Team:Waterloo/Math Book/sRNA

From 2014.igem.org

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<div class="anchor" id="MRSSB">
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    <h2> Model Reduction and Steady State Behaviour </h2>
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<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>
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<p>In this way, the steady state concentration of Hfq is then: </p>
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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/2014/6/60/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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</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:
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<!--IMAGES 5!>
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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>
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<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>
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<img src="https://static.igem.org/mediawiki/2014/7/73/Waterloo_sRNA_model_apply_quasi.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>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> The Least Squares Solution to this equation is:</p>
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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>
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<img src="https://static.igem.org/mediawiki/2014/7/7b/Waterloo_sRNA_2nd_Matrix.png" />
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<p> Which means that the flux term is approximated as:</p>
 
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<img src="https://static.igem.org/mediawiki/2014/7/7b/Waterloo_sRNA_2nd_Matrix.png" />
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<p> Where: </p>
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<p>Which means that the flux term is approximated as: </p>
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<!--IMAGES 7!>
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<img src="https://static.igem.org/mediawiki/2014/2/21/Waterloo_sRNA_J%28m%2Cs%29.png"/>
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<p>The solutions of this equation describe how the expression rates of Hfq and sRNA control the steady state concentration of target mRNA. </p>
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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>
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</div>
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<img src="https://static.igem.org/mediawiki/2014/9/9f/Waterloo_sRNA_lim_f%28m%2Cs%29.png"/>
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<p>So a more precise definition of J is: </p>
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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>
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<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>
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<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>
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<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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<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>
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<img src="https://static.igem.org/mediawiki/2014/e/ea/Waterloo_sRNA_vector_hht_1.png"/>
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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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<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>
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<h4 class="centerUpper">sRNA Parameters from Literature</h4>  
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<h4 class="centerUpper">Tabulated parameters, their descriptions and citations</h4>  
<table id="sRNAParams" class="blueBorders">
<table id="sRNAParams" class="blueBorders">
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<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>
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<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
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</tr>
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<tr>
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<td>&beta;<sub>H<sub>m</sub></sub>,&beta;<sub>s</sub>,&beta;<sub>m</sub></td>
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<td>2.31*10<sup>-3</sup> (s)<sup>-1</sup></td>
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                <td>degradation rates of Hfq mRNA, sRNA, mRNA</td>
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<td>[22]</a></td>
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</tr>
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<tr>
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<td>&beta;<sub>H</sub>,&beta;<sub>M<sub></td>
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<td>6.42*10<sup>-5</sup> (s)<sup>-1</sup></td>
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                <td>Degradation rate of Hfq and Target Protein (YFP)</td>
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>
<td><a href="http://genesdev.cshlp.org/content/24/23/2621"> Fender et al.</a></td>

Revision as of 02:38, 18 October 2014

Math Book: Silencing RNA (sRNA)

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

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

Model Formation

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.

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

Our solution to these problems was to simple provide Staphylococcus aures 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 of Figure X.

Applying the usual mass action to the reaction network in Figure X, 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:

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