Tokyo/Works/Formulation

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<br>[[Tokyo/Works|Works top]]　　0.[[Tokyo/Works/Hybrid promoter|Hybrid promoter]]　　1.[[Tokyo/Works/Formulation |Formulation]]　　2.[[Tokyo/Works/Assay |Assay1]]　　3.[[Tokyo/Works/Simulation |Simulation]]　　4.[[Tokyo/Works/Assay2 |Assay2]]　　5.[[Tokyo/Works/Future works |Future works]]


== [[Tokyo/Formulation/1.toggle model |Step1. toggle model]] ==
<br>First, we saw the qualitative nature implied by simple dimentionless equations and found that the number of the stable points is one or two. Since our model must have A and B states, the number of its stable points should be two. 
<br>For the parameters required for this bistability, we have found that the production rates and Hill coefficients are critical.

== [[Tokyo/Formulation/2.toggle model with hybrid promoter |Step2. toggle model with hybrid promoter ]] ==
Factors of the hybrid promoter should be incorporated into the equations; that is, the term of the repression of toggle by LacI and that of the activation by AHL should be added. By these additions, the phase changes dependent on the amount of AHL. Lower concentration of AHL, monostable; while, higher one gives bistable.
<br>[[Image:expression2-4.jpg|300px|]][[Image:AHLresponse2-2.jpg|300px|]]




<br>[[Image:AHLresponse2-3.jpg|330px|]][[Image:AHLresponse2-4.jpg|300px|]][[Image:AHLresponse2-5.jpg|300px|]]

== [[Tokyo/Formulation/3.AHL-experssing model|Step3. using cell-produced AHL]] ==
Now develop this system to the one with cell-produced AHL. 
This time, the nullcrine is assymetric and the phaseplane is unconventionally shaped.
Here the new parameter λ is introduced whether monostalbe or bistable of the system depends on the velues of several parameters. 
[[Tokyo/Formulation/3.AHL-experssing model|for more detail]]
<br>[[Image:expression3-1.jpg|300px|]]
<br>[[Image:Phaseplane3-1.jpg|300px|]] [[Image:Phaseplane3-2.jpg|300px|]] [[Image:Phaseplane3-3.jpg|300px|]]

== [[Tokyo/Formulation/4.population model|Step4. population model]] ==
Here the concentration of AHL is assumed the same inside and outside of a cell according to the description that AHL is freely permiable through cell membrane in the referenced articles. In the phaseplane analysis here is mede possible by focusing on an individual in the whole. In this case, the parameter n for the number of the cells is introduced. 

However, all the individuals behave in the same way in this deterministic model. To see the bahavior different from each individual, it is necessary to use stocastic simulation.

== [[Tokyo/Formulation/5.poisson stochastic differential equation model |Step5. poisson stochastic differential equation model ]] ==