Team:Technion HS Israel/Modelling/Equations

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.
  • 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

equations 1-8 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 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.