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

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.