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. <i>Yersinia enterocolitica</i> 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 <i>Y.enterocolitica</i> or not. Being able to coexist with <i>Y.enterocolitica</i> was 6 times as effective as the case where the Bactissile could only surround the <i>Y.enterocolitica</i>, 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 <i>Y.enterocolitica</i>.</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 <i>Y.enterocolitica</i> 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 <i>Y.enterocolitica</i> 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 <i>Y.enterocolitica</i> 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 <i>Y.enterocolitica</i> 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 <i>Y.enterocolitica</i> 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("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.<sup><a href="#reference1">[1]</a></sup> 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.<sup><a href="#reference2">[2]</a></sup> 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.<sup><a href="#reference2">[2]</a></sup> 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 <sup><a href="#reference3">[3]</a></sup>, 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.<sup><a href="#reference3">[3]</a></sup> 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("tab3").innerHTML = '<br><p>In this section you can read about how we tested our system. We test to variate the values of parameters that could be optimized and see what effect it would have on our system.</p><h2>Setting the initial OHHL concentrations</h2><p>To find out what levels the concentration of OHHL was starting at we ran the program with no bactissiles. This enables the <i>Y.enterocolitica</i> concentration to be stable and for the OHHL production to reach steady-state with the degradation. As seen in figure 1 the OHHL steady-state value is about 2*10^12 molecules/mL after about 3 hours incubation of a constant value of <i>Y.enterocolitica</i>.</p><img class="schedule" src="https://static.igem.org/mediawiki/2014/7/7f/Populationtest1_Uppsala14.jpg"></img><p><i>Figure 1: A: Shows the maximum density in the intestine of OHHL over time</i></p><h2>Coexistence</h2><p>Another issue is the question whether the Bactissile would stick or fit in the middle or on top of a colony of <i>Y.enterocolitica</i>. We asked a two professors that work with biofilms and gut bacteria, but they didn’t know. After searching in literature we drew the conclusion that this was unknown. Instead we formed simulations to determine what effect the case of “sticking” would have on our bacteria.</p><br><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 <i>Y.enterocolitica</i> 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 <i>Y.enterocolitica</i> does not coexist. 10% remains after 600s.</i></p></td></tr></table><br><p>In figure 2 we have allowed our Bactissile to co-exist at the same place as <i>Y.enterocolitica</i>. As can be seen the killing is very effective and the entire <i>Y.enterocolitica</i> colony is wiped out after about 100 seconds. However when not letting the Bactissile co-exist with <i>Y.enterocolitica</i> we get a dramatic change of effectivity as seen in figure 3. The <i>Y.enterocolitica</i> is not affected until after about 250 seconds, meaning that the time to build up sufficient amounts of colicin is significantly increased. Further on about 10% of the <i>Y.enterocolitica</i> still remain after six times as large time frame. We can see the killing occurring in steps, this is probably due to Bactissiles and OHHL trying to move into the system reaching one step at the time.<br><br>The observed instability is due to the timesteps taken. If more frequent timestep were to be taken this instability would be reduced. The computer power to efficiently study our system with higher accuracy was not obtained.</p><h2>Optimization of parameters</h2><p>There are three parameters that we believe can be improved by optimizing expression. These are colicin production, OHHL threshold and diffusion coefficient. The OHHL threshold can be modified via changing the yenR expression, while diffusion coefficient can be modified via cheZ expression. In our simulations we try to find out which expression is most important, and how large the impact will be for optimizing respective parameter? For each parameter we tested to increase the value by multiplying and decrease by dividing. The time for the entire colony of <i>Y.enterocolitica</i> to die out was then determined by measuring approximately the time from the graphs, see appendix for graphs. The results were then summarized in tables that were then fitted into graphs via Microsoft Office Excel 2010 via the trendline fitting tool, as seen in graphs 1-3 for coexistence model and graph 4-6 for non coexistence.</p><br><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/1/1a/Populationtest4_Uppsala14.png "></td><td><p><i>Graph 1:  Coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original OHHL threshold.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/4/48/Populationtest5_Uppsala14.png"></td><td><p><i>Graph 2: Coexist model,  Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original diffusion coefficient.</i></p></td></tr><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>Graph 3: Coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p></table><br><p>As seen from the graphs it is clear that the production of colicin is the rate limiting step in the coexistence model. A 10-fold increase in diffusion coefficient would only result in a 5% reduction in time.The OHHL threshold seems to follow a similar pattern, but is also halted at a maximum of 120 seconds. However when the OHHL threshold is set to the same value as the OHHL initial concentration, the time is increased to over 200 seconds , see figure 4. This indicates that the value needs to be lower than OHHL initial, but reducing it further is not very effective. Just a doubling of production would result in a reduction of time by half, and even more significant if the production is reduced instead.</p><br><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/0/02/Populationtest7_Uppsala14.jpg"></td><td><p><i>Figure 4:  Coexist model, OHHL threshold = OHHL initial concentration (2*10^12 molecules/mL), Y-axis: Fraction of Y.enterocolitica alive, X-axis: Time passed in seconds.</i></p></td></tr></table><table><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/9/93/Populationtest8_Uppsala14.png"></td><td><p><i>Graph 4: Non coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original diffusion coefficient.</i></p></td></tr><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>Graph 5: Non coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original production rate.</i></p><tr><td><img class="main_pic_left" style="padding-right: 20px;" src="https://static.igem.org/mediawiki/2014/8/8b/Populationtest10_Uppsala14.png"></td><td><p><i>Graph 6: Non coexist model, Y-axis: Time to eliminate the initial Y.enterocolitica colony. X-axis: Multiplier of the original OHHL threshold.</i></p></table><h2>Future</h2><p>There is very much interesting work that can be done to improve on the model. Many of the parameters used are very rough estimates and would need to be examined further. This can be done as with the three modules, by variating the value and measuring time differences. This would give an indication of what variables are most important, how exact our model is and could give insights of what parameters are most important to find exact values on.</p><h2>Summary</h2><p>The model was split into two scenarios depending if the Bactissile could coexist on the <i>Y.enterocolitica</i> or not. Being able to coexist with <i>Y.enterocolitica</i> was 6 times as effective as the case where the Bactissile could only surround the <i>Y.enterocolitica</i>. Both models showed that the production of colicin is the rate limiting step, but 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 <i>Y.enterocolitica</i>.</p><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.</p><br><p>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 <i>Y.enterocolitica</i> 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><h2>MATLAB code</h2><p>MATLAB code. <a hrf=https://static.igem.org/mediawiki/2014/3/33/BactissilePDEGraph.m>Runfile</a>,<a hrf=https://static.igem.org/mediawiki/2014/d/d2/UppsalaiGEMPDEGraph_initial.m>initial values</a> and <a hrf=https://static.igem.org/mediawiki/2014/5/5f/BactissileWorkspace.zip>workspace</a>.</p>';

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