Tokyo/Works/Simulation

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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]]
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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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== Numerical analysis and kinetic simulations for cell population ==
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<!--==ここまでわかったこと==-->
<!--==ここまでわかったこと==-->
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Multi-cell simulations with Hill coefficients of the promoters were carried out to find ranges of other parameters that lead the system to balanced differentiation.  The differential equations for N cells were constructed to describe the cell population interacting with each other(Ex3.1).
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[[Image:expression4-1.jpg|300px|thumb|left|Ex3.1 the differential equations of population model]]
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<br><br>
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<!--Here, the concentration of AHL outside of the cells was assumed to be the same concentration of AHL inside the cell because AHL is freely permeable through the membrane of each cell. First, the phase plane analysis for the equations was carried out by focusing on an individual cell.-->Here the concentrations of AHL inside and outside of the cell are assumed to be the same since AHL is freely permeable though cell membranes. <!--[引用文献番号]-->
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<br><br>First, the phase plane analysis focused on individual cells.
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<br>
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<br><br><br><br><br><br><br><br>
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To analyze the behavior of the cell population, the above differential equations for N cells were extended to the stochastic differential equations(Ex3.2). '''([[Tokyo/Formulation/4.population_model|see detailed derivation ]])''' In order to carry out the simulations with the stochastic model, Poisson random variables were introduced into the differential equations. As a result of the stochastic simulation, the different behaviors of individual cells were observed.
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<br>[[Image:expression5-1.jpg|400px|thumb|none|Ex3.2 the stochastic differential equations of the population model]]
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In the simulations, Hill coefficients, coefficients of repression and activation of AHL and LacI(n2,n3,k2,k3) determined by the wet experiments were used. The values of other parameters were determined by reference to some journals.
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(Kobayashi et al, 2004;Tian T et al, 2006;Garcia-Ojalvo J et al, 2004; Tomioka R et al, 2004)
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<!--
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<br>n2 = 2.08 (-)
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<br>K2 = 4.05 (μM)
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<br>n3 = 2.47 (-)
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<br>K3 = 0.295 (μM)
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-->
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== What we have found so far ==
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{| border="1"
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Wet experiments have determined Hill coefficients, coefficients of repression and activation of AHL and LacI(n2,n3,k2,k3) as follows.
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| n2 || 2.08 (-)
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|-
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| K2 || 4.05 (uM)
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|-
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| n3 || 2.47 (-)  
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|-
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| K3 || 0.295 (uM)
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|}
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<br>'''n2 = 2.08 (-)'''
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==<!--共存安定になるようなパラメータ範囲の提示-->Determining the range of parameter which satisfy balanced differentiation ==
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<br>'''K2 = 4.05 (μM)'''
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<!--===The example of coexistent stable state共存安定状態の例=== -->
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<br>'''n3 = 2.47 (-)'''
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Depending on the value of parameter lambda, three different patterns were observed: the pattern with coexistence state of A and B states, the pattern with only A state, and the pattern with only B. The simulation results for their patterns are shown in Fig.1 to 3.  
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<br>'''K3 = 0.295 (μM)'''
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==<!--共存安定になるようなパラメータ範囲の提示-->Determining the range of parameter which satisfy coexistent stability==
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<!--パラメータlambdaの値によって,AとBが共存して安定する場合と,Aのみ,Bのみになってしまう3つのパターンが生じる.
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===The example of coexistent stable state<!--共存安定状態の例-->===
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<!--パラメータλの値によって,AとBが共存して安定する場合と,Aのみ,Bのみになってしまう3つのパターンが生じる.
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それぞれのパターンについてシミュレーション結果を示すとFig.1-3のようになる.-->
それぞれのパターンについてシミュレーション結果を示すとFig.1-3のようになる.-->
[[Image:3D_start.JPG|200px|left|thumb|Fig.1.A   t=0.0(min)]]
[[Image:3D_start.JPG|200px|left|thumb|Fig.1.A   t=0.0(min)]]
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[[Image:3D_t30_a.JPG|200px|none|thumb|Fig.1.C   t=30(min) only A]]
[[Image:3D_t30_a.JPG|200px|none|thumb|Fig.1.C   t=30(min) only A]]
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===[[Tokyo/Works/Simulation/A movie| movie here!!]]===
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===[[Tokyo/Works/Simulation/A movie| ==> movie here!!]]===
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[[Image:3D_t3_ab.JPG|200px|left|thumb|Fig.2.B   t=3.0(min)]]
[[Image:3D_t3_ab.JPG|200px|left|thumb|Fig.2.B   t=3.0(min)]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:arrow2.JPG|50px|left|thumb]]
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[[Image:3D_t30_ab.JPG|200px|none|thumb|Fig.2.C   t=30(min) A and B are coexistence!!]]
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[[Image:3D_t30_ab.JPG|200px|none|thumb|Fig.2.C   t=30(min) '''A-B balanced differentiation !!''' ]]
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===[[Tokyo/Works/Simulation/AB movie| movie here!!]]===
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===[[Tokyo/Works/Simulation/AB movie| ==> movie here!!]]===
[[Image:3D_start.JPG|200px|left|thumb|Fig.3.A   t=0.0(min)]]
[[Image:3D_start.JPG|200px|left|thumb|Fig.3.A   t=0.0(min)]]
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[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:arrow2.JPG|50px|left|thumb]]
[[Image:3D_t30_b.JPG|200px|none|thumb|Fig.3.C   t=30(min) only B]]
[[Image:3D_t30_b.JPG|200px|none|thumb|Fig.3.C   t=30(min) only B]]
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===[[Tokyo/Works/Simulation/B movie| movie here!!]]===
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===[[Tokyo/Works/Simulation/B movie| ==> movie here!!]]===
<br>
<br>
<!--3パターンについて,nullclinesの交点が2つになる瞬間の相平面はFig.4.A-Cのようになる.-->
<!--3パターンについて,nullclinesの交点が2つになる瞬間の相平面はFig.4.A-Cのようになる.-->
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For the three patterns, the phase portraits at the moment when the number of the intersection of nullclines become two are shown in Fig.4.A-C respectively.
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The nullclines changed dependent on the state of the cell population. When there are only B cells, the nullclines had one intersection in the phase plane. As change of the state of the cell population, the number of the intersections of nullclines turned into two. Fig.4.A-C shows the appearance of an additional intersection, indicating an unstable equilibrium point, for the three patterns, respectively.
<!--相平面:phase portrait
<!--相平面:phase portrait
nullclinesの交点:the intersection of the nullclines-->
nullclinesの交点:the intersection of the nullclines-->
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[[Image:2Dpp-ab-tt.JPG|260px|left|thumb|Fig.4.B]]
[[Image:2Dpp-ab-tt.JPG|260px|left|thumb|Fig.4.B]]
[[Image:2Dpp-b-tt.JPG|260px|none|thumb|Fig.4.C]]
[[Image:2Dpp-b-tt.JPG|260px|none|thumb|Fig.4.C]]
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<br>Fig.4.A shows that when the two equilibrium points appeared, cells were existing closer to A-side than the unstable equilibrium point. Oppositely,in Fig.4.C,when the two equilibrium points appeared, cells were existing closer to A-side than the unstable equiliblium point. Since when the number of the intersection of nullclines becomes two, stable point appears on B-side, at the moment of Fig.4, cells on the A-side cannot transit to B-side across unstble equilibrium point. As a result, all cells transit to A-side.  
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<br>
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===[[Tokyo/Works/Simulation/A phase plane movie| ==> movie about Fig.4.A here!!]]===
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===[[Tokyo/Works/Simulation/A and B phase plane movie| ==> movie about Fig.4.B here!!]]===
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===[[Tokyo/Works/Simulation/B phase plane movie| ==> movie about Fig.4.C here!!]]===
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<br><!--In the case of Fig. 4B, the cell population distributed around the unstable equilibrium point. Right after the moment shown in the Fig. 4B, some cells moved to the A state and the other go in the direction of the B state. The division of the direction may have occurred due to the distribution of the population over the both sides of a “watershed” which is a division line lying over the unstable equilibrium point.  In another possibility, fluctuation of the cell state might allow coming or going over the division line.-->
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In Fig. 4B, the cell population was distributed around the unstable equilibrium point. Right after the moment shown in the Fig. 4B, some cells moved to the A state while the others go toward the B state. The division of the direction may have occurred due to the distribution of the population over the both sides of a “watershed,” a division line lying over the unstable equilibrium point. In another possibility, fluctuation of the cell state might allow going across the division line.  
<!--Fig.4.Aでは,平衡点が二つになった瞬間,cellsは中心付近の不安定平衡点よりA側に存在している.逆に,Fig.4.Cでは,平衡点が二つになった瞬間,cellsは中心付近の不安定平衡点よりもB側に存在している.
<!--Fig.4.Aでは,平衡点が二つになった瞬間,cellsは中心付近の不安定平衡点よりA側に存在している.逆に,Fig.4.Cでは,平衡点が二つになった瞬間,cellsは中心付近の不安定平衡点よりもB側に存在している.
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次に,共存安定するFig.4.Bを見ると,不安定平衡点ができた瞬間にcelllsはその不安定平衡点付近に存在している.そしてB側にも安定点が生じた瞬間に,あるcellsはA側の安定点にシフトし,あるcellsはB側の安定点にシフトする.-->
次に,共存安定するFig.4.Bを見ると,不安定平衡点ができた瞬間にcelllsはその不安定平衡点付近に存在している.そしてB側にも安定点が生じた瞬間に,あるcellsはA側の安定点にシフトし,あるcellsはB側の安定点にシフトする.-->
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===[[Tokyo/Works/Simulation/A phase plane movie| ⇒ movie about Fig.4.A here!!]]===
 
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===[[Tokyo/Works/Simulation/A and B phase plane movie| ⇒ movie about Fig.4.B here!!]]===
 
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===[[Tokyo/Works/Simulation/B phase plane movie| ⇒ movie about Fig.4.C here!!]]===
 
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In the case of Fig. 4C, all the cells existed on the B-side across the dividing line from the B-side. Therefore, all the cells were not able to cross over the dividing line and moved to the B state. When all the cells existed on the A-side not the B-side ( Fig 4.A), all the cells moved to the A state, similarly.
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===The range of parameter λ <!--範囲の提示-->===
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===The range of parameter lambda <!--範囲の提示-->===
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<!--共存安定になるパラメータlambdaの範囲を調べるため,lambdaを変化させてシミュレーションをした.その結果,lambdaの値とAとBが共存安定する確率の関係はFig.5のようになった.-->
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To determine the range of the value of parameter lambda for balanced differentiation, further simulations were carried out by changing the value of lambda. As a result, the relationship between the value of lambda and the probability of balanced differentiation was determined as shown in Fig.5.
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<!--共存安定になるパラメータλの範囲を調べるため,λを変化させてシミュレーションをした.その結果,λの値とAとBが共存安定する確率の関係はFig.5のようになった.-->To determine the range of parameter λ, several simulations were conducted with different values of λ. As a result, the relationship between λ value and the probability of coexistence is determined as shown in Fig.5.
 
[[Image:range_of_lambda.JPG|400px|thumb|none|Fig.5]]
[[Image:range_of_lambda.JPG|400px|thumb|none|Fig.5]]
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<!--よって,パラメータλをこの範囲の中に入るように調節すれば,我々が目的としているモデルを作り出すことができる!-->
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<!--よって,パラメータlambdaをこの範囲の中に入るように調節すれば,我々が目的としているモデルを作り出すことができる!-->
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'''By using parameter λ in this range, we can construct our model.'''
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'''By using parameter lambda in this range, we can construct the artificial genetic circuit for balanced differentiation!''' It would not be difficult to adjust AHL production efficiency within this range by adjustment of promoter and RBS sequences.
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 +
<br>
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In addition, from the result of simulation, it was found that when the value of parameter lambda was fixed, the value of parameter ALPHA2 is needed to be greater than that of ALPHA1, for the balanced differentiation of the system.(ALPHA1,ALPHA2 represent the expression rate of Ra and Rb respectively)
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[[Image:Balanced2.JPG|400px|thumb|left|Fig.6]]
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'''[[Tokyo/Works/Simulation/supplementary|supplementary data Click here!!]]'''
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<!--[[Image:Parameter set before.JPG]]-->
<!--[[Image:Parameter set before.JPG]]-->

Latest revision as of 05:35, 27 October 2007


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

Numerical analysis and kinetic simulations for cell population

Multi-cell simulations with Hill coefficients of the promoters were carried out to find ranges of other parameters that lead the system to balanced differentiation. The differential equations for N cells were constructed to describe the cell population interacting with each other(Ex3.1).

Ex3.1 the differential equations of population model



Here the concentrations of AHL inside and outside of the cell are assumed to be the same since AHL is freely permeable though cell membranes.

First, the phase plane analysis focused on individual cells.









To analyze the behavior of the cell population, the above differential equations for N cells were extended to the stochastic differential equations(Ex3.2). (see detailed derivation ) In order to carry out the simulations with the stochastic model, Poisson random variables were introduced into the differential equations. As a result of the stochastic simulation, the different behaviors of individual cells were observed.


Ex3.2 the stochastic differential equations of the population model

In the simulations, Hill coefficients, coefficients of repression and activation of AHL and LacI(n2,n3,k2,k3) determined by the wet experiments were used. The values of other parameters were determined by reference to some journals. (Kobayashi et al, 2004;Tian T et al, 2006;Garcia-Ojalvo J et al, 2004; Tomioka R et al, 2004)


n2 2.08 (-)
K2 4.05 (uM)
n3 2.47 (-)
K3 0.295 (uM)

Determining the range of parameter which satisfy balanced differentiation

Depending on the value of parameter lambda, three different patterns were observed: the pattern with coexistence state of A and B states, the pattern with only A state, and the pattern with only B. The simulation results for their patterns are shown in Fig.1 to 3.

Fig.1.A  t=0.0(min)
Arrow2.JPG
Fig.1.B  t=3.0(min)
Arrow2.JPG
Fig.1.C   t=30(min) only A

==> movie here!!

Fig.2.A  t=0.0(min)
Arrow2.JPG
Fig.2.B  t=3.0(min)
Arrow2.JPG
Fig.2.C   t=30(min) A-B balanced differentiation !!

==> movie here!!

Fig.3.A  t=0.0(min)
Arrow2.JPG
Fig.3.B  t=3.0(min)
Arrow2.JPG
Fig.3.C   t=30(min) only B

==> movie here!!


The nullclines changed dependent on the state of the cell population. When there are only B cells, the nullclines had one intersection in the phase plane. As change of the state of the cell population, the number of the intersections of nullclines turned into two. Fig.4.A-C shows the appearance of an additional intersection, indicating an unstable equilibrium point, for the three patterns, respectively.

Fig.4.A
Fig.4.B
Fig.4.C


==> movie about Fig.4.A here!!

==> movie about Fig.4.B here!!

==> movie about Fig.4.C here!!


In Fig. 4B, the cell population was distributed around the unstable equilibrium point. Right after the moment shown in the Fig. 4B, some cells moved to the A state while the others go toward the B state. The division of the direction may have occurred due to the distribution of the population over the both sides of a “watershed,” a division line lying over the unstable equilibrium point. In another possibility, fluctuation of the cell state might allow going across the division line.


In the case of Fig. 4C, all the cells existed on the B-side across the dividing line from the B-side. Therefore, all the cells were not able to cross over the dividing line and moved to the B state. When all the cells existed on the A-side not the B-side ( Fig 4.A), all the cells moved to the A state, similarly.

The range of parameter lambda

To determine the range of the value of parameter lambda for balanced differentiation, further simulations were carried out by changing the value of lambda. As a result, the relationship between the value of lambda and the probability of balanced differentiation was determined as shown in Fig.5.

Fig.5

By using parameter lambda in this range, we can construct the artificial genetic circuit for balanced differentiation! It would not be difficult to adjust AHL production efficiency within this range by adjustment of promoter and RBS sequences.


In addition, from the result of simulation, it was found that when the value of parameter lambda was fixed, the value of parameter ALPHA2 is needed to be greater than that of ALPHA1, for the balanced differentiation of the system.(ALPHA1,ALPHA2 represent the expression rate of Ra and Rb respectively)

Fig.6

supplementary data Click here!!