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 Y_in
its amount inside all the cells together (its total amount inside the cells) is denoted by Y_sum
its amount outside all the cells (its external amount) is denoted by Y_out

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.
RNA_TetR - 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.
RNA_ccdb - 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,
- 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 Equations

equations 1-9 of our model

equations 10-17 of our model

Initial conditions

AHL_out - 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.

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-<h1>Full equations </h1>
+<h1><font color="#008080">Full equations</font></h1>
 <h2>1 Notations</h2>
@@ Line 15: / Line 14: @@
 <ul>
-<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
-   inside the cells) is denoted by Y_{sum}</li>
+   inside the cells) is denoted by Y<sub>sum</sub></li>
 <li> its amount outside all the cells (its external amount) is
-   denoted by Y_{out}</li>
+   denoted by Y<sub>out</sub></li>
 </ul>
@@ Line 36: / Line 35: @@
 <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
 together.</li>
@@ Line 42: / Line 41: @@
 <li>aa - Aiia (a AHL-lactonase).</li>
-<li>a_{1}
+<li>a<sub>1</sub>
 - plasmids with an unactivated LuxR promotor.</li>
-<li>a_{2}
+<li>a<sub>2</sub>
 - plasmids with an activated LuxR promotor.</li>
+<li>RNA<sub>TetR</sub>
+  - RNA strands of the TetR gene.</li>
-<li>TRLV -<font color="red">NOTICE IM EMPTY??</font> </li>
+<li>TetR - Tet Repressor protein we use.</li>
-<li>b_{1}
+<li>b<sub>0</sub>
-- plasmids with an unactivated Tet promotor.</li>
+ - plasmids with an unactivated Tet promotor.</li>
-<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
@@ Line 68: / Line 72: @@
 <li>N - number of bacteria. The bacteria are divided to two groups</li>
-<li>   N^{+}
+<li>   N<sup>+</sup>
   - bacteria with our plasmid.</li>
-<li>   N^{-}
+<li>   N<sup>-</sup>
   - bacteria without our plasmid (in other words,
 bacteria that lost the plasmids we introduced into them).</li>
@@ Line 77: / Line 81: @@
 <li>V - volume of the relevant scale. That means,
 <ul>
-<li>   V_{out}
+<li>   V<sub>out</sub>
   - 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>
@@ Line 91: / Line 95: @@
 <li>C1 - C18 - different reaction constants.</li>
-<li>T^{+}
+<li>T<sup>+</sup>
   - plamid positive generation time.</li>
-<li>T^{-}
+<li>T<sup>-</sup>
   - plamid free generation time.</li>
@@ Line 102: / Line 106: @@
 </ul>
-<h2>3 Reactions</h2>
+<h2>3 Equations</h2>
+<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>
-<div lang="latex">\frac{dA_{out}}{dt}=-D(\frac{A_{out}}{V_{out}}-\frac{A_{sum}}{V_{sum}})(N^{+}+N^{-})</div>
+<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>
-\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})
+<h3>Initial conditions</h3>
+<ul>
-\frac{d(a_{0,in}+a_{1,in})}{dt}=0
+<li>AHL<sub>out</sub>
-\frac{da_{1,in}}{dt}=c_{7}a_{0,in}LA_{2,in}-c_{8}a_{1,in}
+- how much AHL we put.</li>
-\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})
+<li>a0 - initial number of strands (probably plasmid number).</li>
+<li>a1 - 0.</li>
-\frac{d(b_{0,in}+b_{1,in})}{dt}=0
+<li>b0 - initial number of strands (probably plasmid number). Sounds
-\frac{db_{1,in}}{dt}=c_{1}b_{0,in}TRLV_{in}-c_{11}b_{1,in}
+equal to a<sub>0</sub>(t=0)
+.</li>
-\frac{dccdb_{sum}}{dt}=B_{RBS}\cdot(b_{0,sum}u_{0}+b_{1,sum}u_{1})-c_{12}ccdb_{sum}
+<li>b1 - 0.</li>
-\frac{dx_{tot}}{dt}=c_{13}N^{+}
+<li>N<sup>+</sup>
+- the number of cells we have at the beginning.</li>
+<li>N<sup>-</sup>
+- 0.</li>
-\frac{dL_{sum}}{dt}=c_{14}N^{+}-c_{15}L_{sum}-(c_{3}L_{in}\cdot A_{sum}-c_{4}LA_{sum})
+<li>all the rest - 0.</li>
+</ul>
-\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
-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.
-b0 - initial number of strands (probably plasmid number). Sounds
-equal to a_{0}(t=0)
-.
-b1 - 0.
-N^{+}
-- the number of cells we have at the beginning.
-N^{-}
-- 0.
-all the rest - 0.
-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
-  this:
-\frac{dTRLV}{dt}=
-• Validity of the plasmid loss computations.
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