis, and many alterations are possible to improve upon the model(7).The KS model controls the movement via sensing a change in concentration of the sensed molecule, thereby following a gradient.A genetically controlled system can only return a expression result depending upon the rate of the chemical signal, or activates/deactivates once a threshold concentration has been reached. Our model can therefore not be based upon the KS model.</p><h2>Movement and target module</h2><p>Since we could not use the KS model we had to start from scratch, and the first step was the modeling of a random walk. Random walk was modeled as a diffusion via Ficks law of diffusion, as is the convention when modeling large number of cells.<br><br>To model the activation of our system when a threshold concentration of chemical signal( K_o) is reached, an additional bacteria density (b_a) for attacking Bactissiles was introduced. This enables a visual representation of the amount of attacking Bactissiles. This also solves the problem to turn off the random walk when the Bactissiles are attacking. The activation and de-activation of bacteria is controlled via a heavieside-step function, notated as theta.</p><img src="https://static.igem.org/mediawiki/2014/c/c8/Populationdesig1_Uppsala14.PNG"></img><h2>The <i>Y.enterocolitica</i></h2><p>The <i>Y.enterocolitica</i>, noted p, has been shown to be immobile at 37 degrees Celsius (8), and does therefore not need a term describing its random walk. Drift will be neglected since most bacteria attach to the intestinal wall and might therefore not be influenced by drift nor intestinal movement. Neither will growth of the <i>Y.enterocolitica</i> be included since the generation time is about 30min in optimal conditions for most bacteria. Considering the rough competitive climate in the intestine, it would be hard for the <i>Y.enterocolitica</i> to grow. It is therefore not worth to add a growth term due to its complexity.<br><br>The death of <i>Y.enterocolitica</i> due to sufficient concentrations of the colicin will be included in the model via a heaviside-step function. We can control when the <i>Y.enterocolitica</i> should be eliminated by setting up a threshold, K_c. When the colicin is above K_c the prey will be eliminated via the -p term, as seen in equation below.</p><img src="https://static.igem.org/mediawiki/2014/8/88/Populationdesign2_Uppsala14.PNG"></img><br><h2>The sensing module</h2><p>To incorporate the sensing of <i>Y.enterocolitica</i> we added a PDE to explain how the concentration of OHHL change over time. OHHL is produced by <i>Y.enterocolitica</i> at a constant production rate,( beta_p), in our formulas. The diffusion of the molecule is modeled via Ficks law with the diffusion coefficient D_o. Since the OHHL is not stable in the intestine we also added a degradation constant, eta_o. Lastly we added consumption of the molecule by our Bactissile with a consumption rate, alpha_b.</p>';
document.getElementById("tab3").innerHTML = '<p>Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et rit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.</p>';
document.getElementById("tab3").innerHTML = '<p>Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et rit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.</p>';

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