Difference between revisions of "Team:Technion HS Israel/Modelling/Equations"

 
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<h1>Full equations </h1>
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<h1><font color="#008080">Full equations</font></h1>
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<h2>1 Notations</h2>
 
<h2>1 Notations</h2>
  
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<ul>
 
<ul>
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<li> its amount inside a single cell is denoted by Y_{in}</li>
+
<li> its amount inside a single cell is denoted by Y<sub>in</sub></li>
  
 
<li> its amount inside all the cells together (its total amount  
 
<li> its amount inside all the cells together (its total amount  
−
   inside the cells) is denoted by Y_{sum}</li>
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   inside the cells) is denoted by Y<sub>sum</sub></li>
  
 
<li> its amount outside all the cells (its external amount) is  
 
<li> its amount outside all the cells (its external amount) is  
−
   denoted by Y_{out}</li>
+
   denoted by Y<sub>out</sub></li>
  
 
</ul>
 
</ul>
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<li>LA - the complex LuxR and AHL form together.</li>
 
<li>LA - the complex LuxR and AHL form together.</li>
  
−
<li>LA_{2}
+
<li>LA<sub>2</sub>
 
- the dimer we get when two LuxR-AHL complexes bind  
 
- the dimer we get when two LuxR-AHL complexes bind  
 
together.</li>
 
together.</li>
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<li>aa - Aiia (a AHL-lactonase).</li>
 
<li>aa - Aiia (a AHL-lactonase).</li>
  
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<li>a_{1}
+
<li>a<sub>1</sub>
 
- plasmids with an unactivated LuxR promotor.</li>
 
- plasmids with an unactivated LuxR promotor.</li>
  
−
<li>a_{2}
+
<li>a<sub>2</sub>
 
- plasmids with an activated LuxR promotor.</li>
 
- plasmids with an activated LuxR promotor.</li>
 +
<li>RNA<sub>TetR</sub>
 +
  - RNA strands of the TetR gene.</li>
  
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<li>TRLV -<font color="red">NOTICE IM EMPTY??</font> </li>
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<li>TetR - Tet Repressor protein we use.</li>
  
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<li>b_{1}
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<li>b<sub>0</sub>
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- plasmids with an unactivated Tet promotor.</li>
+
- plasmids with an unactivated Tet promotor.</li>
  
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<li>b_{2}
+
<li>b<sub>1</sub>
−
- plasmids with an activated Tet promotor.</li>
+
- plasmids with an activated Tet promotor.</li>
 +
<li>RNA<sub>ccdb</sub>
 +
  - RNA strands of the ccdb gene.</li>
  
−
</li>ccbd - Toxin we use to kill the cell.</li>
+
 
 +
<li>ccbd - Toxin we use to kill the cell.</li>
  
 
<li>X - any gene we want to measure the amount of it that will be  
 
<li>X - any gene we want to measure the amount of it that will be  
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<li>N - number of bacteria. The bacteria are divided to two groups</li>
 
<li>N - number of bacteria. The bacteria are divided to two groups</li>
  
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<li>   N^{+}
+
<li>   N<sup>+</sup>
 
  - bacteria with our plasmid.</li>
 
  - bacteria with our plasmid.</li>
  
−
<li>   N^{-}
+
<li>   N<sup>-</sup>
 
  - bacteria without our plasmid (in other words,  
 
  - bacteria without our plasmid (in other words,  
 
bacteria that lost the plasmids we introduced into them).</li>
 
bacteria that lost the plasmids we introduced into them).</li>
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<li>V - volume of the relevant scale. That means,
 
<li>V - volume of the relevant scale. That means,
 
<ul>
 
<ul>
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<li>   V_{out}
+
<li>   V<sub>out</sub>
 
  - the volume of the space outside the cells.</li>
 
  - the volume of the space outside the cells.</li>
−
 
+
 <li>  V<sub>sum</sub>
−
 <li>  V_{sum}
+
  - the volume of the total space inside all the cells.</li></ul>
−
  - the volume of the total space inside all the cells.</li></ul></li>
+
</li>
  
 
<li>w - width of the cell membrane.</li>
 
<li>w - width of the cell membrane.</li>
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<li>C1 - C18 - different reaction constants.</li>
 
<li>C1 - C18 - different reaction constants.</li>
  
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<li>T^{+}
+
<li>T<sup>+</sup>
 
  - plamid positive generation time.</li>
 
  - plamid positive generation time.</li>
  
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<li>T^{-}
+
<li>T<sup>-</sup>
 
  - plamid free generation time.</li>
 
  - plamid free generation time.</li>
  
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</ul>
 
</ul>
  
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<h2>3 Reactions</h2>
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<h2>3 Equations</h2>
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 +
<p><img src="https://static.igem.org/mediawiki/2015/2/25/Technion_HS_2015_eq1.png" alt="equations 1-9 of our model" width="1074px" height="641px" ></img></p>
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<p><img src="https://static.igem.org/mediawiki/2015/5/54/Technion_HS_2015_eq2.png" alt="equations 10-17 of our model" width="1074px" height="680px" ></img></p>
  
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<p><img src="https://static.igem.org/mediawiki/2015/5/5e/Technion_HS_Israel_eq1.png" alt="equations 1-8 of our model" width="933px" height="600px" ></img></p>
 
  
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<p><img src="https://static.igem.org/mediawiki/2015/1/1c/Technion_HS_Israel_eq2.png" alt="equations 9-13 of our model" width="920px" height="347px" ></img></p>
 
  
 
<h3>Initial conditions</h3>
 
<h3>Initial conditions</h3>
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<ul>
 
<ul>
  
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<li>AHL_{out}
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<li>AHL<sub>out</sub>
 
- how much AHL we put.</li>
 
- how much AHL we put.</li>
  
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<li>b0 - initial number of strands (probably plasmid number). Sounds  
 
<li>b0 - initial number of strands (probably plasmid number). Sounds  
−
equal to a_{0}(t=0)
+
equal to a<sub>0</sub>(t=0)
 
.</li>
 
.</li>
  
 
<li>b1 - 0.</li>
 
<li>b1 - 0.</li>
  
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<li>N^{+}
+
<li>N<sup>+</sup>
 
- the number of cells we have at the beginning.</li>
 
- the number of cells we have at the beginning.</li>
  
−
<li>N^{-}
+
<li>N<sup>-</sup>
 
- 0.</li>
 
- 0.</li>
  
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</ul>
 
</ul>
  
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Ways to compute things
 
−
 
−
\alpha^{+}=\frac{1-p}{T^{+}}+\frac{p}{T^{-}}
 
−
 
−
 
−
\mu=1-\frac{ln(2-x)}{ln2}
 
−
 
−
 
−
\alpha^{-}=\frac{2^{\frac{T^{+}}{T^{-}}}-1}{T^{-}}
 
−
 
−
 
−
Sub_{sum}=N^{+}Sub_{in}
 
−
 
−
 
−
V_{out}\sim V_{tot}
 
−
 
−
 
−
Things to talk about
 
−
 
−
• The way I took into account the plasmid-less bacteria.
 
−
 
−
• Mistakes in first equation and what used to be the last one.
 
−
 
−
• Meaningful names.
 
−
 
−
• RNA transcription. In other places, they replace (5-7) with
 
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  this:
 
−
\frac{dTRLV}{dt}=
 
−
 
−
 
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• Validity of the plasmid loss computations.
 
  
  
  
 
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Latest revision as of 00:09, 19 September 2015

Technion 2015 HS Team's Wiki

Full equations

1 Notations

1.1 Notation principles

Every relevant substance in the cell is denoted with uppercase letters which describes the substance, and a subscript which encodes the scale in which the amount of the substance is measured by the variable. For example, if we have a substance Y,

  • its amount inside a single cell is denoted by Yin
  • its amount inside all the cells together (its total amount inside the cells) is denoted by Ysum
  • its amount outside all the cells (its external amount) is denoted by Yout

2 A list of all the notations we used

Substances:

  • A - AHL (The auto inducer, a short for N-Acyl homoserine lactone).
  • L - LuxR (a transciptional activator protein)
  • LA - the complex LuxR and AHL form together.
  • LA2 - the dimer we get when two LuxR-AHL complexes bind together.
  • aa - Aiia (a AHL-lactonase).
  • a1 - plasmids with an unactivated LuxR promotor.
  • a2 - plasmids with an activated LuxR promotor.
  • RNATetR - RNA strands of the TetR gene.
  • TetR - Tet Repressor protein we use.
  • b0 - plasmids with an unactivated Tet promotor.
  • b1 - plasmids with an activated Tet promotor.
  • RNAccdb - RNA strands of the ccdb gene.
  • ccbd - Toxin we use to kill the cell.
  • X - any gene we want to measure the amount of it that will be produced by the bacteria colony. For example, it might represent the amount of a certain drug the bacteria produce.

Other quantitie of interest:

  • N - number of bacteria. The bacteria are divided to two groups
  •    N+ - bacteria with our plasmid.
  •    N- - bacteria without our plasmid (in other words, bacteria that lost the plasmids we introduced into them).
  • V - volume of the relevant scale. That means,
    •    Vout - the volume of the space outside the cells.
      •  
    •   Vsum - the volume of the total space inside all the cells.
  • w - width of the cell membrane.

Constants

  • C1 - C18 - different reaction constants.
  • T+ - plamid positive generation time.
  • T- - plamid free generation time.
  • p - the chance to loose a plasmid.
  • D- AHL diffusion constant.

3 Equations

equations 1-9 of our model

equations 10-17 of our model

Initial conditions

  • AHLout - how much AHL we put.
  • a0 - initial number of strands (probably plasmid number).
  • a1 - 0.
  • b0 - initial number of strands (probably plasmid number). Sounds equal to a0(t=0) .
  • b1 - 0.
  • N+ - the number of cells we have at the beginning.
  • N- - 0.
  • all the rest - 0.