2012 第１回数理生物学サマーレクチャーコース@RIKEN CDB 2012/7/10

CDB サマーレクチャーコース2012：拡散現象柴田達夫
拡散現象
柴田達夫
理化学研究所発生・再生科学総合研究センター

分子の確率的な運動

Jennifer Lippincott-Schwartz @ NIH

大阪大学、上田昌宏氏が撮影、感謝
! Ueda, M. & Shibata, T. Stochastic signal processing and transduction in chemotactic response of eukaryotic cells. Biophysical Journal 93, 11–20 (2007).

1次元ランダムウォーク
-3δ -2δ -δ 0 δ 2δ 3δ
1. 各分子は速度vでx秒毎に距離δ=vxだけ右か左に移動する
2. 各ステップで右に行く確率は1/2であり、左に行く確率は1/2
分子は水の分子と相互作用をすると、最後のステップでどちら
へ動いたのか忘れてしまう
3. 各分子は他の全ての粒子と無関係に動く

0
20
40
60
80
100
-10 -5 0 5 10
timestep
position
x(t)
x(0)

-3δ -2δ -δ 0 δ 2δ 3δ
離散時刻t=nτ x(t) =
n
i=1
i位置：
x(t) =
n
i=1
i = 0位置の期待値：
i = 1, or 1ただし，
確率1/2 1/2

結論
• 粒子は平均としては動かない
-3δ -2δ -δ 0 δ 2δ 3δ

-3δ -2δ -δ 0 δ 2δ 3δ
n
i=1
i位置：
x(t) =
n
i=1
確率1/2 1/2
0
1
平均２乗変位：
（時刻tにおける位置の分散）
x(t)2 =
n⇤
i=1
i
⇥2
= 2
n⇤
i=1
2
i + 2
n⇤
i=1
n⇤
j=i+1
i j
⇥2
= 2
n⇤
i=1
2
i + 2
n⇤
i=1
n⇤
j=i+1
i j
2
= n 2
=
2
⇥
t

-3δ -2δ -δ 0 δ 2δ 3δ
n
i=1
i位置：
x(t) =
n
i=1
確率1/2 1/2
平均２乗変位： x(t)2 =
2
⇥
t
x(t)2 = 2Dt D =
2
2⇥
拡散係数

0
20
40
60
80
100
-10 -5 0 5 10
timestep
position
|x(t) x(0)|2 = 2Dt
x(t)
x(0)
|x(t) x(0)|
|x(t) x(0)|2 =
⇤
2Dt
平均２乗変位

結論
• 粒子は平均としては動かない
• 粒子の分布の広がりは時間の平方根に比例して大きくなる。
-3δ -2δ -δ 0 δ 2δ 3δ

0
t=1
t=4
t=16
0
t=1
0
t=1
t=4
分子の位置の確率分布
(t) =
q
⌦
|x(t) x(0)|2
↵
時刻t=0で位置x=0から出発する
時刻t=1,4,16で位置xにある確率
分布の幅=標準偏差

0
t=1
t=4
t=16
0
t=1
0
t=1
t=4
分子の位置の確率分布
(t) =
q
⌦
|x(t) x(0)|2
↵
時刻t=0で位置x=0から出発する
時刻t=1,4,16で位置xにある確率
|x(t) x(0)|2 = 2Dt

米沢富美子「ブラウン運動」共立出版

Fig. 2. Recorded random walk trajectories by Jean Baptiste Perrin [72]. Left part: three designs obtained by tracing
a small grain of putty (mastic, used for varnish) at intervals of 30 s. One of the patterns contains 50 single points. Right
part: the starting point of each motion event is shifted to the origin. The "gure illustrates the pdf of the travelled distance
r to be in the interval (r, r#dr), according to (2 ) exp(!r/[2 ])2 rdr, in two dimensions, with the length variance
. These gures constitute part of the measurement of Perrin, Dabrowski and Chaudesaigues leading to the determina-
tion of the Avogadro number. The result given by Perrin is 70.5;10. The remarkable wuvre of Perrin discusses all
possibilities of obtaining the Avogadro number known at that time. Concerning the trajectories displayed in the left part
of this gure, Perrin makes an interesting statement: `Si, en e!et, on faisait des pointeH s de seconde en seconde, chacun de
ces segments rectilignes se trouverait remplaceH par un contour polygonal de 30 co( teH s relativement aussi compliqueH que le
dessin ici reproduit, et ainsi de suitea. [If, veritably, one took the position from second to second, each of these rectilinear
segments would be replaced by a polygonal contour of 30 edges, each itself being as complicated as the reproduced
design, and so forth.] This already anticipates LeH vy's cognisance of the self-similar nature, see footnote 9, as well as of the
non-di!erentiability recognised by N. Wiener.
A. Fick set up the di!usion equation in 1855 [68]. Subsequently, the detailed experiments by Gouy
proved the kinetic theory explanation given by C. Weiner in 1863. After attempts of nding
a stochastic footing like the collision model by von NaK geli and John William Strutt, Lord
Rayleigh's results, it was Albert Einstein who, in 1905, unied the two approaches in his treatises on
8 R. Metzler, J. Klafter / Physics Reports 339 (2000) 1}77
e Nobel Prizes
d Lists
Prize in Physics
el Prizes in Physics
on the Nobel Prize in
s
warder for the Nobel
n Physics
ation and Selection of
s Laureates
Medal for Physics
in Physics
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ize in Chemistry
ize in Medicine
ize in Literature
eace Prize
Economic Sciences
ureates Have Their Say
ize Award Ceremonies
on and Selection of
ureates
1901 2010
Sort and list Nobel Prizes and Nobel Laureates Prize category:
1926
Jean Baptiste Perrin
Physics
The Nobel Prize in Physics 1926
The Nobel Prize in Physics 1926
The Nobel Prize in Physics 1926 was awarded to Jean Baptiste Perrin for his
work on the discontinuous structure of matter, and especially for his discovery of
sedimentation equilibrium.
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米沢富美子「ブラウン運動」共立出版 ! Metzler, R. Klafter, J. The random walk's guide to
anomalous diffusion: a fractional dynamics approach. Phys
Rep 339, 1–77 (2000).

拡散
• 水中の微粒子の拡散定数 D 10
-5
cm
2
/sec
• 10-4cm（バクテリアのサイズ）を拡散するのに
t x2/2D=5x10-4sec
• 1cm（試験官の径）を拡散するのに
t x2/2D=5x104sec
• 粒子が2倍の距離を動こうとすれば4倍、10倍の
距離を動こうとすれば100倍の時間がかかる。
|x(t) x(0)|2 =
⇤
2Dt

分子が細胞を拡散するのに何秒かかるか

FIG . 1. Snapshots from photobleaching and photoactivation experiments. In each column the Ærst row shows the cell before the laser pulse. T he next three images
show the cellular Øuorescence distribution at subsequent times after the laser pulse. Columns A , C, E , and F show photobleaching (G FP Ælter set, false color green).
Columns B and D show photoactivation (rhodamine Ælter set, false color red). Columns A to D show two different DH 5a cells expressing G FP (A and B show cell
1; C and D show cell 2). Columns E and F show a cephalexin-treated DH 5a cell, expressing G FP, being bleached Ærst at the pole (E ) and then at the center (F). T ime
points are as follows (t 5 0 is set arbitrarily as the end of the laser pulse). (A ) 2 0.42, 0.05, 0.18, 0.32, and 4.3 s. (B) 2 0.08, 0.08, 0.35, 0.62, and 4.7 s. (C) 2 0.5, 0.03,
0.10, 0.23, and 0.83 s. (D) 2 0.1, 0.03, 0.23, 0.63, and 1.7 s. (E ) 2 0.57, 0.03, 0.43, 0.77, and 2.8 s. (F) 2 0.57, 0.03, 0.20, 0.37, and 1.8 s. Bar 5 4 mm.
細胞内で拡散を測る
D=L2
/t (1~10μm2
/s)
(L: the size of the bleached area, t: recovery time)
ﬂurorescence recovery
after photobleaching
(FRAP)
Elowitz, M. B., M. G. Surette, et al. (1999).
Protein mobility in the cytoplasm of
Escherichia coli. J Bacteriol 181(1): 197-203.

• 分子が細胞内にすっかり行き渡るのにかかる時間
tmix=L2
/D L : 細胞サイズ(1 10μm)
tmix 10 0.1 sec
• L : 胚サイズ(100 1000μm)
tmix 103 105 sec
細胞中のタンパク分子の拡散定数
水中バクテリア真核細胞ミトコンドリア
拡散定数D
μm2
/sec
100程度 10程度 30程度 20~30程度

２つの分子が出会うのに
どれぐらいの時間がかかるのか？
A
B V=1μm３
A: nucleotide, etc.
DA ~500 μm2/s
RA ~0.0005-0.001 μm
B: protien
DA ~10 μm2/s
RA ~0.002-0.01 μm
t ~ 0.02 s
t / V/DR
(半径)
(拡散係数)D = DA + DB
R = RA + RB
t = V/4⇡DR衝突までにかかる平均時間
! Mikhailov, A. Hess, B. Fluctuations in Living Cells and Intracellular Trafﬁc. J Theor Biol 176, 185–192 (1995).
! Hess, B. Mikhailov, A. Microscopic Self-Organization in Living Cells - a Study of Time Matching. J Theor Biol 176, 181–184 (1995).

1次元ランダムウォークを確率分布で考える
-3δ -2δ -δ 0 δ 2δ 3δ
P(i,n) 粒子が離散時刻t=nτに位置x=δiにある確率
P(i,n +1) = P(i,n)+
1
2
P(i −1,n)+
1
2
P(i +1,n)− 2
1
2
P(i,n)
P(x,t +τ ) = P(x,t)+
1
2
P(x −δ,t)+
1
2
P(x +δ,t)− 2
1
2
P(x,t)
∂P(x,t)
∂t
=
δ 2
2τ
∂2
P(x,t)
∂x2
時刻t+τ=(n+1)τの確率を時刻t=nτの確率で表わす
τ,δの十分小さい連続

1次元ランダムウォークは拡散過程と等価
D =
2
2⇥
拡散係数∂P(x,t)
∂t
= D
∂2
P(x,t)
∂x2
P(x,t) =
1
4π Dt
e
−
x2
4 Dt
0
t=1
t=4
t=16
0
t=1
0
t=1
t=4
(t) =
q
⌦
|x(t) x(0)|2
↵
|x(t) x(0)|2 = 2Dt

Fickの法則
J = −D
∂C
∂x
∂C
∂x
0
∂C
∂x
0
C(x):位置xにおける濃度
J(x): 位置xにおける流れ密度
ものは濃度の低い方
に流れる

物質の保存則
反応なし
(Cの時間変化率)=(輸送による正味の流入)
反応あり
(Cの時間変化率)=(正味の生成率)
+(輸送による正味の流入)
∂C
∂t
= −
∂J
∂x
∂C
∂t
= f −
∂J
∂x

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

反応拡散方程式
J = −D
∂C
∂x
Fickの法則
∂C
∂t
= f −
∂J
∂x
物質の保存則
∂C(x,t)
∂t
= f + D
∂2
C(x,t)
∂x2

濃度勾配をつくる
C(x) = C0e
− D
γ x
∂C(x,t)
∂t
= D
∂2
C(x,t)
∂x2
−γ C(x,t)
濃度の時間変化拡散分解反応
反応拡散方程式
source of morphogen
λ = D / γ (µm)
特徴長さ

勾配形成
Dynamics of Dpp Signaling and
Proliferation Control
O. Wartlick,1
* P. Mumcu,2
* A. Kicheva,1
*† T. Bittig,2
* C. Seum,1
F. Jülicher,2
‡ M. González-Gaitán1
‡
Morphogens, such as Decapentaplegic (Dpp) in the fly imaginal discs, form graded concentration
profiles that control patterning and growth of developing organs. In the imaginal discs,
proliferative growth is homogeneous in space, posing the conundrum of how morphogen
concentration gradients could control position-independent growth. To understand the
mechanism of proliferation control by the Dpp gradient, we quantified Dpp concentration and
signaling levels during wing disc growth. Both Dpp concentration and signaling gradients scale
with tissue size during development. On average, cells divide when Dpp signaling levels have
increased by 50%. Our observations are consistent with a growth control mechanism based on
temporal changes of cellular morphogen signaling levels. For a scaling gradient, this mechanism
generates position-independent growth rates.
G
rowth regulation of the Drosophila wing
imaginal disc critically depends on the
Dpp morphogen gradient (1–7). Dpp mu-
tant imaginal discs fail to grow, and ectopic
expression of Dpp in clones of wing cells or-
ganizes growth and elicits the formation of an
ectopic winglet (7). Growth of imaginal discs is
spatially homogeneous. How a graded Dpp sig-
nal can control homogeneous tissue growth is an
open question for which a number of models
have been proposed: For example, it has been
suggested that the steepness of the gradient (5, 8)
and/or mechanical feedback (9, 10) control pro-
liferation. However, little quantitative data sup-
ports these models. To address this, we quantified
spatial and temporal changes of Dpp concentra-
tion, signaling activity, and disc growth param-
eters during development.
The Dpp gradient scales with wing size. We
used a functional green fluorescent protein–Dpp
(GFP-Dpp) fusion (11, 12) expressed in the en-
dogenous Dpp source to quantify GFP-Dpp pro-
files as a function of distance x from the source at
different times t during larval development (Fig.
1, A to C), both with and without expression of
the endogenous Dpp gene (13) (fig. S1). During
the growth period, the Dpp gradient expands:
Both the gradient amplitude C0 (i.e., the concen-
tration at the source boundary) and the decay
length l (the distance l over which the gradient
decays) increase significantly (Fig. 1, D and E).
The decay length, l, is proportional to the target
constant; Fig. 2, A and B; n = two independent
data sets with l/L = 0.107 (n1 = 98 discs) and
l/L = 0.116 (n2 = 60 discs); table S3]. Further
analysis of Dpp gradient profiles, C(r,t), where
r = x/L is the relative distance to the source,
revealed that the relative concentration gradient,
C(r,t)/C0(t), is invariant during development (Fig.
2A); the gradient scales with the growing tissue.
Gradient scaling behaviors have been reported
in this and other systems (14–17), and possible
mechanisms have been discussed (18, 19) [sup-
porting online material (SOM) text S1.2]. Note
that the gradient of another morphogen, Hedgehog
(Hh), does not scale (fig. S2).
Decreasing degradation accounts for gradi-
ent expansion. Gradient expansion is not due to
stretching of the gradient by wing growth, be-
cause the Dpp degradation rate is much larger
than the disc growth rate; the gradient renews
itself faster than the tissue grows (SOM text S1.1).
Hence, gradient expansion is due to changes in
Dpp production (n), diffusion (D), or degradation
(k) (12, 20) (SOM text S1.1). Estimation of these
parameters by fluorescence recovery after photo-
bleaching (FRAP) (12) and a reporter assay [SOM
quantitative procedures (QP) 3] showed that Dpp
production and diffusion vary only slightly dur-
ing the growth phase (Fig. 2, C and D), whereas
the degradation rate decreases substantially as
k ~ 1/A with increasing posterior compartment
area A (Fig. 2E). This decrease of the degrada-
tion rate could account for the constant scaling
tors determine the cellular Dpp concentration:
changes of the gradient profile (Fig. 1) and
changes in cell position, xcell(t), in the growing
tissue. Proliferation is approximately homoge-
neous in space (22, 23) (figs. S3A and S4A), so
the relative position of a cell, rcell = xcell(t)/L(t),
remains constant as the tissue grows (fig. S3A;
SOM QP5). Because rcell is constant and the
relative concentration gradient C(r,t)/C0(t) is in-
variant (Fig. 2A), the relative cellular concentra-
tion, C(rcell,t)/C0(t), is constant during development.
Therefore, the average cellular Dpp concentra-
tion, Ccell(t) = C(rcell,t), increases proportionally
to the gradient amplitude, C0(t) (fig. S4C).
The Dpp concentration increases, on aver-
age, by 40% during each cell cycle. Does the
increase in cellular Dpp concentration correlate
with changes in the proliferation rate? We de-
termined the proliferation rate (fig. S5; SOM QP4)
from the area growth rate, g ¼ A˙=A, where A˙ is
the time derivative of the area A. This is a good
approximation for the cellular proliferation rate
because the cell density only shows a minor in-
crease during wing growth (fig. S5, B and D).
During the growth phase, the growth rate (g)
decreases (fig. S5D), which reflects an increasing
cell doubling time q (q ≈ ln2/g; SOM QP1),
mostly because of a lengthening of the G2 phase
(24) (fig. S6).
We found that area growth correlates with the
increase of the gradient amplitude by a power
law (Fig. 2G)
C0(t) ~ A(t)b
where b = 0.59 (n = two data sets; table S3). The
average cellular Dpp concentration, Ccell, is pro-
portional to the amplitude C0 (see above) and there-
fore, Ccell(t) ~ A(t)b
. Derivation of this expression
with respect to time reveals a correlation of the
average growth rate (g ¼ A˙=A) with average tem-
poral changes in the Dpp level (C˙cell) perceived
by cells: C˙cell=Ccell ¼ C˙0=C0 ¼ bðA˙=AÞ ¼ bg.
Because the area growth rate and the cellular
proliferation rate gcell are approximately equal
(see above), it follows that
gcell ≈
1
b
C˙
cell
Ccell
ð2Þ
i.e., the proliferation rate is proportional to rel-
ative temporal changes of Dpp.
To estimate the relative increase of the cellular
RESEARCH ARTICLES
(1)
onMay30,2011www.sciencemag.orgDownloadedfrom
1.! Wartlick O et al. (2011) Dynamics of Dpp signaling and proliferation control. Science 331:1154–1159.
Mad (P-Mad) (29), P-Mad/Medea complex forma-
tion, and brk and dad transcription [Fig. 3A and
fig. S7; SOM experimental procedures (EP) 1]
(30, 31). Of these, we systematically analyzed nu-
clear red fluorescent protein expressed under control
of the dad enhancer (dad-nRFP) as a transcriptional
readout reflecting cellular signaling activity, Scell.
With time-lapse analysis, we confirmed that
Dpp signaling increases in living wing discs (Fig.
3B and movie S1). Consistent with Eq. 2, relative
changes in signaling, S˙=S, are larger at early
times of development, when growth is faster.
Quantification of dad-nRFP profiles, S(r,t) (Fig.
3C), in fixed discs showed that (i) the signaling
gradient scales (Fig. 3D), i.e., the scaling ratio
ls/L is constant (Fig. 3E); and (ii) the amplitude
cellular signaling level is proportional to the
amplitude (Scell ~ S0). The power-law relation
between amplitude S0 and area A (Fig. 3F) in-
dicates that the proliferation rate correlates with
the average relative temporal increase of Dpp
signal, S˙cell=Scell ¼ S˙0=S0 (as in Eq. 2):
gcell ≈
ln2
as
S˙cell
Scell
ð4Þ
Here, as = 48% implies that the cellular Dpp
signaling level Scell increases by about 50% dur-
ing each cell cycle. On the basis of Eq. 4, we
propose a model of growth control where the cell
cycle length is determined by how fast an in-
crease of cellular Dpp signal by 50% is achieved.
growth, we analyzed three conditions with changed
Dpp source and/or transport parameters (SOM
EP2): (i) haltere discs, where we found that Dpp
production, diffusion, and degradation are smaller
(32, 33) (Fig. 2, C to F; SOM QP3.2); (ii) wing
discs with a Dpp source of haltere histotype
(dppUbx) (32, 33); and (iii) wing discs with a
constant one-cell-wide source [limiting Hh sig-
naling range to one cell with membrane-tethered
Hh (Hh-CD2)] (34).
In these tissues, the decay time of the growth
rate, the growth period, and final size differ from
that of the wild-type wing disc (table S2 and fig.
S8). However, growth and Dpp signaling still are
related by the same features: (i) Gradients scale
with tissue size. The scaling ratio l/L is constant,
Fig. 1. Dpp gradient parameters.
(A) dpp-Gal4/UAS-GFP-Dpp wing
(Wi), leg (Le), and haltere (Ha) discs
at different developmental times;
w, source width, L, target width. (B)
Images of GFP-Dpp gradients, cor-
responding to boxed areas in (A).
(C) Quantification of GFP-Dpp con-
centration as a function of the dis-
tance to the source (x). (D and E)
(D) Amplitude, C0 and (E) decay
length, l, over time. At the end of
the growth phase, in prepupal discs
(t 140 hours), C0 again decreases.
Error bars correspond to standard
errors (SEM) of averages frombinned
data, and one data set per graph is
shown. For fit functions, parame-
ters, number of data sets, and num-
ber of discs per data set, see tables
S1 to S3 and SOM QP1. For ex-
tended versions of figure legends,
see SOM.
µ
λµ
A B C
D E
onMay30,2011www.sciencemag.orgDownloadedfrom

拡散方程式で表わされる現象は自由拡
散に限らない
C1 C2
∂C(x,t)
∂t
= f + D
∂2
C(x,t)
∂x2

2012 第１回数理生物学サマーレクチャーコース@RIKEN CDB 2012/7/10

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