Tokyo/Formulation/3.AHL-experssing model

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Step.3 Single cell model with hybrid promoter and cell-produced AHL


The differential equaitons of the system considering AHL produced by E.coli themselves were given as


Ex 3-1
Table 3


These equations were normalized as follows:


Ex 3-2


In the steady state,time derivatives are zero.As a result,the nullclines of this system were derived as


Ex 3-3


By substituting the third equation into the second,the nullclines for Ra and Rb were obtained as


Ex 3-4


Therefore, the phase plane of this system can be plotted as Fig.3.1.A-D and the number of equilibrium points were decided by the value of the parameters:


Figure 3.1.A

Figure 3.1.B

Figure 3.1.C

Figure 3.1.D

Comparison between Fig3.1.A and B indicated that Hill coefficients are critical parameters even in the cell-produced AHL model. In the case of N2=1, N3=1, and λ=3, the phase plane was monostable. In contrast, in the case of N2=2, N3=2, and λ=3, the phase plane was bistable.
In the cases of λ=1 (Fig.3.1.C and D), the system can not take bistability even if the values of Hill coefficients are changed. For the implementation of the circuit in a cell, the parameter λ should be controlled by changing the RBSs and/or promoter sequences of LuxR.


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