Difference between revisions of "Team:Technion HS Israel/Modelling/Equations"
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<h1>Full equations </h1> | <h1>Full equations </h1> | ||
Revision as of 06:34, 16 September 2015
'
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.
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.
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.
- LA_{2} - the dimer we get when two LuxR-AHL complexes bind together.
- aa - Aiia (a AHL-lactonase).
- a_{1} - plasmids with an unactivated LuxR promotor.
- a_{2} - plasmids with an activated LuxR promotor.
- TRLV -NOTICE IM EMPTY??
- b_{1} - plasmids with an unactivated Tet promotor.
- b_{2} - plasmids with an activated Tet promotor. 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 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^{+}