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1.0hexagonally packed during pupal development. Processed images of(A–G) The wing cartoons highlight the region of the wing. Epithelia become
Cadherin-stained wings at (A) larval and (B) prepupal stages and at five different pupal times (TP1–TP5) from (C–G) Waddington stages P2B–P2C.
0.8
ci,jfor1each cell, determined by using Cellenger automated
The number of neighbors
image analysis software, is color coded on the images. All scalebars are 10 mm.
0.6 Consistent with Euler’s formula (F + V 2 E = 2), which describes the(H) Percentage of 3-fold and 4-fold vertices during pupal stages TP1–TP5.
mathematical relationship between the total number of faces (F), vertices (V), and edges (E), the percentage of 3-fold vertices increases as the
0.4epithelium becomes more hexagonal.
(I) Percentage of cells with four, five, six, seven, or eight neighbors (color coded as indicated) at different developmental times.
0.2(J) Variability in the total cell perimeter and individual cell edge length decreases between TP1 and TP4. Standard deviations are expressed asa percentage of the average perimeter (or average cell edge length) to control
for cell size differences. Approximately 2500 cells per data point
0.0
were averaged0to obtain the data in (H)–(J). Only intervein cells on the dorsal side of the wing surface were analyzed.
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i 1,j in Packing
i,j
i+1,j
Figure 1. Changesof Wild-Type
Wing Epithelia during Developmentc<ylatexitsha1_base64="8ZexNb4Hh7MaKO7xa4I+z4zi6CY=">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</latexit><latexit

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cj

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sha1_base64="xLDAuWcjsTHCM5P0nV25fLEcgDU=">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</latexit>

latexit

ハエの脳における分化の伝搬現象に対する数理モデルの数値計算結果：
xx<

sha1_base64="ZElzLLv0+Wjvf6NDgYuVGn1+K/Y=">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</latexit>

応⽤例：連続モデルでも離散構造を再現できる．
yy
積分⽅程式の重み関数を適当にとる
t=0.0
t=1.0
ことで，連続モデルでも四⾓格⼦の離散的なパターンを再現できる．
(a)<latexitcΛ6j

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latexit<sha1_base64="xLDAuWcjsTHCM5P0nV25fLEcgDU=">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</latexit>

latexit<sha1_base64="ZElzLLv0+Wjvf6NDgYuVGn1+K/Y=">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</latexit><latexitcΛ1j<

sha1_base64="xLDAuWcjsTHCM5P0nV25fLEcgDU=">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</latexit>

latexitx<

sha1_base64="ZElzLLv0+Wjvf6NDgYuVGn1+K/Y=">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</latexit><latexitysha1_base64="LQb9DcfkYmGoxCLGv61CYTuiXmU=">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</latexit>

0

latexitsha1_base64="LQb9DcfkYmGoxCLGv61CYTuiXmU=">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</latexit>

latexit

<

ハエの複眼

<

sha1_base64="xLDAuWcjsTHCM5P0nV25fLEcgDU=">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</latexit>

latexit<sha1_base64="ZElzLLv0+Wjvf6NDgYuVGn1+K/Y=">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</latexit><latexit

sha1_base64="71LHThys+WDDl19ryF2c2lRkCIs=">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</latexit><latexit

t=2.0

yeight-sided cells occur also (Figure 1I and
Figures S1H
t=3.0
t=6.0
smaller decreases in the number of four-, seven-, andand S1I). Consistent with Euler’s formula, which defines
the relationship between faces, edges, and vertices of

(b)
0.6

0.4

x
0.2

0.0

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分化度
polyhedra, the number of four-cell vertices decreases,
1.0 the number of three-cell vertices increases (Figand
ure 1H) as the average neighbor number grows from
0.8
5.55
to 5.96. This means that there is a net increase in

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latexit<8
u1,t = f (uN , u1 , u2 ) + g(u1 ),
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ui,t = f (ui 1 , ui , ui+1 ) + g(ui ),
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uN,t = f (uN 1 , uN , u1 ) + g(uN ),sha1_base64="bOpeMiFa+HEhRojhqNYgDP4RaVk=">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</latexit>

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い．⼀⽅で解析が困難なことが多い．格⼦の⼤きさを0に極限をとって連続化することが多いが，離散構造が消失してしまうことがある．

<latexit

mined the patterning of retinal cells mutant for
mall clones領域を細胞や格⼦で分割して,その上で構築した数理モデル．
of cells mutant for DE-cadherin were
多細胞⽣物の発⽣現象等に対して，良い再現性をもち，実験と相性がよ
tly after the
onset of retinal differentiation. The

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articles・空間離散*モデルを同値変形で空間連続**モデルに変換する⼿法を開発した
ns junction of all cell–cell interfaces (Fig. 3a–c).
・空間離散モデルに連続モデルの解析⼿法や，数理モデリングの⼿法が適⽤可能となり，様々な応⽤が期待できるs also expressed in the retina but is restricted from

【研究の概要】<latexit

Developmental Cell
806

⇡

<

複雑系コース ⽥中 吉太郎 准教授が2020年度⽇本数学会応⽤数学研究奨励賞を受賞：
⾮局所相互作⽤による細胞や格⼦の⼤きさと形状を保存する空間離散モデルの連続化法共同研究者：【理論班】栄伸⼀郎、⽯井宙志（北海道⼤学）、【⽣物実験班】佐藤純 、⼋杉徹雄、Miaoxing Wang （⾦沢⼤学）

s (Fig. 3d–f). Within cone cells, it is localized to thetion of cone–cone interfaces. These localization
nsistent with the homophilic interactions typical of

空間離散モデル
本連続化法

平⾏移動作⽤素や積分作⽤素を⽤いて離散モデルから同値変形で，細胞や格⼦の⼤きさと形状を保存したまま離散モデルを連続化できる
平⾏移動と特性関数を⽤いて同値変形
連続モデル(PS ) ut = f (u(x0<"⌧1l, t), u, u(x + l, t)) + g(u)u , u" ⇤ ⇢",l ) + g(u" ),"積分⽅程式で近似可能＊は重み関数との積分

ci,j+1

応⽤例： 球⾯上に離散モデルを拡張できる

A0

⽤いた重み関数

ハエの脳における分化の伝搬
現象の写真(右)と数値計算結果（左）．
本⼿法を⽤いることで，解析しやすい形でより現実的な数
理モデリングが可能となる．

*離散：⾃分の周りを切り取る区間を⼩さくしていくと⾃分以外含まれなくなる集合（数直線上の整数）**連続：⾃⾝の周りを切り取る区間を⼩さくしても，⾃⾝以外に元がある集合（数直線）