Modle

Figure One, The concentraiton of the cells (demonstrated by the induced expression of the antibiotic resistant genes) again time

Figure Two, expression of antibiotic resistant genes again time and the concentration of AHL/BHL.

The expression of the antibiotic resistance proteins(red for chloramphenicol and green for kanamycin resistant gene ) are infulenced by the amount of BHL/AHL added into the culture. The curve will disply a different look when the concentration of AHL/BHL passes the threshold of 7uM/m.

Figure Tree, the concentraiton of antibiotic resistant protiens against time and its degration rate.

Assume that the initial concentration of the two antibiotics (kanamycin and chloramphenicol) are the same, and the amount of the antibiotic resistant proteins comsumed per minute is [s]. When [s] is below 0.5uM, the expressions of the two antibiotic resistant genes will be enhanced when AHL or BHL is added into the culture, and enter to the muatualism phase where the survival of one strain is beneficial for the other. However, when [s] passes the threshold of 0.5μL, the extinction of one strain may happen before the two enter the muatualism stage.

Figure Four, the concentration of cells against time.
As is shown in the fig from the starting point to the third hour the gene circle starts to work and finally get to a stable stage. And during this process there is some amount of chloramphenicol resistant protein and the kanamycin resistant protein accumulated. However the expression of the antibiotic resistant protein has not reached the highest level. 

The third hour is the critical point for the AHL for the concentration of AHL will get down to the same level as the BHL within 24 minutes and keep on this cooperation. When the plux and prhl promoters have been activate to their greatest capacity there will be a small peak because of the leakiness of the PBad/araC promoter. And in this situation the expression of the antibiotic resistant protein get to the highest level.

By adding arabinose or IPTG at the eighth hour there still will be a small peak. However both of them will fall until the restarting of the PBad/araC promoter.

From the starting point to the twelfth hour the whole mechanism can be seen as a completely regulating process which shows the shifting from the competent state to the mutualism state.

The robustness of the whole system depend on several key factors listed below:

• The different inherent behaviors of the promoters determines the performance of the toggle switch.
• The degradation rates of the AHL（BHL）and arabinose(IPTG) will have some impact in the next control.

Figure Five, The expression of kanamycin resistant genes against time and Hill coefficient(Xm) of the promoter Plux and Prhi.

As is shown, the graph takes a look of Figure 1 only when Xm is within the range of 2.6-2.84.
The choice of the parameters will affect the result of modeling to a great existent, which also offers a resource of reference for our experiments, the choice of the biobrickes, especially.

The assumptions of the molding
1The concentration of the signing molecules is even in all regions of the culture, as well as within the cells.
2 The amounts of the antibiotic resistant proteins are enough to make sure the cells’ survival.
3 The amounts of IPTG or arabinose added to the culture are regarded as enough for cells’ survival.

4 The amount of antibiotics resistant proteins consumed per minute [s] is a constant when the concentration of antibiotics is stable, and will vary with the variance of the concentration of antibiotics. A strain is assumed to be extinguished when the amount of antibiotic resistant protein expressed is negative.

5The inducers IPTG or arobinose will function for long, and their concentration won’t be affected significantly when they combine the repressors.

6The growth of the cells is proportional to the concentration of antibiotic resistant genes, and won’t be restricted by space and nutrition.

7The survival ability of a cell is demonstrated by the amount of antibiotic resistant proteins expressed.

8The capacity of the promoters Plux and Prhl presents a sudden change around the critical concentration of regulators (the complexes of AHL and LuxR or the complexes of BHL and RhlR ), so will the degradation capacity of aiiA.