Team:Waterloo/Math Book/sRNA

From 2014.igem.org

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       <li><a href="#MRSSB">Model Reduction & Steady State Behaviour</a></li>
       <li><a href="#MRSSB">Model Reduction & Steady State Behaviour</a></li>
       <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="MCA">Metabolic Control Analysis</a></li>
       <li><a href="#Conclusion">Conclusion</a></li>
       <li><a href="#Conclusion">Conclusion</a></li>
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      <h2> Sensitivity Analysis</h2>
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<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>
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<img class="floatRight half-column" src="https://static.igem.org/mediawiki/2014/e/ee/UWaterloo_-_sRNA_Control_Coefficient.png" />
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<p>The most notable of these are the flux control coefficients for the following parameters:
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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>
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<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>
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     <div class="anchor" id="Results">
     <div class="anchor" id="Results">
       <h2> Results</h2>
       <h2> Results</h2>
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<div class="anchor" id="MCA">
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    <h2> Metabolic Control Analysis</h2> </div>
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<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 below are the Concentration Control Coefficients for the steady state concentration of YFP for each parameter in the model:</p>
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<img class="floatRight half-column" src="https://static.igem.org/mediawiki/2014/e/ee/UWaterloo_-_sRNA_Control_Coefficient.png" />
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<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>
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<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>
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<div class="anchor" id="Conclusion">
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    <h2> Conclusion</h2> </div>
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<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>
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<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.
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</p>
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<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>
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Revision as of 03:02, 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:

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.

Tabulated parameters, their descriptions and citations

Parameter Value Description Reference
αm 1/600 (nM*s)-1 transcription mRNA Fender et al.
αs 1/600 (nM*s)-1 transcription sRNA Fender et al.
αM 1/600 (nM*s)-1 translation of the mRNA Fender et al.
k-1 0.7*10-4s-1 dissociation constant of Hs to Hfq and sRNA Fender et al.
k1 106 s-1 association constant of Hfq and sRNA Fender et al.
k2 3.5*106 (M*s) association constant of Hs and mRNA Fender 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 et al.
βHm,βs,βm 2.31*10-3 (s)-1 degradation rates of Hfq mRNA, sRNA, mRNA [22]
βH,βM 6.42*10-5 (s)-1 Degradation rate of Hfq and Target Protein (YFP) Fender et al.

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 below 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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[15]Jianfang Jia and Hong Yue. “Sensitivity Analysis and Parameter Estimation of Signal Transduction Pathways Model”. In: Proceedings of the 7th Asian Control Conference (Aug. 2009), pp. 1357–1362.
[16]Fi-John Chang and J. W. Delleur. “Systematic Parameter Estimation Of Watershed Acidification Model”. In: Hydrological Processes 6. (1992), pp. 29–44. doi: 10.1002/hyp.3360060104.
[17]Aiba, H. (2007). Mechanism of RNA silencing by Hfq-binding small RNAs. Current opinion in microbiology, 10 (2), 134-139.
[18]Horstmann, N., Orans, J., Valentin-Hansen, P., Shelburne, S. A., & Brennan, R. G. (2012). Structural mechanism of Staphylococcus aureus Hfq binding to an RNA A-tract. Nucleic acids research, gks809.
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[21]Fender, A., Elf, J., Hampel, K., Zimmermann, B., & Wagner, E. G. H. (2010). RNAs actively cycle on the Sm-like protein Hfq. Genes & Development, 24 (23),2621-2626.
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