2013年8月28−29日に理化学研究所 発生・再生科学総合研究センター（CDB）の連携大学院の大学院生向けレクチャーの柴田の講義のスライドです。

1. 細胞や組織における時間
空間パターン形成の数理
柴田達夫
理化学研究所 発生・再生科学総合研究センター
2013 理研 CDB-連携大学院 集中レクチャー 2013/8/28

2. 細胞中の時空間パタン形成
Figure 4. AurA Regulates the Subunit Composition of the Par
Complex
(A and B) Cortical release of Lgl regulates the localization of Baz to the
posterior lateral cortex. Baz-GFP was coexpressed with either Lgl-RFP
or His-RFP in pupal SOP cells. NEBD is t = 0. Anterior is oriented toward
the left. (A) In prophase, Baz-GFP localizes to the posterior lateral cortex,
as Lgl-RFP is released from this side. (B) Posterior lateral localization of
Baz-GFP fails in aurA37/37
mutants.
(C and D) AurA promotes and Lgl inhibits the assembly of the Baz com-
plex. Immunoprecipitates (IP) from larval brains expressing Baz-GFP
were analyzed. (C0
) Quantification of (C). The IP signal was adjusted to
the corresponding input signal and normalized for wild-type (WT) (set
to 1). Averages and standard deviations are shown (n = 5). Differences
to WT are significant (p < 0.05). (D) Immunoprecipitates from brains
expressing Baz-GFP alone (control) or together with either LglWT
-myc or
Lgl3A
-myc were analyzed. (D0
) Quantification of (D). The IP signal was
adjusted to the corresponding input signal and normalized for control
(set to 1). Averages and standard deviations are shown (n = 3 for Baz;
n = 6 for Lgl). Baz levels are significantly different from the control, and
Lgl levels are significantly different from each other (p < 0.05).
immunoprecipitates from aurA mutants contained an excess
of Lgl at the expense of Baz (Figure 4C). This was phenocop-
ied by expression of Lgl3A
(Figure 4D), demonstrating that
entry of Baz into the Par complex requires AurA to initiate
the phosphorylation-dependent release of Lgl from the cell
cortex. Thus, AurA triggers a remodeling of the Par complex
! Wirtz-Peitz F, Nishimura T, Knoblich JA (2008) Linking
cell cycle to asymmetric division: Aurora-A
phosphorylates the Par complex to regulate Numb
localization. Cell 135:161–173.
neuroblast
Figure 4. PAR domain maintenance does not require an intact actin cytoskeleton. (A and B) Tr
leads to rapid disruption of the actomyosin cortex as visualized with NMY-2–GFP (A) or LifeAc
cortical plane taken before and 2–3 min after drug treatment. (C) Treatment of permeable embr
(red) with CD or latrunculin A does not lead to loss of PAR domains. Select wide-field images
(D) PAR distributions several minutes after treatment with CD are similar to untreated embryos (com
to posterior profiles). Similar measurements for latrunculin A are provided in Fig. S2. (E) The reco
Published April 25, 2011
Goehring NW, Hoege C, Grill SW, Hyman AA (2011)
PAR proteins diffuse freely across the anterior-
posterior boundary in polarized C. elegans
embryos. The Journal of Cell Biology
193:583–594.
C. elegans
(Dictyostelium)
Arai Y, Shibata T et al. (2010) PNAS
107:12399–12404.
Chemotaxis cell
PIP3/PTEN
(Asymmetric cell division) (Asymmetric cell division)
Cell polarity and asymmetry

3. 組織の中のパタン形成
0
0
0.5
1
1.5
2
pSmad1@nucleus(AU)
Cont.
Venus/pSmad/DAPI
A’
a'G G’
the morphogen. In Fig. G and H, the BMP
by the nuclear phospho-Smad1 (pSmad
腹
背
背
! Inomata, H., et al. Cell 153, 1296–1311 (2013).
Fig S2

4. Wolpert’s “French flag”model (1969)
• Concentration gradient can convey
positional information.
• Such a molecule is called
“morphogen”
• Question:
What mechanism can produce a
gradient ?
s required for maximal DPP
aginal disc, the glass bottom
) gene is expressed through-
disc (Fig. 2a, top), and the
n hypomorphic gbb mutant
mall reduction of dpp activ-
tes DPP signaling within the
ysis of TGF␤ receptor func-
t be integrated downstream
heteromeric TGF␤ receptor
types of serine–threonine
ng, the type II kinase phos-
hich then transduces the sig-
ts22
. While the type II recep-
pe I receptor, Thick veins
DPP signaling23–27
, a second
SAX), is also necessary for
sue. In the embryo, SAX is
t levels of DPP signaling to
s, and in the wing disc SAX
normal levels of DPP signal-
P activity gradient23–25,28
.
bypass the requirement for
the two receptors could use
ransduction machinery.
w clarified the interrelation-
ds and receptors. In embryos
ssion, injection of increasing
ng a mutated, constitutively
receptor (TKV-A) induces
cell fates in a dose-depend-
downstream of TKV reca-
ponse to DPP (Ref. 3). In
f a constitutively activated
biological effect3,4
. However,
ected with low or moderate
Reviewsrpretation
FIGURE 1. The ‘French flag’ model of positional information
(Top) A morphogen, produced in a restricted domain within a field of cells (left panel, black stripe),
mediates the organization of the entire field into a set of discrete domains (right panel, red, white and
blue stripes), which could represent either differentiation of particular cell types or the expression
patterns of individual genes. (Bottom) The morphogen conveys positional information by forming an
extracellular gradient (curved black line) as the result of diffusion from its source and subsequent
French FlagFrenchFlag
trends in Genetics
morphogen
gradient
position
1.! Wolpert, L. Positional information and the
spatial pattern of cellular differentiation. J
Theor Biol 25, 1–47 (1969).

5. 1.! Crick, F. Diffusion in embryogenesis. Nature 225, 420–422 (1970).

6. 米沢富美子「ブラウン運動」共立出版
インクは拡散する
拡散 (Diffusion)

7. Movie S1
たんぱく質は組織中を拡散する
拡散 (Diffusion)
! Inomata, H., et al. Cell 153, 1296–1311 (2013).

8. ものは濃度の低い方に流れる
(Fickの法則)
J = −D
∂C
∂x
流れ 濃度の勾配
D > 0 : 係数
• 流れ J は濃度の勾配の低い方向に流れる
∂C
∂x
> 0
∂C
∂x
< 0
C C

9. 物質の保存則
(Cの時間変化率)=(輸送による正味の流入)
∂C
∂t
= −
∂J
∂x
x x+Δxx-Δx
C(x,t)
J(x − Δx,t) J(x,t)
C(x,t + Δt)−C(x,t)
Δt
= −
J(x,t)
Δx
+
J(x − Δx,t)
Δx
or
C(x):位置xにおける濃度
J(x): 位置xにおける流れ密度

10. 拡散方程式
J = −D
∂C
∂x
Fickの法則
∂C
∂t
= −
∂J
∂x
物質の保存則
∂C(x,t)
∂t
= D
∂2
C(x,t)
∂x2
D : 拡散係数 (μm2/sec)

11. 反応がある場合の物質の保存則
(Cの時間変化率)=(正味の生成率)+(輸送による正味の流入)
x x+Δxx-Δx
C(x,t)
J(x − Δx,t) J(x,t)
C(x):位置xにおける濃度
J(x): 位置xにおける流れ密度
f(x) : 位置xにおける反応
∂C
∂t
= f −
∂J
∂x
C(x,t + Δt)−C(x,t)
Δt
= f (x,t)−
J(x,t)
Δx
+
J(x − Δx,t)
Δx
or
f (x,t)

12. 反応拡散方程式
∂C
∂t
= f −
∂J
∂x
物質の保存則
∂C(x,t)
∂t
= f (x,t)+ D
∂2
C(x,t)
∂x2
J = −D
∂C
∂x
Fickの法則
D : 拡散係数 (μm2/sec)

13. 離散的に考える
∂C(x,t)
∂t
= f (x,t)+ D
∂2
C(x,t)
∂x2
C(x,t +Δt)−C(x,t)
Δt
= f (x,t)+ D
C(x +Δx,t)−C(x,t)[ ]− C(x,t)−C(x −Δx,t)[ ]
(Δx)2
= f (x,t)+ D
C(x +Δx,t)+C(x −Δx,t)−2C(x,t)
(Δx)2
C(x,t +Δt) = C(x,t)+Δt f (x,t)+ D
C(x +Δx,t)+C(x −Δx,t)−2C(x,t)
(Δx)2
⎛
⎝
⎜
⎞
⎠
⎟
反応拡散方程式
微分を離散化する
C(x,t) C(x + Δx,t)C(x − Δx,t)
x x+Δxx-Δx
or

14. 離散的に考える
∂C(x,t)
∂t
= f (x,t)+ D
∂2
C(x,t)
∂x2
C(x,t +Δt) = C(x,t)+Δt f (x,t)+ D
C(x+Δx,t)+C(x −Δx,t)−2C(x,t)
(Δx)2
⎛
⎝
⎜
⎞
⎠
⎟
反応拡散方程式
t
t+Δt
C(x,t) C(x + Δx,t)C(x − Δx,t)
x x+Δxx-Δx
C(x,t + Δt)
x x+Δxx-Δx
f (x,t)

15. 拡散方程式で表わされる現象は
「拡散」に限らない

16. 細胞の相互作用
C1 C2
細胞1 細胞2

17. 細胞の相互作用
C1 C2
細胞1 細胞2
∂C1(t)
∂t
=αC2 (t)− βC1(t)
細胞2の中の濃度C2に応じて
細胞1の発現量が決まる
∂C2 (t)
∂t
=αC1(t)− βC2 (t)
同様に

18. Ci Ci+1
細胞i 細胞i+1
細胞の相互作用
∂Ci (t)
∂t
=αCi−1(t)+αCi+1(t)−βCi (t)
隣の細胞の中の濃度に応じて
細胞iの発現量が決まる

19. Ci Ci+1
細胞i 細胞i+1
細胞の相互作用
∂Ci (t)
∂t
=αCi−1(t)+αCi+1(t)−βCi (t)
=α Ci−1(t)+Ci+1(t)−2Ci (t)( )−(β +2α)Ci (t)
隣の細胞の中の濃度に応じて
細胞iの発現量が決まる
∂C(x,t)
∂t
=α(Δx)2 ∂2
C(x,t)
∂x2
−(β +2α)Ci (t)
~細胞サイズ

20. まとめ１
• 拡散方程式は物質の拡散を表わす
• 反応拡散方程式は反応効果が加わっている
• 細胞間の相互作用によって組織中を拡がる
反応も反応拡散方程式で表わされる

21. 濃度勾配をつくる
モルフォゲン(morphogen)の生成
∂C(x,t)
∂t
= D
∂2
C(x,t)
∂x2
−λC(x,t)
濃度の時間変化 拡散 分解反応
反応拡散方程式
 = D λ (µm)C(x) = C0e−x/濃度の空間分布 特徴長さ
拡散距離
• 特徴長さ ℓ は、分子が分解されるまでに移動する平均距離

22. まとめ２
• 指数関数で表わされる勾配が作られる
• 因子がある領域で作られる
• 拡散によって拡がる
• 分解される
• 空間の濃度勾配は拡散係数と分解レートで
決まる

23. What is the mechanism that dynamically
adjusts the DV pattern to embryonic size?
! Inomata, H., et al. Cell 153, 1296–1311 (2013).