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

### From OpenWetWare

### In the absence of IPTG:

**Equation 1:** Equation describing the rate of transcription of LacI MRNA (M_{LacI}):

k_{mlacI} is the transcription rate of M_{lacI} (a measure of promoter strength) and d_{mlacI} is the degradation rate.
At steady state so

**Equation 2:** Equation describing the rate of translation of LacI protein (P_{lacI})

At steady state ,
where k_{plac} is the translation rate of lacI protein and d_{placI} is the degradation rate of P_{lacI}.

Equations 3 and 4 describe the transcription and translation of the protein of interest P_{out}.

**Equation 3:** Transcription of P_{out}
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 P_{lacI} amounts fall below threshold. Such assumption can be revised in the light of contradicting experimental data.

At steady state: where k_{leak} is the lac promoter leakiness factor, K is the switching threshold of P_{lacI}} concentration needed to repress, n is the hill exponent k_{mout} is the transcription rate of Mout and d_{mout} is the degradation rate.

**Equation 4:** Equation describing the rate of translation of protein of interest P_{out}:

At steady state so
which relates to the initial prediction that when the Lac promoter is weak and there is not enough P_{lacI} , we don’t get sufficient repression of production of our protein of interest. When levels of P_{lacI} 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]: