2010 理研 CDB-連携大学院集中レクチャー 2010/8/19 2010

2010/8/19 理研 CDB-連携大学院集中レクチャー

対称性を破る数理的メカニズム

キーワード：力学系、双安定振動、分岐、興奮性、安定性、不安定性、アトラクタ

ー、多アトラクター、時空間、対称性 ,

力学系=Dynamical system（動力学系）(力学=Mechanics)

柴田達夫広島大学理学研究科　数理分子生命理学専攻フィジカルバイオロジー研究室、CDB(10月より)


Nature Cell Biology - 5, 346 - 351 (2003) Published online: 10 March 2003; | doi:10.1038/ncb954

Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2Joseph R. Pomerening, Eduardo D. Sontag & James E. Ferrell Jr

Figure 1. Expected behaviours of several plausible Cdc2-APC circuits.a–c, Three ways that Cdc2 could respond to different concentrations of non-degradable cyclin in the absence of the APC. The Michaelian response (a) would be expected if cyclin directly activated Cdc2. The ultrasensitive response (b) could arise from multistep activation mechanisms, from stoichiometric inhibitors or from saturation effects. The bistable response (c) could arise from a combination of ultrasensitivity and positive feedback.


トグルスイッチ

R eporterR epressor 1R epressor 2Promoter 1

Promoter 2

Inducer 2

Inducer 1

Figure 1 Toggleswitch design. Repressor1 inhibits transcription from Promoter1 and isinduced by Inducer 1. Repressor2 inhibits transcription from Promoter 2 and is inducedby Inducer 2.

細胞はメモリーを持つことが出来る

Nature 403, 339-342 (20 January 2000) | doi:10.1038/35002131; Received 15 September 1999; Accepted 23 November 1999

Construction of a genetic toggle switch in Escherichia coliTimothy S. Gardner1,2, Charles R. Cantor1 & James J. Collins1,2


微分方程式で細胞現象を表現する

• 微分方程式から発現量の変化を知るために必要な情報

• 時刻t=0における、P(t)の値（初期条件)

• 反応速度定数、濃度など（パラメータ）

• 境界条件、（境界の形状、膜上(2D)か、細胞質中(3D)かなど、）

東京大学,前多裕介さん,佐野雅己さんに感謝

dP(t)dt

= α − γ P(t)

P(t):GFP強度（発現量）

生成分解・希釈変化速度


調節

リプレッサー

X

• リプレッサーは蛋白質の合成速度を遅くする

リプレッサー

X

0 2 4 6 8 10 120

5

10

15

y

x

リプレッサーの量

発現量

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

転写・翻訳分解・希釈

[X] = Vγ

1([Y] /φ)n +1

d[X]dt

= 0

ヌルクライン(nullcline)

d[X]dt

変化量の大きさ


d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

X

Y

X

Y

0 2 4 6 8 10 120

5

10

15

y

x

0 5 10 150

2

4

6

8

10

12

xy

d[X]dt

= 0

d[Y]dt

= 0

nullcline

nullcline


0

5

10

15

0 2 4 6 8 10 12

x

y

0 5 10 150

2

4

6

8

10

12

xy

X

Y

X

Y

Xと Yの発現量はどのように決まるか？

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]d[X]dt

= V 1([Y] /φ)n +1

− γ [X]


05

1015

024681012

x

y

0 5 10 150

2

4

6

8

10

12

xy

X

Y

X

Y


d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]d[X]dt

= V 1([Y] /φ)n +1

− γ [X]


0 5 10 150

2

4

6

8

10

12

xy

0 5 10 150

2

4

6

8

10

12

x

yXと Yの発現量はどのように決まるか？

X

Y

X

Y

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

2010/8/19 理研 CDB-連携大学院集中レクチャーXと Yの発現量はどのように決まるか？

X

Y

X

Y

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

0 5 10 150

2

4

6

8

10

12

x

y

0 5 10 150

2

4

6

8

10

12

x

y


0 5 10 150

2

4

6

8

10

12

x

y


X

Y

X

Y

d[X]dt

= 0

d[Y]dt

= 0

nullcline

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

接ベクトルd[X]dt

, d[Y]dt

⎛⎝⎜

⎞⎠⎟


0.05 0.10 0.50 1.00 5.00 10.000.01

0.1

1

10

x

y

0 5 10 150

2

4

6

8

10

12

x

y


X

Y

X

Y

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

[X]

[Y]

[X]

[Y]

発現量は2種類の状態を取ることが出来る

（双安定, bistable）

安定

安定

不安定

固定点、平衡点、fixed point


0 5 10 150

2

4

6

8

10

12

x

y

Xと Yの発現量の軌跡

X

Y

X

Y

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

0 5 10 150

2

4

6

8

10

12

x

y位相空間

(phase space)

解軌道


0 5 10 150

2

4

6

8

10

12

x

y

Xと Yの発現量の軌跡

X

Y

X

Y

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

位相空間(phase space)

解軌道

separatrix

アトラクター

アトラクター


d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

φ =ψ = γ = η = 1n = m = 2v = 10V = 0→ 40

分岐


d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

φ =ψ = γ = η = 1n = m = 2v = 10V = 0→ 40

分岐

分岐点分岐点サドル・ノード分岐

0 10 20 30 40 500

10

20

30

40

50

a1

x0 10 20 30 40 50

0.001

0.01

0.1

1

10

a1

x

V

V




Figure 1. Expected behaviours of several plausible Cdc2-APC circuits.a–c, Three ways that Cdc2 could respond to different concentrations of non-degradable cyclin in the absence of the APC. The Michaelian response (a) would be expected if cyclin directly activated Cdc2. The ultrasensitive response (b) could arise from multistep activation mechanisms, from stoichiometric inhibitors or from saturation effects. The bistable response (c) could arise from a combination of ultrasensitivity and positive feedback.


d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

φ =ψ = γ = η = 1n = m = 2v = 10V = 0→ 40

ヒステリシス

Vを一定速度で増やした後、減らすVの値が同じでも履歴に依存してX

は異なる値をとるX

インデューサー

Y

0 10 20 30 40 500

10

20

30

40

50

a1

x

Vを増加

Vを減少

VV


[Y]

[X]X

Y

[X]

[Y]どうやればコントロールできるか？


[Y]

[X]

[Y]

[X]X

Y

X


Y

[X]

[Y]どうやればコントロールできるか？


トグルスイッチ

R eporterR epressor 1R epressor 2Promoter 1

Promoter 2

Inducer 2

Inducer 1

Figure 1 Toggleswitch design. Repressor1 inhibits transcription from Promoter1 and isinduced by Inducer 1. Repressor2 inhibits transcription from Promoter 2 and is inducedby Inducer 2.

細胞はメモリーを持つことが出来る

Nature 403, 339-342 (20 January 2000) | doi:10.1038/35002131; Received 15 September 1999; Accepted 23 November 1999

Construction of a genetic toggle switch in Escherichia coliTimothy S. Gardner1,2, Charles R. Cantor1 & James J. Collins1,2




Figure 1. Expected behaviours of several plausible Cdc2-APC circuits.a–c, Three ways that Cdc2 could respond to different concentrations of non-degradable cyclin in the absence of the APC. The Michaelian response (a) would be expected if cyclin directly activated Cdc2. The ultrasensitive response (b) could arise from multistep activation mechanisms, from stoichiometric inhibitors or from saturation effects. The bistable response (c) could arise from a combination of ultrasensitivity and positive feedback. d–i, Three ways that a Cdc2–APC circuit could respond to a constant rate of cyclin synthesis. If the response of Cdc2 to cyclin is Michaelian or ultrasensitive (as in a and b) and the regulation of the APC by Cdc2 is direct, then the system will always approach a stable steady state (d, e). Adding an intermediate enzyme (such as Plx1) between Cdc2 and the APC can turn a monostable system (as in d and e) into a negative feedback oscillator (f, g). Adding positive feedback to make the response of Cdc2 to cyclin bistable (as in c) can turn the system into a relaxation oscillator, with explosive spikes of Cdc2 activity (h–i). Details of the modelling can be found in Supplementary Information, Part 3.


Science 4 July 2008:Vol. 321. no. 5885, pp. 126 - 129DOI: 10.1126/science.1156951

REPORTSRobust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback LoopsTony Yu-Chen Tsai,1* Yoon Sup Choi,1,2* Wenzhe Ma,3,4 Joseph R. Pomerening,5 Chao Tang,3,4 James E. Ferrell, Jr.1


X


Y

時間対称性の破れrelaxation oscillator

Vを増加

Vを減少

V

X

Y

Z Z

X0 10 20 30 40 50

0

10

20

30

40

50

a1

x

d[X]dt

= V 1([Y] /φ)n +1

− γ [X]

d[Y]dt

= ν 1([X] /ψ )m +1

−η[Y]

φ =ψ = γ = η = 1n = m = 2v = 10V = 0→ 40

d[X]dt

=[Z]

[Y]2 +1− [X]

d[Y]dt

= 10 1[X]2 +1

− [Y]

d[Z]dt

=1τ

50[X]2 +1

− [Z ]⎛⎝⎜

⎞⎠⎟

0 200 400 600 800 10000

10

20

30

40


X

Y

Z

0 10 20 30 40 500

10

20

30

40

50

a1

x

0 200 400 600 800 10000

10

20

30

40 Z

X

0 10 20 30 40 500

10

20

30

40

50

a1x

0 200 400 600 800 10000

10

20

30

40

0 200 400 600 800 10000

10

20

30

40

τ=100

τ=10

τ=1

half life time of gene Z

relaxation oscillator

0 10 20 30 40 500

10

20

30

40

50

a1

x


0 200 400 600 800 10000

10

20

30

40

Z

X0 200 400 600 800 10000

10

20

30

40

0 200 400 600 800 10000

10

20

30

40

τ=100τ=10

τ=1

half life time of gene Z

X

Y

Z

分岐


X

Y

Z興奮する遺伝子発現

0 50 100 150 200 250 3000

10

20

30

40

t

X,Y,Z

蛋白質Xを少しだけ増やす（刺激）と、Xはますます増える

Z

X

Y

d[X]dt

=[Z]

[Y]2 +1− [X]

d[Y]dt

= 10 1[X]2 +1

− [Y]

d[Z]dt

=120

40[X]2 +1

− [Z ]⎛⎝⎜

⎞⎠⎟

興奮系(excitable)


X

Y

Z興奮する遺伝子発現

0 50 100 150 200 250 3000

10

20

30

40

t

X,Y,Z

蛋白質Xを少しだけ増やす（刺激）と、Xはますます増える

Z

X

Y

d[Z]dt

= 0

0 10 20 30 40 500.001

0.01

0.1

1

10

zx

d[X]dt

= 0

d[X]dt

=[Z]

[Y]2 +1− [X]

d[Y]dt

= 10 1[X]2 +1

− [Y]

d[Z]dt

=120

40[X]2 +1

− [Z ]⎛⎝⎜

⎞⎠⎟

安定

Education

2010 理研 CDB-連携大学院 集中レクチャー 2010/8/19 2010

2010 理研 CDB-連携大学院集中レクチャー 2010/8/19 2010