Team:Imperial College London/Drylab/M1/Protein production/Analysis/detailed

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 <math>\frac{d[M_{lacI}]}{dt} = 0 </math> so <math>M_{lacI} =\frac{k_{mlacI}}{d_{mlacI}}</math>

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

At steady state <math>P_{lacI}= \frac{k_{mlac} k_{plac}}{d_{mlac} d_{plac}}</math> , where k_{http://openwetware.org/skins/common/images/button_bold.pngplac} 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: <math>Mout=\frac{k_{mout}}{d_{mout}}[k_{leak}+(1-k_{leak})\frac {K^n}{K^n+P_{lacI}^n }]</math> 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 <math> P_{out} = \frac{k_{pout}}{d_{pout}}M_{out}</math> so <math>P_{out}= \frac{k_{mout}k_{pout}}{d_{mout}d_{pout}}[k_{leak}+(1-k_{leak})\frac {K^n}{K^n+P_{lacI}^n }]</math> 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]: