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| − | {{Heidelberg/header}}
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| − | {{#tag:html|
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| − | <script>
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| − | {{Heidelberg/title|Phage Titer}}
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| − | </script>
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| − | {{Heidelberg/navbar
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| − | {{Heidelberg/main|
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| − | Modeling|
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| − | Phage Titer|
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| − | https://static.igem.org/mediawiki/2017/3/38/T--Heidelberg--2017_Background_Owl.jpg|blue|
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | While developing <a href="https://2017.igem.org/Team:Heidelberg/Predcel">PREDCEL</a> in the lab, we simultaneously developed it <i>in silico</i> so that both sides could benefit from each other. One of the most important parameter of phage assisted directed evolution experiments like PREDCEL and PACE is the phage titer itself. If the phage titer drops washout can occur and the experiment has to be restarted with the disadvantage of loosing library complexity. If the phage titer increases too much, the multiplicity of infection (MOI), that means the amount of phage relative the the amount of <i>E. coli</i> rises too. If fore example the MOI is 10 and an <i>E. coli</i> can only be infected by one phage, nine out of ten phages will not infect an <i>E. coli</i> and thus will not evolve, but still make up most of the phage population.
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/c/cd/T--Heidelberg--2017_phage_titer_fig1.png|{{#tag:html|Figure 1: Basic logarithmic phage and <i>E. coli</i> titer plot with 20 % wildtype fitness}}|{{#tag:html| The blue lines correspond to the different <i>E. coli</i> populations. Exponential growth of <i>E. coli</i> and constant fitness of 20% of the wildtype, equal in all phages was assumed. After ten minutes infected <i>E. coli</i> start producing phage, corresponding to a drop in infected <i>E. coli</i> and an increase in phage concentration.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/3/38/T--Heidelberg--2017_phage_titer_fig2.png|{{#tag:html|Figure 2: Basic logarithmic phage and <i>E. coli</i> titer plot with 100 % wildtype fitness.}}|{{#tag:html| The blue lines correspond to the different <i>E. coli</i> populations. Exponential growth of <i>E. coli</i> and constant fitness of 100% of the wildtype, equal in all phages was assumed. After ten minutes infected <i>E. coli</i> start producing phage, corresponding to a drop in infected <i>E. coli</i> and an increase in phage concentration.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | Comparing the plots for 20 % and 100 % fitness show that a higher fitness increases the final phage titer and decreases the amount of uninfected <i>E. coli</i> earlier.
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| − | <h1>Modeling concentrations in one Lagoon</h1>
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| − | Here the concentrations \(c\) of uninfected <i>E. coli</i>, infected <i>E. coli</i> and phage producing <i>E. coli</i> as well as the <i>M13</i> phage are modeled. They are denoted with the subscripts \(_{u}\), \(_{i}\), \(_{p}\) and \(_{P}\). If the whole <i>E. coli</i> population is referred to, \(c_{E}\) is used. If an arbitrary <i> E. coli</i> population is meant, the subscript \(_{e}\) is used. The phage concentration \(c_{P}\) refers to the free phage only, phage that are contained in an <i>E. coli</i> they infected are not included.
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| − | The used parameters include the time \(t\), the affinity of phage for <i>E. coli</i> \(k\), the duration between infection of an <i>E. coli</i> and the first phage leaving the <i>E. coli</i> \(t_{P}\). The three different <i>E. coli</i> populations each have a generation time \(t\) that is denoted with their subscript. The fitness of a phage population is \(f\).
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| − | {{Heidelberg/templateus/Tablebox|
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| − | Table 1: Variables and Parameters used in this model |
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| − | {{#tag:html|
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| − | <table class="table table-bordered mdl-shadow--4dp" XSSCleaned="overflow-x: scroll !important">
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| − | <thead>
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| − | <tr>
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| − | <th>Symbol</th>
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| − | <th>Name in source code</th>
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| − | <th>Value and Unit</th>
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| − | <th>Explanation</th>
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| − | </tr>
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| − | </thead>
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| − | <tbody>
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| − | <tr>
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| − | <td>\(c \)</td>
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| − | <td>-</td>
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| − | <td>[cfu] or [pfu] </td>
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| − | <td>colony forming units for <i> E. coli</i> (cfu) or plaque forming units (pfu) for M13 phage</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _u\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript for uninfected <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _i\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript for infected <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _p\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript for phage-producing <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _e\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript any the of <i>E. coli</i> populations on its own</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _E\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript for all populations of <i>E. coli</i> together</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( _P\)</td>
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| − | <td>-</td>
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| − | <td> - </td>
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| − | <td>Subscript for M13 phage</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(c_{c} \)</td>
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| − | <td><code class="python">capacity</code></td>
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| − | <td>[cfu/ml]</td>
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| − | <td>Maximum concentration of <i>E. coli</i> possible under given conditions, important for logistic growth</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(t\)</td>
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| − | <td><code class="python">t</code></td>
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| − | <td>[min]</td>
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| − | <td>Duration since the experiment modeled was started</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(t_{u} \)</td>
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| − | <td><code class="python">tu</code></td>
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| − | <td>\(20\) min</td>
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| − | <td>Duration one division of uninfected <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(t_{i} \)</td>
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| − | <td><code class="python">ti</code></td>
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| − | <td>\(30\) min</td>
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| − | <td>Duration one division of infected <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(t_{p} \)</td>
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| − | <td><code class="python">tp</code></td>
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| − | <td>\(40\) min</td>
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| − | <td>Duration one division of phage producing <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( t_{P}\)</td>
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| − | <td><code class="python">tpp</code></td>
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| − | <td>[min]</td>
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| − | <td>Duration between an <i>E. coli</i> being infected by an M13 phage and releasing the first new phage</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(g_{e} \)</td>
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| − | <td><code class="python">e_growth_rate</code></td>
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| − | <td>[cfu/min]</td>
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| − | <td>Growth rate of <i>E. coli</i>, depending on the type of growth (either logistic or exponential), the current concentration \(c_{e}\), the maximum concentration \(c_{c}\), and the generation time \(t_{e}\)</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( k\)</td>
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| − | <td><code class="python">k</code></td>
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| − | <td>\(3 \cdot 10^{-11}\frac{1}{cfu \cdot pfu \cdot ml \cdot min}\)</td>
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| − | <td>Affinity of M13 phage for <i>E. coli</i></td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( \mu_{max}\)</td>
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| − | <td><code class="python">mumax</code></td>
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| − | <td>\(16.67 \frac{cfu}{min \cdot ml \cdot cfu}\)</td>
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| − | <td>Wildtype M13 phage production rate</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\( f\)</td>
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| − | <td><code class="python">f</code></td>
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| − | <td>?</td>
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| − | <td>Fitnessvalue, fraction of actual \(\mu\) and \(\mu_{max}\)</td>
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| − | </tr>
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| − | </tbody>
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| − | </table>
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| − | }}|
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| − | List of all paramters and variables used in this model. When possible values are given.
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | Each term describing the change of an <i>E. coli</i> concentration contains its growth, \(g_{e}\). The growth rate of an <i>E. coli</i> population can be modeled by exponential growth or by logistic growth. Especially, when long durations per lagoon are modeled, the logistic growth model is more exact. [source].
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| − | In the exponential case the growth rate \(g_{e}\) is modeled as
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| − | $$
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| − | g_{e} (t_{e}) = c_{e} \cdot \frac{log(2)}{t_{e} }
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| − | $$
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| − | Note that the growth rate in the model increases over time, while in the modeled culture, the nutrient concentration decreases.
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| − | That makes the logistic model more plausible, it models \(g_{e}\) as
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| − | $$
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| − | g_{e} (t_{e}, \: c_{e}(t), \: c_{c}) = \frac{c_{c} - c_{e} (t)}{c_{c} } \cdot \frac{log(2)}{t_{e} }
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| − | $$
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| − | In this case the learning rate decreases as the current concentration \(c_{e}\) approaches the maximum capacity for <i>E. coli</i> in the given setup \(c_{c}\). With this model \(c_{e} \leq c_{c}\) is true for any point in time.
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| − | <b>Change of concentration of uninfected <i>E. coli</i>, \(\frac{\partial c_{u} }{\partial t} \: [cfu/min]\)</b>
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| − | $$
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| − | \frac{\partial c_{u} }{\partial t}(t) = g_{u} (t_{u}, \: c_{u}(t), \: c_{c})
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| − | - k \cdot c_{u}(t) \cdot c_{p}(t)
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| − | $$
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| − | In addition to the growth term, the concentration of uninfected <i>E. coli</i> is described by a term for infection that takes into account the concentration of uninfected <i>E. coli</i> and the concentration of free phage and reduces the conentration of uninfected <i>E. coli</i>.
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| − |
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| − | <b>Change of concentration of uninfected <i>E. coli</i>, \(\frac{\partial c_{i} }{\partial t} \: [cfu/min]\)</b>
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| − | $$
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| − | \frac{\partial c_{i} }{\partial t}(t) = \begin{cases}
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| − | g_{i} (t_{i}, \: c_{i}(t), \:c_{c})
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| − | + k \cdot c_{i}(t) \cdot c_{p}(t)
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| − | - c_{i}(t - t_{P}),
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| − | \quad \text{for} \: t > t_{P} \\
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| − | g_{i} (t_{i}, \: c_{i}(t), \: c_{c})
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| − | + k \cdot c_{i}(t) \cdot c_{p}(t),
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| − | \quad \text{otherwise}
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| − | \end{cases}
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| − | $$
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| − | Until \(t > t_{P}\) the concentration of infected <i>E. coli</i> increases by growth and infection of previouly uninfected <i>E. coli</i>. When \(t > t_{P}\), a third term describing that infected <i>E. coli</i> turn into phage-producing <i>E. coli</i> is subtracted.
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| − |
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| − | <b>Change of concentration of phage producing <i>E. coli</i>, \(\frac{\partial c_{p} }{\partial t} \: [cfu/min]\)</b>
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| − |
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| − | $$
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| − | \frac{\partial c_{p} }{\partial t}(t) = \begin{cases}
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| − | g_{p} (t_{p}, \: c_{p}(t), \: c_{c}) -
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| − | c_{i}(t - t_{P}),
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| − | \quad \text{for} \: t > t_{P} \\
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| − | g_{p} (t_{p}, \: c_{p}(t), \: c_{c}),
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| − | \quad \text{otherwise}
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| − | \end{cases}
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| − | $$
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| − | The population of phage producing E. coli only increases by growth until \(t > t_{P}\). When infected <i>E. coli</i> drop their first phage they turn into producing <i>E. coli</i> as described by the second term.
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| − |
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| − | <b>Change of concentration of <i>M13</i> phage, \(\frac{\partial c_{P} }{\partial t} \: [cpu/min]\)</b>
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| − |
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| − | $$
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| − | \frac{\partial c_{P} }{\partial t}(t) = c_{P}(t) \cdot \mu_{max} \cdot f - k \cdot c_{u}(t)\cdot c_{P}(t)
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| − | $$
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| − | The phage concentration is only increased by phage that leave phage-producing <i>E. coli</i>, which happens at a rate of \(f \cdot \mu_{max}\) per time unit, with f being the fitness, a value between 0 and 1, equal to the share of the wildtype <i>M13</i> phages fitness and \(\mu_{max}\) being the wildtype phages production rate. We assume that the only negative influence on the free phage titer is phage infecting <i>E. coli</i>, which depends on both the phage titer \(c_{P}\) and the titer of uninfected <i>E. coli</i>, \(c_{i}\).
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| − |
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| − | When the fitness is assumed to be the same for all phages, it is modeled to be constant during the time in one lagoon.
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| − | <br>
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| − |
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| − | If a fitness distribution is assumed, the fitness does not necessarily stay constant in one lagoon. The distribution is initialised with 100 % of the phages having the starting fitness. First changes in the fitness distribution occur after the first <i>E. coli</i> start to release phages.
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| − | <br>
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| − | The fitness distribution is modeled to change by mutation and selection. The <b>selection</b> is implemented in the following equation:
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| − |
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| − | $$
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| − | s_{i}(t_{n + 1}) = f_{i}(t_{n}) \cdot s_{i}(t_{n})
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| − | $$
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| − |
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| − | \(f_{i}\) is one of \(N\) fitness values, \(s_{i}\) is the share of phages with that fitness value relative to the total phage population. Since the fitness is interpreted as a percentage of the wildtype fitness in this model, the new shares can be calculated als the product of the fitness value \(f_{i}\) and its old share \(s_{i}(t_{n})\). The resulting values are then normalized by divsion with
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| − |
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| − | $$
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| − | \sum_{i = 0}^{N} f_{i}(t_{n + 1})
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| − | $$
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| − |
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| − | in order to always have shares that sum up to 1.
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| − | <br>
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| − | The <b>mutation</b> is implemented as a function that manipulates the fitness shares and applied to the fitness shares after selection.
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| − | Here, \(r_{M}\) is the mutation rate, that means the amount of the fitness share \(i\) that changed during the mutation. \(w\) is the width of a fitness bin:
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| − |
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| − | $$
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| − | w = \frac{1}{N - 1}
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| − | $$
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| − |
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| − | The new shares are calculated as
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| − |
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| − | $$
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| − | s_{i}(t_{n + 1}) = \big(1 - r_{M}\big) \cdot s_{i}(t_{n}) + r_{M} \sum_{i = 0}^{N} \int_{f_{i} - \frac{1}{2} \cdot w}^{f_{i} + \frac{1}{2} \cdot w} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx
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| − | $$
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| − |
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| − | <b>Special cases:</b><br>
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| − | For \(f_{0}\) the integral is changed to
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| − |
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| − | $$
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| − | \int_{- \infty }^{f_{0} + \frac{1}{2} \cdot w} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx
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| − | $$
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| − |
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| − | For \(f_{0}\) the integral is changed to
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| − |
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| − | $$
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| − | \int_{f_{N} - \frac{1}{2} \cdot w}^{\infty} \mathcal{N}_{\mu=f_{i}, \: \sigma}(x) \: dx
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| − | $$
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| − |
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| − | to have the integrals over the normal distribution add up to 1.
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| − |
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| − | The normal distribution \(\mathcal{N}\) is used to model that many mutations have only small effects on the fitness, while some have a larger effect. \(\sigma\) is a parameter of the model that influences how much the new scores differ from the previous ones. In this model it is theoretically possible that mutation turns any given fitness into any other fitness, however small values for can \(\sigma\) prevent this.
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/0/02/T--Heidelberg--2017_phage_titer_fig3.png|{{#tag:html|Figure 3: Constant fitness assumed to be equal for all phages in the lagoon}}|{{#tag:html|The color indicates the share of phages that have a specific fitness (y-axis) at a given time (x-axis). With the constant non-distributional mode the complete phage population has the initial fitness for the whole time.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/7/7d/T--Heidelberg--2017_phage_titer_fig4.png|{{#tag:html|Figure 4: A distribution of fitness assumed with 10 % of new phages being subjected to changes in fitness.}}|{{#tag:html|The color indicates the share of phages that have a specific fitness (y-axis) at a given time (x-axis). With the distributional fitness mode after 10 minutes variants with a higher fitness occur.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | When the the concentrations are modeled over a longer period of time the fitness of the wildtype is reached. To more accurate model evolution in which the wildtype fitness is never reached, \(\mu_{max}\) can be set to a smaller value. This value will be reached by most of the phages, if enough time is given. Finally almost no phages with the lowest fitness can be found in the lagoon.
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/b/be/T--Heidelberg--2017_phage_titer_fig5.png|{{#tag:html|Figure 5: <i>E. coli</i> and phage titer modled over 6.5 h with a normal distribution based fitness distribution.}}|{{#tag:html|Blue lines correspond to <i>E. coli</i> titers, the red line is the total phage titer.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/5/55/T--Heidelberg--2017_phage_titer_fig6.png|{{#tag:html|Figure 6: A distribution of fitness is assumed with 10 % of new phages being subjected to changes in fitness over 6.5 h.}}|{{#tag:html|The color indicates the share of phages that have a specific fitness (y-axis) at a given time (x-axis). After 4 h the system goes in a steay state where mutation and selection cancel each other out.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/a/ac/T--Heidelberg--2017_phage_titer_fig7.png|{{#tag:html|Figure 7: Shares of the phage-producing <i>E. coli</i> producing phages with different fitness values.}}|{{#tag:html|Modelled with normal distribution based mutation. Initially all phage-producing <i>E. coli</i> produce phage with 20 % of the wildtype fitness, they make up the largest share for 1 h after which fitter phage-producing <i>E. coli</i> take over the lagoon.
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| − | <br>
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| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | Since most mutations decrease fitness in reality, a symmetric function like the normal distributions density function may not be the best choice. Consequently we also implemented a mutation function with a skew normal distribution \(\mathcal{S}\), that replaces \(\mathcal{N}\). It has an additional parameter \(\alpha\) that modulates the skewness.
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| − |
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| − | $$
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| − | s_{i}(t_{n + 1}) = \big(1 - r_{M}\big) \cdot s_{i}(t_{n}) + r_{M} \sum_{i = 0}^{N} \int_{f_{i} - \frac{1}{2} \cdot w}^{f_{i} + \frac{1}{2} \cdot w} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx
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| − | $$
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| − |
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| − |
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| − | <b>Special cases:</b><br>
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| − | For \(f_{0}\) the integral is changed to
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| − |
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| − | $$
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| − | \int_{- \infty }^{f_{0} + \frac{1}{2} \cdot w} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx
| |
| − | $$
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| − |
| |
| − | For \(f_{0}\) the integral is changed to
| |
| − |
| |
| − | $$
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| − | \int_{f_{N} - \frac{1}{2} \cdot w}^{\infty} \mathcal{S}_{\mu=f_{i}, \: \sigma, \: \alpha}(x) \: dx
| |
| − | $$
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| − |
| |
| − | to have the integrals over the normal distribution add up to 1.
| |
| − | <br>
| |
| − |
| |
| − | The equation for \(\frac{\partial c_{P} (t)}{\partial t}\) is changed into the following to incorporate the distributed fitness.
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| − |
| |
| − | $$
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| − | \frac{\partial c_{P} (t)}{\partial t} = -k \cdot c_{u}(t) \cdot c_{P} (t)
| |
| − | + \sum_{i = 0}^N f_{i} \cdot s_{i} \cdot \mu \cdot c_{p} (t)
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| − | $$
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| − |
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| − |
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/c/cc/T--Heidelberg--2017_phage_titer_fig8.png|{{#tag:html|Figure 8: Shares of the phage-producing <i>E. coli</i> producing phages with different fitness values.}}|{{#tag:html|Modeled with skew normal distribution based mutation biased towards lower fitness scores. Initially all phage-producing <i>E. coli</i> produce phage with 20 % of the wildtype fitness, they make up the largest share for 1 h after which fitter phage-producing <i>E. coli</i> take over the lagoon.
| |
| − | <br>
| |
| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | Other assumptions for the mutation function are possible, for example a function that randomly samples from a normal distribution but is biased towards mutations that decrease fitness.
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| − | <br>
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| − | Since this model of fitness and mutation is heavily abstracted it is most likely limeted to qualitative statements. Still it provides an intuitive understanding on how directed evolution can work and illustrates the role of selection and mutation. If the model without mutation (Fig. 3) is compared to one with mutation and selection (Fig. 6) role of mutation, diversificating the pool of phages can be seen. When comparing a model without selection (Fig. 9) to the one with mutation and selection (Fig. 6) the role of selection can be described as channeling the variants caused by mutation into a specific direction.
| |
| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | {{Heidelberg/templateus/Imagebox|https://static.igem.org/mediawiki/2017/c/c6/T--Heidelberg--2017_phage_titer_fig9.png|{{#tag:html|Figure 9: Effect of missing selection}}|{{#tag:html|Without selection fitness does not show a clear trend until the boundary values show higher shares. This is most likely due to the equation that gives the boundary values fitness that leaves the interval [0, 1]. The color indicates the share of phages that have a specific fitness (y-axis) at a given time (x-axis).
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| − | <br>
| |
| − | <a href="#parameters">See the full list of parameters.</a>
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| − | }}
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| − | }}
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| − | }}
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| − | }}
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| − |
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | <h2>Modeling concentrations over multiple Lagoons</h2>
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| − | When transfer from one volume to the next is performed, new lagoon can be modeled with starting values calculated from the last lagoons end values.
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| − | For each concentration from the previous lagoon \(c_{t}\), the concentration in the next lagoon \(c_{t+1}\) is calculated as
| |
| − | $$
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| − | c_{t+1} = \frac{v_{t} }{v_{l} } \cdot c_{t}
| |
| − | $$
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| − | with \(v_{l}\), the volume of a lagoon and \(v_{t}\), the volume that is transferred.
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| − | If the transfered volume is spinned down before it is added to the new lagoon, the initial value for \(c_{P}\) is calculated this way. The initial concentration of uninfected <i>E. coli</i> is set to the initial cell density. Initial concentrations of infected and phage-producing <i>E. coli</i> are set to zero, because before the transfer, no phages are present in the new lagoon.
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| − | If the transfer volume is not spinned down, the concentration of infected and phage-producing <i>E. coli</i> are calculated, using the above formula. The initial concentration of uninfected <i>E. coli</i> is the calculated the same way, but the initial cell density is added.
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| − |
| |
| − | In directed evolution the fitness should increase over time. A linear increase in fitness between to given values was implemented to show this. The problem with this approach is its basic assumption being that all phage-producing <i>E. coli</i> are infected by phages with the same fitness.
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| − | To make the model more plausible, a distribution of fitness was introduced. For a set of discrete fitness values each fitness values share of the phage-producing <i>E. coli</i> population is calculated.
| |
| − | That changes the equation for the change in the concentration of phage-producing <i>E. coli</i> to
| |
| − |
| |
| − | The calculation is for \(N\) different fitness values \(f_{i}\) and their share of the total phage-producing <i>E. coli</i> population \(s_{i}\).
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| − |
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | <h2>Numeric solutions</h2>
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| − |
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| − | The problem described above is a system of four differential equations, of which two ( \(\frac{\partial c_{i} }{\partial t} \:, \: \frac{\partial c_{p} }{\partial t}\) ) are so called delayed differential equations. They contain a term that needs to be evaluated at a timepoint in the past \(t - t_{P}\). A custom script was used to solve the problem numerically, using the explicit Euler method.
| |
| − | The basic idea is that from a point in time with all values and all derivatives values given, the next point in time can be calculated by assuming a linear progress between the two points.
| |
| − | $$
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| − | f(t_{n+1}) = f(t_{n}) + (t_{n+1} - t_{n}) \cdot f'(t_{n})
| |
| − | $$
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| − | This is performed for \(c_{u}(t)\), \(c_{i}(t)\), \(c_{p}(t)\) and \(c_{P}(t)\) rotatory, to always have the needed values from \(t_{n}\) ready for \(t_{n+1}\).
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| − |
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| − | To explore, how unprecise parameters and noise influence the outcome of the model, a mode was implemented, that adds gaussian noise to all parameters. It uses the function \(n\) that makes a value \(v\) noisy with a random parameter \(r\).
| |
| − | $$
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| − | n(v) = \big(1 - 2r\big) \cdot \sigma_{G} \cdot \sigma_{v} \cdot v, \quad r \in (0, 1)
| |
| − | $$
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| − | Here, \(\sigma_{G}\) is a factor that is the same for all \(v\), \(\sigma_{v}\) is specific for \(v\). This way, it is possible to have one parameter being noisier than another, while being able to tune the noise globally.
| |
| − | [Results]
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| − | }}
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| − | {{Heidelberg/templateus/Tablebox|
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| − | Table 2: Additional Variables and Parameters used in the numeric solution of the model |
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| − | {{#tag:html|
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| − | <table class="table table-bordered mdl-shadow--4dp" XSSCleaned="overflow-x: scroll !important">
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| − | <thead>
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| − | <tr>
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| − | <th>Symbol</th>
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| − | <th>Name in Source code</th>
| |
| − | <th>Value and Unit</th>
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| − | <th>Explanation</th>
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| − |
| |
| − | </tr>
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| − | </thead>
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| − | <tbody>
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| − | <tr>
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| − | <td>\(v_{l}\)</td>
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| − | <td><code class="python">vl</code></td>
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| − | <td>[ml]</td>
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| − | <td>Volume of lagoon</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(t_{l} \)</td>
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| − | <td><code class="python">tl</code></td>
| |
| − | <td>[min]</td>
| |
| − | <td>Duration until transfer to the next lagoon</td>
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| − | </tr>
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| − | <tr>
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| − | <td>\(c_{u}(t_{0})\)</td>
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| − | <td><code class="python">ceu0</code></td>
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| − | <td>[cfu]</td>
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| − | <td>Concentration of <i>E. coli</i> in a lagoon when M13 phages are transfered to it</td>
| |
| − | </tr>
| |
| − | <tr>
| |
| − | <td>\(c_{P}(t_{0})\)</td>
| |
| − | <td><code class="python">cp0</code></td>
| |
| − | <td>[pfu]</td>
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| − | <td>Initial concentration of M13 phage in the first lagoon</td>
| |
| − | </tr>
| |
| − | <tr>
| |
| − | <td>\(n\)</td>
| |
| − | <td><code class="python">epochs</code></td>
| |
| − | <td>-</td>
| |
| − | <td>Number of epochs that are modeled, one epoch being everything that happens in one particular lagoon</td>
| |
| − | </tr>
| |
| − | <tr>
| |
| − | <td>\(s\)</td>
| |
| − | <td><code class="python">tsteps</code></td>
| |
| − | <td>-</td>
| |
| − | <td>Number of time steps for which numeric solutions are calculated, counted per epoch</td>
| |
| − | </tr>
| |
| − | <tr>
| |
| − | <td>\(c_{P}^{min}\)</td>
| |
| − | <td><code class="python">min_cp</code></td>
| |
| − | <td>[pfu]</td>
| |
| − | <td>Lower threshold for valid phage titers</td>
| |
| − | </tr>
| |
| − | <tr>
| |
| − | <td>\(c_{P}^{max}\)</td>
| |
| − | <td><code class="python">max_cp</code></td>
| |
| − | <td>[pfu]</td>
| |
| − | <td>Upper threshold for valid phage titers</td>
| |
| − | </tr>
| |
| − | </tbody>
| |
| − | </table>
| |
| − | }}|
| |
| − | List of all additional paramters and variables used in the numeric solution of this model. When possible values are given.
| |
| − | }}
| |
| − | {{Heidelberg/templateus/Imagebox|
| |
| − | https://static.igem.org/mediawiki/2017/8/88/T--Heidelberg--2017_modeling-graphical-abstract.svg|
| |
| − | Fig: 1a Numeric solution calculated with explicit Euler approach|
| |
| − | {{#tag:html|
| |
| − | Logarithmic plot of the concentrations of all <i>E. coli</i> populations cE, uninfected <i>E. coli</i> ceu, infected <i>E. coli</i> cei, phage-producing <i>E. coli</i> cep and M13 phage cP
| |
| − | }}|
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| − | pos = left
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|
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| − | https://static.igem.org/mediawiki/2017/8/88/T--Heidelberg--2017_modeling-graphical-abstract.svg|
| |
| − | Fig: 1b Numeric solution calculated with explicit Euler approach|
| |
| − | {{#tag:html|
| |
| − | Non-logarithmic plot of the derivatives of concentrations of all <i>E. coli</i> populations cE, uninfected <i>E. coli</i> ceu, infected <i>E. coli</i> cei, phage-producing <i>E. coli</i> cep and M13 phage cP
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| − | }}|
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| − | pos = left
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| − | }}
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| − | {{Heidelberg/templateus/Imagebox|
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| − | https://static.igem.org/mediawiki/2017/8/88/T--Heidelberg--2017_modeling-graphical-abstract.svg|
| |
| − | Fig: 1c First derivative of concentrations calculated with explicit Euler approach|
| |
| − | {{#tag:html|
| |
| − | Logarithmic plot of the concentrations of all <i>E. coli</i> populations cE, uninfected <i>E. coli</i> ceu, infected <i>E. coli</i> cei, phage-producing <i>E. coli</i> cep and M13 phage cP
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| − | }}|
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| − | pos = left
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| − | }}
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| − | {{Heidelberg/templateus/Imagesection|
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| − | https://static.igem.org/mediawiki/2017/8/88/T--Heidelberg--2017_modeling-graphical-abstract.svg|
| |
| − | Fig: 2 Numeric solution for a range of values for \(t_{l}\) and for \(v_{t}\)|
| |
| − | {{#tag:html|
| |
| − | All combinations of setups for the two ranges were calculated. The number of epochs plotted is counted until either the phage titer is less than a minimal threshold (orange) or larger than a maximum threshold (blue)
| |
| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/templateus/Contentsection|
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| − | {{#tag:html|
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| − | <h1 id="parameters">Parameters used for the figures</h1>
| |
| − | {{Heidelberg/accpanel|#005493|
| |
| − | {{Heidelberg/accord|Figure 1, 3 - 20 % fitness|{{#tag:html|<code>
| |
| − | 'capacity': 1000000000.0,
| |
| − | 'ceu0': 100000000.0,
| |
| − | 'cp0': 100000000.0,
| |
| − | 'epochs': 1,
| |
| − | 'f0': 0.2,
| |
| − | 'f_prec': 21,
| |
| − | 'fend': 1.0,
| |
| − | 'fitnessmode': 'dist',
| |
| − | 'ftype': 'const',
| |
| − | 'growth_mode': 'exp',
| |
| − | 'k': 3e-11,
| |
| − | 'max_cp': 2000000000.0,
| |
| − | 'min_cp': 100000.0,
| |
| − | 'mumax': 16.667,
| |
| − | 'mutation_dist': 'norm',
| |
| − | 'noisy': 0.0,
| |
| − | 'phageonly': 'True',
| |
| − | 'plot_dist': 'True',
| |
| − | 'sigma': 0.0,
| |
| − | 'skewness': 1.0,
| |
| − | 'ti': 30,
| |
| − | 'tl': 60,
| |
| − | 'to_mutate': 0.0,
| |
| − | 'tp': 40,
| |
| − | 'tpp': 10,
| |
| − | 'tsteps': 100,
| |
| − | 'tu': 20,
| |
| − | 'vl': 20,
| |
| − | 'vt': 1
| |
| − | </code>}}
| |
| − | |fig1}}
| |
| − | {{Heidelberg/accord|Figure 2 - 100 % fitness|{{#tag:html|<code>
| |
| − | 'capacity': 1000000000.0,
| |
| − | 'ceu0': 100000000.0,
| |
| − | 'cp0': 100000000.0,
| |
| − | 'epochs': 1,
| |
| − | 'f0': 1.0,
| |
| − | 'f_prec': 21,
| |
| − | 'fend': 1.0,
| |
| − | 'fitnessmode': 'const',
| |
| − | 'ftype': 'const',
| |
| − | 'growth_mode': 'exp',
| |
| − | 'k': 3e-11,
| |
| − | 'max_cp': 2000000000.0,
| |
| − | 'min_cp': 100000.0,
| |
| − | 'mumax': 16.667,
| |
| − | 'mutation_dist': 'norm',
| |
| − | 'noisy': 0.0,
| |
| − | 'phageonly': 'True',
| |
| − | 'plot_dist': 'False',
| |
| − | 'sigma': 0.001,
| |
| − | 'skewness': 1.0,
| |
| − | 'ti': 30,
| |
| − | 'tl': 60,
| |
| − | 'to_mutate': 0.0,
| |
| − | 'tp': 40,
| |
| − | 'tpp': 10,
| |
| − | 'tsteps': 100,
| |
| − | 'tu': 20,
| |
| − | 'vl': 20,
| |
| − | 'vt': 1
| |
| − | </code>}}
| |
| − | |fig2}}
| |
| − | {{Heidelberg/accord|Figure 4 - Distributional fitness 1 h|{{#tag:html|<code>
| |
| − | 'capacity': 1000000000.0,
| |
| − | 'ceu0': 100000000.0,
| |
| − | 'cp0': 100000000.0,
| |
| − | 'epochs': 1,
| |
| − | 'f0': 0.2,
| |
| − | 'f_prec': 21,
| |
| − | 'fend': 1.0,
| |
| − | 'fitnessmode': 'dist',
| |
| − | 'ftype': 'const',
| |
| − | 'growth_mode': 'exp',
| |
| − | 'k': 3e-11,
| |
| − | 'max_cp': 2000000000.0,
| |
| − | 'min_cp': 100000.0,
| |
| − | 'mumax': 16.667,
| |
| − | 'mutation_dist': 'norm',
| |
| − | 'noisy': 0.0,
| |
| − | 'phageonly': 'True',
| |
| − | 'plot_dist': 'True',
| |
| − | 'sigma': 0.0,
| |
| − | 'skewness': 1.0,
| |
| − | 'ti': 30,
| |
| − | 'tl': 60,
| |
| − | 'to_mutate': 0.0,
| |
| − | 'tp': 40,
| |
| − | 'tpp': 10,
| |
| − | 'tsteps': 100,
| |
| − | 'tu': 20,
| |
| − | 'vl': 20,
| |
| − | 'vt': 1
| |
| − | </code>}}
| |
| − | |fig4}}
| |
| − | {{Heidelberg/accord|Figure 5, 6, 7 - Distributional fitness 6.5 h|{{#tag:html|<code>
| |
| − | 'capacity': 1000000000.0,
| |
| − | 'ceu0': 100000000.0,
| |
| − | 'cp0': 100000000.0,
| |
| − | 'epochs': 1,
| |
| − | 'f0': 0.2,
| |
| − | 'f_prec': 21,
| |
| − | 'fend': 1.0,
| |
| − | 'fitnessmode': 'dist',
| |
| − | 'ftype': 'const',
| |
| − | 'growth_mode': 'logistic',
| |
| − | 'k': 3e-11,
| |
| − | 'max_cp': 2000000000.0,
| |
| − | 'min_cp': 100000.0,
| |
| − | 'mumax': 16.667,
| |
| − | 'mutation_dist': 'norm',
| |
| − | 'noisy': 0.0,
| |
| − | 'phageonly': 'True',
| |
| − | 'plot_dist': 'True',
| |
| − | 'sigma': 0.1,
| |
| − | 'skewness': 1.0,
| |
| − | 'ti': 30,
| |
| − | 'tl': 400,
| |
| − | 'to_mutate': 0.1,
| |
| − | 'tp': 40,
| |
| − | 'tpp': 10,
| |
| − | 'tsteps': 2000,
| |
| − | 'tu': 20,
| |
| − | 'vl': 20,
| |
| − | 'vt': 1
| |
| − | </code>}}
| |
| − | |fig5_6_7}}
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| − | }}
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| − | }}
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| − | }}
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| − | }}
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| − | {{Heidelberg/references2
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| − | }}
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| − | {{Heidelberg/footer
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| − | }}
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