Team:Waterloo/Modeling/Silencing

From 2014.igem.org

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</head>
</head>
<body>
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 +
<h1>CRISPR Interference Model</h1>
<h2>Motivation</h2>
<h2>Motivation</h2>
-
We decided to create a model of the CRISPR system for two main reasons
+
We decided to create a model of the CRISPR system for two main reasons:
<ul>
<ul>
   <li>Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency</li>
   <li>Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency</li>
-
   <li>To approximate time-series mecA repression data for use in modelling the overall vulnerability of a S. auerus population</li>
+
   <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>
</ul>
</ul>
<h2>Model Construction</h2>
<h2>Model Construction</h2>
-
After a literature review we were able to construct a network of a CRISPR interference system. 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:
+
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:
-
<h3>Definitions</h3>
+
<h3>IMG: reaction netowrk</h3>
-
<h3>Conventions</h3>
+
-
<h3>Chemical Network</h3>
+
-
Using standard mass-action kinetics, the network can be translated into the following set of differential equations:
+
Using standard mass-action kinetics, the network simplifies into the following set of differential equations:
<h3>Differential System Here</h3>
<h3>Differential System Here</h3>
-
The choice of a largely first-order model is supported by the findings of several recent studies <cite ref="Sternberg2014"></cite><cite ref="Qi2013"></cite>. To simplify the model further, we made a quasi-steady-state assumption (QSSA) about the formation of  of the dCas9-sgRNA complex $b$. That is, we assume that dCas9 and sgRNA associate on a faster timescale than the other reactions (i.e. transcription, translation and the binding of the complex to the DNA). Therefore, we disregard the kinetics of the complex formation reaction and assume that it is always at steady-state relative to the other time-dependent species concentrations.
+
The choice of a largely first-order model is supported by findings of several recent studies <cite ref="Sternberg2014"></cite><cite ref="Qi2013"></cite>. To simplify the model further, we made a quasi-steady-state assumption (QSSA) about the formation of  of the dCas9-sgRNA complex <b>b</b>. That is, we assume that dCas9 and sgRNA associate on a faster timescale than the other reactions (i.e. transcription, translation and the binding of the complex to the DNA). Therefore, we disregard the kinetics of the complex formation reaction and assume that it is always at steady-state relative to the other time-dependent species concentrations.
-
Under the QSSA, the quantity of the complex is given by:
+
Under the QSSA, the concentration of the complex is given by:
see latex 4 equation
see latex 4 equation
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see latex 4 equation
see latex 4 equation
-
The leaky expression of YFP mRNA originates from incomplete repression of mRNA production by the dCas9-sgRNA complex. We considered two possible mechanisms for leaky repression: either RNA polymerase is sometimes able to push past or dislodge the bound complex (which should be represented by an $\alpha_0$ basal expression term) or the complex is unable to bind efficiently (which should be captured by the $K_a$ dissociation constant). Several studies <cite ref="Qi2013"></cite><cite ref="Bikard2013"></cite> have found that almost 100% repression can be achieved if dCas9 is targeted at the promoter, preventing transcription initiation, while targets downstream of the promoter lead to at most 40% repression. Since the structure of the DNA at the promoter is not chemically distinct from the DNA in the rest of the gene, these findings support the "`dislodging"' leaky expression hypothesis.
+
<h3>Modeling Incomplete Repression</h3>
 +
<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>
 +
<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>
 +
<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>
 +
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 the following term:
-
From this analysis, we did not tune the dissociation constant $K_a$. In addition, rather than keeping a separately-defined $\alpha_0$ term, we modelled the complex as being able to affect a certain maximum percentage of the production from the promoter. This leads to a new equation for YFP mRNA:
+
<!--
 +
The leaky expression of YFP mRNA originates from incomplete repression of mRNA production by the dCas9-sgRNA complex. We considered two possible mechanisms for leaky repression: either RNA polymerase is sometimes able to push past or dislodge the bound complex (which is represented by an <b>&alpha;<sub>0</sub></b> basal expression term) or the complex is unable to bind efficiently (which is captured by the <b>K<sub>a</sub></b> dissociation constant). Several studies <cite ref="Qi2013"></cite><cite ref="Bikard2013"></cite> have found that almost 100% repression can be achieved if dCas9 is targeted at the promoter, preventing transcription initiation, while targets downstream of the promoter lead to at most 40% repression. Since the structure of the DNA at the promoter is not chemically distinct from the DNA in the rest of the gene, these findings support the 'dislodging' leaky expression hypothesis.
 +
 
 +
From this analysis, we did not tune the dissociation constant <b>K<sub>a</sub></b>. In addition, rather than keeping a separately-defined <b>&alpha;<sub>0</sub></b> term, we modeled the complex such that it limits a maximum percentage of the production from the promoter. This leads to a new equation for YFP mRNA:
see latex 4 equation
see latex 4 equation
 +
-->
When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed <em>sarA</em> promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from <em>sarA</em>.
When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed <em>sarA</em> promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from <em>sarA</em>.
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     <tr>
     <tr>
       <td></td>
       <td></td>
-
       <td>0.0011 nM min^-1</td>
+
       <td>0.0011 nM min<sup>-1</sup></td>
-
       <td>mRNA production from <em>SarA</em> P1 Promoter</td>
+
       <td>mRNA production from <em>sarA</em> P1 Promoter</td>
       <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>
       <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>
     </tr>
     </tr>
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Details on the more roundabout estimations are given below:
Details on the more roundabout estimations are given below:
-
Determining production of dCas9 from dCas9 mRNA
+
<h3>dCas9 Production Rates from mRNA</h3>
-
 
+
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. 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.
-
We were unable to find a peptide chain elongation rate for \textit{S. aureus}, 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 \textit{Streptomyces coelicolor}, 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. We were unable to find ribosome densities in \textit{S. aureus}, 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.
+
-
 
+
-
Degradation Rate of dCas9
+
-
 
+
-
We were unable to find any specific data on dCas9 degradation, so instead we used a protein half-life of \textit{SarA} measured in \textit{S. Aureus} by Michelik et al. <cite ref="Michalik2012"></cite>. We chose \texit{SarA} rather than a protein more chemically similar to \texit{dCas9} because data on \textit{SarA} was readily available and because \texit{dCas9} is transcribed using the \textit{SarA} promoter, which allows us to at least capture sensitivity of the degradation rate to production.
+
-
Determining production from the sarA promoter
+
<h3>Degradation Rates of dCas9</h3>
 +
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.
-
We used the time-series data given by Cheung et al. <cite ref="Cheung2008"></cite> to estimate the rate of production from the \textit{sarA} P2-P3-P1 promoter in S. aureus. The figure from their paper is reproduced below. After diluting 1:100, the S. aureus strains were serially monitored for $OD_650$. We used data from the SarA+ strain, as that's more like a wild-type S. aureus strain.
+
<h3>Production Rates from the <em>sarA</em> Promoter</h3>
 +
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 <b>OD_650</b>. We used data from the <em>sarA</em>+ strain, as that's more like a wild-type S. aureus strain.
-
Using the laboratory-conditions doubling time of 24 minutes given in Using the laboratory-conditions doubling time of 24 minutes given in <cite ref="Domingue1996"></cite>, we found that the bacteria would re-enter stationary phase after 2.5 hours. For this reason, we considered only timepoints after 3 hours, after the bacteria would re-enter stationary phase and the number of number of \textit{sarA} genes producing fluorescence could be assumed as constant. 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 \textit{sarA} genes, considered the relative change in number of molecules to be representative of the per-promoter rate.
+
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.
-
We were interested, however, in changes of concentration rather than changes in the raw number of molecules. The diameter of a USA300 \textit{S. aureus} cell has previously been measured as $1.1 {\mu}m$ <cite ref="Cheng2014"></cite> and Staphylococci are named for their spherical shape, so we assumed the cells to be spherical and found the overall cell volume to be $5.575 \times 10^-15 L$. 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 below:
+
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 &mu;&bull;m <cite ref="Cheng2014"></cite> resulting in the overall cell volume to be calculated as 5.575&bull;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 below:
-
This resulted in a exponential model $a e^bt$ with a $b$ rate constant of 0.0011 nM/min.
+
This resulted in a exponential model <b>a&bull;e<sup>bt</sup></b> with a <b>b</b> rate constant of 0.0011 nM/min.
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Latest revision as of 19:51, 16 October 2014

CRISPR Interference Model

Motivation

We decided to create a model of the CRISPR system for two main reasons:
  • Identifying the parts of the network that could be targeted by our lab team to improve repression efficiency
  • To approximate time-series mecA repression data for use in modelling the overall vulnerability of a S. aureus population

Model Construction

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 . The bound complex prevents transcription elongation by RNA polymerase, repressing YFP mRNA expression . The chemical network is shown below:

IMG: reaction netowrk

Using standard mass-action kinetics, the network simplifies into the following set of differential equations:

Differential System Here

The choice of a largely first-order model is supported by findings of several recent studies . To simplify the model further, we made a quasi-steady-state assumption (QSSA) about the formation of of the dCas9-sgRNA complex b. That is, we assume that dCas9 and sgRNA associate on a faster timescale than the other reactions (i.e. transcription, translation and the binding of the complex to the DNA). Therefore, we disregard the kinetics of the complex formation reaction and assume that it is always at steady-state relative to the other time-dependent species concentrations. Under the QSSA, the concentration of the complex is given by: see latex 4 equation This is the same assumption made by previous teams . Our model then simplifies to: see latex 4 equation

Modeling Incomplete Repression

A recent study by Bikard et al. 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.

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.

We assumed that the incomplete repression is accounted for by the first mechanism. This assumption was based on several studies 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.

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 the following term: When the concentration of the complex is zero, YFP mRNA is produced at the rate expected from the unrepressed sarA promoter. At a large concentration of the complex, the YFP mRNA is produced at only 60% of the possible rate from sarA.

Parameter Search and Fitting

We turned to the literature to find parameters for our model. We first looked for exact parameter values in S. aureus. If these 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 endeavour, but those parameters about which we are very uncertain are marked with asterisks. A general rationale is given for each parameter, but details on the more circuitously estimated parameters are given after the table.
Parameter Value Description Source/Rationale
0.0011 nM min-1 mRNA production from sarA P1 Promoter Determined based on linear fitting to the time-series fluorescence measurements from YFP/P2-P3-P1 fusion, as reported in and fluorescence per molecule from
The only model parameters without some basis in the literature are the association rates for dCas9 and sgRNA. However, since the model is based on the QSSA that those dynamics are much faster than the others in the model, we were able to define a range for those parameters based on the other. Details on the more roundabout estimations are given below:

dCas9 Production Rates from mRNA

We were unable to find a peptide chain elongation rate for S. aureus, so instead we used the values reported in BioNumber 107869 which gives a range of 0.59-3.17 amino acids per second per ribosome in Streptomyces coelicolor, another gram-positive bacteria. Freiburg's dCas9 part from last year is composed of 1372 amino acids. This translates to a range of 0.0258 to 0.1386 dCas9 molecules per minute per ribosome. We were unable to find ribosome densities in S. aureus, but found two different estimates for ribsosome density in Bionumbers: 0.22 ribosomes per 100 codons (i.e. per 3 nt coding sequence) and 3.46 ribosomes per 100 codons . 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.

Degradation Rates of dCas9

We were unable to find any specific data on dCas9 degradation, so instead we used a protein half-life of sarA measured in S. Aureus by Michelik et al. . We chose sarA rather than a protein more chemically similar to dCas9 because data on sarA was readily available and because dCas9 is transcribed using the sarA promoter, which allows us to at least capture sensitivity of the degradation rate to production.

Production Rates from the sarA Promoter

We used the time-series data given by Cheung et al. to estimate the rate of production from the sarA P2-P3-P1 promoter in S. aureus. The figure from their paper is reproduced below. After diluting 1:100, the S. aureus strains were serially monitored for OD_650. We used data from the sarA+ strain, as that's more like a wild-type S. aureus strain. Using the laboratory-conditions doubling time of 24 minutes given in given in , 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 sarA 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 and, using our assumption of a fixed number of active sarA genes, considered the relative change in number of molecules to be representative of the per-promoter rate. We were interested, however, in the changes of concentration rather than the changes in the raw number of molecules. As the name suggests, Staphylococcus aureus are spherical in shape. Assuming that all S. aureus are spheres, the volume of the cell can be determined. The diameter of a USA300 S. aureus cell was previously measured as 1.1 μ•m resulting in the overall cell volume to be calculated as 5.575•10-15 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 below: This resulted in a exponential model a•ebt with a b rate constant of 0.0011 nM/min.

References

[1]D. Bikard et al. “Programmable repression and activation of bacterial gene expression using an engineered CRISPR-Cas system”. In: Nucleic Acids Res. 41.15 (Aug. 2013), pp. 7429–7437.
[2]Florian Brandt et al. “The Native 3D Organization of Bacterial Polysomes”. In: Cell 136.2 (2009), pp. 261 –271. issn: 0092-8674. doi: 10.1016/j.cell.2008.11.016.
[3]A. G. Cheng, D. Missiakas, and O. Schneewind. “The giant protein Ebh is a determinant of Staphylococcus aureus cell size and complement resistance”. In: J. Bacteriol. 196.5 (2014), pp. 971–981.
[4]A. L. Cheung, K. Nishina, and A. C. Manna. “SarA of Staphylococcus aureus binds to the sarA promoter to regulate gene expression”. In: J. Bacteriol. 190.6 (Mar. 2008), pp. 2239–2243.
[5]G. Domingue, J. W. Costerton, and M. R. Brown. “Bacterial doubling time modulates the effects of opsonisation and available iron upon interactions between Staphylococcus aureus and human neutrophils”. In: FEMS Immunol. Med. Microbiol. 16.3-4 (Dec. 1996), pp. 223–228.
[6]S. Michalik et al. “Life and death of proteins: a case study of glucose-starved Staphylococcus aureus”. In: Mol. Cell Proteomics 11.9 (Sept. 2012), pp. 558–570.
[7]R. Milo et al. “BioNumbers-the database of key numbers in molecular and cell biology”. In: Nucleic Acids Res. 30 (Jan. 2010), pp. D750–D753. url: http://bionumbers.hms.harvard.edu/bionumber.aspx?id=107869}.
[8]L. S. Qi et al. “Repurposing CRISPR as an RNA-guided platform for sequence-specific control of gene expression”. In: Cell 152.5 (Feb. 2013), pp. 1173–1183.
[9]C. Roberts et al. “Characterizing the effect of the Staphylococcus aureus virulence factor regulator, SarA, on log-phase mRNA half-lives”. In: J. Bacteriol. 188.7 (Apr. 2006), pp. 2593–2603. doi: 10.1128/JB.188.7.2593-2603.2006
[10]Marlena Siwiak and Piotr Zielenkiewicz. “Transimulation - Protein Biosynthesis Web Service”. In: PLoS ONE 8.9 (Sept. 2013), e73943. doi: 10.1371/journal.pone.0073943.
[11]S.H. Sternberg et al. “DNA interrogation by the CRISPR RNA-guided endonuclease Cas9”. In: Nature 7490 (2014), 6267. doi: 10.1038/nature13011. url: http://www.nature.com/nature/journal/v507/n7490/full/nature13011.html.
[12]Freiburg iGEM Team. dCas9. BBa K1150000 Standard Biological Part. 2013. url: http://parts.igem.org/Part:BBa_K1150000.
[13]UCSF iGEM Team. Operation CRISPR: Decision Making Circuit Model. 2013. url: https://2013.igem.org/Team:UCSF/Modeling.
[14]Jian-Qiu Wu and Thomas D. Pollard. “Counting Cytokinesis Proteins Globally and Locally in Fission Yeast”. In: Science 310.5746 (2005), pp. 310–314. doi: 10.1126/science.1113230.
[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.
[19]Eyraud, A., Tattevin, P., Chabelskaya, S., & Felden, B. (2014). A small RNA controls a protein regulator involved in antibiotic resistance in Staphylococcus aureus. Nucleic acids research, gku149.
[20]Shimoni, Y., Friedlander, G., Hetzroni, G., Niv, G., Altuvia, S., Biham, O., & Margalit, H. (2007). Regulation of gene expression by small non‐coding RNAs: a quantitative view. Molecular Systems Biology, 3 (1)
[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.
[22] Swain, P. S. (2004). Efficient attenuation of stochasticity in gene expression through post-transcriptional control. Journal of molecular biology, 344 (4),965-976.
[23] Hussein, R., & Lim, H. N. (2012). Direct comparison of small RNA and transcription factor signaling. Nucleic acids research, 40 (15), 7269-7279.
[24] Levin, B.R., Stewart, F.M. and Rice, V.A. 1979. “The Kinetics of Conjugative Plasmid Transmission: Fit of a Simple Mass Action Model.” In: Plasmid. 2. pp. 247-260.
[25]Projan, S.J. and Archer, G.L. 1989. “Mobilization of the Relaxable Staphylococcus aureus Plasmid pC221 by the Conjugative Plasmid pGO1 Involves Three pC221 Loci.” In: Journal of Bacteriology. pp. 1841-1845.
[26]Phornphisutthimas, S., Thamchaipenet, A., and Panijpan, B. 2007. “Conjugation in Escherichia coli.” In: The International Union of Biochemistry and Molecular Biology. 35. 6. pp. 440-445.
[27]Phornphisutthimas, S., Thamchaipenet, A., and Panijpan, B. 2007. “Conjugation in Escherichia coli.” In: The International Union of Biochemistry and Molecular Biology. 35. 6. pp. 440-445.
[28]P Chung P., McNamara P.J., Campion J.J., Evans M.E. 2006. “Mechanism-based pharmacodynamic models of fluoroquinolone resistance in Staphylococcus aureus.” In: In: Antimicrobial Agents Chemotherapy. 50. pp. 2957-2965.
[29] Chang H., Wang L. “A Simple Proof of Thue's Theorem on Circle Packing” In: arXiv:1009.4322v1.

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