View source for Team:Heidelberg/Model/Lagoon Contamination

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    Modeling|
    Lagoon contamination|
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            <a href="https://2017.igem.org/Team:Heidelberg/Model/Contamination">For this model an interactive webtool is available.</a>
            <br>
            Contaminations with bacteria other than the host cells that are part of the experiment, funghi or other microorganisms can survive in the lagoons, it is necessary to take precautions. Since these can be costly (e.g. fungicides) an simulation of contamination growth can save resources. If the constant dilution rate of the lagoons is sufficiently high that the overall change in population size of the contamination is negative, the contamination will wash out and not be a lasting problem.
            <br>
            <b>Note:</b> contamination with viruses such as bacteriophages is not modelled here, because exponential growth of the contamination is assumed. This is plausible for bacteria and funghi since the lagoons are constantly diluted so that there is always fresh medium and space available.

            The change in concentration of the contamination can be described as

            $$
            \frac{\partial c_{X}(t)}{\partial t} = -\Phi_{L} \cdot c_{X}(t) + \frac{\ln{2} }{T_{X} } \cdot c_{X}(t)
            $$
            
            The concentration of the contamination \(x_{X}\) is assumed only to depend on the time, since the constant dilution reduces effects such as spent resources or growht inhibition by waste. The growth factor \(\frac{ln(2)}{t_{X} }\) is the factor by which the current concentration and its derivative are proportional.
            There is the dilution term containing the factor \(\Phi_{L}\) that is the flow rate trough the lagoon in volumes per hour and the growth term of the contamination based on the contaminations generation time \(t_{X}\). Solving this equation yields

            $$
            c_{X}(t) = c_{X}(t_{0}) \cdot e^{\big(\frac{ln(2)}{t_{X} } - \Phi_{L}\big) \cdot t}
            $$
            
            Here \(c_{X}(t_{0})\) is introduced, which is the initial concentration of the contamination.
            <br>
            When working with the model it is more practical to calculate relative changes in the concentration:
            
            $$\frac{c_{X}(t)}{c{X}(t_{0})}$$

            That simplifies the above equation to 

            $$
            c_{X}(t) = e^{\big(-\Phi_{L} + \frac{\ln{2} }{t_{X} }\big) \cdot t}
            $$
            
            Differentiation after \(t\) gives

            $$
            \frac{\partial c_{X}(t)}{\partial t} = \Big(\frac{ln(2)}{t_{X} } - \Phi_{L}\big) e^{\big(\frac{ln(2)}{t_{X} } - \Phi_{L}\Big) \cdot t}
            $$
            
            <b>The following statements can be made:</b>
            <br>
            The contamination expands if the flow rate \(\Phi_{L}\) is greater than the growth factor, 
            $$
            c_{X}(t) > 0, \quad if \quad \Phi_{L} > \frac{ln(2)}{t_{X} } 
            $$
            it remains constant if the flow rate is exactly the growth factor
            $$
            c_{X}(t) = 0, \quad if \quad \Phi_{L} = \frac{ln(2)}{t_{X} }
            $$   
            and it diminishes, when the flow rate is higher than the growth factor.
            $$
            c_{X}(t) < 0, \quad if \quad \Phi_{L} < \frac{ln(2)}{t_{X } }
            $$
            <br>
            
            
                {{Heidelberg/templateus/Tablebox|
                        Table 1: Additional Variables and Parameters used for the calculation of the number of mutated sequences|
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                            <table class="table table-bordered mdl-shadow--4dp" XSSCleaned="overflow-x: scroll !important">
                                <thead>
                                    <tr>
                                        <th>Symbol</th>
                                        <th>Value and Unit</th>
                                        <th>Explanation</th>
                                    </tr>
                                </thead>
                                <tbody> 
                                    <tr>
                                        <td>\(t \)</td>
                                        <td>[h]</td>
                                        <td>Time</td>
                                    </tr> 
                                    <tr>
                                        <td>\(c_{X} \)</td>
                                        <td>>relative units</td>
                                        <td>Contamination concentration </td>
                                    </tr>  
                                    <tr>
                                        <td>\(\Phi_{L}\)</td>
                                        <td>[volumes/h]</td>
                                        <td>Flow rate of the lagoons</td>
                                    </tr>  
                                    <tr>
                                        <td>\(T_{X}\)</td>
                                        <td>[min]</td>
                                        <td>Generation time of contamination, time in which it doubles</td>
                                    </tr>  
                                    
                                </tbody>
                            </table>
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                    List of all paramters and variables used in the analytical solution of this model and for calculations with the <a href="https://2017.igem.org/Team:Heidelberg/Model/Contamination">interactive webtool</a>. When possible values are given.}}
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