# IGEM:IMPERIAL/2009/M1/Modelling/M1 1/detailed

The E.ncapsulator: the disease killer

### In the absence of IPTG:

Equation 1: Equation describing the rate of transcription of LacI MRNA (MLacI):

kmlacI is the transcription rate of MlacI (a measure of promoter strength) and dmlacI is the degradation rate. At steady state $\frac{d[M_{lacI}]}{dt} = 0$ so $M_{lacI} =\frac{k_{mlacI}}{d_{mlacI}}$

Equation 2: Equation describing the rate of translation of LacI protein (PlacI)

At steady state $P_{lacI}= \frac{k_{mlac} k_{plac}}{d_{mlac} d_{plac}}$ , where kplac is the translation rate of lacI protein and dplacI is the degradation rate of PlacI.

Equations 3 and 4 describe the transcription and translation of the protein of interest Pout.

Equation 3: Transcription of Pout Unlike in the previous case, the output promoter is inducible. In the absence of further information, we model the effect of LacI on transcription/ POPS activity with a Hill function, which represses when amounts are above the threshold K, and activates when PlacI amounts fall below threshold. Such assumption can be revised in the light of contradicting experimental data.

At steady state: $Mout=\frac{k_{mout}}{d_{mout}}[k_{leak}+(1-k_{leak})\frac {K^n}{K^n+P_{lacI}^n }]$ where kleak is the lac promoter leakiness factor, K is the switching threshold of PlacI} concentration needed to repress, n is the hill exponent kmout is the transcription rate of Mout and dmout is the degradation rate.

Equation 4: Equation describing the rate of translation of protein of interest Pout:

At steady state $P_{out} = \frac{k_{pout}}{d_{pout}}M_{out}$ so $P_{out}= \frac{k_{mout}k_{pout}}{d_{mout}d_{pout}}[k_{leak}+(1-k_{leak})\frac {K^n}{K^n+P_{lacI}^n }]$ which relates to the initial prediction that when the Lac promoter is weak and there is not enough PlacI , we don’t get sufficient repression of production of our protein of interest. When levels of PlacI go below the repression threshold, we get a “bump” in production.

### When IPTG is introduced

When IPTG is added into the system, LacI can bind to it, forming an intermediate complex [IPTG-LacI]: