Tokyo/Formulation/3.AHL-experssing model

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<br>大腸菌の中からAHLを産生している系を考えてみる．すると，微分方程式がもう一本増え3連立の微分方程式になる．
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Taking into consideration the system with AHL produced by E. coli themselves, it is necessary to add another differential equation; therefore, the whole system is expressed by three simultaneous equations.  
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<br>[[Tokyo/Works|Works top]]&nbsp;&nbsp;&nbsp;0.[[Tokyo/Works/Hybrid promoter|Hybrid promoter]]&nbsp;&nbsp;&nbsp;'''1.[[Tokyo/Works/Formulation |Formulation]]'''&nbsp;&nbsp;&nbsp;2.[[Tokyo/Works/Assay |Assay1]]&nbsp;&nbsp;&nbsp;3.[[Tokyo/Works/Simulation |Simulation]]&nbsp;&nbsp;&nbsp;4.[[Tokyo/Works/Assay2 |Assay2]]&nbsp;&nbsp;&nbsp;5.[[Tokyo/Works/Future works |Future works]]
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<br><br>[[Tokyo/Formulation/1.toggle model |Step1]]&nbsp;&nbsp;&nbsp;[[Tokyo/Formulation/2.toggle model with hybrid promoter |Step2]]&nbsp;&nbsp;&nbsp;[[Tokyo/Formulation/3.AHL-experssing model|Step3]]　　
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<br>
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<br> [[Image:expression3-4.jpg|400px|]][[Image:parameter3-1.jpg|200px|]]
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== Step.3 Single cell model with hybrid promoter and cell-produced AHL ==
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<br>demenionless にすると，
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<br>The differential equations of the system considering AHL produced by E.coli themselves were given as
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<br>[[Image:expression3-1.jpg|300px|]]
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<br> [[Image:expression3-4.jpg|400px|left|thumb|Ex 3-1]][[Image:parameter3-1.jpg|200px|none|thumb|Table 3]]
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<br>定常状態では，
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<br>These equations were normalized as follows:
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In the steady state,
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<br>[[Image:expression3-2.jpg|300px|]]
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<br>[[Image:expression3-1.jpg|300px|none|thumb|Ex 3-2]]
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<br>となる．すると，3本目の式を2本目の式に代入することができ，結局RaとRbの2変数になる．これより，今までと同様RaとRbの相平面が描ける．
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<br>In the steady state,time derivatives are zero.As a result,the nullclines of this system were derived as
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The third equation is substituted into the second. Now the variants being two, Ra and Rb, it is possible to draw the same type of phaseplane of Ra and Rb.
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<br>[[Image:expression3-3.jpg|300px|]]
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<br>[[Image:expression3-2.jpg|300px|none|thumb|Ex 3-3]]
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<br>今回も，パラメータの選び方により，安定点が一つのときと二つのときが出てくる．
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<br>By substituting the third equation into the second,the nullclines for Ra and Rb were obtained as
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In this case as well, the parameter choice can result in one or two equibrium points.
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<br>[[Image:Phaseplane3-1.jpg|300px|]] [[Image:Phaseplane3-2.jpg|300px|]] [[Image:Phaseplane3-3.jpg|300px|]]
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<br>[[Image:expression3-3.jpg|300px|none|thumb|Ex 3-4]]
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<br>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:
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<br>[[Image:step3-3.JPG|300px|left|thumb|Figure 3.1.A]]
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<br>[[Image:step3-4.JPG|300px|thumb|none|Figure 3.1.B]]
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<br>[[Image:step3-5.JPG|300px|thumb|left|Figure 3.1.C]]
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<br>[[Image:step3-6.JPG|300px|thumb|none|Figure 3.1.D]]
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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.
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<br>  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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<!--
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<br>[[Image:Phaseplane3-1.jpg|300px|left|thumb|Figure 3.1.A]] [[Image:Phaseplane3-2.jpg|290px|left|thumb|Figure 3.1.B]] [[Image:Phaseplane3-3.jpg|270px|none|thumb|Figure 3.1.C]]
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-->
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== ==
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[[Tokyo/Formulation/3.AHL-experssing model|Step.3]] >> [[Tokyo/Works/Formulation|Formulation top]]

Latest revision as of 05:00, 27 October 2007


Works top   0.Hybrid promoter   1.Formulation   2.Assay1   3.Simulation   4.Assay2   5.Future works

Step1   Step2   Step3　　

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


The differential equations 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.


Step.3 >> Formulation top

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