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| | <!--Content below images--> | | <!--Content below images--> |
| - | <div 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 list-style-type=" none"><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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| - |
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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 style="list-style-type: circle;">Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency</li>
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| - | <li style="list-style-type: circle;">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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| - | <h3> 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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| - | <h3>MISSINGEQUATION</h3>
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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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| - | <h3>MISSINGEQUATION</h3>
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| - |
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| - | <p>Our model then simplifies to:</p>
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| - |
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| - | <h3>MISSINGEQUATION</h3>
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| - |
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| - | This is the same assumption made by previous teams.
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| - |
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| - | <h4> Modelling Incomplete Repression </h4>
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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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| - | <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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| - | <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 behaviour 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 modelled 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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| - | <h3>MISSINGEQUATION</h3>
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| - | 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>.
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| - |
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| - | <h3> Parameters </h3>
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| - | <table id="CRISPRParams">
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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>, &alpha<sub>r}</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 $\cdot$ 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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| - | <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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| - | <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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| - | <td>&gamma<sub>c</sub>, &gamma<sub>b</sub></td>
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| - | <td>-5.6408e<sup>-04</sup> min\textsuperscript{-1}</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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| - | <td>K<sub>a</sub></td>
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| - | <td>0.28 nM</td>
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| - | <td>Dissociation constant for <code>(C:R)</code> 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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| - | <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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| - | <td>k<sub>+</sub></td>
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| - | <td>0.01-1.0 nM</td>
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| - | <td>Rate of association of $C$ and $R$ to form $(C:R)</td>
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| - | <td>Range defined relative to other parameters (one order of magnitude times based on QSSA</td>
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| - | <tr>
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| - | <td>k<sub>-</sub></td>
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| - | <td>0.01-1.0 nM</td>
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| - | <td>Rate of dissociation of $(C:R)</td>
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| - | <td>Range defined relative to other parameters based on QSSA</td>
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| - | <tr>
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| - | <td>Fold Reduction</td>
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| - | <td>6-35</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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| - | <h4> Production of dCas9 from dCas9 mRNA </h4>
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| - | <h4> Degradation rate of dCas9 </h4>
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| - | <h4> mRNA production from the <em>sarA</em> promoter </h4>
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| - | <h3> Initial Model Results </h3>
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| - | <h3> Updating mRNA Production Rates </h3>
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| - | <h3> Sensitivity Analysis </h3>
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| - | </div>
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| - |
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| - | <!------------------- CONJUGATION 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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| | </div> | | </div> |
| | </div> | | </div> |
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