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

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  • 1. 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)
  • 2. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 Cdc2 Joseph 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.
  • 3. 2010/8/19 理研 CDB-連携大学院 集中レクチャー トグルスイッチ ReporterRepressor 1Repressor 2 Promoter 1 Promoter 2 Inducer 2 Inducer 1 Figure 1 Toggleswitch design. Repressor1 inhibits transcription from Promoter1 and is induced by Inducer 1. Repressor2 inhibits transcription from Promoter 2 and is induced by 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 coli Timothy S. Gardner1,2, Charles R. Cantor1 & James J. Collins1,2
  • 4. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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
  • 5. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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
  • 6. リプレッサー X 0 2 4 6 8 10 12 0 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] = V γ 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
  • 7. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 12 0 5 10 15 y x 0 5 10 15 0 2 4 6 8 10 12 x y d[X] dt = 0 d[Y] dt = 0 nullcline nullcline
  • 8. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 051015 0 2 4 6 8 10 12 x y 0 5 10 15 0 2 4 6 8 10 12 x y 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]
  • 9. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 051015 0 2 4 6 8 10 12 x y 0 5 10 15 0 2 4 6 8 10 12 x y 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]
  • 10. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0 5 10 15 0 2 4 6 8 10 12 x y0 5 10 15 0 2 4 6 8 10 12 x y 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]
  • 11. 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 15 0 2 4 6 8 10 12 x y 0 5 10 15 0 2 4 6 8 10 12 x y
  • 12. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0 5 10 15 0 2 4 6 8 10 12 x y XとYの発現量はどのように決まるか? 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 ⎛ ⎝⎜ ⎞ ⎠⎟
  • 13. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.1 1 10 x y 0 5 10 15 0 2 4 6 8 10 12 x y 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] [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、fixedpoint
  • 14. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0 5 10 15 0 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 15 0 2 4 6 8 10 12 x y 位相空間 (phase space) 解軌道
  • 15. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0 5 10 15 0 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
  • 16. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 = 1 n = m = 2 v = 10 V = 0 → 40 分岐
  • 17. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 = 1 n = m = 2 v = 10 V = 0 → 40 分岐 分岐点 分岐点 サドル・ノード分岐 0 10 20 30 40 50 0 10 20 30 40 50 a1 x 0 10 20 30 40 50 0.001 0.01 0.1 1 10 a1 x V V
  • 18. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 Cdc2 Joseph 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.
  • 19. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 = 1 n = m = 2 v = 10 V = 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 50 0 10 20 30 40 50 a1 x Vを増加 Vを減少 VV
  • 20. 2010/8/19 理研 CDB-連携大学院 集中レクチャー [Y] [X] X Y [X] [Y] どうやればコントロールできるか?
  • 21. 2010/8/19 理研 CDB-連携大学院 集中レクチャー [Y] [X] [Y] [X] X Y X インデューサー Y [X] [Y] どうやればコントロールできるか?
  • 22. 2010/8/19 理研 CDB-連携大学院 集中レクチャー トグルスイッチ ReporterRepressor 1Repressor 2 Promoter 1 Promoter 2 Inducer 2 Inducer 1 Figure 1 Toggleswitch design. Repressor1 inhibits transcription from Promoter1 and is induced by Inducer 1. Repressor2 inhibits transcription from Promoter 2 and is induced by 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 coli Timothy S. Gardner1,2, Charles R. Cantor1 & James J. Collins1,2
  • 23. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 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 Cdc2 Joseph 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. 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.
  • 24. 2010/8/19 理研 CDB-連携大学院 集中レクチャー Science 4 July 2008: Vol. 321. no. 5885, pp. 126 - 129 DOI: 10.1126/science.1156951 REPORTS Robust, Tunable Biological Oscillations from Interlinked Positive and Negative Feedback Loops Tony 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
  • 25. 2010/8/19 理研 CDB-連携大学院 集中レクチャー X インデューサー Y 時間対称性の破れ relaxation oscillator Vを増加 Vを減少 V X Y Z Z X 0 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 = 1 n = m = 2 v = 10 V = 0 → 40 d[X] dt = [Z] [Y]2 +1 − [X] d[Y] dt = 10 1 [X]2 +1 − [Y] d[Z] dt = 1 τ 50 [X]2 +1 − [Z] ⎛ ⎝⎜ ⎞ ⎠⎟ 0 200 400 600 800 1000 0 10 20 30 40
  • 26. 2010/8/19 理研 CDB-連携大学院 集中レクチャー X Y Z 0 10 20 30 40 50 0 10 20 30 40 50 a1 x 0 200 400 600 800 1000 0 10 20 30 40 Z X 0 10 20 30 40 50 0 10 20 30 40 50 a1 x 0 200 400 600 800 1000 0 10 20 30 40 0 200 400 600 800 1000 0 10 20 30 40 τ=100 τ=10 τ=1 half life time of gene Z relaxation oscillator 0 10 20 30 40 50 0 10 20 30 40 50 a1 x
  • 27. 2010/8/19 理研 CDB-連携大学院 集中レクチャー 0 200 400 600 800 1000 0 10 20 30 40 Z X0 200 400 600 800 1000 0 10 20 30 40 0 200 400 600 800 1000 0 10 20 30 40 τ=100 τ=10 τ=1 half life time of gene Z X Y Z 分岐
  • 28. 2010/8/19 理研 CDB-連携大学院 集中レクチャー X Y Z 興奮する遺伝子発現 0 50 100 150 200 250 300 0 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 = 1 20 40 [X]2 +1 − [Z] ⎛ ⎝⎜ ⎞ ⎠⎟ 興奮系(excitable)
  • 29. 2010/8/19 理研 CDB-連携大学院 集中レクチャー X Y Z 興奮する遺伝子発現 0 50 100 150 200 250 300 0 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 50 0.001 0.01 0.1 1 10 z x d[X] dt = 0 d[X] dt = [Z] [Y]2 +1 − [X] d[Y] dt = 10 1 [X]2 +1 − [Y] d[Z] dt = 1 20 40 [X]2 +1 − [Z] ⎛ ⎝⎜ ⎞ ⎠⎟ 安定