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| | <!--Content below images--> | | <!--Content below images--> |
| | <div id="MathBook" class="content"> | | <div id="MathBook" class="content"> |
| | + | <div class="tabcontents"> |
| | + | <p>Our <b>Math Book</b> is meant to be the mathematical modeling equivalent of a lab book, where we store everything another team might need to recreate our models. You can access code related to the models can be accessed from <a href="https://github.com/alexanian/uwaterloo-igem-2014">this GitHub page</a>.</p> |
| | + | <h3>We hope you enjoy learning more about our model on the subpages!</h3> |
| | + | <p>We created mathematical models of the three main aspects of the Staphylocide system: <a href="https://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi">CRISPR Interference</a>, <a href="https://2014.igem.org/Team:Waterloo/Math_Book/sRNA">RNA Interference</a> and <a href="https://2014.igem.org/Team:Waterloo/Math_Book/Conjugation">Conjugation</a> and have created sub-pages for each of them that detail all the steps in our analysis.</p> |
| | + | <p> </p> |
| | <ul class="tabs"> | | <ul class="tabs"> |
| - | <li><a href="#view0">CRISPR</a></li> | + | <li><a href="https://2014.igem.org/Team:Waterloo/Math_Book/CRISPRi">CRISPRi</a></li> |
| - | <li><a href="#view1">sRNA</a></li> | + | <li><a href="https://2014.igem.org/Team:Waterloo/Math_Book/sRNA">RNAi</a></li> |
| - | <li><a href="#view2">Conjugation</a></li> | + | <li><a href="https://2014.igem.org/Team:Waterloo/Math_Book/Conjugation">Conjugation</a></li> |
| | + | <li><a href="https://2014.igem.org/Team:Waterloo/Math_Book/References">Bibliography</a></li> |
| | </ul> | | </ul> |
| - | <div class="tabcontents">
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| - | This page gathers the detailed process information for the mathematical models created by the team this year. Code related to the models can be accessed from <a href="https://github.com/alexanian/uwaterloo-igem-2014">this github page</a>.
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| - |
| |
| - | <!------------------- CRISPR SECTION --------------------------------->
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| - | <div class="anchor" id="view0">
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| - | <h2>CRISPR</h2>
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| - | <p>We decided to create a model of the CRISPR system for two main reasons:</p>
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| - | <ul>
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| - | <li>Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency</li>
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| - | <li>To approximate time-series <em>mecA</em> repression data for use in modelling the overall vulnerability of a <em>S. aureus</em> population</li>
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| - | </ul>
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| - | <p>The steps we followed are detailed below, but were:</p>
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| - | <ol>
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| - | <li><a href="#C_ModelFormation">Model Formation</a></li>
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| - | <li><a href="#C_ParameterFinding">Parameter Finding</a></li>
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| - | <li><a href="#C_InitialModelling">Model Formation</a></li>
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| - | <li><a href="#C_ParameterAdjustment">Parameter Adjustment</a></li>
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| - | <li><a href="#C_Sensitivity">Sensitivity Analysis</a></li>
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| - | </ol>
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| - | <h3 id="C_ModelFormation"> Model Formation </h3>
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| - | <p>After a literature review we were able to construct the CRISPR interference system network. The targeted single guide RNA (sgRNA) associates with nuclease-deficient Cas9 protein (dCas9) to form a complex that binds with the DNA complementary to the sgRNA target <cite ref="Qi2013"></cite>. The bound complex prevents transcription elongation by RNA polymerase, repressing YFP mRNA expression <cite ref="Bikard2013"></cite>. The chemical network is shown below:</p>
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| - |
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| - | <img src="https://static.igem.org/mediawiki/2014/c/c9/CRISPRNetwork.png" alt="CRISPR Network Diagram" />
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| - |
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| - | <p>Using standard mass-action kinetics, the network simplifies into the following set of differential equations:</p>
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| - |
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| - | <img class="equation" src="https://static.igem.org/mediawiki/2014/b/b7/Waterloo_CRISPR_DEs1.png" alt="Initial CRISPR DEs" />
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| - |
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| - | <p>We chose the model kinetics to be largely first-order; this decision was supported by the findings of several recent studies <cite ref="Sternberg2014"></cite><cite ref="Qi2013"></cite>. To simplify the model, we assumed that the formation of the dCas9-sgRNA complex ($b$ in Figure xyz) is in made a quasi-steady-state. That is, we assume that the association/dissociation of dCas9 and sgRNA occurs on a faster timescale than the other reactions in the network (i.e. transcription, translation and the binding of the complex to the DNA), allowing us to assume that the complex is always at steady-state, relative to the other time-dependent species concentrations. This same assumption was made in previous modelling efforts, e.g. <cite ref="UCSF2013"></cite>.</p>
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| - | <p>Under this quasi-steady state assumption, the differential expression for the complex is given by:</p>
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| - |
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| - | <img class="equation" src="https://static.igem.org/mediawiki/2014/5/5a/Waterloo_CRISPR_QSSA.png" alt="QSSA Assumption changes complex DE" />
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| - |
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| - | <p>Our model then simplifies to:</p>
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| - |
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| - | <img class="equation" src="https://static.igem.org/mediawiki/2014/b/b7/Waterloo_CRISPR_DEs2.png" alt="CRISPR DEs Updated with QSSA" />
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| - | <h3>QUICKLY EXPLAIN CONVENTIONS</h3>
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| - | <p>This is the same assumption made by previous teams.</p>
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| - |
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| - | <h4> Modelling Incomplete Repression </h4>
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| - |
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| - | <p>A recent study by Bikard et al. <cite ref="Bikard2013"></cite>found that maximal repression (on the order of 100 fold) was achieved when the promoter was targeted. However, targeting the promoter is not viable in this project since an essential promoter from elsewhere in the genome has been harnessed to produce the fluorescent promoter. Instead, we model the incomplete repression (ranging from 6-fold to 35-fold) observed when the off-promoter regions, specifically on the non-coding strand, are targeted.</p>
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| - |
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| - | <p>There are two possible approaches for modelling the incomplete repression, each reflecting a different physical mechanism that allows leaky YFP expression. In the first mechanism, RNA polymerase is sometimes able to cleave the bound dCas9-sgRNA complex from the DNA. In the second mechanism, the complex binds inefficiently and is sometimes separated from the DNA, permitting transcription to continue.</p>
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| - |
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| - | <p>We assumed that the incomplete repression is accounted for by the first mechanism. This assumption was based on several studies <cite ref="Qi2013"></cite><cite ref="Bikard2013"></cite> showing radically different repression rates if the complex targets the promoter, preventing transcription initiation, rather than targeting the DNA further downstream and impeding transcription elongation. The differences in the system behavior depending on whether or not RNA polymerase has the opportunity to bind suggest that the “cleavage” mechanism may more closely resemble the chemical reality.</p>
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| - | <p>Consequently, we modeled incomplete repression using a leaky expression term proportional to the expected YFP expression when the complex is saturated. The differential equation model was updated with a repression term dependent on the fold reduction <code>FR</code> and the initial concentration of YFP mRNA, <code>Y<sub>0</sub></code>:</p>
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| - |
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| - | <img class="equation" src="https://static.igem.org/mediawiki/2014/c/cf/Waterloo_CRISPR_mRNADE.png" alt="Updated YFP DE with Incomplete Repression" />
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| - |
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| - | <p>This equation was derived using two boundary conditions. Before repression, when the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the sarA promoter, <code>α</code>. After repression has reached its steady state, the YFP mRNA production has been reduced by <code>FR</code> fold, to <code>Y<sub>0</sub>/FR</code>.</p>
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| - |
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| - |
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| - | <h3 id="C_ParameterFinding"> Parameter Finding </h3>
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| - | <p>We turned to the literature to find parameters for our model, given in the Table below. We first looked for parameter values that had been measured in S. aureus. In cases where those could not be found, we next looked for ways to to estimate the parameters using other available data for S. aureus and finally searched for the parameters in other gram-positive bacteria. Aggregating parameters from many experiments across the literature is by nature a somewhat uncertain endeavor; those parameters about which we are very uncertain are marked with asterisks. An explanation for how we arrived at each parameter is given in the table, but details on the more circuitously estimated parameters are given after the table.</p>
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| - |
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| - | <table id="CRISPRParams" class="blueBorders">
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| - | <thead>
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| - | <tr>
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| - | <td>Parameter</td>
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| - | <td>Value</td>
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| - | <td>Description</td>
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| - | <td>Source/Rationale</td>
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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>α<sub>m<sub>y</sub></sub>, α<sub>r</sub></td>
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| - | <td>0.0011 nM • min<sup>-1</sup></td>
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| - | <td>mRNA production from SarA P1 Promoter</td>
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| - | <td>Determined based on linear fitting to the time-series fluorescence measurements from YFP/P2-P3-P1 fusion, as reported in <cite ref="Cheung2008"></cite> and fluorescence per molecule from <cite ref="Wu2005"></cite></td>
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| - | </tr>
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| - | <tr>
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| - | <td>α<sub>m<sub>c</sub></sub></td>
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| - | <td>0.0011 nM • min<sup>-1</sup></td>
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| - | <td>mRNA production from Xylose Promoter</td>
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| - | <td>Same as SarA rate since the addition of the Xylose-inducible promoter was to simplify labwork and thus for modelling we assume it is fully induced.</td>
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| - | </tr>
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| - | <tr>
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| - | <td>β<sub>c</sub></td>
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| - | <td>0.0057-0.4797 protein • transcript<sup>-1</sup> min<sup>-1</sup></td>
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| - | <td>dCas9 protein synthesis rate from dCas9 mRNA</td>
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| - | <td> Estimated from peptide elongation rates in <em>Streptomyces coelicolor</em> <cite ref="Milo2010"></cite>, the dCas9 BioBrick from <cite ref="Freiburg2013"></cite> and ribosome density from <cite ref="Brandt2009, Siwiak2013} </td>
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| - | </tr>
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| - | <tr>
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| - | <td>γ<sub>m<sub>y</sub></sub>, γ<sub>m<sub>c</sub></sub>, γ<sub>r</sub></td>
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| - | <td> 0.1734 min<sup>-1</sup></td>
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| - | <td>mRNA degradation rates</td>
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| - | <td><cite ref="Roberts2006"></cite> report log-phase mRNA half-lives in <em>{S. aureus}. An approximate average value of 4 minutes leads to this degradation rate.</td>
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| - | </tr>
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| - | <tr>
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| - | <td>γ<sub>c</sub>, γ<sub>b</sub></td>
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| - | <td>-5.6408e<sup>-04</sup> min<sup>-01</sup></td>
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| - | <td>dCas9/complex degradation rate</td>
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| - | <td>Based off half-life of SarA protein in <em>S. aureus</em> as reported in <cite ref="Michalik2012"></cite> </td>
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| - | </tr>
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| - | <tr>
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| - |
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| - | <td>K<sub>a</sub></td>
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| - | <td>0.28 nM</td>
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| - | <td>Dissociation constant for complex and DNA (given by k<sub>2</sub>/k<sub>1</sub>)</td>
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| - | <td><cite ref="Sternberg2014"></cite> found this dissociation rate for dCas9 and a single-stranded DNA substrate. </td>
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| - | </tr>
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| - | <tr>
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| - | <td>n</td>
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| - | <td>2.5</td>
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| - | <td>Hill Constant for Repression</td>
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| - | <td>UCSF iGEM 2013 <cite ref="UCSF2013"></cite> </td>
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| - | </tr>
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| - | <tr>
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| - | <td>k<sub>+</sub>, k<sub>-</sub></td>
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| - | <td>0.01 to 1.0 nM</td>
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| - | <td>Rate of dissociation of dCas9-sgRNA to form complex</td>
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| - | <td>Range defined relative to other parameters, using the QSSA assumption that these dynamics are fast</td>
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| - | </tr>
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| - | <tr>
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| - | <td>Fold Reduction</td>
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| - | <td>6 to 35</td>
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| - | <td>Maximum percent repression achievable with CRISPRi system</td>
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| - | <td>Based on the relative fluorescence measurements observed when the non-coding strand was targeted by dCas9 in <cite ref="Bikard2013"></cite></td>
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| - | </tr>
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| - | </tbody>
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| - | </table>
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| - |
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| - | <p>The only model parameters without some basis in the literature are the association rates for dCas9 and sgRNA. However, we have made a quasi-steady state assumption for that reaction, which requires that it reach equilibrium on a much faster time scale than the rest of the system. We thus defined a range for the possible values based on the other model parameters</p>
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| - | <p>Details on the more roundabout estimations are given below:</p>
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| - |
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| - | <h4> Production of dCas9 from dCas9 mRNA </h4>
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| - | <p>We were unable to find a peptide chain elongation rate for <em>S. aureus</em>, so instead we used the values reported in BioNumber 107869 <cite ref="Milo2010"></cite> which gives a range of 0.59-3.17 amino acids per second per ribosome in <em>Streptomyces coelicolor</em>, another gram-positive bacteria. Freiburg's dCas9 part from last year <cite ref="Freiburg2013" ></cite> is composed of 1372 amino acids. This translates to a range of 0.0258 to 0.1386 dCas9 molecules per minute per ribosome.</p>
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| - | <p>We were unable to find ribosome densities in <em>S. aureus</em>, but found two different estimates for ribsosome density in Bionumbers: 0.22 ribosomes per 100 codons (i.e. per 3 nt coding sequence) <cite ref="Brandt2009"></cite> and 3.46 ribosomes per 100 codons <cite ref="Siwiak2013"></cite>. Using our assumption of 3 nt:1 amino acid, we then multiply to get the 0.0057-0.4797 range of dCas9 molecules per minute.</p>
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| - |
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| - | <h4> Degradation rate of dCas9 </h4>
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| - | <p>We were unable to find any specific data on dCas9 degradation, so instead we used a protein half-life of <em>sarA</em> measured in <em>S. Aureus</em> by Michelik et al. <cite ref="Michalik2012"></cite>. We chose <em>sarA</em> rather than a protein more chemically similar to dCas9 because data on <em>sarA</em> was readily available and because dCas9 is transcribed using the <em>sarA</em> promoter, which allows us to at least capture sensitivity of the degradation rate to production.</p>
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| - |
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| - | <h4> mRNA production from the <em>sarA</em> promoter </h4>
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| - |
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| - | <p>We used the time-series data given by Cheung et al. <cite ref="Cheung2008"></cite> to estimate the rate of production from the <em>sarA</em> P2-P3-P1 promoter in <em>S. aureus</em>. The figure from their paper is reproduced below. After diluting 1:100, the <em>S. aureus</em> strains were serially monitored for <code>OD_650</code>. We used data from the <em>sarA</em>+ strain, as that's more like a wild-type S. aureus strain.</p>
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| - |
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| - | <img class="floatLeft half-column" src="https://static.igem.org/mediawiki/2014/8/89/Waterloo_CRSPR_ExpFit2Cheung.png" alt="Exponential fit to SarA Promotor production rates from A. L. Cheung et al."/>
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| - | <p>Using the laboratory-conditions doubling time of 24 minutes given in given in <cite ref="Domingue1996"></cite>, we found that the bacteria would re-enter stationary phase after 2.5 hours; for time-points after 3 hours, the number of number of <em>sarA</em> genes producing fluorescence could be assumed as constant. For this reason, we excluded time-points prior to 3 hours. We then converted from fluorescence units to number of fluorescent molecules using the quantization measurements provided by Wu & Pollard <cite ref="Wu2005"></cite> and, using our assumption of a fixed number of active <em>sarA</em> genes, considered the relative change in number of molecules to be representative of the per-promoter rate.</p>
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| - | <p>We were interested, however, in the changes of concentration rather than the changes in the raw number of molecules. As the name suggests, <em>Staphylococcus aureus</em> are spherical in shape. Assuming that all <em>S. aureus</em> are spheres, the volume of the cell can be determined. The diameter of a USA300 <em>S. aureus</em> cell was previously measured as 1.1 μ•m <cite ref="Cheng2014"></cite> resulting in the overall cell volume to be calculated as 5.575•10<sup>-15</sup> L. The number of molecules were thus converted to units of molar concentration in the cell, specifically nanomoles per litre (nM). The exponential fit used to find the rate constant is shown beside the figure from Cheung et al. above.</p>
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| - | <p>This resulted in a exponential model <code>a•e<sup>bt</sup></code> with a <code>b</code> rate constant of 0.0011 nM/min.</p>
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| - |
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| - | <h3 id="C_InitialModelling"> Initial Model Results </h3>
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| - | <p>Using these estimated parameters, we simulated our differential model of the CRISPR system. It was immediately clear that these parameters, scraped from assorted publications, did not provide an accurate system model once combined. Below are two simulations of the system dynamics using the parameters found in the literature. On the left the system is shown with CRISPR repression active and on the right is a simulation where:</p>
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| - |
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| - | <img src="https://static.igem.org/mediawiki/2014/6/69/CRISPR_mRNAYNoInteraction.png" alt="mRNA Y production without CRISPR interference" />
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| - |
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| - | <p>or rather, the system simulated without any repression of YFP Transcription by CRISPR. Both plots use 6 as the expected final fold reduction.</p>
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| - |
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/e/e4/Waterloo_OriginalCRISPRDynamics.png" alt="CRISPR Model with parameters from the literature"/>
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/b/bc/Waterloo_OriginalCRISPRDynamics_noRepression.png" alt="CRISPR Model with parameters from the literature and no CRISPR interference" />
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| - |
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| - | <p>The difference between the two plots is negligible. This unrealistic behavior emerges directly from the parameter values. The mRNA degradation dynamics prevent sgRNA from reaching any significant concentration, so there is never enough dCas9-sgRNA complex to influence YFP expression. However, in both plots the YFP mRNA levels plummet from the high initial concentration because the degradation rate is so much higher than the production rate.</p>
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| - | <p>These plots do not fit with the observations of CRISPR interference systems reported in the literature <cite ref="Qi2013"></cite><cite ref="Sternberg2014}"></cite>. To ensure that our parameters were at fault, rather than the fact that we were examining CRISPR in <em>S. aureus</em> instead of <em>E. coli</em>, we consulted fluorescence results from the laboratory, which showed that unrepressed YFP continued to fluoresce after several hours.</p>
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| - |
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| - |
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| - | <h3 id="C_ParameterAdjustment"> Parameter Adjustment </h3>
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| - | <img class="floatRight half-column" src="https://static.igem.org/mediawiki/2014/7/7f/Waterloo_CRISPR_fmincon_fit.png" alt="Fit results acheived with fmincon optimization" />
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| - |
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| - | <p>Confident that the initial model was inaccurate, we had to update some of the model parameters. We were least sure of our estimates of the mRNA production rates from the literature and manual fiddling showed that CRISPR interference in the model with adjustments to the production rates only. Accordingly, we used <code>MATLAB fmincon</code> to find updated mRNA production rates and fixed all other parameters.</p>
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| - | <p>In the <code>MATLAB fmincon</code> parameter search, we generated an error function by comparing the sum of squares error of the model YFP dynamics to time-series data derived Qi et al. <cite ref="Qi2013"></cite>, who measured repression with CRISPRi and found a short delay followed by exponential decay with a 35-minute half life. We assumed that the delay and rate of decay would be similar in our system, but fixed the final level of repression according to the two extremes reported by Bikard et al. (6-Fold and 35-Fold) <cite ref="Bikard2013"></cite>. The simulated time-series data and the model output after the parameter search are compared below.</p>
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| - |
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| - | <p>Satisfied with these fits, we simulated the system again using the fitted values for the mRNA production rates. The model output using the novel parameter sets are shown below for both the 6-fold and 35-fold simulations. The results with and without CRISPR interference are contrasted as before, though now a difference is observed when CRISPR inteacts with YFP</p>
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| - |
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| - | <p style="text-align: center"><b>Simulated System Dynamics with 6-Fold Repression</b></p>
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/1/1d/Waterloo_6FoldCRISPRDynamics.png" alt="Final dynamics of CRISPRi system with fit to 6-fold repression profile"/>
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/c/c4/6FoldCRISPRDynamics_noRepression.png" alt="System dynamics with fit to 6-fold repression profile and no CRISPR interference" />
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| - |
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| - | <p style="text-align: center"><b>Simulated System Dynamics with 35-Fold Repression</b></p>
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/e/e9/35FoldCRISPRDynamics.png" alt="Final dynamics of CRISPRi system with fit to 35-fold repression profile"/>
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| - | <img class="half-column floatLeft" src="https://static.igem.org/mediawiki/2014/c/c2/35FoldCRISPRDynamics_noRepression.png" alt="System dynamics with fit to 35-fold repression profile and no CRISPR interference" />
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| - |
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| - |
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| - |
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| - | <h3 id="C_Sensitivity"> Sensitivity Analysis </h3>
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| - | </div>
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| - |
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| - | <!------------------- sRNA SECTION --------------------------------->
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| - | <div class="anchor" id="view1">
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| - | <h2>sRNA</h2>
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| - | <h3> Relevant Biology </h3>
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| - | <h3> Model Formation </h3>
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| - | <h3></h3>
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| - |
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| - | </div>
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| | | | |
| - | <!------------------- CONJUGATION SECTION --------------------------------->
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| - | <div class="anchor" id="view2">
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| - | <h2>Conjugation</h2>
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| - | </div>
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