˜ ms œ ù m Ç V ê s„ ÆX c l Þ 5 “ Ó 2² Ž ù± Žõ u § T “ Ó Þ” ¢ M X  Ö «V R Ë ¿ P < T 8 ý 4 • « Œ º; c 6 ” X ¢ Ž ì ŏ Œ

‰

˜ ms œ ù m Ç V ê s„ ÆX c l Þ 5 “ Ó 2² Ž ù± Žõ u § T “ Ó Þ” ¢ M X  Ö «V R Ë ¿ P <  T 8 ý 4 • « Œ º; c 6 ” X ¢ Ž ì ŏ Œ

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y

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 s. ‘:rƒ½¨\"fH>hZ é>ߖÑþ˜|9`¦sÖ© 



HéߖÑþ˜|9õ ©œ ñŒ•6 x H ”1lx–Ð"f çߖÅÒôÇ. sQôÇ éߖÑþ˜|9[þt s_ ©œ ñŒ•6 x“Ér 4Ÿ¤¸úš> W1àÔ 0

>ß¼ (complex network)–Ð
è­q ú e”. ‘:r 7HëH\"f ĺoHéߖÑþ˜|9 œ ©ñŒ•6 x W1àÔ0>ß¼ (protein interaction network)\"f l0px$íçH (functional class)[þt_ 1lxlo\ @/ #Œ ·ú˜˜Ð€¤Œ. Õªo“¦ yŒ•yŒ• _

 l0px$íçH_ 1lxlo\¦„½Óܼ–Ð l0px$çíHs_ >8£x½¨›¸ (hierarchy)\¦&ñ_ l 0A #Œ B>h Ã

º rlink\¦•¸{9 %i. #Œl\"f rlinkH ²DG™è&h 1lxlo_ &ñ•¸\¦
?/HÁº s. s\¦s6 x #Œ é

ߖÑþ˜|9 ©œ ñŒ•6 xW1àÔ0>ß¼_ $í|9õ l0px$íçH_ >8£x&h“ ½¨›¸s_aº'›$í\ @/ #Œ 7H_Ùþ¡.

PACS numbers: 87.16.Yc,89.75.Fb,05.45.Xt Keywords: 4Ÿ¤¸úš> W1àÔ0>ß¼, Òqt^‰ W1àÔ0>ß¼

I. " e  ] Ø

éß

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)

a. sQôÇ éߖÑþ˜|9 ©œ ñŒ•6x \ ›'aôÇ ~½Ó@/ôÇ X<s'H yeast two-hybrid z´+«> 1px`¦ :Ÿx"Kf %3`¦Ãº e” [1,2]. Òqt Ó

ü

t†<Æ&h ©œ ñŒ•6 x1àWÔ0>ß¼_ „^‰&h“ ½¨›¸ (large-scale topology)H€ªœôÇ 1lx%i†<Æ&h :£¤$[íþtõ ÒqtÓütÆ&<†h îߖ&ñ$í 1

p

x %\ò†¾Ó`¦Î~ÃH [3,4]. éߖј|þ9 ©œ ñŒ•6 x_ :£¤$í ×æ 

H °ú “Érl0x$píçH\ 5ÅqÇ éôߖÑþ˜|9[þts "f–Ð ~1> ©œ ñ



Œ

•6x` ¦ H†â¾Ó`¦˜Ð“Hכ s [5]. "f éߖÑþ˜|9



©œ ñŒ•6 xW1àÔ0>ß¼ (protein-protein interaction network;

PIN)©œ\"f °ú “Érl0px$íçH\ 5ÅqôÇ éߖÑþ˜|9[þtzoH"f–Ð Ó

ü

æ5g4R e”H½¨›¸\¦°úH [5]. sQôÇ l0px$íçH“Ér PIN

\

"f_ modular structure\¦Ä»µ1ÏôÇ. ´ú§“Ér 7áxÀÓ_ PIN

½

¨›¸\"f €ªœôÇ ~½ÓZO\ _ôÇ modular structure\¦&ñ_

9Hr•¸ e”#QM®o [6–9]. :£¤y ‚ÃГ¦ëH‰³ [6,7,9] \

"

fH coupled Kuramoto oscillator [10,11]_ 1lx%i†<Æ&h :£¤

$í

`¦s6 x #Œ modular structure x9 Õª[þtçߖ_ >8£x½¨›¸

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¦ƒ½¨ %Üi¼ 9, sHY>Y> 4Ÿ¤¸úš> W1àÔ0>ß¼_ ½¨›¸ +þA

$í

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H 1lx%i†<Æ>ü< x9]XôÇ ƒ›a's e”6£§`¦Ð#˜Œïr. ‘:rƒ

½

¨\"fH PIN_ modular structure s_a›'>\¦sK

l 0A #Œ yŒ•yŒ•_ é–Ñßþ˜|9`¦ 
_ phase oscillator–Ð ç

ߖÅÒôÇ. s\¦ phase oscillator_ 1lxlo ‰&³©œ`¦s6 x 

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Œ yŒ•yŒ•_ l0px$íçH s_ >8£x½¨›¸_ :£¤$`í¦¶ú˜(R ^¦

כ s.

Ä

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#Œ ¶ú˜(R˜Ð•¸2Ÿ¤ . \V\¦ [þt#Q ĺo 3lu îߖ\"f 8AH А

o~ÃÌs
 d”©œ 1px_ ©œl[þts 1lxlo\¦ ÀsÒ 9 Œ•6 x 

“

¦ e” [12]. ¢¸ôÇ ƒ\"f ´ú§sa›'¹1Ï÷&H) ̺p[þt _

 Ö¦6£§™èo {9u #Œ [þt2;~t [13], <ʓÉrìøÍ`± Ô¦s [

þ

t_ µ1ÏFg ÅÒl {9u H [14] 1כ pxs @/³ð&h“ 1lxl



o ‰&³©œ s. "f–Ð ©œ ñ Œ•6 x H ”1lx[t“þÉr \Oi± ˜Ð

€

 Bĺ çߖéߖôÇ rÛ¼%7›“ כ °ú tëߖ, ÅÒ €ªœ “¦ <ɪ p

–Ðîr‰&³©œ[þt`¦
·p. sQôÇ 1lxlo ‰³&©œ`¦[O"î

H ©œ çߖéߖôÇ —¸+þAܼ–Ð Kuramoto model`¦ ÒqtyŒ•½+É Ã

º e”. Kuramoto model“Ér e”__ rçߖ t\"f ”1lx i_ 0A©œ (phase) θi(t)_ rçߖ\ Éro 6£§õ °ú 

“

Ér d”ܼ–Ð ÅÒ#Q”.

dθi

dt = ωi+ λ

N

X

j=1

Aijsin(θj− θi) i = 1, 2, · · · , N (1)

#

Œl"f ωiH ”1lx i_ “¦Ä»”1lxú\¦
?/“¦ [-π,π]

\

"f uniform > ìrŸí÷&#Q e”“¦ &ñ %i. λH ” 1

l

x iü< ƒ)a”1xl[þtõ_ ½+Ë>ú (coupling coef- ficient)\¦
·p. AijH ”1lx iü< ”1lx j ƒ

÷

&#Q e”€ 1s“¦, ÕªXOt ·ú§€ 0ܼ–Ð ïr. Õªo“¦ -45-

Table 1. Identification (ID) number of protein functional classes and the number of proteins in each functional classes.

Functional class ID Functional class Number of proteins

1 metabolism 177

2 energy 32

3 cellcycle and DNA processing 261

4 transcription 277

5 protein synthesis 104

6 protein fate 251

7 protein with binding function or cofactor requirement 303 8 regulation of metabolism and protein function 78 9 cellular transport, transport facilities and transport routes 161 10 cellular communication / signal transduction mechanism 65

11 cell rescue, defence and virulence 92

12 interaction with enviroment 100

13 cell fate 99

14 development 17

15 biogenesis of cellular components 211

16 cell type differentition 142

17 unclassified protein 8

N “Ér 8úx ”1lx_ >hú\¦
·p. Kuramoto model_ 1

l

xlo „s‰&³©œ`¦lÕüt l 0A #Œ

re^iϕ= 1 N

N

X

j=1

e^iθj (2)

–

Ð &ñ_÷&H |9"fë“B6§£ú (order parameter)\¦6 xôÇ



. #Œl"f r“Ér 0 ≤ r≤ 1_ °úכ`¦t 9 0A©œ_ ´ú6£§

`

¦ 8£¤&ñ H°úכs“¦, ϕHçî¨H0A©œ`¦
·p. r=1 “

â

ĺ —¸ŽH 0A©œs 1lxlo )a phase\¦
?/ 9 r=0 “Ér incoherent phase\¦
·p. ·ú¡\"f ƒ/åL)a ü< °ú s

‚ Ã

Г¦ëH‰³ [6, 7]\"fH 4¤Ÿ¸úš> W1àÔ0>ß¼\"f_ modular structure ¢¸H yŒ• modules_ >8£x½¨›¸\¦ Kuramoto model_ 1lxlo ‰&³©œ`¦s6x  #Œ ˜Ð{9 ú e%”3. ‘:rƒ

½

¨\"fH ‚ÃГ¦ëH‰³ [15]\ "f lF÷&%3~ yeast PIN\"f _

 modular structure_ ©œ ñ›'a>\¦ Kuramoto model`¦



6 x #Œ [O"î “¦ ôÇ.

II. ‰ ˜ ms œ ù m Ç V ê s„ ÆX c l “ Ó Þ 5 2² Ž ù± Ž

Ä

ºoH éߖÑþ˜|9 ©œ ñ•6Œ x W1àÔ0>ß¼\ @/ #Œ ƒ½¨  l

 0A #Œ ·ú¡\"f ƒ/åLÙ¡~þ yeast PIN`¦“¦9½+É כ s [15]. Kuramoto model_ 1lxlo „s ‰³&©œ`¦s6 x l 0

A #Œ PIN\"f ©œ H 9þtQÛ¼' (largest cluster; LC) ë

ߖ ÒqtyŒ• %i. Õª sÄ»H 1lxlo ‰&³©œs {9#Q
l 0A K

"f “]X”¸×¼ü< ©œ ñŒ•6 xs e”#Q tëߖ –Ð ìro

)

a 9þtQÛ¼'[tþs ”>rFôÇ€ „^‰&h“ 1lxlo sÀÒ#Q

|9

 ú \Ol M:ëHs. ‘:r½ƒ¨\"f 6 x)aPIN_ largest cluster_ ß¼l N“Ér 778s. s W1àÔ0>ß¼\¦sÀҍHyŒ•yŒ•

Fig. 1. Distribution of transcriptional proteins in PIN.

Black circles represents the proteins belong to the tran- scriptional class.

_

 éߖÑþ˜|9“Ér‚ÃГ¦ëH‰³ [16]_ «Ñ\¦„½Óܼ–Ð #Œ 17>h _

 l0px$íçHܼ–Ð ìrÀÓ %i (Table 1 ‚ÃЛ¸). 
_ éߖ Ñþ

˜|9“Ér ôÇ t_ l0px$íçH\ëߖ Ÿí†<Ê÷&H כ s m“¦

#

ŒQ t l0px$íçH\ Ÿí†<Ê|¨cú e”. ‚ÃГ¦ëH‰³ [5]\ _

€ PIN\"f 1lx{9ôÇ l0px$íçH\ 5ÅqôÇ éߖј|þ9[þts "f–Ð Ó

ü

æ5ge”H ↾Ós e”6§£s ·ú˜94R e”. Fig. 1“Ér sQôÇ

â

†¾Ó`¦˜Ð#ŒÅҍH ôÇ \Vs. #Œl\"f Ž“ÉrҐo ”¸×¼[þt“Ér

Fig. 2. Plot of m^f(d) against d.

„

çH (transcriptional class)\ K{©œ HéߖÑþ˜|9`¦




·p. ÕªaË>\"f ^¦ú e”H ü< °ú s Ž“Ér¸×”¼[þt“ÉrŽ

“ É

r”¸×¼[þts\"f 8 ´ú§“Érœ ©ñŒ•6 x`¦“ ¦ e”6£§`¦ ^¦ Ã

º e”. sü< °ú “Ér PIN\"f_ ‰&³©œ“Érl0px$íçH\ Ér ì

ro (functional segregation) ‰&³©œÜ¼–Ð [O"î½+É Ãº e”“¦, s

כ “Ér „çH ü@_ Ér l0px$íçH\"f•¸ 1lx{9 > ›'a

¹1

Ï ÷&%3 [5]. sQôÇ l0px$íçH\ Ér ìro‰&³©œ`¦&ñ|¾Ó

&

hܼ–Ð
?/l 0A #Œ

m^f_i(d) = M_i^f(d)

M_i(d) (3) ü

< °ú s segregation function`¦ _ñ& %i [5]. #Œl\

"

f Mi^f(d)HéߖÑþ˜|9 i\"fÂÒ' o d ëߖ
pu b#Q” f _ l

0px$íçH\ 5ÅqK e”H–Ñßéþ˜|9_ >hú\¦
?/ 9 Mⁱ(d)



HéߖÑþ˜|9 i\"fÂÒ' o d ëߖu
pb#Q” éߖÑþ˜|9_ >hú

\

¦
·p. —¸ŽH i\ @/ôÇ m ^f_i(d)_ ¨îçH m^f(d)\¦ ½¨

†

<Êܼ–Ð+‹ yŒ•yŒ•_ l0px$íçH[þts #QbG> d\  ìrŸí÷&

#

Qe”Ht\¦ ¶ú˜(R ^¦Ãº e”. sM: dH PIN\"f ÅÒ#Q”

¿

º éߖÑþ˜|9 s_ þj™è ƒ‚ ú\¦
·p. Fig. 2\

"

f ˜ÐH כ õ °ú s o d 7£x†<Ê\  functional segregation°úכs yŒ™™è H כ `¦ ^¦ ú e”. 7£¤, îr

o\"fH @/ÂÒìr °ú “Ér l0px$í çHܼ–Ð sÀÒ#Q” éߖÑþ˜

|9

[þts e”Hכ s“¦, sכ ܼ–Ð ˜Ð °ú “Ér0lpx$íçH`¦

t“¦ e”H–Ñßéþ˜|9[þtzo "f–Ð Óüæ5ge”H½¨›¸&h“ :£¤

$í

`¦&ñ|¾Ó&hܼ–ЉSX“½+É Ãº e”.

III. Ž Ò ÞM × D ü z > [­ Ž' [

l

”>r_ ƒ½¨õ[þt“Ér, Aij§'Ÿ>=_ element[þts éߖíHy

ƒ

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?/t ·§“ú¦ ×æu\¦t> ÷&Hâ Ä

º „^‰ 1lxlo ×æu\ %ò†¾Ó`¦ ~ÃÎ6£§`¦˜Ð%i [17].

‘

:r 7HëH\"f ĺoH PIN\"f ©œ ñŒ•6 x`¦ H éߖÑþ˜|9 [

þ

ts \O
 1lx{9ôÇ l0px`¦ ´ú§s /BNÄ» Ö¼
\  

×

æu\¦&ñ %i. s\¦0AK 6£§õ °ú “Ér×æu 'Ÿ§>= W \¦•¸{9 %i.

W_ij =

 ₁

2(1 + nij) éߖÑþ˜|9 iü< j ƒ÷&#Q e”`¦M: 0 éߖÑþ˜|9 iü< j ƒ÷&#Q e”t ·ú§`¦M:

(4)

#

Œl"f n^ij“Ér6£§ °õú s &ñ_÷&#Q ”.

n_ij = 1 2(O_ij

ni

+O_ji nj

) (5)

d”

 (5)\"f Oij(=O_ji)Héߖј|þ9 iü< éߖÑþ˜|9 j °ú s Ÿí

†

<Ê “¦ e”H 0lpx$íçH_ >hús 9 nⁱ, njH •yyŒŒ• éߖÑþ˜

|9

 iü< éߖÑþ˜|9 j Ÿí†<Ê “¦ e”H 8úx l0px$íçH_ >hú s

. 7£¤, éߖј|þ9 i_ l0px$íçHs sÖ© H éߖÑþ˜|9 j_ l

0px$íçHõ \O
 1lx{ 9 Ö¼
\  n^ijכs°ú &ñ÷&

“

¦ ¢-a„y 1lx{9 € 1, ÕªXtO ·ú§€ 0_ °úכ`¦ °úH



. "f WijH PIN\"f iü< j ƒ“Ér ÷&#Qe”ܼ
 /

B

N:Ÿx)a l0px$íçH`¦ tt ·ú§H âĺ (O^ij = Oji = 0) Wij=1/2s ÷&“¦ O^ij 7£x†<Ê\  W^ijH 1\  0

>t>)a. ¢¸ôÇ &ñ_ (5)\ _ € n_ij = n_ji ÷&Ù¼–Ð Wij = Wji\¦–7ßëá¤ôÇ. Fig. 3\"fH"f–Ð ƒ)a(©œ ñ



Œ

•6 x`¦ H) éߖÑþ˜|9 iü< éߖÑþ˜|9 j ¿º>h_ /BN:Ÿx)a l 0

px$íçH`¦t“¦ e”`¦M: (Oij = O_ji= 2) A_ij°úכ`¦&ñ_

H\V\¦˜Ð#Œïr. #Œl\"f nⁱ=5s“¦ n^j=3s € s

–ÐÂÒ' A^ij=0.7655\¦%3`¦º eÔ. d” (4)\¦s6 x #Œ y

Œ

• ƒ‚_ ×æ u Ér W1àÔ0>ß¼\"f_ Kuramoto model“Ér

dθ_i

dt = ω_i+ λ

N

X

j=1

A_ijW_jisin(θ_j− θi) (6)

–

Ð ³ð‰&³|¨c ú e”. ‘:r ½ƒ¨\"fH d (6)`” ¦ Û¦l 0AK

"

f 4  Runge-Kutta method\¦6 x% i. s]j yŒ•yŒ• _

 l0px$íçH\ 5ÅqKe”H ”1lx_ ²DG™è&h 1lxlo\¦ &ñ

|

¾Ó&hܼ–Ð
?/l 0A #Œ Dh–Ðîr ú rlink\¦ •¸{9 

% i [18].

rlink= 1 2L

N

X

i,j=1

Γij

∆t→∞lim 1

∆t

Z t_r+∆t t_r

e^{i[θi(t)−θj(t)]}dt     (7)

#

Œl"f L“Ér 8úx ƒ‚ ú\¦
·p. 'Ÿ§>= D_ ele- ment\¦ s6 x #Œ 1lxloa) 9þtQÛ¼'\¦ ¹1Ô?/l 0AK

"

f 6£§õ °ú s Dij¦\&ñ_ %i.

D_ij = A_ij    

lim

∆t→∞

1

∆t

Z t_r+∆t t_r

e^{i[θi(t)−θj(t)]}dt    

(8)

Fig. 3. An example to assign weights ωi and ωj when proteins i and j are connected.

Fig. 4. Plot of ρ(λ) against λ for 17 functional classes.

#

Œl\"f ĺoH 0õ 1s_ e”__ °úכ`¦°úH thresh- old Tr`¦ •¸{9 #Œ Dij > Trs€ iü< jH "f–Ð 1lxlo

)

a ”1lx–Ð çߖÅÒ “¦ ”1lx iü< ”1lx j\¦ƒôÇ.

s

ü< °ú s ƒ)a 1lxloa) ”1lx Š©œ_ >hú_ qÖ¦ s

 rlinkü< 1lx{9 •¸2Ÿ¤ Tr¦` ›¸]X†<Êܼ–Ð+‹ Dij\¦ filter- ing ½+É Ãº e”. s\¦ :Ÿx #Œ "f–Ð 1xllo)a1”lxzo Ó

ü

#Œe”H 1lxlo 9þtQÛ¼'\¦ &ñ_½+É Ãº e” [18]. PIN _

 âĺ\H  “ú°Ér 9þtQÛ¼'\ 5qÅKe”H éߖÑþ˜|9[þt“Ér #Q

‹

"

 e”__ ÒqtÓtü†<Æ&h lŒ•`¦0AK až?1lx “¦ e”“¦ &ñ ô

Ç. "f 1lxlo)a 9þtQÛ¼'[t ×þ恩œH 9 þtQÛ¼' î

ߖ\ yŒ•yŒ•_ l0px$íçHs \O
 Ÿí†<Ê÷&#Q e”Ht\¦ ›¸



K ˜Ð€ #Q*‹ôÇ l0px$íçH[þts ÅÒ#Q” ÒqtÓtü†<Æ&h lŒ•

`

¦0AK až?1xl Ht\ @/ôÇ éߖ"f\¦·ú˜ ú e”ܼo \V©œ

)

a. λ°úכ`¦ or(”\  1lxlo 9þtQÛ¼'_ ß¼l

 ØÔ>

“¦ yŒ•yŒ•_ λ°כ\ú Ér 1lxlo 9þtQÛ¼

Fig. 5. Hierarchical structure between functional classes based on the order to reach a specific value of ρ. (a) ρ = 0.4, (b) ρ = 0.5, (c) ρ = 0.6.

'

 Òqt$í)a. Õªo“¦ PIN\"f

H 1lxlo)a 9þt

QÛ¼'[þt ×æ\"f ©œH 9 þtQÛ¼' (largest cluster)\¦“¦

9ôÇ. ĺ‚ f l0x$píçH\ 5ÅqKe”Héߖј|þ9 „^‰_ >hú n_{f T}õ s×æ\"f largest cluster\ 5qÅKe”H>hú nf L_ q

Ö¦ ρ = nf L/nf T\¦ λ\¦orv 9 8£¤&ñ %i. Fig.

4\"f ^¦ ú e”1pws e_”_ ÅÒ#Q” ρ \ •¸²ú˜ > ÷&H λHl0px$íçHf\  ØÔ>
èߖ.

Fig. 5H e”__ {9&ñôÇ ρ\ •¸²ú˜ H©œ Œ•“Ér λ\¦

tHl0px$çíHÂÒ' YV–Ð Óü“Érõ\¦ ˜Ð#Œïr. Õª aË

>\"f ^¦Ãº e1”pws e”__ “¦&ñ)aρ\ •¸²ú˜ H íH"f



H ρ\ ©œ›'a\Os _ 1lx{9ôÇ ↾Ó`¦˜Ð“. sHÅÒ#Q

”

 e”__ ÒqtÓtü†<Æ&h lŒ•`¦ ú'Ÿ l 0A #Œ {9&ñôÇ q Ö

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. ¢¸ôÇ éߖÑþ˜|9çߖ_ œ ©ñŒ•6 x“Ér phase oscillator[þt_ ©œ  

ñŒ•6 x`¦Hr~´ ú e”6£§`¦˜Ð#Œïr.

é

ߖÑþ˜|9_ l0px$íçHs_ >8£x½¨›¸\¦›¸ l 0A 

#

Œ, e”__ ÅÒ#Q” ρ = const.\ •¸²ú˜ H λ°úכ_ s



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Study of the Hierarchy Among the Functional Classes in Protein-protein Interaction Networks

Young-Jin Ko, Soon-Hyung Yook and Yup Kim

Department of Physics and Research Institute for Basic Science, Kyung Hee University, Seoul 130-701 (Received 21 November 2007)

Detecting community structures and hierarchy among communities has been one of the most attractive research topics in complex network studies. In this study, we regard each protein as an oscillator that interacts with its neighboring proteins. The interaction between the proteins produces a complex network. First, we study the synchronizability of the proteins in the same functional classes of the protein-protein interaction network (PIN). In order to define the hierarchy among the functional classes based on the synchronizability of each functional class, we introduce a parameter rlink. Here, rlink represents the fraction of all possible links whose end nodes are synchronized.

Some possible relationships between the observed hierarchical structure of the functional classes and the properties of the PIN are also discussed.

PACS numbers: 87.16.Yc,89.75.Fb,05.45.Xt Keywords: Complex networks, Biological network