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^{+} 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.