Tokyo/Formulation/1.toggle model

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== 1.toggle model ==
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__NOTOC__
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First,we obtain the ordinary differential equations(ODEs) of the toggle switch.
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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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== 1.Single cell model:mutual inhibition ==
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First, the ordinary differential equations (ODEs) of the toggle switch were derived as
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<br>[[Image:expression1-1.jpg|200px|]]  [[Image:parameter1-1.jpg|200px|]]  
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<br>[[Image:expression1-1.jpg|200px|left|thumb|Ex1-1 ]]  [[Image:parameter1-1.jpg|200px|]]  
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<br>And,we normalize these expressions to analyze easily.So,ODEs become dementionless.
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<br>These equations were normalized as follows:
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なるってbecomeであってる？？
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<br>[[Image:expression1-2.jpg|200px|]]
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<br>[[Image:expression1-2.jpg|200px|none|thumb|Ex1-2 ]]
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<br>if the system goes to the steady state,time variation equal to zero．So we solve righe-hand side=0.
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<br>In the steady state, time derivatives are zero:
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<br>[[Image:Siki2.jpg|200px|]]
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<br>[[Image:expression1-3.jpg|80px|none|thumb|Ex1-3]]
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<br>These graph are below.The lines of graph are nullcline,and the intersection of nullclines is the equillibrium point.
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<br>As a result, the nullclines of this system were derived as
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<br>About parameters,we use three sets of parameters.
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<br>[[Image:Siki2.jpg|200px|none|thumb|Ex1-4]]
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<br>　　1)the maximum expression rate of repressor A and repressor B is balanced,and hill coefficient of both A and B is three.
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<br>　　2)the maximum expression rate of repressor A and repressor B is equal,and hill coefficient of A is one.
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<br>　　3)the maximum expression rate of repressor A and repressor B is not balanced,and hill coefficient of both A and B is three.
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<br>[[Image:parameter1-2.jpg|150px|]][[Image:parameter1-3.JPG|150px|]][[Image:parameter1-4.JPG|150px|]]
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<br>[[Image:toggle1.jpg|303px|]][[Image:toggle2.jpg|300px|]][[Image:Toggle1-4.jpg |300px|]]
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<br>we correlate phaseplane analysis and simulation results.
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<br>which indicate the nullclines of the system shown in Fig 1.1.A-C. Where about parameters, we use three sets of parameters.
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<br>First,we simulate about the phaseplane of two stable equilibrium points(the upper left figure) and use three kinds of initial values.
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<br>  1. (Ra:low , Rb:high) 2. (Ra:high , Rb:low) 3. (Ra:middle , Rb:middle)
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<br>[[Image:toggle3.jpg|200px|]] [[Image:toggle4.JPG|200px|]] [[Image:toggle5.JPG|200px|]][[Image:toggle1-1.jpg|200px|]]
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<br>　　A)the maximum expression rate of repressor A and repressor B is balanced,and hill coefficient of both A and B is three.
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<br>[[Image:parameter1-2.jpg|150px|center|thumb|Table1.A]]
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<br>　　B)the maximum expression rate of repressor A and repressor B is equal,and hill coefficient of A is one.
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<br>[[Image:parameter1-3.JPG|150px|center|thumb|Table1.B]]
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<br>　　C)the maximum expression rate of repressor A and repressor B is not balanced,and hill coefficient of both A and B is three.
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<br>[[Image:parameter1-4.JPG|150px|center|thumb|Table1.C]]
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<br>安定点B付近から始めるとB状態で安定し，安定点A付近から始めるとA状態で安定しているのが分かる．
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[[Image:toggle1.jpg|260px|left|thumb|Figure 1.1.A]]
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不安定点付近から始めるとどちらかで安定化する．
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[[Image:toggle2.jpg|270px|left|thumb|Figure 1.1.B]]
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[[Image:Toggle1-4.jpg|270px|none|thumb|Figure 1.1.C]]
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<br>First,we carried out kinetic simulations in the condition of Fig 1.1.A. The results are shown in Fig 1.2.A-C.
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<br>Next,
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[[Image:toggle3.jpg|200px|left|thumb|Figure 1.2.A  (Ra(0),Rb(0))=(0.0,2.5)]] [[Image:toggle4.JPG|200px|left|thumb|Figure 1.2.B  (Ra(0),Rb(0))=(2.5,0.0)]]
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次に，安定点が一つしかない場合のシミュレーション結果は下のようになる．
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[[Image:toggle5.JPG|200px|left|thumb|Figure 1.2.C  (Ra(0),Rb(0))=(1.5,1.3)]]
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[[Image:toggle1-1.jpg|200px|none|thumb|Figure 3  bistable]]
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<br>[[Image:toggle6.JPG|200px|]] [[Image:Toggle7.JPG|200px|]] [[Image:toggle8.JPG|200px|]][[Image:toggle1-2.jpg|200px|]]
 
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<br>安定点が一つしかない場合は，安定点B付近から始めてもA状態で安定化してしまうのが分かる．
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<br>Fig 1.2.A-C indicate that when the initial condition is (Ra,Rb)=(0.0,2.5), which is near the stable equilibrium point B, the values of Ra and Rb go to stable equilibrium point B, and when the initial condition is (Ra,Rb)=(2.5,0.0), which is near the stable equilibrium point A, the values of Ra and Rb go to the stable equilibrium point B.
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<br>'''As a result,taking two stable status need the phaseplane of two stable equilibrium points and we have to set proper parameters.'''
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<br>Next, when the number of stable equilibrium point is one(Fig 1.1.B), the result of simulation are shown in Fig 1.4.A-C.
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[[Image:toggle6.JPG|200px|left|thumb|Figure 1.4.A  (Ra(0),Rb(0))=(0.0,2.5)]] [[Image:toggle7.JPG|200px|left|thumb|Figure 1.4.B  (Ra(0),Rb(0))=(2.5,0.0)]]
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[[Image:toggle8.JPG|200px|left|thumb|Figure 1.4.C  (Ra(0),Rb(0))=(1.5,1.3)]]
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[[Image:toggle1-2.jpg|220px|none|thumb|Figure 1.5  monostable]]
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<br><br>
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<br>
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<br>Fig 4.A-C indicate that the value of Ra and Rb go to the stable equilibrium point A regardless of an initial value in case of monostable state.
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<br>'''As a result, taking two stable states needs the phaseplane of two stable equilibrium points and Hill coefficients was very important.'''
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== ==
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[[Tokyo/Formulation/1.toggle model|Step.1]] >> [[Tokyo/Formulation/2.toggle model with hybrid promoter|Step.2]]

Latest revision as of 02:45, 27 October 2007


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

Step1   Step2   Step3　　

1.Single cell model:mutual inhibition

First, the ordinary differential equations (ODEs) of the toggle switch were derived as


Ex1-1
Parameter1-1.jpg


These equations were normalized as follows:


Ex1-2


In the steady state, time derivatives are zero:


Ex1-3


As a result, the nullclines of this system were derived as


Ex1-4


which indicate the nullclines of the system shown in Fig 1.1.A-C. Where about parameters, we use three sets of parameters.


　　A)the maximum expression rate of repressor A and repressor B is balanced,and hill coefficient of both A and B is three.


Table1.A


　　B)the maximum expression rate of repressor A and repressor B is equal,and hill coefficient of A is one.


Table1.B


　　C)the maximum expression rate of repressor A and repressor B is not balanced,and hill coefficient of both A and B is three.


Table1.C
Figure 1.1.A
Figure 1.1.B
Figure 1.1.C


First,we carried out kinetic simulations in the condition of Fig 1.1.A. The results are shown in Fig 1.2.A-C.

Figure 1.2.A (Ra(0),Rb(0))=(0.0,2.5)
Figure 1.2.B (Ra(0),Rb(0))=(2.5,0.0)
Figure 1.2.C (Ra(0),Rb(0))=(1.5,1.3)
Figure 3 bistable



Fig 1.2.A-C indicate that when the initial condition is (Ra,Rb)=(0.0,2.5), which is near the stable equilibrium point B, the values of Ra and Rb go to stable equilibrium point B, and when the initial condition is (Ra,Rb)=(2.5,0.0), which is near the stable equilibrium point A, the values of Ra and Rb go to the stable equilibrium point B.


Next, when the number of stable equilibrium point is one(Fig 1.1.B), the result of simulation are shown in Fig 1.4.A-C.

Figure 1.4.A (Ra(0),Rb(0))=(0.0,2.5)
Figure 1.4.B (Ra(0),Rb(0))=(2.5,0.0)
Figure 1.4.C (Ra(0),Rb(0))=(1.5,1.3)
Figure 1.5 monostable





Fig 4.A-C indicate that the value of Ra and Rb go to the stable equilibrium point A regardless of an initial value in case of monostable state.


As a result, taking two stable states needs the phaseplane of two stable equilibrium points and Hill coefficients was very important.

Step.1 >> Step.2

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