Tokyo/Works/Formulation

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Latest revision as of 05:09, 27 October 2007

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

Numerical analysis for single cell

Step1. Single cell model:mutual inhibition

First, the qualitative nature was analyzed by simple dimensionless differential equations for this mutual inhibition system. The results of phase plane analysis for the equations suggested that the values of Hill coefficients were especially important in order that the system had the bistability corresponding to A and B states. ==> see more

Step2. Single cell model with hybrid promoter

Second, the effect of hybrid promoter was introduced into the single cell model. In the case of the model with hybrid promoter, the contribution from the repression of the promoter by LacI and that from the activation by AHL should be considered.; that is, the factor of AHL contribution was added to the differential equations for RB, resulting in the dependence of the phase plane on the AHL concentration. As a result of phase plane analysis for the equations, the system was monostable in the case of lower AHL concentration; whereas it was bistable in the case of higher AHL concentration. The values of Hill coefficients were important for the system bistability even in this case. ==> see more

Step3.Single cell model with hybrid promoter and cell-produced AHL

Third, the system with hybrid promoter was expanded into the system using cell-produced AHL. Here, the new parameter λ was introduced to represent the producing rate of AHL by the promoter A. As a result of phase plane analysis, the values of Hill coefficients were important for the system bistability even in this case. ==> see more
The results of the numerical analysis from Step1 to Step3 suggest the importance of the values of the Hill coefficients in this system. However, adjustments of Hill coefficients by DNA sequence modification are much more difficult than those of the other parameters (α1, α2, and λ); modifications in RBS -35 box, or -10 box allow change of those three parameters. We thus practically determined the values in the next wet experiments to confirm feasibility of our model.

　

　Before(Hybrid promoter)　<<　Formulation >>　Formulation Step.1　>>>>　Next(Assay1)

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 __NOTOC__
-<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]]
+<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]]
-== [[Tokyo/Formulation/1.toggle model |Step1.toggle model]] ==
+<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]]
-<!--First,we analysis the simple dimentionless toggle model.
+<br>
-We use the phaseplane analysis to understand the quantitative behavior of the toggle switch.
-As a result,the toggle model has three kinds of phaseplanes,from one to three equilibrium points. when one equilibrium point,only one stable equilibrium point
-And, we need three equilibrium points for the toggle to be bistable.-->
-＜イントロ＞
-まずは，次元をもっていない単純な式で定性的な振る舞いをみてみる．
-その結果，安定点が１つのときと２つのときがある．我々が求めているモデルはA状態とB状態をとる必要があるので，
-安定点が二つのときである．bistalbeになるには，パラメータにおいて，合成rateの強さ比とヒル係数が重要であることが分かった．
-<!--案１
+== Numerical analysis for single cell ==
-<br>Introduction
-<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/1.toggle model |Step1. Single cell model:mutual inhibition ]] ==
-上記の英語訳のPhaseplane, Threeは下の日本語訳にないです。
--->
+First, the qualitative nature was analyzed by simple dimensionless differential equations for this mutual inhibition system. The results of phase plane analysis for the equations suggested that the values of Hill coefficients were especially important in order that the system had the bistability corresponding to A and B states.
-<!--案2
+'''[[Tokyo/Formulation/1.toggle model | ==> see more]]'''
-<br>Introduction
+<br>
-<br>First, according to the simple dimentionless equations of our toggle, the number of equibrium points is one or two. To takes two states of A and B, our model should have two equibrium points.
+<br>    [[Image:expression1-2.jpg|200px]]
+<!--
+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 expression rates and Hill coefficients are critical.
 -->
-== [[Tokyo/Formulation/2.toggle model with hybrid promoter |Step2.toggle model with hybrid promoter ]] ==
-＜イントロ＞
+== [[Tokyo/Formulation/2.toggle model with hybrid promoter |Step2. Single cell model with hybrid promoter]] ==
-ハイブリットプロモータを式に入れ込む必要がある．
-トグルの抑制項に加えてAHLによってactivateされる項が加わる．
-これにより，AHL量によって相平面変化するようになる．
-AHL量が少ないときはmonoで多いときトグルと同じになりbiになる．
-<!--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. With few AHL, ??? -->
-<br>[[Image:expression2-4.jpg|300px|]][[Image:AHLresponse2-2.jpg|300px|]]
+Second, the effect of hybrid promoter was introduced into the single cell model. In the case of the model with hybrid promoter, the contribution from the repression of the promoter by LacI and that from the activation by AHL should be considered.; that is, the factor of AHL contribution was added to the differential equations for RB, resulting in the dependence of the phase plane on the AHL concentration. As a result of phase plane analysis for the equations, the system was monostable in the case of lower AHL concentration; whereas it was bistable in the case of higher AHL concentration. The values of Hill coefficients were important for the system bistability even in this case.
+'''[[Tokyo/Formulation/2.toggle model with hybrid promoter | ==> see more]]'''
+<br>
+<br>[[Image:expression2-4.jpg|280px|]][[Image:AHLresponse2-2.jpg|300px|]]
+<br>[[Image:step2-4.JPG|300px|]][[Image:step2-5.JPG|300px|]][[Image:step2-6.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.AHL-experssing model]] ==
+<!--
+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.
+-->
+== [[Tokyo/Formulation/3.AHL-experssing model|Step3.Single cell model with hybrid promoter and cell-produced AHL]] ==
+Third, the system with hybrid promoter was expanded into the system using cell-produced AHL. Here, the new parameter λ was introduced to represent the producing rate of AHL by the promoter A. As a result of phase plane analysis, the values of Hill coefficients were important for the system bistability even in this case. '''[[Tokyo/Formulation/3.AHL-experssing model| ==> see more]]'''
+<br>The results of the numerical analysis from Step1 to Step3 suggest the importance of the values of the Hill coefficients in this system.  However, adjustments of Hill coefficients by DNA sequence modification are much more difficult than those of the other parameters (α1, α2, and λ); modifications in RBS -35 box, or -10 box allow change of those three parameters. We thus practically determined the values in the next wet experiments to confirm feasibility of our model.
-＜イントロ＞
-今度は，大腸菌の中からAHLを作り出す系に拡張する．
-すると，nullclineが非対称になり，従来のトグルとは異なった相平面になる．
-この際，新たなパラメータλが入ってくる．
-パラメータセットによって，monoになったり，biになったりする．
-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 mono or bi of the system depends on the parameter sets
-[[Tokyo/Formulation/3.AHL-experssing model|for more detail]]
 <br>[[Image:expression3-1.jpg|300px|]]
+<br>[[Image:Step3-3.JPG|300px|]][[Image:Step3-4.JPG|300px|]]
+<!--
 <br>[[Image:Phaseplane3-1.jpg|300px|]] [[Image:Phaseplane3-2.jpg|300px|]] [[Image:Phaseplane3-3.jpg|300px|]]
+-->
+<!--
+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/4.population model|Step4. population model]] ==
+Here the concentration of AHL outside of the cells was assumed to be the same concentration of AHL inside the cell according to the description that AHL is freely permiable through cell membrane in the referenced articles.The phase plane analysis was made possible by focusing on  an indivisual cell.The parameter N was assumed to be the number of the cells.
+However, all the individuals behaved in the same way in this deterministic model. To see the bahavior of each individual cell, it is necessary to carry out stocastic simulation.
+'''[[Tokyo/Formulation/4.population model| ==> see more]]'''
+<br>[[Image:expression4-1.jpg|400px]]
+== [[Tokyo/Formulation/5.stochastic differential equation model with poisson random variables|Step5. stochastic differential equation model with poisson random variables]] ==
+we introduced the differential equations of step4 into Poisson random variables to simulate the stochastic model.
+<br>By using simulation with stochastic model, every cell can take different behavior.
-== [[Tokyo/Formulation/4.population model|Step4.population model]] ==
+<br>[[Image:expression5-1.jpg|400px]]
-＜イントロ＞
-１個体から複数個体に拡張．AHLの式は，AHLの移動がfreelyと論文にあるので大腸菌内と外とを区別していない
-ここでも，相平面解析を行った．ｎ個体の振る舞いであっても，その中の１個体に着目することで相平面解析を可能にした．
-この際，新たなパラメータとして個体数ｎがはいってくる．
-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.
-ただし，deterministicなため全個体同じ動き．我々のモデルはこれではみれない．
+<!--
-stochasticなシミュレーションが要求される．
+'''A preview of the latter part of this Wiki:'''-->
-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.
+<!--With parameters from our [[Tokyo/Works/Assay |wet experiments]], we confirmed that our circuit with the new part can show “stable coexistence”'''see more リンク　to simulation PAGE'''.-->
+<!--
+[[Image:3d-2.7-0.2.JPG|200px|left|thumb|Figure 5.2.A  　t=0.2(min)]]
+[[Image:3d-2.7-30.JPG|200px|none|thumb|Figure 5.2.C 　 t=30(min)　　'''success!!          coexistence''']]
+.-->
+<!--
+===[[Tokyo/Formulation/5.stochastic differential equation model with poisson random variables/ A and B coexistence movie/ |Click!! movie here!!]]===
+.-->
-== [[Tokyo/Formulation/5.poisson stochastic differential equation model |Step5.poisson stochastic differential equation model ]] ==
+==　==
+<br>　[[Tokyo/Works/Hybrid promoter|Before(Hybrid promoter)]]　<<　[[Tokyo/Works/Formulation |Formulation]] >>　[[Tokyo/Formulation/1.toggle model|Formulation Step.1]]　>>>>　[[Tokyo/Works/Assay|Next(Assay1)]]

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