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.^{<a href="#reference1">[1]</a>} 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.^{<a href="#reference2">[2]</a>} 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.^{<a href="#reference2">[2]</a>} 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 ^{<a href="#reference3">[3]</a>}, 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><img src="https://static.igem.org/mediawiki/2014/0/09/Populationdesign3_Uppsala14.PNG"></img><h2>The killing module</h2><p>Finally, we needed the killing module. Since the killing is determined by the concentration of colicin we added a PDE for the colicin. The colicin is produced by attacking Bactissiles with a production rate of, beta_c. As with the OHHL we also introduce both diffusion with diffusion constant, D_c, and degradation with constant, eta_c. However we have no consumption because the <i>Y.enterocolitica</i> will probably lyse upon death and the molecules can be reused.</p><img src="https://static.igem.org/mediawiki/2014/e/e5/Populationdesign4_Uppsala14.PNG"></img><h2>Domain and boundary</h2><p>Further, we need to define a domain for our system to be active in. The domain will consist of a rectangle representing the intestinal wall cut open and stretched out. This creates the advantageous 2D domain that could easily be studied. The intestine is folded over and over again, which creates opportunities for bacteria and cells to interact between the walls folded close together. Interactions between areas of the fold will however be neglected, due to high complexibility. Since most bacteria is attached to the intestinal wall they cannot diffuse over to the other wall and neglecting that diffusion will not be unrealistic.  The main loss is the diffusion of molecules over the surface. If the diffusion is fast enough the molecules would be able to reach the wall on the other side and trigger both detection and death of cells. If this would happen in real life it would not make our system less useful since the real life result will be better then the result proposed by the model.<br><br>The boundary of the domain was defined by setting the concentration of all molecules and bacteria to zero at the edges leading out of the defined part of the intestine. This is an approximation of the greater area outside the small intestine which will lead to concentrations approaching zero as the distance from the diffusion source increases. If the diffusion source would be set close to the domain boundaries this approximation would not be realistic, instead the domain should then be expanded to allow diffusion calculations to take place.</p><h2>Initial values</h2><p>To start our model we need to set up initial values, a time = 0 scenario, that is the moment before the medicine reached the intestine. We want to place a colony of <i>Y.enterocolitica</i> that has already built up a steady-state of OHHL around itself, in our model. Therefore we need to define the location x_p, y_p of the <i>Y.enterocolitica</i> and the radius of the colony, r_p. This can then be inserted via an if-statement into MATLAB initial code. The concentration of OHHL in the location can be calculated via running a simulation on a large time frame, eg. 3 hours, and then observe at what value OHHL concentrations converges over time.<br><br>The position of the <i>Y.enterocolitica</i> colony was set to the middle of the system to make sure that the boundary conditions assumption would not effect our model too much. The radius of the colony was set to 3.16 (sqrt(10)) mm since no good data was found. The estimation originates from a test run to get nice results and internal guess.<br><br>The initial density of <i>Y.enterocolitica</i> was set to 1*10^8 cells/mL since it is known that they are constantly in their exponential phase during host infection.^{<a href="#reference3">[3]</a>} The initial concentration of our Bactissile was set to 1*10^7 cells/mL since it is likely that it is lower than the <i>Y. enterocolitica</i> concentration. One might suggest that the density should be set even lower, but due to restricted capacity to run long simulations we decided to go with 1*10^7 cells/mL.</p><h2>Parameters</h2><p>Some of the parameter values were found in articles while others were estimated, see table 1. A few parameters needed to be reworked, or did not precisely fit our system and will be discussed further.</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: Values of parameters used in the PDE model.</i><br><br>The diffusion coefficient value for all parameters had been calculated in water. These values are well studied and are therefore very reliable. However the real intestinal  wall and the the intestinal contents are probably more viscous leading to a lower diffusion coefficient for all parts.<br><br>The threshold value for the OHHL was calculated from the threshold concentration of the Lux-system documented in the iGEM parts registry. The production rate was recalculated from production rates of another colicin molecule. No good value on <i>Y.enterocolitica</i>s production of OHHL was found and the OHHL production rate was set to a low range guess.<br><br>Degradation of the OHHL was taken from a pH graph since it is degraded at fast rates at the pH present in the small intestine. However the activity of enzymes from both human and microbial cells are not included since not much is known about their effect. The colicin degradation was set to the same as the OHHL since no good values could be found. Once again we hope that this is a lower limit.</p><h2>Discussion and limitations</h2><p>It is always difficult to make a good mathematical model of complex biological systems. The gut is one of the most complex environments to work with and many parts of its unique and complex environment is therefore lost in models. One major part that is not included in this model is the effect of the microbial landscape of the gut that will most likely affect the system significantly. Since the microbial landscape is so complex we believe that most models will only give a false representation. Instead one should be aware of this fact, and acknowledge it when validating the prospects of this system.<br><br>Another concern is the fact that bacteria need to be anchored to the intestine wall in order not to be flushed out of the system. This means that it is likely that our Bactissile will not be able to move once it has arrived in the gut. However since the targeting module does not have a major impact on our system, as seen in the simulations, the movement should perhaps just be removed.</p><ul class="reference"><li><a id="reference1">[1]</a> Joy Sinha, Samuel J Reyes, Justin P Gallivan Reprogramming bacteria to seek and destroy an herbicide. Nature Chemical Biology 6, 464{470 (2010)</li><li><a id="reference2">[2]</a> Dirk Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Jahresbericht der Deutschen Mathematiker-Vereinigung, 105 (2003) 3, p. 103-165</li><li><a id="reference3">[3]</a> De Berardis, B., Torresini, G., Brucchi, M., Marinelli, S., Mattucci, S., Schietroma, M., Vecchio, L., Carlei, F. <i>Yersinia enterocolitica</i> intestinal infection with ileum perforation: report of a clinical observation. Acta bio-medica: Atenei Parmensis, 75(1): 77-81. (2004).</li></ul>';

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.^{<a href="#reference1">[1]</a>} 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.^{<a href="#reference2">[2]</a>} 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.^{<a href="#reference2">[2]</a>} 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 ^{<a href="#reference3">[3]</a>}, 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><img src="https://static.igem.org/mediawiki/2014/0/09/Populationdesign3_Uppsala14.PNG"></img><h2>The killing module</h2><p>Finally, we needed the killing module. Since the killing is determined by the concentration of colicin we added a PDE for the colicin. The colicin is produced by attacking Bactissiles with a production rate of, beta_c. As with the OHHL we also introduce both diffusion with diffusion constant, D_c, and degradation with constant, eta_c. However we have no consumption because the <i>Y.enterocolitica</i> will probably lyse upon death and the molecules can be reused.</p><img src="https://static.igem.org/mediawiki/2014/e/e5/Populationdesign4_Uppsala14.PNG"></img><h2>Domain and boundary</h2><p>Further, we need to define a domain for our system to be active in. The domain will consist of a rectangle representing the intestinal wall cut open and stretched out. This creates the advantageous 2D domain that could easily be studied. The intestine is folded over and over again, which creates opportunities for bacteria and cells to interact between the walls folded close together. Interactions between areas of the fold will however be neglected, due to high complexibility. Since most bacteria is attached to the intestinal wall they cannot diffuse over to the other wall and neglecting that diffusion will not be unrealistic.  The main loss is the diffusion of molecules over the surface. If the diffusion is fast enough the molecules would be able to reach the wall on the other side and trigger both detection and death of cells. If this would happen in real life it would not make our system less useful since the real life result will be better then the result proposed by the model.<br><br>The boundary of the domain was defined by setting the concentration of all molecules and bacteria to zero at the edges leading out of the defined part of the intestine. This is an approximation of the greater area outside the small intestine which will lead to concentrations approaching zero as the distance from the diffusion source increases. If the diffusion source would be set close to the domain boundaries this approximation would not be realistic, instead the domain should then be expanded to allow diffusion calculations to take place.</p><h2>Initial values</h2><p>To start our model we need to set up initial values, a time = 0 scenario, that is the moment before the medicine reached the intestine. We want to place a colony of <i>Y.enterocolitica</i> that has already built up a steady-state of OHHL around itself, in our model. Therefore we need to define the location x_p, y_p of the <i>Y.enterocolitica</i> and the radius of the colony, r_p. This can then be inserted via an if-statement into MATLAB initial code. The concentration of OHHL in the location can be calculated via running a simulation on a large time frame, eg. 3 hours, and then observe at what value OHHL concentrations converges over time.<br><br>The position of the <i>Y.enterocolitica</i> colony was set to the middle of the system to make sure that the boundary conditions assumption would not effect our model too much. The radius of the colony was set to 3.16 (sqrt(10)) mm since no good data was found. The estimation originates from a test run to get nice results and internal guess.<br><br>The initial density of <i>Y.enterocolitica</i> was set to 1*10^8 cells/mL since it is known that they are constantly in their exponential phase during host infection.^{<a href="#reference3">[3]</a>} The initial concentration of our Bactissile was set to 1*10^7 cells/mL since it is likely that it is lower than the <i>Y. enterocolitica</i> concentration. One might suggest that the density should be set even lower, but due to restricted capacity to run long simulations we decided to go with 1*10^7 cells/mL.</p><h2>Parameters</h2><p>Some of the parameter values were found in articles while others were estimated, see table 1. A few parameters needed to be reworked, or did not precisely fit our system and will be discussed further.</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: Values of parameters used in the PDE model.</i><br><br>The diffusion coefficient value for all parameters had been calculated in water. These values are well studied and are therefore very reliable. However the real intestinal  wall and the the intestinal contents are probably more viscous leading to a lower diffusion coefficient for all parts.<br><br>The threshold value for the OHHL was calculated from the threshold concentration of the Lux-system documented in the iGEM parts registry. The production rate was recalculated from production rates of another colicin molecule. No good value on <i>Y.enterocolitica</i>s production of OHHL was found and the OHHL production rate was set to a low range guess.<br><br>Degradation of the OHHL was taken from a pH graph since it is degraded at fast rates at the pH present in the small intestine. However the activity of enzymes from both human and microbial cells are not included since not much is known about their effect. The colicin degradation was set to the same as the OHHL since no good values could be found. Once again we hope that this is a lower limit.</p><h2>Discussion and limitations</h2><p>It is always difficult to make a good mathematical model of complex biological systems. The gut is one of the most complex environments to work with and many parts of its unique and complex environment is therefore lost in models. One major part that is not included in this model is the effect of the microbial landscape of the gut that will most likely affect the system significantly. Since the microbial landscape is so complex we believe that most models will only give a false representation. Instead one should be aware of this fact, and acknowledge it when validating the prospects of this system.<br><br>Another concern is the fact that bacteria need to be anchored to the intestine wall in order not to be flushed out of the system. This means that it is likely that our Bactissile will not be able to move once it has arrived in the gut. However since the targeting module does not have a major impact on our system, as seen in the simulations, the movement should perhaps just be removed.</p><ul class="reference"><li><a id="reference1">[1]</a> Joy Sinha, Samuel J Reyes, Justin P Gallivan Reprogramming bacteria to seek and destroy an herbicide. Nature Chemical Biology 6, 464{470 (2010)</li><li><a id="reference2">[2]</a> Dirk Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Jahresbericht der Deutschen Mathematiker-Vereinigung, 105 (2003) 3, p. 103-165</li><li><a id="reference3">[3]</a> De Berardis, B., Torresini, G., Brucchi, M., Marinelli, S., Mattucci, S., Schietroma, M., Vecchio, L., Carlei, F. <i>Yersinia enterocolitica</i> intestinal infection with ileum perforation: report of a clinical observation. Acta bio-medica: Atenei Parmensis, 75(1): 77-81. (2004).</li></ul>';

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