Team:Aberdeen Scotland/Equations

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<h1>Equations</h1>
<h1>Equations</h1>
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<p>Here we define the equation and parameters that describe the genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of protein production.  For the purpose of testing, we tagged the proteins with markers, green fluorescent protein(GFP) for one and cyan fluorescent protein (CFP) for the other. This allows us to easily measure the protein expression by measuring the concentrations of GFP and CFP.  For the sake of simplicity, we refer to the combination of protein and marker by the marker name (i.e. GFP and CFP).  This synthetic biological circuit is represented in Fig 1.</p>  
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 +
<p>Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.</p>
 +
<br>
<center>
<center>
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg">
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg">
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<p>Fig 1</p>
 
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</center>
 
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<p>We regulate the system by adding galactose or copper.  Galactose binds to the GAL promoter and activates the transcription of mRNA1 (M1), allowing the system to express GFP.  Copper binds to the CUP1 promoter, activating the transcription of mRNA2 (M2) and leading to the expression of CFP.
 
<br>
<br>
<br>
<br>
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Fig 1 shows the mutual inhibition of the translation of the two mRNAs by the proteins binding to the corresponding stem loop structures on the opposing construct.
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<p style="font-size:10px">Figure 1: Translation of DNA to mRNA.</p>
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</center>
<br>
<br>
 +
 +
<p>We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system  instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.</p>
<br>
<br>
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This means that if GFP is being expressed, the proteins will bind onto the M2 stem loops, thus preventing M2 from producing CFP.  CFP exhibits the same behaviour, inhibiting the production of GFP.</p>  
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<p>From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.</p>
<br>
<br>
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<h3>Equation Terms</h3>
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<p>For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP. </p>
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<p>Each equation is composed of three terms: <b>generation</b>, <b>degradation</b>, and <b>base rate</b>
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<br>
<br>
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<p>The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.</p>
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<br>
<br>
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<b> Generation:</b> There are two forms of the generation term: one for the mRNAs and one for the proteins (GFP and CFP).
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<h3>Equation 1</h3>
<br>
<br>
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For the mRNAs, the generation term is in the form of the <a href="http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics">Michaelis-Menten equation</a> with <a href="http://en.wikipedia.org/wiki/Cooperative_binding">Hill coefficients</a> to model the cooperativity of the binding affinities of the stem loops.
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<div align="center">
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<table>
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  <tr>
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    <td>
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    <div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div>
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    </td>
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    <td>
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    <div align="right"><p><b>(1)</b></p></div>
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    </td>
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  </tr>
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</table>
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</div>
<br>
<br>
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For the proteins (GFP and CFP), the Michaelis-Menten equation is modified to take into account the <a href="http://en.wikipedia.org/wiki/Non-competitive_inhibition">inhibition</a> of one protein on the other.  This describes how GFP inhibits the generation of CFP and vice-versa.
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<p>This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.</p>
<br>
<br>
 +
<p>[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.
<br>
<br>
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<b>Degradation:</b> This term describes the degradation the component within the cell and is a function of the <a href="http://en.wikipedia.org/wiki/Rate_equation">reaction kinetics</a> for the breakdown of the component over time and the dilution that occurs as the cell divides.
 
<br>
<br>
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[M1] is the concentration of mRNA that translates the N-peptide and GFP.</p>
<br>
<br>
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<b>Base Rate:</b> This is the concentration of molecules present in the cell when the promoter or inhibitor is not activated.</p>
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<div align="center">
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<table>
 +
  <tr>
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    <td>
 +
    <div align="right"><b><p>Parameter<p></b></div>
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    </td>
 +
    <td>
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    <p><b>Description</b></p>
 +
    </td>
 +
  </tr>
 +
    <tr>
 +
    <td>
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    <div align="right"><b><p>λ<sub style="font-size:10px">1</sub>:<p></b></div>
 +
    </td>
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    <td>
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    <p>Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP</p>
 +
    </td>
 +
  </tr>
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  <tr>
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    <td>
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    <div align="right"><b><p>μ<sub style="font-size:10px">1</sub>:</p></b></div>
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    </td>
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    <td>
 +
    <p>Constant representing rate of degradation of mRNA</p>
 +
    </td>
 +
  </tr>
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  <tr>
 +
    <td>
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    <div align="right"><b><p>n<sub style="font-size:10px">1</sub>:<p></b></div>
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    </td>
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    <td>
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    <p>Hill coefficient for the association between the galactose and the GAL promoter<p>
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    </td>
 +
  </tr>
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  <tr>
 +
    <td>
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    <div align="right"><b><p>K<sub style="font-size:10px">1</sub>:</p></b></div>
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    </td>
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    <td>
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    <p>Dissociation constant for the GAL promoter</p>
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    </td>
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  </tr>
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  <tr>
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    <td>
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    <div align="right"><b><p>T:</p></b></div>
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    </td>
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    <td>
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    <p>Time constant representing rate of cellular division</p>
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    </td>
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  </tr>
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</table>
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</div>
<br>
<br>
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<h3>Equations and Parameters</h3>
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<h3>Equation 2</h3>
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<p><div align="center"><b>Equation (1)</b> describes the rate of change of the mRNA (mRNA1) that is transcribed from the galactose promoter.</div></p>
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<br>
<div align="center">
<div align="center">
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   <tr>
   <tr>
     <td>
     <td>
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     <div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div>
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     <div align="center"><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"></div>
     </td>
     </td>
     <td>
     <td>
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     <div align="right"><p><b>(1)</b></p></div>
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     <div align="right"><p><b>(2)</b></p></div>
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    </td>
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  </tr>
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</table>
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</div>
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<br>
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<p>This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.</p>
 +
<br>
 +
<p>[M1] is the concentration of mRNA that translates the N-peptide GFP.
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<br>
 +
<br>
 +
[GFP] represents the concentration of N-peptide and GFP.
 +
<br>
 +
<br>
 +
[CFP] represents the concentration of the MS2-protein and CFP.</p>
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<br>
 +
<div align="center">
 +
<table>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>Parameter:<p></b></div>
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    </td>
 +
    <td>
 +
    <p><b>Description</b></p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>λ<sub style="font-size:10px">2</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>μ<sub style="font-size:10px">2</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of degradation of the GFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>n<sub style="font-size:10px">2</sub>:<p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Hill coefficient of the CFP/MS2 stem loop association<p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>K<sub style="font-size:10px">2</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Dissociation constant for the MS2-CFP protein to MS2 loop</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>T:</p></b></div>
 +
    </td>
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    <td>
 +
    <p>Time constant representing rate of cellular division</p>
     </td>
     </td>
   </tr>
   </tr>
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</div>
</div>
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<br>
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<h3>Equation 3</h3>
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<br>
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<div align="center">
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<table>
 +
  <tr>
 +
    <td>
 +
    <div align="center"><img src="https://static.igem.org/mediawiki/2010/2/23/Dm2.jpg"></div>
 +
    </td>
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    <td>
 +
    <div align="right"><p><b>(3)</b></p></div>
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    </td>
 +
  </tr>
 +
</table>
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</div>
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<br>
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<div table>
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<p>This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.</p>
-
<b>[GAL]:</b>
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-
represents the concentration of galactose that is added to the system; when galactose is added, it binds to the promoter and activates the transcription of M1
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<br>
<br>
-
<b>[M1]:</b>
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<p>[Cu<sup style="font-size:10px">2+</sup>] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.
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is the concentration of mRNA that translates the N-peptide and GFP
+
<br>
<br>
-
<b>λ1:</b>
 
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constant representing the rate of transcription of the DNA that encodes for the production of N-peptide and GFP
 
<br>
<br>
-
<b>μ1:</b>
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[M2] represents the concentration of mRNA that translates the MS2-protein and CFP. </p>
-
constant representing the rate of degradation of mRNA
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<br>
<br>
-
<b>n1:</b>
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<div align="center">
-
Hill coefficient
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<table>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>Parameter</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p><b>Description</b></p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>λ<sub style="font-size:10px">3</sup>:<p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of transcription of the DNA that encodes for the production of the MS2-protein and CFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>μ<sub style="font-size:10px">3</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of degradation of mRNA</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>n<sub style="font-size:10px">3</sub>:<p></b></div>
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    </td>
 +
    <td>
 +
    <p>Hill coefficient of the association between copper and the CUP1 promoter<p>
 +
    </td>
 +
  </tr>
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  <tr>
 +
    <td>
 +
    <div align="right"><b><p>K<sub style="font-size:10px">3</sub>:</p></b></div>
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    </td>
 +
    <td>
 +
    <p>Dissociation constant for Copper promoter</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>T:</p></b></div>
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    </td>
 +
    <td>
 +
    <p>Time constant representing rate of cellular division</p>
 +
    </td>
 +
  </tr>
 +
</table>
 +
</div>
 +
 
<br>
<br>
-
<b>k1:</b>
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constant of association between galactose and DNA
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<h3>Equation 4</h3>
<br>
<br>
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<b>T:</b>
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<div align="center">
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time constant representing rate of cellular division
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<table>
 +
  <tr>
 +
    <td>
 +
    <div align="center"><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"></div>
 +
    </td>
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    <td>
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    <div align="right"><p><b>(4)</b></p></div>
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    </td>
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  </tr>
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</table>
</div>
</div>
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<br>
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 +
<p>This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.</p>
 +
<br>
 +
<p>[M2] is the concentration of mRNA that translates to MS2-protein and CFP.
 +
<br>
 +
<br>
 +
[GFP] represents the concentration of the N-peptide and GFP.
 +
<br>
 +
<br>
 +
[CFP] represents the concentration of the MS2-protein and CFP.</p>
 +
<br>
 +
<div align="center">
 +
<table>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>Parameters<p></b></div>
 +
    </td>
 +
    <td>
 +
    <p><b>Description</b></p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>λ<sub style="font-size:10px">4</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>μ<sub style="font-size:10px">4</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Constant representing rate of degradation of the CFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>n<sub style="font-size:10px">4</sub>:<p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Hill coefficient of the GFP/Bbox stem loop association<p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>K<sub style="font-size:10px">4</sub>:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>T:</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p>time constant representing rate of cellular division</p>
 +
    </td>
 +
  </tr>
 +
</table>
 +
</div>
 +
<br>
 +
<h1>Parameter Study</h1>
 +
<p>The parameter values were first estimated based on the literature <a href="#ref1"><sup style="font-size:10px">[1]</sup></a> and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Parameter Space Analysis</a> and <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution">Directed Evolution</a>.</p>
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<br>
 +
<h1>Modification of the construct</h1>
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 +
<p>Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:<p>
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<br>
 +
<center>
 +
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png">
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<br><br>
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<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png">
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</center>
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<br>
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<p>The behaviour of the switch can then be summarise in the following table:</p>
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<br>
 +
<div align="center">
 +
<table>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><b><p>What is present in the system</p></b></div>
 +
    </td>
 +
    <td>
 +
    <p><b>Protein(s) produced</b></p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><p>Galactose and Methionine</p></div>
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    </td>
 +
    <td>
 +
    <p>GFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><p>Galactose only</p></div>
 +
    </td>
 +
    <td>
 +
    <p>GFP, CFP (doses dependent)</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><p>Methionine only</p></div>
 +
    </td>
 +
    <td>
 +
    <p>No GFP or CFP</p>
 +
    </td>
 +
  </tr>
 +
  <tr>
 +
    <td>
 +
    <div align="right"><p>No Galactose and no Methionine</p></div>
 +
    </td>
 +
    <td>
 +
    <p>CFP</p>
 +
    </td>
 +
  </tr>
 +
</table>
 +
</div>
 +
<br>
 +
 +
<h1>References</h1>
 +
<a name="ref1"></a>
 +
<p><sup style="font-size:10px">[1]</sup> Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, <i>Mol Cell Proteomics.</i>, Vol. 3, No.11, pp. 1083-1092.</p>
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<br>
 +
<p><sup style="font-size:10px">[2]</sup> Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall. </p>
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<br><br>
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<hr>
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<table class="nav">
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<tr>
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<td>
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<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Modeling"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to the Modelling Summary</a>
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</td>
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<td align="right">
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<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points">Continue to Nullclines and Fixed Points&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a>
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</td>
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</tr>
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</table>
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<div align="center"><p><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"/></p></div>
 
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<p>This equation describes the rate of change of GFP that is translated from mRNA1.</p>
 
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<div align="center"><p><img src="https://static.igem.org/mediawiki/2010/2/23/Dm2.jpg"/></p></div>
 
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<p>This equation describes the rate of change of the mRNA (mRNA2) that is transcribed from the copper promoter.</p>
 
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<div align="center"><p><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"/></p></div>
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</html>
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<p>This equation describes the rate of change of CFP translated from mRNA2.</p>
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Latest revision as of 20:29, 27 October 2010

University of Aberdeen - ayeSwitch - iGEM 2010

Equations

Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.




Figure 1: Translation of DNA to mRNA.


We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.


From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.


For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP.


The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.


Equation 1

(1)

This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.


[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.

[M1] is the concentration of mRNA that translates the N-peptide and GFP.


Parameter

Description

λ1:

Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP

μ1:

Constant representing rate of degradation of mRNA

n1:

Hill coefficient for the association between the galactose and the GAL promoter

K1:

Dissociation constant for the GAL promoter

T:

Time constant representing rate of cellular division

Equation 2

(2)

This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.


[M1] is the concentration of mRNA that translates the N-peptide GFP.

[GFP] represents the concentration of N-peptide and GFP.

[CFP] represents the concentration of the MS2-protein and CFP.


Parameter:

Description

λ2:

Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP

μ2:

Constant representing rate of degradation of the GFP

n2:

Hill coefficient of the CFP/MS2 stem loop association

K2:

Dissociation constant for the MS2-CFP protein to MS2 loop

T:

Time constant representing rate of cellular division

Equation 3

(3)

This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.


[Cu2+] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.

[M2] represents the concentration of mRNA that translates the MS2-protein and CFP.


Parameter

Description

λ3:

Constant representing rate of transcription of the DNA that encodes for the production of the MS2-protein and CFP

μ3:

Constant representing rate of degradation of mRNA

n3:

Hill coefficient of the association between copper and the CUP1 promoter

K3:

Dissociation constant for Copper promoter

T:

Time constant representing rate of cellular division

Equation 4

(4)

This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.


[M2] is the concentration of mRNA that translates to MS2-protein and CFP.

[GFP] represents the concentration of the N-peptide and GFP.

[CFP] represents the concentration of the MS2-protein and CFP.


Parameters

Description

λ4:

Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP

μ4:

Constant representing rate of degradation of the CFP

n4:

Hill coefficient of the GFP/Bbox stem loop association

K4:

Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop

T:

time constant representing rate of cellular division

Parameter Study

The parameter values were first estimated based on the literature [1] and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see Parameter Space Analysis and Directed Evolution.


Modification of the construct

Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:





The behaviour of the switch can then be summarise in the following table:


What is present in the system

Protein(s) produced

Galactose and Methionine

GFP

Galactose only

GFP, CFP (doses dependent)

Methionine only

No GFP or CFP

No Galactose and no Methionine

CFP

References

[1] Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, Mol Cell Proteomics., Vol. 3, No.11, pp. 1083-1092.


[2] Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall.



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