Tokyo/Formulation/5.stochastic differential equation model with poisson random variables

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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]]
<br><br>[[Tokyo/Formulation/1.toggle model |Step1]]　　[[Tokyo/Formulation/2.toggle model with hybrid promoter |Step2]]　　[[Tokyo/Formulation/3.AHL-experssing model|Step3]]　　[[Tokyo/Formulation/4.population model|Step4]]　　[[Tokyo/Formulation/5.stochastic differential equation model with poisson random variables|Step5]]
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we introduced the terms of [[Tokyo/Formulation/4.population model|Ex 4-1]] into a stochastic process to simulate the sthochastic behavior.we used Poisson random variables as a sthochastic process. Threfore,a stochastic differential equations were given as

<br>[[Image:expression5-1.jpg|500px|none|thumb|Ex 5-1]]
<!--[[Image:parameter4-1.jpg|150px|none|thumb|Table 5]]-->



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<br>The values of parameters in the right table were used and the results of simulation were shown in Fig 5.1-3.


<br>where α2 = 1(μM) in Fig 5.1,α2 = 2.7(μM) in Fig 5.2,α2 = 4(μM) in Fig 5.3. and it has been estimated that 1(μM) = 1000 molecules (count).

<br>Fig.5.1 indicates that all cells shift to A state in the steady state and Fig 5.3 indicates that all cells shift to B state in the steady state.These results doesn't represent coexistence stable.
<br>Fig.5.2 indicates that a portion of cells shift to A state and the others shift to B state in steady state;that is,individual cells are stable under stable coexistence.

[[Image:3d-1-0.2.JPG|200px|left|thumb|Figure 5.1.A  　t=0.2(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3d-1-0.8.JPG|200px|left|thumb|Figure 5.1.B  　t=0.8(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3d-1-30.JPG|200px|none|thumb|Figure 5.1.C 　 t=30(min)　only A]]
===[[Tokyo/Formulation/5.stochastic differential equation model with poisson random variables/only A movie/|movie here!!]]===

<br>
[[Image:3d-2.7-0.2.JPG|200px|left|thumb|Figure 5.2.A  　t=0.2(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3d-2.7-0.8.JPG|200px|left|thumb|Figure 5.2.B 　 t=0.8(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[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/ |movie here!!]]===

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[[Image:3d-4-0.2.JPG|200px|left|thumb|Figure 5.3.A 　 t=0.2(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3d-4-0.8.JPG|200px|left|thumb|Figure 5.3.B 　 t=0.8(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3d-4-30.JPG|200px|none|thumb|Figure 5.3.C 　 t=30(min)　　'''   only B''']]

===[[Tokyo/Formulation/5.stochastic differential equation model with poisson random variables/only B movie/|movie here!!]]===




<br>Next，the relation between the results of simulation and phase plane are shown in Fig.5.4.1-3,where the nullclines of this system were [[Tokyo/Formulation/4.population model|Ex 4-3]]

[[Image:phase plane and simulation-1.JPG|270px|left|thumb|Figure 5.4.1]]
[[Image:phase plane and simulation-2.7.JPG|270px|left|thumb|Figure 5.4.2]]
[[Image:phase plane and simulation-4.JPG|270px|none|thumb|Figure 5.4.3]]

<br>Fig.5.4.1-3 indicate that the distribution of the cells corresponds with the stable equilibrium points.

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