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Team:Bologna/Modeling
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Mathematical Model
Let us pose: 
The genetic circuit in Figure 1 can be modeled with the following equations:
Where:
In the model we distinguish between LacI protein binded to repressor IPTG 
and protein free 
.
Since 
and considering the law of mass action 
we can write 
.
Posing:
The equations (1.1) and (1.2) can be written in adimensional form:
Equibrium conditions
If we assume that:
- to be under conditions of equilibrium
;
- all the entries are void
;
-
the affinity of the repressor for the operator site is high, 
for as we have been defined it will surely be smaller of one so 
;
- the cooperativity is the same both for the LacI and for TetR and it is worth 2;
then the equilibrium equations are:
The figure 2 has been gotten with both the aces in logarithmic staircase and hypothesizing that ]
. There are three point of equilibrium, the blue points to the extreme are stable while the red point to the center is unstable. If we maintain constant a parameter (
or 
) and we make to vary the other is possible to notice that all of a sudden a bifurcation is had, because from three points of equilibrium us neither only one stable.
If it is set 
it is possible to get figure 3 in which an only point of stable equilibrium is had. This means that is present so much TetR inside the cell and there is little LacI because the gene of LacI is inhibited from the protein TetR that has been synthesized thanks to the entry of IPTG inside the cell.
If instead it is hypothesized that File:F018.jpg it is possible to get the figure 4 in which an only point of stable equilibrium is had. This means that has been produced a lot of LacI that has gone to inhibit the gene that synthesizes TetR after having induced the cell with the UVc. This new point of equilibrium represents the state in which LacI is ON while TetR is OFF.
Bifurcation lines are illustrated in figure 6 for three different values of cooperativity (1.5, 2, 2.5). It is supposed that the cooperativity is always the same one both for LacI and for TetR. The figure 6 is always gotten in logarithmic staircase in function of ( and ). The bistable region lies inside of each pair of curves.
Placing:
The dimensionless equations are:
Equibrium conditions
Hypothesizing: to be under conditions of equilibrium;
all the entries are void (); the affinity of the repressor for the operator site is high, for as we have been defined it will surely be smaller of one so ;the cooperativity is the same both for the LacI and for TetR and it is worth 2; the equations become:
