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 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.

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

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 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

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

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]

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
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
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

0 5 10 15
0
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 10.00
0.01
0.1
1
10
x
y
0 5 10 15
0
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、fixedpoint

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

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

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

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

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

[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？

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.

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

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
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

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

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

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)

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

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

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