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

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Full equations  
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<h1><font color="#008080">Full equations</font></h1>
 +
<h2>1 Notations</h2>
  
−
1 Notations
+
<h3>1.1 Notation principles</h3>
  
−
1.1 Notation principles
+
<p>Every relevant substance in the cell is denoted with uppercase  
−
 
+
−
Every relevant substance in the cell is denoted with uppercase  
+
 
letters which describes the substance, and a subscript which  
 
letters which describes the substance, and a subscript which  
 
encodes the scale in which the amount of the substance is  
 
encodes the scale in which the amount of the substance is  
−
measured by the variable. For example, if we have a substance Y,  
+
measured by the variable. For example, if we have a substance Y, </p>
  
−
• its amount inside a single cell is denoted by Y_{in}
+
<ul>
−
.
+
<li> its amount inside a single cell is denoted by Y<sub>in</sub></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}
+
   inside the cells) is denoted by Y<sub>sum</sub></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}
+
   denoted by Y<sub>out</sub></li>
−
.
+
  
−
2 A list of all the notations we used
+
</ul>
 +
<h2>2 A list of all the notations we used</h2>
  
−
Substances:
+
<h3>Substances:</h3>
−
 
+
<ul>
−
A
+
<li>A
 
  - AHL (The auto inducer, a short for N-Acyl homoserine  
 
  - AHL (The auto inducer, a short for N-Acyl homoserine  
−
lactone).
+
lactone).</li>
  
−
L - LuxR (a transciptional activator protein)
+
<li>L - LuxR (a transciptional activator protein)</li>
  
−
LA - the complex LuxR and AHL form together.
+
<li>LA - the complex LuxR and AHL form together.</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.
+
together.</li>
  
−
aa - Aiia (a AHL-lactonase).
+
<li>aa - Aiia (a AHL-lactonase).</li>
  
−
a_{1}
+
<li>a<sub>1</sub>
−
- plasmids with an unactivated LuxR promotor.
+
- plasmids with an unactivated LuxR promotor.</li>
  
−
a_{2}
+
<li>a<sub>2</sub>
−
- plasmids with an activated LuxR promotor.
+
- plasmids with an activated LuxR promotor.</li>
 +
<li>RNA<sub>TetR</sub>
 +
  - RNA strands of the TetR gene.</li>
  
−
TRLV -  
+
<li>TetR - Tet Repressor protein we use.</li>
  
−
b_{1}
+
<li>b<sub>0</sub>
−
- plasmids with an unactivated Tet promotor.
+
- plasmids with an unactivated Tet promotor.</li>
  
−
b_{2}
+
<li>b<sub>1</sub>
−
- plasmids with an activated Tet promotor.
+
- plasmids with an activated Tet promotor.</li>
 +
<li>RNA<sub>ccdb</sub>
 +
  - RNA strands of the ccdb gene.</li>
  
−
ccbd - Toxin we use to kill the cell.
 
  
−
X - any gene we want to measure the amount of it that will be  
+
<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  
 
produced by the bacteria colony. For example, it might represent  
 
produced by the bacteria colony. For example, it might represent  
−
the amount of a certain drug the bacteria produce.
+
the amount of a certain drug the bacteria produce.</li>
 +
</ul>
  
−
Other quantitie of interest:
+
<h3>Other quantitie of interest:</h3>
 +
<ul>
  
−
N - number of bacteria. The bacteria are divided to two groups
+
<li>N - number of bacteria. The bacteria are divided to two groups</li>
  
−
   N^{+}
+
<li>   N<sup>+</sup>
−
  - bacteria with our plasmid.
+
  - bacteria with our plasmid.</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).
+
bacteria that lost the plasmids we introduced into them).</li>
−
 
+
−
V - volume of the relevant scale. That means,
+
−
 
+
−
   V_{out}
+
−
- the volume of the space outside the cells.
+
−
 
+
−
   V_{sum}
+
−
- 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 Reactions
+
−
 
+
−
 
+
−
 
+
−
 
+
−
\frac{dA_{out}}{dt}=-D(\frac{A_{out}}{V_{out}}-\frac{A_{sum}}{V_{sum}})(N^{+}+N^{-})
+
−
 
+
−
\frac{dA_{sum}}{dt}=D(\frac{A_{out}}{V_{out}}-\frac{A_{sum}}{V_{sum}})N^{+}-\frac{c_{1}aa_{in}A_{sum}}{c_{18}+A_{in}}+c_{4}LA_{sum}-(c_{3}L_{in}\cdot A_{sum})-c_{2}A_{sum}
+
−
 
+
−
\frac{dLA_{sum}}{dt}=c_{3}L_{in}\cdot A_{sum}-c_{4}LA_{sum}-2(c_{5}LA_{sum}LA_{in}-c_{6}LA_{2,sum})
+
−
 
+
−
\frac{dLA_{2,sum}}{dt}=c_{5}LA_{sum}LA_{in}-c_{6}LA_{2,sum}-(c_{7}a_{0,in}LA_{2,sum}-c_{8}a_{1,sum})
+
−
 
+
−
 
+
−
\frac{d(a_{0,in}+a_{1,in})}{dt}=0
+
−
\frac{da_{1,in}}{dt}=c_{7}a_{0,in}LA_{2,in}-c_{8}a_{1,in}
+
−
 
+
−
\frac{dTRLV_{sum}}{dt}=A_{RBS}\cdot(a_{0,sum}v_{0}+a_{1,sum}v_{1})-c_{9}TRLV_{sum}-(c_{1}b_{0,in}TRLV_{sum}-c_{11}b_{1,in})
+
−
 
+
−
 
+
−
\frac{d(b_{0,in}+b_{1,in})}{dt}=0
+
−
\frac{db_{1,in}}{dt}=c_{1}b_{0,in}TRLV_{in}-c_{11}b_{1,in}
+
−
 
+
−
\frac{dccdb_{sum}}{dt}=B_{RBS}\cdot(b_{0,sum}u_{0}+b_{1,sum}u_{1})-c_{12}ccdb_{sum}
+
−
 
+
−
\frac{dx_{tot}}{dt}=c_{13}N^{+}
+
−
 
+
−
 
+
−
\frac{dL_{sum}}{dt}=c_{14}N^{+}-c_{15}L_{sum}-(c_{3}L_{in}\cdot A_{sum}-c_{4}LA_{sum})
+
−
 
+
−
 
+
−
\frac{daa_{sum}}{dt}=c_{16}N^{+}-c_{17}aa_{sum}
+
−
 
+
−
 
+
−
\frac{dN^{+}}{dt}=\frac{ln(2-p)}{T^{+}}N^{+}(1-\frac{N^{+}+N^{-}}{N_{max}})
+
−
 
+
−
 
+
−
\frac{dN^{-}}{dt}=\frac{ln2}{T^{-}}N^{-}(1-\frac{N^{+}+N^{-}}{N_{max}})+\frac{ln2-ln(2-p)}{T^{+}}N^{+}
+
−
 
+
−
 
+
−
With some assumptions
+
−
 
+
−
Assumptions
+
−
 
+
−
4 section
+
−
 
+
−
\frac{dA_{out}}{dt}=-(\frac{A_{out}}{V_{out}}-\frac{A_{sum}}{V_{sum}})(N^{+}+N^{-})c_{20}Area_{in}
+
−
 
+
−
\frac{dA_{sum}}{dt}=(\frac{A_{out}}{V_{out}}-\frac{A_{sum}}{V_{sum}})N^{+}c_{20}Area_{in}-\frac{c_{1}aa_{in}A_{sum}}{c_{18}+A_{in}}-(c_{3}L_{in}\cdot A_{sum})
+
−
 
+
−
\frac{dLA_{sum}}{dt}=c_{3}L_{in}\cdot A_{sum}-2(c_{5}LA_{sum}LA_{in}-c_{6}LA_{2,sum})
+
−
 
+
−
\frac{dLA_{2,sum}}{dt}=c_{5}LA_{sum}LA_{in}-c_{6}LA_{2,sum}
+
−
 
+
−
 
+
−
\frac{d(a_{0,in}+a_{1,in})}{dt}=0
+
−
\frac{da_{1,in}}{dt}=c_{7}a_{0,in}LA_{2,in}-c_{8}a_{1,in}
+
−
 
+
−
\frac{dTRLV_{sum}}{dt}=A_{RBS}\cdot(a_{0,sum}v_{0}+a_{1,sum}v_{1})
+
−
 
+
−
 
+
−
\frac{d(b_{0,in}+b_{1,in})}{dt}=0
+
−
\frac{db_{1,in}}{dt}=c_{1}b_{0,in}TRLV_{in}-c_{11}b_{1,in}
+
−
 
+
−
\frac{dccdb_{sum}}{dt}=B_{RBS}\cdot(b_{0,sum}u_{0}+b_{1,sum}u_{1})-c_{12}ccdb_{sum}
+
−
 
+
−
\frac{dx_{tot}}{dt}=c_{13}(N^{+}+N^{-})
+
−
 
+
−
 
+
−
\frac{dL_{sum}}{dt}=c_{14}N^{+}-(c_{3}L_{in}\cdot A_{sum}-c_{4}LA_{sum})
+
−
 
+
−
 
+
−
\frac{daa_{sum}}{dt}=c_{16}N^{+}
+
−
 
+
−
 
+
−
\frac{dN^{+}}{dt}=\alpha^{+}N^{+}(1-\frac{N^{+}+N^{-}}{N_{max}})(1-\mu)
+
−
 
+
−
 
+
−
\frac{N^{-}}{dt}=\alpha^{-}N^{-}(1-\frac{N^{+}+N^{-}}{N_{max}})+\alpha^{+}N^{+}(1-\frac{N^{+}+N^{-}}{N_{max}})(1-\mu)
+
−
 
+
−
 
+
−
Initial conditions
+
−
 
+
−
-----
+
−
 
+
−
AHL_{out}
+
−
- how much AHL we put.
+
−
 
+
−
a0 - initial number of strands (probably plasmid number).
+
  
−
a1 - 0.
+
<li>V - volume of the relevant scale. That means,
 +
<ul>
 +
<li>   V<sub>out</sub>
 +
- the volume of the space outside the cells.</li>
 +
 <li>  V<sub>sum</sub>
 +
- the volume of the total space inside all the cells.</li></ul>
 +
</li>
  
−
b0 - initial number of strands (probably plasmid number). Sounds
+
<li>w - width of the cell membrane.</li>
−
equal to a_{0}(t=0)
+
</ul>
−
.
+
  
−
b1 - 0.
+
<h3>Constants</h3>
 +
<ul>
  
−
N^{+}
+
<li>C1 - C18 - different reaction constants.</li>
−
- the number of cells we have at the beginning.
+
  
−
N^{-}
+
<li>T<sup>+</sup>
−
- 0.
+
- plamid positive generation time.</li>
  
−
all the rest - 0.
+
<li>T<sup>-</sup>
 +
- plamid free generation time.</li>
  
−
Ways to compute things
+
<li>p - the chance to loose a plasmid.</li>
  
−
\alpha^{+}=\frac{1-p}{T^{+}}+\frac{p}{T^{-}}
+
<li>D- AHL diffusion constant.</li>
 +
</ul>
  
 +
<h2>3 Equations</h2>
  
−
\mu=1-\frac{ln(2-x)}{ln2}
+
<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>
  
 +
<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>
  
−
\alpha^{-}=\frac{2^{\frac{T^{+}}{T^{-}}}-1}{T^{-}}
 
  
  
−
Sub_{sum}=N^{+}Sub_{in}
+
<h3>Initial conditions</h3>
  
 +
<ul>
  
−
V_{out}\sim V_{tot}
+
<li>AHL<sub>out</sub>
 +
- how much AHL we put.</li>
  
 +
<li>a0 - initial number of strands (probably plasmid number).</li>
  
−
Things to talk about
+
<li>a1 - 0.</li>
  
−
• The way I took into account the plasmid-less bacteria.
+
<li>b0 - initial number of strands (probably plasmid number). Sounds
 +
equal to a<sub>0</sub>(t=0)
 +
.</li>
  
−
• Mistakes in first equation and what used to be the last one.
+
<li>b1 - 0.</li>
  
−
• Meaningful names.
+
<li>N<sup>+</sup>
 +
- the number of cells we have at the beginning.</li>
  
−
• RNA transcription. In other places, they replace (5-7) with
+
<li>N<sup>-</sup>
−
  this:
+
- 0.</li>
−
\frac{dTRLV}{dt}=
+
  
 +
<li>all the rest - 0.</li>
 +
</ul>
  
−
• Validity of the plasmid loss computations.
 
  
  
  
 
</html>
 
</html>

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.