Team:Waterloo/Math Book/sRNA

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<div id="MathBook" class="content">
<div id="MathBook" class="content">
   <ul class="tabs">
   <ul class="tabs">
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    <li><a href="#view0">Overview</a></li>
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       <li><a href="#MF">Formation</a></li>
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       <li><a href="#MF">Model Formation</a></li>
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       <li><a href="#MRSSB">Reduction</a></li>
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       <li><a href="#MRSSB">Model Reduction & Steady State Behaviour</a></li>
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       <li><a href="#Parameters">Parameters</a></li>
       <li><a href="#Parameters">Parameters</a></li>
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       <li><a href="#Results">Results</a></li>
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       <li><a href="#Results">Results</a></li>    
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      <li><a href="#SA">Sensitivity Analysis</a></li>
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       <li><a href="#Conclusion">Conclusion</a></li>
       <li><a href="#Conclusion">Conclusion</a></li>
   </ul>
   </ul>
   <div class="tabcontents">
   <div class="tabcontents">
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    <div class="anchor" id="view0">
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    </div>
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     <div class="anchor" id="view0">
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     <div class="anchor" id="MF">
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<p> The ordinary differential equation model for small ribonucleic acid (sRNA) gene silencing was formulated for the purpose of:</p>
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<p> The ordinary differential equation model for ribonucleic acid interference (RNAi) gene silencing was formulated for the purpose of:</p>  
<ol>
<ol>
<li> Model Formation</li>
<li> Model Formation</li>
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<li>Parameters</li>
<li>Parameters</li>
<li>Results</li>
<li>Results</li>
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<li>Sensitivity Analysis</li>
 
<li>Conclusion</li>
<li>Conclusion</li>
</ol>
</ol>
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    </div>
 
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    <div class="anchor" id="MF">
 
     <h2> Model Formation </h2>
     <h2> Model Formation </h2>
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<p>Inspiration for the model came from the metabolic pathway reported in the literature by Abia in 2007 [17]. In the network, sRNA binds to by Hfq, a chaperone protein which increases the binding rate between sRNA and its target mRNA substantially. Once bound, the Hfq-sRNA-mRNA complex is broken down by a degradosome, a specialized quaternary structure in sRNA-regulated gene expression.</p>
+
<p>Inspiration for the model came from the metabolic pathway reported in the literature by Abia in 2007 [17]. In the network, small RNA (sRNA) binds to Hfq, a chaperone protein which increases the binding rate between sRNA and its target mRNA substantially. Once bound, the Hfq-sRNA-mRNA complex is broken down by a degradosome, a specialized quaternary structure in sRNA-regulated gene expression.</p>
-
<p>At least, this is how the pathway works in <em>E. coli.</em> A major difficulty is that Hfq in <em>S. aures</em> doesn’t seem to play any major physiological role [18]. To make matters more difficult, the existence of a chaperone protein for sRNA in <em>S. aures</em> has yet to be discovered [19]. Additionally, the proteins that make up the degradosome in <em>E. coli</em> are not present in <em>S. aures.</em>.</p>
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<p>At least, this is how the pathway works in <em>E. coli.</em> A major difficulty is that Hfq in <em>S. aureus</em> does not seem to play any major physiological role [18]. To make matters more difficult, the existence of a chaperone protein for sRNA in <em>S. aureus</em> has yet to be discovered [19]. Additionally, the proteins that make up the degradosome in <em>E. coli</em> are not present in <em>S. aureus.</em></p>
-
<p>Our solution to these problems was to simple provide <em>Staphylococcus aures</em> the Hfq present in <em>E. coli.</em> In this way, a model of sRNA gene-regulation could be implemented to aid with laboratory design, and respond to the purposes of the model. Since Hfq would need to be expressed in the target cell, the reaction network took the form of Figure X.</p>
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<p>Our solution to these problems was to simple provide <em>S. aureus</em> the Hfq present in <em>E. coli.</em> In this way, a model of sRNA gene-regulation could be implemented to aid with laboratory design, and respond to the purposes of the model. Since Hfq would need to be expressed in the target cell, the reaction network took the form shown below.</p>
<img src="https://static.igem.org/mediawiki/2014/b/b0/Waterloo_model_network_with_caption.PNG" />
<img src="https://static.igem.org/mediawiki/2014/b/b0/Waterloo_model_network_with_caption.PNG" />
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     <p>Applying the usual mass action to the reaction network in Figure X, we arrive at the model equations:<p>
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     <p>Applying the usual mass action to the reaction network in the figure below, we arrive at the model equations:<p>
<img src="https://static.igem.org/mediawiki/2014/f/f7/Waterloo_sRNA_Model_Equations.png" />
<img src="https://static.igem.org/mediawiki/2014/f/f7/Waterloo_sRNA_Model_Equations.png" />
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<div class="anchor" id="MRSSB">
 +
    <h2> Model Reduction and Steady State Behaviour </h2>
 +
<p> In our model equations presented previously, if we define the total amount of Hfq present in the cell as H<sub>T</sub> = H + H<sub>s</sub> + H<sub>ms</sub>, we find:</p>
 +
<img src="https://static.igem.org/mediawiki/2014/3/33/Wiki_page_srna_fig_1.JPG"/>
 +
<p>In this way, the steady state concentration of Hfq is then: </p>
 +
<img src="https://static.igem.org/mediawiki/2014/e/e6/Wiki_page_srna_fig_2.JPG"/>
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<p>This model somewhat lends itself well to the idea of simplification, primarily between all of the complexes. In particular, we will apply a quasi-steady state assumption to the movement between all of the complexes. Recall the dynamics of the complexes seen in the network diagram, we can simplify this a little by first considering the dynamics of the complexes as a single, isolated system, showcased in the figure below. </p>
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<p>Applying a quasi-steady state approximation on the last three equations in the model yields a system of linear equations: </p>
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<img src="https://static.igem.org/mediawiki/2014/a/a6/Waterloo_sRNA_system_network_2.png" />
 
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<p>If we assume no source and no degradation of H (for now), the above system is modeled as: </p>
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<img src="https://static.igem.org/mediawiki/2014/2/29/Wiki_page_srna_fig_3.JPG"/>
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<img src="https://static.igem.org/mediawiki/2014/6/60/Waterloo_sRNA_model_equations_-2nd.png" />
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<p> Or, equivalently:</p>
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<p>One simplifying assumption would be if, rather than a Michaelis-Menton term, we had a mass-action term. We don't need to make this assumption, but it makes the rest of the calculations a whole lot easier (as in, will fit between the margins easier). Consider the first-order approximation to the Michaelis-Menton Term:
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<img src="https://static.igem.org/mediawiki/2014/1/14/Wiki_page_srna_fig_4.JPG"/>
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</p>
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<img src="https://static.igem.org/mediawiki/2014/1/16/Waterloo_sRNA_f%28x%29_for_the_model.png" />
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<p>This system has least-squares solution: </p>
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<p>With this approximation (which is valid for small H <sub>ms</sub> - something we can look into), the model reduces nicely. In particular the flux term is approximated as:
+
<img src="https://static.igem.org/mediawiki/2014/2/29/Wiki_page_srna_fig_5.JPG"/>
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</p>
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-
<img src="https://static.igem.org/mediawiki/2014/0/02/Waterloo_sRNA_J%28hms%29.png" />
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<p>The significance of this flux is that it represents the steady state breakdown rate of mRNA and sRNA. What we would really like is this flux term as a function of m and s so that we don't have to keep track of the entire Hfq-complex pathway in our model.</p>
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-
<p>Since we are assuming that flux through this pathway happens much faster than transcription and translation, let's also consider the total amount of Hfq to be constant and given by H<sub>T</sub> = H + H<sub>s</sub> + H<sub>ms</sub>, then we apply the Quasi-Steady State Assumption to our model: </p>
+
<p>We can then substitute these expressions into the first two equations of the model to (ultimately) arrive at a reduced model. After simplification: </p>
-
<img src="https://static.igem.org/mediawiki/2014/7/73/Waterloo_sRNA_model_apply_quasi.png" />
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<img src="https://static.igem.org/mediawiki/2014/c/ca/Wiki_page_srna_fig_6.JPG"/>
-
<p>Written more compactly: </p>
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-
<img src="https://static.igem.org/mediawiki/2014/6/62/Waterloo_1st_matrix.png" />
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<p> Where, V<sub>m</sub> = k<sub>3</sub>, K<sub>1</sub> = k<sub>3</sub>/k<sub>2</sub> and K<sub>m</sub> = (k<sub>-1</sub>k<sub>3</sub>) / (k<sub>1</sub>k<sub>2</sub>). We could use this simplified model to explore a phase space, however, it is much more valuable to explore the steady state behaviour of the model.<p>
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<p> The Least Squares Solution to this equation is:</p>
 
-
<img src="https://static.igem.org/mediawiki/2014/7/7b/Waterloo_sRNA_2nd_Matrix.png" />
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<p> Inspired by [20], where the authors examined the steady state concentration of target mRNA exposed to sRNA regulation as a function of sRNA transcription, we also seek the steady state concentration of mRNA. The major difference is that the steady state expression of mRNA in this case will be controlled by two expressions, those of Hfq as well as sRNA, as opposed to simply sRNA. In our simplified model, it can be shown that this steady state concentration  of mRNA obeys the cubic equation:</p>
-
<p> Which means that the flux term is approximated as:</p>
+
<img src="https://static.igem.org/mediawiki/2014/e/eb/Wiki_page_srna_fig_8.JPG"/>
-
<img src="https://static.igem.org/mediawiki/2014/7/7b/Waterloo_sRNA_2nd_Matrix.png" />
 
-
<p>Which means that the flux term is approximated as: </p>
+
<p> Where: </p>
-
<img src="https://static.igem.org/mediawiki/2014/2/21/Waterloo_sRNA_J%28m%2Cs%29.png"/>
+
<img src="https://static.igem.org/mediawiki/2014/4/4a/Wiki_page_srna_fig_9.JPG"/>
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<p> Where K = k<sub>3</sub>/(k<sub>1</sub>k<sub>2</sub>) and note that k<sub>3</sub>H<sub>T</sub> plays a similar (identical, actually) role to V<sub>max</sub> in the Michaelis-Menton formula. Note, however, that J(0,0)is not defined. Observe that:</p>
+
<p>The solutions of this equation describe how the expression rates of Hfq and sRNA control the steady state concentration of target mRNA. </p>
-
<img src="https://static.igem.org/mediawiki/2014/9/9f/Waterloo_sRNA_lim_f%28m%2Cs%29.png"/>
+
</div>
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+
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<p>So a more precise definition of J is: </p>
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-
 
+
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<img src="https://static.igem.org/mediawiki/2014/4/46/Waterloo_sRNA_J%28m%2Cs%29_piecewise.png"/>
+
-
 
+
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<p>With all of these simplifying assumptions, the model reduces to: </p>
+
-
 
+
-
<img src="https://static.igem.org/mediawiki/2014/a/a4/Waterloo_sRNA_Model_Simplified_again.png"/>
+
-
 
+
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<p>Here, we've lumped all of the Hfq protein complexes (and, indeed, Hfq) into one species. Notice that the last two differential equations are independent of the rest. Even better - they're Linear! And even better, it's a 2x2 system! Focusing on the last two equations, and rewriting them in vector-matrix form: </p>
+
-
 
+
-
<img src="https://static.igem.org/mediawiki/2014/b/b6/Waterloo_sRNA_ddt_hht.png"/>
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<p>Focusing on the homogeneous part, notice that the matrix is lower triangular. Thus, the eigenvalues appear on the diagonal, &lambda;<sub>1</sub> = - &beta;<sub>H</sub>; &lambda;<sub>2</sub>=-&beta;<sub>h</sub>. The corresponding eigenvectors are:</p>
+
-
 
+
-
<img src="https://static.igem.org/mediawiki/2014/c/cc/Waterloo_sRNA_v1_v2.png"/>
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<p> and thus, the fundamental matrix is:</p>
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-
 
+
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<img src="https://static.igem.org/mediawiki/2014/b/b6/Waterloo_sRNA_Fundamental_Matrix.png"/>
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<p> With the Fundamental Matrix, we can apply Variation of Parameters to get the general solution for H<sub>T</sub>. Since the transcription for Hfq is under the control of our wonderful experimenters, we could chose to have &alpha;<sub>h</sub> be some arbitrary function of time. Regardless, the general solution is:</p>
+
-
<img src="https://static.igem.org/mediawiki/2014/e/ea/Waterloo_sRNA_vector_hht_1.png"/>
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-
 
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<p> Where h<sup>(0)</sup>, and H<sub>T</sub><sup>(0)</sup> are initial conditions on h and H<sub>T</sub>.</p>
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-
 
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-
<img src="https://static.igem.org/mediawiki/2014/8/8c/Waterloo_sRNA_vector_hht_2.png"/>
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<p> This looks pretty crazy, but it gives us a way of solving for H<sub>T</sub>, written explicitly: </p>
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<img src="https://static.igem.org/mediawiki/2014/d/df/Waterloo_sRNA_equation_to_solve_ht.png"/>
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<p> The input is how &alpha;<sub>h</sub> varies with time. If we take the initial conditions to be zero for each we get: </p>
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<img src="https://static.igem.org/mediawiki/2014/b/bb/Waterloo_sRNA_equation_for_ht_solved_2.png"/>
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<p>At any rate, the point here is that we can solve for exactly how H<sub>T</sub> varies with time. We can also see that if our forcing term is constant, then we'll settle down to a steady state concentration of H<sub>T</sub>. Additionally, our original model then reduces to three equations:</p>
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<img src="https://static.igem.org/mediawiki/2014/5/5f/Waterloo_sRNA_model_equations_simplified_final.png"/>
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<p>Notice also that the last equation doesn't really influence the dynamics of the first two. Additionally, if H<sub>T</sub> is constant, then it's possible to observe a phase-portrait of the system. It's also possible to look at possible steady states. In other words, we can actually do some analysis on this model without physically being supercomputers.</p>
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    </div>
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     <div class="anchor" id="Parameters">
     <div class="anchor" id="Parameters">
       <h2> Parameters </h2>
       <h2> Parameters </h2>
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<p>We identified parameters in the literature. The identified parameters and their sources are given in the table below.</p>
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<p>Our parameters, and their citations, are tabulated in the table below.</p>
-
<h4 class="centerUpper">sRNA Parameters from Literature</h4>  
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<table id="sRNAParams" class="blueBorders">
<table id="sRNAParams" class="blueBorders">
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<tbody>
<tbody>
<tr>
<tr>
-
<td>&alpha;<sub>m</sub></td>
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<td>&alpha;<sub>h</sub></td>
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<td>1/600 (nM*s)<sup>-1</sup> </td>
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<td>1/600 nM/s</td>
                 <td>transcription mRNA</td>
                 <td>transcription mRNA</td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
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<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
-
+
</tr>
</tr>
-
<tr>
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<tr>
 +
<td>&beta;<sub>h</sub></td>
 +
<td>2.31*10<sup>-3</sup> s<sup>-1</sup></td>
 +
                <td>degradation mRNA</td>
 +
<td><a href="http://nar.oxfordjournals.org/content/early/2012/05/22/nar.gks439.full"> Hussein, R. & Lim, H.N.</a></td>
 +
</tr>
 +
        <tr>
 +
<td> &alpha;<sub>H</sub></td>
 +
<td>0.9 s<sup>-1</sup></td>
 +
                <td>degradation rates of Hfq mRNA, sRNA, mRNA</td>
 +
<td><a href="http://www.readcube.com/articles/10.1093/nar/gks439">  Hussein, P.S. et al.</a></td>
 +
</tr>
 +
        <tr>
<td>&alpha;<sub>s</sub></td>
<td>&alpha;<sub>s</sub></td>
-
<td>1/600 (nM*s)<sup>-1</sup> </td>
+
<td>1/600 nM/s</td>
                 <td>transcription sRNA</td>
                 <td>transcription sRNA</td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
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<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf">  Fender, A et al.</a></td>
 +
</tr>
 +
        <tr>
 +
<td>&alpha;<sub>m</sub></td>
 +
<td>1/600 nM/s</td>
 +
                <td>transcription Hfq mRNA</td>
 +
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
</tr>
</tr>
-
<tr>
+
        <tr>
<td>&alpha;<sub>M</sub></td>
<td>&alpha;<sub>M</sub></td>
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<td>1/600 (nM*s)<sup>-1</sup> </td>
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<td>1/600 s<sup>-1</sup> </td>
                 <td>translation of the mRNA</td>
                 <td>translation of the mRNA</td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
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<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
</tr>
</tr>
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        <tr>
-
<tr>
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<td>&beta;<sub>Hm</sub>, &beta;<sub>s</sub>, &beta;<sub>m</sub></td>
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<td>k<sub>-1</sub></td>
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<td>2.31*10<sup>-3</sup> s<sup>-1</sup></td>
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<td>0.7*10<sup>-4</sup>s<sup>-1</sup> </td>
+
                 <td>degradation rates of Hfq mRNA, sRNA, mRNA</td>
-
                 <td>dissociation constant of H<sub>s</sub> to Hfq and sRNA</td>
+
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf">  Swain, P.S. et al.</a></td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
+
</tr>
 +
        <tr>
 +
<td>&beta;<sub>H</sub>, &beta;<sub>M</sub>, &beta;<sub>Hs</sub>, &beta;<sub>Hms</sub></td>
 +
<td>6.42*10<sup>-5</sup> s<sup>-1</sup></td>
 +
                <td>Degradation rate of Hfq and Target Protein (YFP)</td>
 +
<<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Swain, P.S. et al.</a></td>
</tr>
</tr>
-
<tr>
+
<tr>
 +
<td>&beta;<sub>ms</sub></td>
 +
<td>2.31*10<sup>-2</sup> s<sup>-1</sup></td>
 +
                <td>transcription mRNA</td>
 +
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> ?</a></td>
 +
</tr>
 +
        <tr>
<td>k<sub>1</sub></td>
<td>k<sub>1</sub></td>
-
<td>10<sup>6</sup> s<sup>-1</sup></td>
+
<td>10<sup>-3</sup> (nM*s)<sup>-1</sup></td>
                 <td>association constant of Hfq and sRNA</td>
                 <td>association constant of Hfq and sRNA</td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
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<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
-
+
</tr>
</tr>
-
<tr>
+
        <tr>
 +
<td>k<sub>-1</sub></td>
 +
<td>0.7*10<sup>-4</sup> s<sup>-1</sup> </td>
 +
                <td>dissociation constant of H<sub>s</sub> to Hfq and sRNA</td>
 +
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf">  Fender, A et al.</a></td>
 +
</tr>
 +
        <tr>
<td>k<sub>2</sub></td>
<td>k<sub>2</sub></td>
-
<td>3.5*10<sup>6</sup> (M*s)</td>
+
<td>3.5*10<sup>6</sup> (nM*s)<sup>-1</sup></td>
                 <td>association constant of H<sub>s</sub> and mRNA</td>
                 <td>association constant of H<sub>s</sub> and mRNA</td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
+
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
-
+
</tr>
</tr>
-
<tr>
+
        <tr>
<td>k<sub>3</sub></td>
<td>k<sub>3</sub></td>
<td>0.7*10<sup>-4</sup> s<sup>-1</sup> </td>
<td>0.7*10<sup>-4</sup> s<sup>-1</sup> </td>
                 <td>dissociation constant of H<sub>ms</sub>, assuming that the Hfq-sRNA binding to the MicC region is independent of the sRNA, mRNA binding. Therefore, k<sub>-1</sub>=k<sub>3</sub></td>
                 <td>dissociation constant of H<sub>ms</sub>, assuming that the Hfq-sRNA binding to the MicC region is independent of the sRNA, mRNA binding. Therefore, k<sub>-1</sub>=k<sub>3</sub></td>
-
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
+
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf"> Fender, A et al.</a></td>
-
+
</tr>
 +
        <tr>
 +
<td>k<sub>4</sub></td>
 +
<td> ? (nM*s)<sup>-1</sup></td>
 +
                <td>production rate of Target mRNA-sRNA complex</td>
 +
<td><a href="http://elflab.icm.uu.se/references/Genes&Dev-2010-Fender.pdf">  ???</a></td>
</tr>
</tr>
</table>
</table>
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</div>
</div>
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     <div class="anchor" id="Sensitivity">
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       <h2> Sensitivity Analysis</h2>
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     <div class="anchor" id="Results">
-
<p>To get a better handle on the dynamics of the system we ran a local sensitivity analysis. This determined what parameters the sRNA system is most sensitive to. The flux control coefficients for the sRNA system can be seen in the figure to the right.</p>
+
       <h2> Results</h2>
 +
    <p>The resulting time-history of the concentration of Yellow Fluorescent Protein (YFP), the target protein in this case, when sRNA gene-silencing is introduced is displayed in the figure below. As can be seen from the graph, the target protein exhibits an exponential decay until it reaches an almost negligible steady state. The approximate time it takes to do this is on the order of 18 hours, which is an artefact of the half life of the protein. The small RNA and Hfq in the cell effectively destroy YFP’s mRNA, turning off expression, forcing the protein concentration dynamics to be mostly governed by the decay.</p>
 +
 
 +
<div class="imageContainer">
 +
<img src="https://static.igem.org/mediawiki/2014/e/e5/Wiki_page_srna_fig_10.JPG"/>
 +
<p>The response of YFP concentration when sRNA is activated at time 0. After approximately 18hours, there is a 99.8% silencing of protein.
 +
</p>
 +
</div>
 +
 
 +
 
 +
 
 +
<p>In using Equation 6 to construct a surface relating sRNA and Hfq transcription to the steady state concentration of target mRNA, we generate the surface pictured in the figure below. </p>
 +
 
 +
 
 +
<img src="https://static.igem.org/mediawiki/2014/2/2b/Wiki_page_srna_fig_11.JPG"/>
 +
 
 +
<p> The figure above seems to indicate, qualitatively, that the steady state concentration of target mRNA is much more sensitive to changes in Hfq expression than sRNA expression. To explore this relationship, we performed a sensitivity analysis on the model to each of the parameters.</p>
 +
<br></br>
 +
 
 +
<h2> Metabolic Control Analysis</h2>  
<img class="floatRight half-column" src="https://static.igem.org/mediawiki/2014/e/ee/UWaterloo_-_sRNA_Control_Coefficient.png" />
<img class="floatRight half-column" src="https://static.igem.org/mediawiki/2014/e/ee/UWaterloo_-_sRNA_Control_Coefficient.png" />
-
<p>The most notable of these are the flux control coefficients for the following parameters:  
+
<br></br>
-
β M(the degradation rate of YFP),  αm (the transcription rate of the YFP mRNA transcript). These large flux control coefficient values demonstrate the system is highly sensitive to changes in these variables. So if we were able to influence these rates we would be able to dramatically change the level of flux through the system. In order for the sRNA system to be most efficient, we would want the flux of YFP through the system to be as low as possible. The most direct way to affect this would be to alter these values. Since we want to use this sRNA system to control a protein whose degradation rate and transcription rate we could not alter, we need to turn our gaze to some of the other factors at play.</p>
+
<br></br>
-
<p>Some other notable flux control coefficients are αS, αM, K-1. These rates are respectively the rate of transcription of sRNA, the rate of translation of YFP, and the rate of dissociation of Hfq-sRNa to Hfq and sRNA. These rates have the least impact on the system and are not good targets for optimization of sRNA. </p>
+
<br></br>
 +
<p> Since a major purpose of this model was to elucidate potential avenues for silencing a target protein, our Metabolic Control Analysis focused primarily on how the parameters of the model and affect the steady state concentration of the target protein (in this case YFP). Listed to the right are the Concentration Control Coefficients for the steady state concentration of YFP for each parameter in the model:</p>
 +
<p> The most notable of these are the concentration control coefficients for the following parameters: &beta;<sub>M</sub> the degradation rate of YFP),  &alpha;<sub>m</sub> (the transcription rate of the YFP mRNA transcript). These relatively large flux control coefficient values demonstrate the system is highly sensitive to changes in these variables. Therefore, these would make particularly good choices for synthetic intervention, as they would induce the largest changes on the steady state concentration of YFP. In order for sRNA gene suppression to be most efficient, a minimal amount of YFP-flux through the system is desirable. However, since we want to use this sRNA system to control a protein whose degradation rate and transcription rate that we could not physically change, we need to turn our attention to some other potential confounding factors.</p>
 +
<p> Some other notable concentration control coefficients are &alpha;<sub>s</sub>, &alpha;<sub>M</sub>, K<sub>-1</sub>. These constants are respectively the rate of transcription of sRNA, the rate constant for translation of YFP, and the rate constant of dissociation of Hfq-sRNa to Hfq and sRNA. These rates have the least impact on the system and are not good targets for optimization of sRNA.</p>
 +
</div>
 +
 
 +
 
 +
<div class="anchor" id="Conclusion">
 +
    <h2> Conclusion</h2>
 +
<br></br>
 +
<br></br>
 +
<p>Using parameters from the literature we were able to construct a model of sRNA gene repression in <em>Staphylococcus aureus</em>, using Hfq from E. coli. In the subsequent analysis of the model, the relationships between the expression rate of sRNA, Hfq, and the subsequent steady state concentration of a target protein, YFP, was elucidated.  </p>
 +
<p>The model was successful in providing an estimate of the amount of suppression as well as the approximate amount of time until maximum suppression was obtained. Additionally, in the Metabolic Control Analysis, we were able to deduce the best regions of the metabolic pathway to target in order to reduce the steady state concentration of YFP; unfortunately, however, these ended up being parameters outside of our control.
 +
</p>
 +
<p> In the end, the model of sRNA repression was able to inform the conjugation model of the magnitude and temporal characteristics of sRNA gene regulation.</p>
 +
<br></br>
 +
<br></br>
 +
<br></br>
</div>
</div>

Latest revision as of 00:22, 28 November 2014

Math Book: Silencing RNA (sRNA)

The ordinary differential equation model for ribonucleic acid interference (RNAi) gene silencing was formulated for the purpose of:

  1. Model Formation
  2. Model Reductions and Steady State Behaviour
  3. Parameters
  4. Results
  5. Conclusion

Model Formation

Inspiration for the model came from the metabolic pathway reported in the literature by Abia in 2007 [17]. In the network, small RNA (sRNA) binds to Hfq, a chaperone protein which increases the binding rate between sRNA and its target mRNA substantially. Once bound, the Hfq-sRNA-mRNA complex is broken down by a degradosome, a specialized quaternary structure in sRNA-regulated gene expression.

At least, this is how the pathway works in E. coli. A major difficulty is that Hfq in S. aureus does not seem to play any major physiological role [18]. To make matters more difficult, the existence of a chaperone protein for sRNA in S. aureus has yet to be discovered [19]. Additionally, the proteins that make up the degradosome in E. coli are not present in S. aureus.

Our solution to these problems was to simple provide S. aureus the Hfq present in E. coli. In this way, a model of sRNA gene-regulation could be implemented to aid with laboratory design, and respond to the purposes of the model. Since Hfq would need to be expressed in the target cell, the reaction network took the form shown below.

Applying the usual mass action to the reaction network in the figure below, we arrive at the model equations:

Model Reduction and Steady State Behaviour

In our model equations presented previously, if we define the total amount of Hfq present in the cell as HT = H + Hs + Hms, we find:

In this way, the steady state concentration of Hfq is then:

Applying a quasi-steady state approximation on the last three equations in the model yields a system of linear equations:

Or, equivalently:

This system has least-squares solution:

We can then substitute these expressions into the first two equations of the model to (ultimately) arrive at a reduced model. After simplification:

Where, Vm = k3, K1 = k3/k2 and Km = (k-1k3) / (k1k2). We could use this simplified model to explore a phase space, however, it is much more valuable to explore the steady state behaviour of the model.

Inspired by [20], where the authors examined the steady state concentration of target mRNA exposed to sRNA regulation as a function of sRNA transcription, we also seek the steady state concentration of mRNA. The major difference is that the steady state expression of mRNA in this case will be controlled by two expressions, those of Hfq as well as sRNA, as opposed to simply sRNA. In our simplified model, it can be shown that this steady state concentration of mRNA obeys the cubic equation:

Where:

The solutions of this equation describe how the expression rates of Hfq and sRNA control the steady state concentration of target mRNA.

Parameters

Our parameters, and their citations, are tabulated in the table below.

<
Parameter Value Description Reference
αh 1/600 nM/s transcription mRNA Fender, A et al.
βh 2.31*10-3 s-1 degradation mRNA Hussein, R. & Lim, H.N.
αH 0.9 s-1 degradation rates of Hfq mRNA, sRNA, mRNA Hussein, P.S. et al.
αs 1/600 nM/s transcription sRNA Fender, A et al.
αm 1/600 nM/s transcription Hfq mRNA Fender, A et al.
αM 1/600 s-1 translation of the mRNA Fender, A et al.
βHm, βs, βm 2.31*10-3 s-1 degradation rates of Hfq mRNA, sRNA, mRNA Swain, P.S. et al.
βH, βM, βHs, βHms 6.42*10-5 s-1 Degradation rate of Hfq and Target Protein (YFP) Swain, P.S. et al.
βms 2.31*10-2 s-1 transcription mRNA ?
k1 10-3 (nM*s)-1 association constant of Hfq and sRNA Fender, A et al.
k-1 0.7*10-4 s-1 dissociation constant of Hs to Hfq and sRNA Fender, A et al.
k2 3.5*106 (nM*s)-1 association constant of Hs and mRNA Fender, A et al.
k3 0.7*10-4 s-1 dissociation constant of Hms, assuming that the Hfq-sRNA binding to the MicC region is independent of the sRNA, mRNA binding. Therefore, k-1=k3 Fender, A et al.
k4 ? (nM*s)-1 production rate of Target mRNA-sRNA complex ???

Results

The resulting time-history of the concentration of Yellow Fluorescent Protein (YFP), the target protein in this case, when sRNA gene-silencing is introduced is displayed in the figure below. As can be seen from the graph, the target protein exhibits an exponential decay until it reaches an almost negligible steady state. The approximate time it takes to do this is on the order of 18 hours, which is an artefact of the half life of the protein. The small RNA and Hfq in the cell effectively destroy YFP’s mRNA, turning off expression, forcing the protein concentration dynamics to be mostly governed by the decay.

The response of YFP concentration when sRNA is activated at time 0. After approximately 18hours, there is a 99.8% silencing of protein.

In using Equation 6 to construct a surface relating sRNA and Hfq transcription to the steady state concentration of target mRNA, we generate the surface pictured in the figure below.

The figure above seems to indicate, qualitatively, that the steady state concentration of target mRNA is much more sensitive to changes in Hfq expression than sRNA expression. To explore this relationship, we performed a sensitivity analysis on the model to each of the parameters.



Metabolic Control Analysis







Since a major purpose of this model was to elucidate potential avenues for silencing a target protein, our Metabolic Control Analysis focused primarily on how the parameters of the model and affect the steady state concentration of the target protein (in this case YFP). Listed to the right are the Concentration Control Coefficients for the steady state concentration of YFP for each parameter in the model:

The most notable of these are the concentration control coefficients for the following parameters: βM the degradation rate of YFP), αm (the transcription rate of the YFP mRNA transcript). These relatively large flux control coefficient values demonstrate the system is highly sensitive to changes in these variables. Therefore, these would make particularly good choices for synthetic intervention, as they would induce the largest changes on the steady state concentration of YFP. In order for sRNA gene suppression to be most efficient, a minimal amount of YFP-flux through the system is desirable. However, since we want to use this sRNA system to control a protein whose degradation rate and transcription rate that we could not physically change, we need to turn our attention to some other potential confounding factors.

Some other notable concentration control coefficients are αs, αM, K-1. These constants are respectively the rate of transcription of sRNA, the rate constant for translation of YFP, and the rate constant of dissociation of Hfq-sRNa to Hfq and sRNA. These rates have the least impact on the system and are not good targets for optimization of sRNA.

Conclusion





Using parameters from the literature we were able to construct a model of sRNA gene repression in Staphylococcus aureus, using Hfq from E. coli. In the subsequent analysis of the model, the relationships between the expression rate of sRNA, Hfq, and the subsequent steady state concentration of a target protein, YFP, was elucidated.

The model was successful in providing an estimate of the amount of suppression as well as the approximate amount of time until maximum suppression was obtained. Additionally, in the Metabolic Control Analysis, we were able to deduce the best regions of the metabolic pathway to target in order to reduce the steady state concentration of YFP; unfortunately, however, these ended up being parameters outside of our control.

In the end, the model of sRNA repression was able to inform the conjugation model of the magnitude and temporal characteristics of sRNA gene regulation.







References

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