Tokyo/Formulation/1.toggle model

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＜中身＞
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__NOTOC__
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（相平面解析）
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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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パラメータによって相平面が変わり，平衡点が１つのときと３つのときがある．
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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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平衡点が３つのときbistableになり，A状態，B状態ができる．
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<br>
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（シミュレーション）
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== 1.Single cell model:mutual inhibition ==
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初期値を安定点付近から始めるとちゃんとそのまま安定してるよ
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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|left|thumb|Ex1-1 ]]  [[Image:parameter1-1.jpg|200px|]]
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<br>[[Image:式-1.JPG|200px|]]
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<br>These equations were normalized as follows:
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<br>定常状態では時間変化が０．つまり，右辺＝０を解けばよい．すると，
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<br>[[Image:expression1-2.jpg|200px|none|thumb|Ex1-2 ]]
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<br>[[Image:Siki2.jpg|200px|]]
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<br>In the steady state, time derivatives are zero:
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<br>となり，グラフにすると下のようになる．この線のことを一般的にnullclineと言い，nullclineの交点が
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<br>[[Image:expression1-3.jpg|80px|none|thumb|Ex1-3]]
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平衡点となる．
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<br>[[Image:toggle1.jpg|200px|]] [[Image:toggle2.jpg|200px|]]
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<br>As a result, the nullclines of this system were derived as
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<br>平衡点には安定な平衡点と不安定な平衡点があり，今回は黒丸が安定な平衡点で白丸が不安定な平衡点である．
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<br>[[Image:Siki2.jpg|200px|none|thumb|Ex1-4]]
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<br>実際にシミュレーションした結果と比較すると明らかである．
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<br>左端のグラフが左上の安定点（B状態）付近を初期値として計算した結果．真ん中のグラフが中心の不安定点付近を初期値として計算した結果．右端のグラフが，右下の安定点（A状態）付近を初期値として計算した結果．
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<br>[[Image:toggle3.jpg|200px|]] [[Image:toggle4.JPG|200px|]] [[Image:toggle5.JPG|200px|]]
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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>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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<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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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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<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>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:toggle1.jpg|260px|left|thumb|Figure 1.1.A]]
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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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[[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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[[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>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>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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