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

time

step

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




n�

i=1

�i位置：

x(t) = �n�

i=1


�i = �1, or 1ただし，確率1/2 1/2

01

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

�2i + 2n⇤

i=1

n⇤

j=i+1

�i� j

⇥2

= �2n⇤

i=1

�2i + 2n⇤

i=1

n⇤

j=i+1

�i� j2

= n�2 =�2

⇥t




n�

i=1

�i位置：

x(t) = �n�

i=1


�i = �1, or 1ただし，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

time

step

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 tracinga small grain of putty (mastic, used for varnish) at intervals of 30 s. One of the patterns contains 50 single points. Rightpart: the starting point of each motion event is shifted to the origin. The "gure illustrates the pdf of the travelled distancer 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 allpossibilities of obtaining the Avogadro number known at that time. Concerning the trajectories displayed in the left partof this "gure, Perrin makes an interesting statement: `Si, en e!et, on faisait des pointeH s de seconde en seconde, chacun deces segments rectilignes se trouverait remplaceH par un contour polygonal de 30 co( teH s relativement aussi compliqueH que ledessin ici reproduit, et ainsi de suitea. [If, veritably, one took the position from second to second, each of these rectilinearsegments would be replaced by a polygonal contour of 30 edges, each itself being as complicated as the reproduceddesign, and so forth.] This already anticipates LeH vy's cognisance of the self-similar nature, see footnote 9, as well as of thenon-di!erentiability recognised by N. Wiener.

$ In the historical context, note that the theory of the continuum formulation of #uid dynamics had already been fullydeveloped at that time. Thus, some of its milestones date back to the 18th and "rst half of the 19th century, such asBernoulli's equation (1738); Euler's equation (1755); Navier's (1827) use as a phenomenological model and Stokes' (1845)derivation of the Navier}Stokes equation. Maxwells' dynamical theory of gases dates back to 1867 and Boltzmann'stransport equation was published in 1872 for the description of collision processes. The latter is the footing for theatomistic random walk approach to Brownian motion.

A. Fick set up the di!usion equation in 1855 [68].$ Subsequently, the detailed experiments by Gouyproved the kinetic theory explanation given by C. Weiner in 1863. After attempts of "ndinga stochastic footing like the collision model by von NaK geli and John William Strutt, LordRayleigh's results, it was Albert Einstein who, in 1905, uni"ed the two approaches in his treatises onthe Brownian motion, a name coined by Einstein although he reportedly did not have access toBrown's original work. Note that a similar description of di!usion was presented by the French

8 R. Metzler, J. Klafter / Physics Reports 339 (2000) 1}77

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1926

Jean Baptiste Perrin

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The Nobel Prize in Physics 1926Jean Baptiste Perrin

The Nobel Prize in Physics 1926

Jean Baptiste Perrin

The Nobel Prize in Physics 1926 was awarded to Jean Baptiste Perrin "for hiswork on the discontinuous structure of matter, and especially for his discovery ofsedimentation 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≃10-5cm2/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


F I G . 1. Snapshots from photobleaching and photoactivation experiments. I n each column the Ærst row shows the cell before the laser pulse. T he next three imagesshow the cellular Øuorescence distribution at subsequent times after the laser pulse. C olumns A , C , E , and F show photobleaching (G F P Ælter set, false color green) .C olumns B and D show photoactivation (rhodamine Ælter set, false color red) . C olumns A to D show two different D H 5a cells expressing G F P (A and B show cell1; C and D show cell 2) . C olumns E and F show a cephalexin-treated D H 5a cell, expressing G F P, being bleached Ærst at the pole (E ) and then at the center (F ) . T imepoints 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. B ar 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/sec100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

BV=1μm３

A: nucleotide, etc.

DA ~500 μm2/s RA ~0.0005-0.001 μmB: protien

DA ~10 μm2/s RA ~0.002-0.01 μm

t ~ 0.02 st / 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).



-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


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 3δP(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)+ 12P(i −1,n)+ 1

2P(i +1,n)− 2 1

2P(i,n)

P(x,t +τ ) = P(x,t)+ 12P(x −δ ,t)+ 1

2P(x +δ ,t)− 2 1

2P(x,t)

∂P(x,t)∂t

= δ 2

2τ∂2P(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 ∂2P(x,t)

∂x2

P(x,t) = 14πDt

e− x2

4Dt

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 ∂2C(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 ∂2C(x,t)

∂x2


濃度勾配をつくる

C(x) = C0e− D

γ x

∂C(x,t)∂t

= D ∂2C(x,t)∂x2

−γC(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 andProliferation ControlO. 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 concentrationprofiles that control patterning and growth of developing organs. In the imaginal discs,proliferative growth is homogeneous in space, posing the conundrum of how morphogenconcentration gradients could control position-independent growth. To understand themechanism of proliferation control by the Dpp gradient, we quantified Dpp concentration andsignaling levels during wing disc growth. Both Dpp concentration and signaling gradients scalewith tissue size during development. On average, cells divide when Dpp signaling levels haveincreased by 50%. Our observations are consistent with a growth control mechanism based ontemporal changes of cellular morphogen signaling levels. For a scaling gradient, this mechanismgenerates position-independent growth rates.

Growth regulation of the Drosophila wingimaginal disc critically depends on theDppmorphogen gradient (1–7). Dppmu-

tant imaginal discs fail to grow, and ectopicexpression of Dpp in clones of wing cells or-ganizes growth and elicits the formation of anectopic winglet (7). Growth of imaginal discs isspatially homogeneous. How a graded Dpp sig-nal can control homogeneous tissue growth is anopen question for which a number of modelshave been proposed: For example, it has beensuggested that the steepness of the gradient (5, 8)and/or mechanical feedback (9, 10) control pro-liferation. However, little quantitative data sup-ports thesemodels. To address this, we quantifiedspatial and temporal changes of Dpp concentra-tion, signaling activity, and disc growth param-eters during development.

The Dpp gradient scales with wing size. Weused 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 atdifferent times t during larval development (Fig.1, A to C), both with and without expression ofthe endogenous Dpp gene (13) (fig. S1). Duringthe growth period, the Dpp gradient expands:Both the gradient amplitude C0 (i.e., the concen-tration at the source boundary) and the decaylength l (the distance l over which the gradientdecays) increase significantly (Fig. 1, D and E).The decay length, l, is proportional to the targettissue width L [the scaling ratio l/L = 0.112 is

constant; Fig. 2, A and B; n = two independentdata sets with l/L = 0.107 (n1 = 98 discs) andl/L = 0.116 (n2 = 60 discs); table S3]. Furtheranalysis of Dpp gradient profiles, C(r,t), wherer = 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 reportedin this and other systems (14–17), and possiblemechanisms have been discussed (18, 19) [sup-porting online material (SOM) text S1.2]. Notethat the gradient of anothermorphogen, Hedgehog(Hh), does not scale (fig. S2).

Decreasing degradation accounts for gradi-ent expansion. Gradient expansion is not due tostretching of the gradient by wing growth, be-cause the Dpp degradation rate is much largerthan the disc growth rate; the gradient renewsitself faster than the tissue grows (SOM text S1.1).Hence, gradient expansion is due to changes inDpp production (n), diffusion (D), or degradation(k) (12, 20) (SOM text S1.1). Estimation of theseparameters by fluorescence recovery after photo-bleaching (FRAP) (12) and a reporter assay [SOMquantitative procedures (QP) 3] showed that Dppproduction and diffusion vary only slightly dur-ing the growth phase (Fig. 2, C and D), whereasthe degradation rate decreases substantially ask ~ 1/Awith increasing posterior compartmentarea A (Fig. 2E). This decrease of the degrada-tion rate could account for the constant scalingratio l/L, because l = ! ¯¯¯D/k and A ~ L2 (12, 21)(SOM text S1.2). Furthermore, the gradient am-plitude, C0, also increases, because of the de-creasing degradation rate and the widening of theDpp source (w) (Fig. 2F) (SOM text S1.1). Thus,changes of the Dpp source width and trafficking(degradation rate) result in Dpp gradient expan-sion during growth.

Cells experience an increase in Dpp concen-tration. Gradient expansion implies that cellularDpp concentration changes over time. Two fac-

tors determine the cellular Dpp concentration:changes of the gradient profile (Fig. 1) andchanges in cell position, xcell(t), in the growingtissue. Proliferation is approximately homoge-neous in space (22, 23) (figs. S3A and S4A), sothe 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 therelative 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 proportionallyto the gradient amplitude, C0(t) (fig. S4C).

The Dpp concentration increases, on aver-age, by 40% during each cell cycle. Does theincrease in cellular Dpp concentration correlatewith changes in the proliferation rate? We de-termined the proliferation rate (fig. S5; SOMQP4)from the area growth rate, g ! A=A, where A isthe time derivative of the area A. This is a goodapproximation for the cellular proliferation ratebecause 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 increasingcell 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 theincrease of the gradient amplitude by a powerlaw (Fig. 2G)

C0(t) ~ A(t)b

where b = 0.59 (n = two data sets; table S3). Theaverage cellular Dpp concentration, Ccell, is pro-portional to the amplitudeC0 (see above) and there-fore, Ccell(t) ~ A(t)b. Derivation of this expressionwith respect to time reveals a correlation of theaverage growth rate (g ! A=A) with average tem-poral changes in the Dpp level (Ccell) perceivedby cells: Ccell=Ccell ! C0=C0 ! b"A=A# ! bg.Because the area growth rate and the cellularproliferation rate gcell are approximately equal(see above), it follows that

gcell "1bCcell

Ccell"2#

i.e., the proliferation rate is proportional to rel-ative temporal changes of Dpp.

To estimate the relative increase of the cellularDpp concentration a = DCcell/Ccell during thecell cycle time q, we combine Eq. 2 with the ap-proximations Ccell=Ccell " "DCcell=q#=Ccell andq " ln 2/gcell, and obtain the following:

a ! DCcell

Ccell" b ln2 "3#

Thus, we find a constant a = 0.41 (n = two datasets; table S3); throughout development, cell di-vision correlates with an increase of Dpp con-centration by 40%.

RESEARCHARTICLES

1Departments of Biochemistry and Molecular Biology, Facultyof Sciences, GenevaUniversity, 30Quai Ernest-Ansermet, 1211Geneva, Switzerland. 2Max Planck Institute for the Physics ofComplex Systems,Nöthnitzer Strasse38, 01187Dresden,Germany.

*These authors contributed equally to this work.†Present address: Developmental Neurobiology, National In-stitute for Medical Research, Medical Research Council, TheRidgeway, Mill Hill, London NW71AA, UK.‡To whom correspondence should be addressed. E-mail:[email protected] (F.J.), [email protected] (M.G.-G.)

(1)

4 MARCH 2011 VOL 331 SCIENCE www.sciencemag.org1154

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from

1.!Wartlick O et al. (2011) Dynamics of Dpp signaling and proliferation control. Science 331:1154–1159.

Dpp signaling activity parallels Dpp concen-tration. Proliferation depends on Dpp signalingactivity, rather than ligand concentration (3, 25–27).We therefore measured Dpp pathway activity atdifferent levels [reviewed in (28)]: phosphorylatedMad (P-Mad) (29), P-Mad/Medea complex forma-tion, and brk and dad transcription [Fig. 3A andfig. S7; SOM experimental procedures (EP) 1](30, 31). Of these, we systematically analyzed nu-clear red fluorescent protein expressed under controlof the dad enhancer (dad-nRFP) as a transcriptionalreadout reflecting cellular signaling activity, Scell.

With time-lapse analysis, we confirmed thatDpp signaling increases in living wing discs (Fig.3B andmovie S1). Consistent with Eq. 2, relativechanges in signaling, S=S, are larger at earlytimes of development, when growth is faster.Quantification of dad-nRFP profiles, S(r,t) (Fig.3C), in fixed discs showed that (i) the signalinggradient scales (Fig. 3D), i.e., the scaling ratiols/L is constant (Fig. 3E); and (ii) the amplitude

S0 increases with A as a power law (Eq. 1),with bs = 0.69 T 0.02 (SEM) (n = four data sets;table S3), corresponding to as = 48% T 2% (Fig.3F). Invariance (scaling) of the relative signalingprofile, S(r,t)/S0(t) (Fig. 3D), implies that thecellular signaling level is proportional to theamplitude (Scell ~ S0). The power-law relationbetween amplitude S0 and area A (Fig. 3F) in-dicates that the proliferation rate correlates withthe average relative temporal increase of Dppsignal, Scell=Scell ! S0=S0 (as in Eq. 2):

gcell !ln2as

Scell

Scell"4#

Here, as = 48% implies that the cellular Dppsignaling level Scell increases by about 50% dur-ing each cell cycle. On the basis of Eq. 4, wepropose a model of growth control where the cellcycle length is determined by how fast an in-crease of cellular Dpp signal by 50% is achieved.

In different growth regimes, as ! 50%. Dppsource and transport parameters contribute to theamplitude S0 (SOM text S1.1) and therefore tocellular signaling levels Scell. To test how therate of increase of the gradient amplitude affectsgrowth, we analyzed three conditions with changedDpp source and/or transport parameters (SOMEP2): (i) haltere discs, where we found that Dppproduction, diffusion, and degradation are smaller(32, 33) (Fig. 2, C to F; SOM QP3.2); (ii) wingdiscs with a Dpp source of haltere histotype(dpp>Ubx) (32, 33); and (iii) wing discs with aconstant one-cell-wide source [limiting Hh sig-naling range to one cell with membrane-tetheredHh (Hh-CD2)] (34).

In these tissues, the decay time of the growthrate, the growth period, and final size differ fromthat of the wild-type wing disc (table S2 and fig.S8). However, growth and Dpp signaling still arerelated by the same features: (i) Gradients scalewith 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) discsat 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) decaylength, l, over time. At the end ofthe growth phase, in prepupal discs(t> 140 hours), C0 again decreases.Error bars correspond to standarderrors (SEM) of averages frombinneddata, and one data set per graph isshown. For fit functions, parame-ters, number of data sets, and num-ber of discs per data set, see tablesS1 to S3 and SOM QP1. For ex-tended versions of figure legends,see SOM.

µ

! µ

A B C

D E

Fig. 2. The Dpp gradi-ent and growth. (A) Rel-ative Dpp concentrationprofiles C(r, t)/C0 from 48hours to 130hours (span-ning the whole growthperiod) with density plot(below). Note the datacollapse of gradient pro-files onto a single curve.(B) Decay length,l, versuscompartment width, L.(C to F) (C) Dpp produc-tion rate, n; (D) diffu-sion coefficient, D; and(E) degradation rate, k,versus posterior compartment area, A; and (F) Dpp source width, w, versus posterior compartment width, L, of wing (black) and haltere (blue) discs during growth,estimated by FRAP (red rectangles, wing) and a reporter assay (circles) (SOM QP3). (G) Gradient amplitude, C0, versus posterior compartment area, A.

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RESEARCH ARTICLES

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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 ∂2C(x,t)∂x2


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