IGEM:IMPERIAL/2009/M1/Modelling/Analysis/Detailed

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 khttp://openwetware.org/skins/common/images/button_bold.pngplac 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]:

k1 and k2 are the dissociation constants of the forward and reverse reactions. Therefore, we can modify Equation 2 to include the effects of IPTG on the LacI system as follows: Note that now, the effects of IPTG have been included. The –k2[Plac][IPTG] term contributes negatively, as it removes PLac from the system. The k1[IPTG-Plac] term contributes positively, as it re-stores the levels of PLac in the system. The equations that describe the evolution of IPTG and IPTG-LacI are: Equation 5: Rate of change of IPTG in the system: $$dIPTG/dt= -k2[PLacI][IPTG]+ k1[IPTG-PLacI]$$

Equation 6: Rate of change of intermediate [IPTG-Plac] complex in the system. $$(d[IPTG-PlacI])/dt= k2[PLacI][IPTG]- k1[IPTG-PLacI]$$

Notice that dIPTG/dt= - (d[IPTG-PlacI])/dt, since when one is destroyed, the other one is formed. Link to table of parameters, also include values: http://openwetware.org/wiki/IGEM:IMPERIAL/2009/M1/Modelling/LacI-IPTG#Legend