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Mathematical Model
The behavior of this device and the conditions for bistability can be understood using the following dimensionless model for the network:
Where:
In the figure is possible to see the relation between alfa, the rate of synthesis of repressor, and the inductor. In the first phase is considered only the amount of inductor intracellular and alfa is the same for the two repressors. From the figure is possible notice that for smaller values of alfa there is a linear relation between alfa and the inductor while for great values of alfa there is a condition of saturation and it is necessary provide the cell a larger amount of the inductor. Furthermore increasing the cooperativity of repression (beta and gamma)there are different curves and grows the amount of inductor to pass the transition (bifurcation) between bistability and monostability. Reducing the cooperativity of repression reduces the size of the bistable region. Based on the characteristics that must have the system is possible to choose the most suitable parameters. For example, to obtain a very stable system, it is important to have high values for alfa and the cooparativity
Bibliography
Mads Kaern, William J. Blake, and J.J. Collins. The Engineering Of Gene Regulatory Networks. Annu.Rev.Biomed.Eng. 2003.5:179-206
Timothy S. Gardner, Charles R. Cantor & James J. Collins : Construction of a genetic toggle switch in Escherichia coli. Nature vol. 403 339-342
