Superfamilies

of

Evolved

and

Designed

Networks

Ron

Milo

et

al.

Science

-LRB-

-RRB-

DOI

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What

led

to

the

pervasiveness

of

hybrids

between

Cx

pipiens

and

Cx

molestus

in

North

America

but

not

in

Europe

and

Africa

still

remains

to

be

determined

In

southernmost

Europe

we

identified

two

populations

with

a

few

hybrid

individuals

as

well

as

populations

with

pure

Cx

pipiens

signatures

and

populations

with

a

mix

of

pure

Cx

pipiens

and

pure

Cx

molestus

signatures

-LRB-

Fig.

-RRB-

Indeed

previous

allozyme

based

studies

indicated

the

existence

of

populations

in

Italy

with

a

mix

of

the

two

forms

-LRB-

-RRB-

but

a

very

low

rate

of

hybridization

-LRB-

-RRB-

probably

because

of

their

different

mating

behaviors

-LRB-

-RRB-

The

rarity

of

southern

European

hybrids

and

our

failure

to

find

hybrids

in

northern

Europe

may

be

due

to

their

low

fitness

and

inability

to

diapause

Importantly

the

introduction

to

the

United

States

of

separate

populations

of

Cx

pipiens

and

Cx

molestus

that

later

hybridized

or

of

hybrids

from

southern

Europe

has

led

to

abundant

and

ubiquitous

hybrid

forms

that

survive

the

rigors

of

northern

winters

It

is

now

clear

that

models

derived

from

the

U.S.

epidemic

of

WNV

-LRB-

-RRB-

may

not

be

applicable

to

Eurasia

and

vice

versa

-LRB-

-RRB-

A

major

factor

in

all

recent

outbreaks

-LRB-

Romania

Russia

and

United

States

-RRB-

is

the

involvement

of

mosquitoes

in

the

Cx

pipiens

complex

as

the

primary

vectors

-LRB-

-RRB-

Unlike

European

Cx

pipiens

U.S.

Cx

pipiens

appears

to

bite

readily

both

avian

hosts

and

humans

-LRB-

-RRB-

Here

we

have

shown

that

across

the

northeastern

United

States

a

large

proportion

of

individuals

are

hybrids

of

humanbiter

and

bird-biter

forms

In

combination

with

susceptible

migrating

birds

and

highly

concentrated

human

populations

in

U.S.

cities

and

suburbs

the

prevalence

of

such

bridge

vectors

that

readily

transmit

the

virus

among

and

between

avian

hosts

and

humans

could

have

created

the

current

epidemic

conditions

The

present

study

suggests

that

changes

in

vectorial

capacity

and

the

creation

of

new

efficient

vectors

may

occur

with

new

introductions

In

particular

the

arrival

of

hybrid

American

forms

in

northern

Europe

has

the

potential

to

radically

change

the

dynamics

of

WNV

in

Europe

References

and

Notes

C.

G.

Hayes

Ann

N.Y.

Acad

Sci

-LRB-

-RRB-

R.

E.

Harbach

C.

Dahl

G.

B.

White

Proc

Entomol

Soc

Wash.

-LRB-

-RRB-

T.

Guillemaud

N.

Pasteur

F.

Rousset

Proc

R.

Soc

London

Ser

B.

-LRB-

-RRB-

K.

L.

Knight

Supplement

to

the

Catalog

of

the

Mosquitoes

of

the

World

-LRB-

Diptera

Culicidae

-RRB-

T.

S.

Foundation

Ed

-LRB-

Entomological

Society

of

America

College

Park

MD

-RRB-

supplement

to

vol

K.

Byrne

R.

A.

Nichols

Heredity

-LRB-

-RRB-

R.

E.

Harbach

B.

A.

Harrison

A.

M.

Gad

Proc

Entomol

Soc

Wash.

-LRB-

-RRB-

P.

F.

Mattingly

et

al.

Trans

R.

Entomol

Soc

Lond

-LRB-

-RRB-

D.

X.

Zhang

G.

M.

Hewitt

Mol

Ecol

-LRB-

-RRB-

N.

Keyghobadi

M.

A.

Matrone

G.

D.

Ebel

L.

D.

Kramer

D.

M.

Fonseca

Mol

Ecol

Notes

-LRB-

-RRB-

D.

M.

Fonseca

C.

T.

Atkinson

R.

C.

Fleischer

Mol

Ecol

-LRB-

-RRB-

F.

H.

Drummond

Trans

R.

Entomol

Soc

Lond

-LRB-

-RRB-

K.

Tanaka

K.

Mizusawa

E.

S.

Saugstad

Contrib

Am

Entomol

Inst

-LRB-

-RRB-

J.

K.

Pritchard

M.

Stephens

P.

Donnelly

Genetics

-LRB-

-RRB-

A.

R.

Barr

Am

J.

Trop

Med

Hyg

-LRB-

-RRB-

A.

J.

Cornel

et

al.

J.

Med

Entomol

-LRB-

-RRB-

S.

Urbanelli

F.

Silvestrini

W.

K.

Reisen

E.

De

Vito

L.

Bullini

J.

Med

Entomol

-LRB-

-RRB-

L.

L.

Cavalli-Sforza

F.

Cavalli-Sforza

The

Great

Human

Diasporas

The

History

of

Diversity

and

Evolution

-LRB-

Addison-Wesley

Reading

MA

-RRB-

J.

de

Zulueta

Parassitologia

-LRB-

-RRB-

S.

Urbanelli

et

al.

in

Ecologia

Atti

I

Congr

Naz

Soc

Ital

Ecol.

A.

Moroni

O.

Ravera

A.

Anelli

Eds

-LRB-

Zara

Parma

Italy

-RRB-

pp.

C.

Chevillon

R.

Eritja

N.

Pasteur

M.

Raymond

Genet

Res

-LRB-

-RRB-

P.

D.

Crook

N.

S.

Crowcroft

D.

W.

Brown

Commun

Dis

Public

Health

-LRB-

-RRB-

Z.

Hubalek

Viral

Immunol

-LRB-

-RRB-

H.

M.

Savage

et

al.

Am

J.

Trop

Med

Hyg

-LRB-

-RRB-

A.

G.

Richards

Entomol

News

-LRB-

-RRB-

We

thank

N.

Becker

C.

Curtis

M.

Carroll

G.

Ebel

C.

Evans

M.

Santa

Ana

Gouveia

L.

Kramer

G.

O'Meara

F.

Noguera

and

C.

Williams

for

providing

invaluable

mosquito

samples

J.

Smith

M.

Matrone

T.

Ganguly

and

the

DNA

Sequencing

Facility

University

of

Pennsylvania

for

technical

assistance

and

A.

Bhandoola

and

four

anonymous

reviewers

for

comments

and

valuable

suggestions

on

an

earlier

version

of

this

manuscript

Supported

by

a

National

Research

Council

Associateship

through

the

Walter

Reed

Army

Institute

of

Research

-LRB-

D.M.F.

-RRB-

by

NIH

grant

nos

and

and

by

NSF

grant

no

This

material

reflects

the

views

of

the

authors

and

should

not

be

construed

to

represent

those

of

the

Department

of

the

Army

or

the

Department

of

Defense

Supporting

Online

Material

Materials

and

Methods

Tables

to

References

and

Notes

December

accepted

January

Superfamilies

of

Evolved

and

Designed

Networks

Ron

Milo

Shalev

Itzkovitz

Nadav

Kashtan

Reuven

Levitt

Shai

Shen-Orr

Inbal

Ayzenshtat

Michal

Sheffer

Uri

Alon

Complex

biological

technological

and

sociological

networks

can

be

of

very

different

sizes

and

connectivities

making

it

difficult

to

compare

their

structures

Here

we

present

an

approach

to

systematically

study

similarity

in

the

local

structure

of

networks

based

on

the

significance

profile

-LRB-

SP

-RRB-

of

small

subgraphs

in

the

network

compared

to

randomized

networks

We

find

several

superfamilies

of

previously

unrelated

networks

with

very

similar

SPs

One

superfamily

including

transcription

networks

of

microorganisms

represents

rate-limited

information-processing

networks

strongly

constrained

by

the

response

time

of

their

components

A

distinct

superfamily

includes

protein

signaling

developmental

genetic

networks

and

neuronal

wiring

Additional

superfamilies

include

power

grids

protein-structure

networks

and

geometric

networks

World

Wide

Web

links

and

social

networks

and

word-adjacency

networks

from

different

languages

Many

networks

in

nature

share

global

properties

-LRB-

-RRB-

Their

degree

sequences

-LRB-

the

number

of

edges

per

node

-RRB-

often

follow

a

long-tailed

distribution

in

which

some

nodes

are

much

more

connected

than

the

average

Departments

of

Molecular

Cell

Biology

Physics

of

Complex

Systems

and

Computer

Science

Weizmann

Institute

of

Science

Rehovot

Israel

To

whom

correspondence

should

be

addressed

at

Department

of

Molecular

Cell

Biology

Weizmann

Institute

of

Science

Rehovot

Israel

E-mail

urialon@weizmann.ac.il

E.

B.

Vinagradova

Culex

pipiens

pipiens

Mosquitoes

Taxonomy

Distribution

Ecology

Physiology

Genetics

Applied

Importance

and

Control

-LRB-

Pensoft

Moscow

-RRB-

A.

Spielman

Ann

N.Y.

Acad

Sci

-LRB-

-RRB-

M.

J.

Turell

M.

L.

O'Guinn

D.

J.

Dohm

J.

W.

Jones

J.

Med

Entomol

-LRB-

-RRB-

K.

A.

Bernard

et

al.

Emerg

Infect

Dis

-LRB-

-RRB-

V.

L.

Kulasekera

et

al.

Emerg

Infect

Dis

-LRB-

-RRB-

D.

J.

Dohm

M.

R.

Sardelis

M.

J.

Turell

J.

Med

Entomol

-LRB-

-RRB-

R.

S.

Nasci

et

al.

Emerg

Infect

Dis

-LRB-

-RRB-

-LRB-

-RRB-

In

addition

natural

networks

often

show

the

small-world

property

of

short

paths

between

nodes

and

highly

clustered

connections

-LRB-

-RRB-

Despite

these

global

similarities

networks

from

different

fields

can

have

very

different

local

structure

-LRB-

-RRB-

It

was

recently

found

that

networks

display

certain

patterns

termed

network

motifs

at

much

higher

frequency

than

expected

in

randomized

networks

-LRB-

-RRB-

In

biological

networks

these

motifs

were

suggested

to

be

recurring

circuit

elements

that

carry

out

key

informationprocessing

tasks

-LRB-

-RRB-

MARCH

VOL

SCIENCE

Downloaded

from

on

May

REPORTS

To

understand

the

design

principles

of

complex

networks

it

is

important

to

compare

the

local

structure

of

networks

from

different

fields

The

main

difficulty

is

that

these

networks

can

be

of

vastly

different

sizes

-LSB-

for

example

World

Wide

Web

-LRB-

WWW

-RRB-

hyperlink

networks

with

millions

of

nodes

and

social

networks

with

tens

of

nodes

-RSB-

and

degree

sequences

Here

we

present

an

approach

for

comparing

network

local

structure

based

on

the

significance

profile

-LRB-

SP

-RRB-

To

calculate

the

SP

of

a

network

the

network

is

compared

to

an

ensemble

of

randomized

networks

with

the

same

degree

sequence

The

comparison

to

randomized

networks

compensates

for

effects

due

to

network

size

and

degree

sequence

For

each

subgraph

i

the

statistical

significance

is

described

by

the

Z

score

-LRB-

-RRB-

Zi

Nreali

<Nrandi >

-RRB-

std

-LRB-

Nrandi

-RRB-

graph

appears

in

the

network

and

Nrandi

and

std

-LRB-

Nrandi

-RRB-

are

the

mean

and

standard

deviation

of

its

appearances

in

the

randomized

network

ensemble

The

SP

is

the

vector

of

Z

scores

normalized

to

length

SPi

Z

i

-LRB-

Z

i

-RRB-

The

normalization

emphasizes

the

relative

significance

of

subgraphs

rather

than

the

absolute

significance

This

is

important

for

comparison

of

networks

of

different

sizes

because

motifs

-LRB-

subgraphs

that

occur

much

more

often

than

expected

at

random

-RRB-

in

large

networks

tend

to

display

higher

Z

scores

than

motifs

in

small

networks

-LRB-

-RRB-

We

present

in

Fig.

the

SP

of

the

possible

directed

connected

triads

-LRB-

triad

significance

profile

TSP

-RRB-

for

networks

from

different

fields

-LRB-

-RRB-

The

TSP

of

these

networks

is

almost

always

insensitive

to

removal

of

of

the

edges

or

to

addition

of

new

edges

at

random

demonstrating

that

it

is

robust

to

missing

data

or

random

data

errors

-LRB-

SOM

Text

-RRB-

Several

superfamilies

of

networks

with

similar

TSPs

emerge

from

this

analysis

One

superfamily

includes

sensory

transcription

networks

that

control

gene

expression

in

bacteria

and

yeast

in

response

to

external

stimuli

In

these

transcription

networks

the

nodes

represent

genes

or

operons

and

the

edges

represent

direct

transcriptional

regulation

-LRB-

-RRB-

Networks

from

three

microorganisms

the

bacteria

Escherichia

coli

-LRB-

-RRB-

and

Bacillus

subtilis

-LRB-

-RRB-

and

the

yeast

Saccharomyces

cerevisiae

-LRB-

-RRB-

were

analyzed

The

networks

have

very

similar

TSPs

-LRB-

correlation

coefficient

c

-RRB-

They

show

one

strong

motif

triad

termed

feedforward

loop

The

feedforward

loop

has

been

theoretically

and

experimentally

shown

Fig.

The

triad

significance

profile

-LRB-

TSP

-RRB-

of

networks

from

various

disciplines

The

TSP

shows

the

normalized

significance

level

-LRB-

Z

score

-RRB-

for

each

of

the

triads

Networks

with

similar

characteristic

profiles

are

grouped

into

superfamilies

The

lines

connecting

the

significance

values

serve

as

guides

to

the

eye

The

networks

are

as

follows

-LRB-

where

N

and

E

are

the

number

of

nodes

and

edges

respectively

-RRB-

-LRB-

-RRB-

-LRB-

i

-RRB-

Direct

transcription

interactions

in

the

bacteria

E.

coli

-LRB-

-RRB-

-LRB-

TRANSC-E

COLI

N

E

-RRB-

and

B.

subtilis

-LRB-

-RRB-

-LRB-

TRANSC-B

SUBTILIS

N

E

-RRB-

and

in

the

yeast

S.

cerevisiae

-LSB-

TRANC-YEAST

N

E

-LRB-

-RRB-

and

N

E

-LRB-

-RRB-

-RSB-

-LRB-

ii

-RRB-

Signal-transduction

interactions

in

mammalian

cells

based

on

the

signal

transduction

knowledge

environment

-LRB-

STKE

-RRB-

-LRB-

SIGNAL-TRANSDUCTION

N

E

-RRB-

transcription

networks

that

guide

development

in

fruit

fly

-LRB-

from

the

GeNet

literature

database

-RRB-

-LRB-

TRANSC-DROSOPHILA

N

E

-RRB-

endomesoderm

development

in

sea

urchin

-LRB-

-RRB-

-LRB-

TRANSC-SEA

URCHIN

N

E

-RRB-

and

synaptic

connections

between

neurons

in

C.

elegans

-LRB-

NEURONS

N

E

-RRB-

-LRB-

iii

-RRB-

WWW

hyperlinks

between

Web

pages

in

the

site

-LRB-

-RRB-

-LRB-

N

E

-RRB-

pages

related

to

literary

studies

of

Shakespeare

-LRB-

-RRB-

-LRB-

N

E

-RRB-

and

pages

related

to

tango

specifically

the

music

of

Piazzolla

-LRB-

-RRB-

-LRB-

N

E

-RRB-

and

social

networks

including

inmates

in

prison

-LRB-

N

E

-RRB-

sociology

freshmen

-LRB-

-RRB-

-LRB-

N

E

-RRB-

and

college

students

in

a

course

about

leadership

-LRB-

N

E

-RRB-

-LRB-

iv

-RRB-

Word-adjacency

networks

of

a

text

in

English

-LRB-

ENGLISH

N

E

-RRB-

French

-LRB-

FRENCH

N

E

-RRB-

Spanish

-LRB-

SPANISH

N

E

-RRB-

and

Japanese

-LRB-

JAPANESE

N

E

-RRB-

and

a

bipartite

model

with

two

groups

of

nodes

of

sizes

and

with

probability

of

a

directed

or

mutual

edge

between

nodes

of

different

groups

being

p

and

q

respectively

and

no

edges

between

nodes

within

the

same

group

-LRB-

BIPARTITE

N

E

-RRB-

SCIENCE

VOL

MARCH

Downloaded

from

on

May

where

Nreali

is

the

number

of

times

the

sub

REPORTS

to

perform

signal-processing

tasks

such

as

persistence

detection

-LRB-

-RRB-

pulse

generation

and

acceleration

of

transcription

responses

-LRB-

-RRB-

Triad

the

is

an

anti-motif

-LRB-

a

significantly

underrepresented

subgraph

-RRB-

corresponding

to

the

shallow

architecture

of

these

networks

which

have

few

long

cascades

-LRB-

-RRB-

These

networks

are

sensory

networks

which

need

to

respond

within

minutes

to

transient

signals

such

as

stresses

and

nutrients

The

minimal

time

required

for

response

-LRB-

for

the

first

proteins

to

be

expressed

-RRB-

is

indeed

on

the

order

of

minutes

If

the

information

needs

to

pass

additional

steps

-LRB-

a

regulator

protein

needs

to

be

expressed

and

cross

its

activation

threshold

to

turn

on

a

gene

-RRB-

then

the

response

time

is

much

longer

The

response

time

in

each

cascade

step

has

been

experimentally

and

theoretically

shown

to

be

on

the

order

of

the

geneproduct

lifetime

-LRB-

-RRB-

often

tens

of

minutes

Thus

these

networks

are

rate-limited

networks

where

the

desired

response

times

are

often

as

short

as

the

response

times

of

the

network

components

In

the

rate-limited

network

superfamily

long

cascades

and

feedback

loops

are

rare

-LRB-

-RRB-

Feedbacks

are

usually

closed

by

protein-protein

interactions

and

not

by

transcription

-LRB-

-RRB-

Purely

transcriptional

feedback

loops

may

be

rare

because

they

are

unstable

and

noisy

due

to

their

delays

-LRB-

-RRB-

or

because

they

can

be

locked

into

an

irreversible

state

-LRB-

-RRB-

both

undesirable

properties

for

sensory

transcription

networks

We

find

a

distinct

superfamily

that

includes

three

types

of

biological

networks

-LRB-

-RRB-

signal-transduction

interactions

in

mammalian

cells

based

on

the

Signal

Transduction

Knowledge

Environment

-LRB-

STKE

-RRB-

-LRB-

-RRB-

developmental

transcription

networks

that

guide

development

in

fruit

fly

-LRB-

-RRB-

and

sea-urchin

-LRB-

-RRB-

and

synaptic

wiring

between

neurons

in

Caenorhabditis

elegans

-LRB-

-RRB-

These

networks

show

triads

and

with

positive

TSPs

and

triads

and

with

negative

TSPs

-LRB-

Fig.

-RRB-

In

contrast

to

the

sensory

transcription

networks

of

microorganisms

these

networks

display

two-node

feedbacks

that

regulate

or

are

regulated

by

a

third

node

-LRB-

triads

-RRB-

and

are

less

biased

against

cascades

-LRB-

triad

-RRB-

The

common

feature

to

this

superfamily

of

informationprocessing

networks

is

that

the

response

time

of

each

step

in

the

network

is

usually

much

shorter

than

the

response

time

required

for

the

biological

function

of

the

network

Protein

signal-transduction

networks

often

need

to

respond

within

an

hour

or

longer

but

each

interaction

can

take

minutes

or

less

Cascade

steps

in

developmental

networks

can

have

response

times

of

tens

of

minutes

but

the

processes

they

control

are

much

slower

on

the

order

of

animal

-LRB-

-RRB-

cell-division

times

that

can

take

several

hours

Neuronal

net

Fig.

The

correlation

coefficient

matrix

of

the

triad

significance

profiles

for

the

directed

networks

in

Fig.

works

in

C.

elegans

typically

need

to

respond

within

a

second

but

neuron

response

times

are

shorter

than

ms.

Thus

it

appears

that

this

superfamily

characterizes

biological

information-processing

networks

that

are

not

rate-limited

Next

we

analyzed

three

WWW

networks

of

hyperlinks

between

Web

pages

related

to

university

literature

or

music

-LRB-

-RRB-

The

TSPs

were

quite

similar

-LRB-

c

to

-RRB-

Triads

and

had

the

highest

TSP

values

and

and

the

lowest

The

overrepresented

triads

have

many

transitive

triplet

interactions

where

if

and

then

-LRB-

table

-RRB-

For

example

the

overrepresented

triad

termed

clique

has

six

transitive

interactions

the

highest

transitivity

possible

in

a

triad

The

less

represented

triads

such

as

and

are

highly

intransitive

We

also

analyzed

three

social

networks

in

which

nodes

represent

people

in

a

group

and

edges

represent

positive

sentiment

directed

from

one

group

member

to

another

based

on

questionnaires

-LRB-

-RRB-

The

TSPs

of

the

three

social

networks

were

very

similar

-LRB-

c

to

-RRB-

Notably

their

TSPs

were

quite

close

to

the

TSP

of

the

WWW

nets

-LRB-

c

to

-RRB-

This

suggests

that

WWW

networks

and

social

networks

may

be

part

of

a

superfamily

The

tendency

of

social

networks

to

display

transitive

interactions

and

transitive

triads

is

well

established

-LRB-

-RRB-

The

similarity

of

WWW

and

social

networks

suggests

that

classical

models

of

social

structural

organization

-LRB-

-RRB-

may

also

be

used

to

understand

WWW

structure

Texts

can

also

be

represented

as

networks

-LRB-

-RRB-

We

studied

word-adjacency

networks

in

which

each

node

represents

a

word

and

a

directed

connection

occurs

when

one

word

directly

follows

the

other

in

the

text

The

TSPs

of

texts

from

different

languages

and

of

different

sizes

were

similar

-LRB-

Fig.

compares

texts

in

English

French

Spanish

and

Japanese

-RRB-

The

main

feature

was

the

underrepresentation

of

triangle-shaped

triads

to

This

is

related

to

the

structure

of

languages

in

which

words

belong

to

categories

and

a

word

from

one

category

tends

to

be

followed

by

one

from

a

different

category

-LRB-

-RRB-

For

example

among

the

most

connected

words

are

prepositions

which

are

usually

followed

by

nouns

or

articles

Figure

also

shows

the

TSP

for

a

model

bipartite

graph

-LRB-

-RRB-

in

which

nodes

belong

to

two

groups

and

connections

are

formed

between

these

groups

and

not

within

the

groups

The

bipartite

model

graph

shows

a

TSP

that

resembles

that

of

the

wordadjacency

networks

The

similarities

between

networks

can

also

be

visualized

by

looking

at

the

correlation

between

the

TSPs

of

different

networks

-LRB-

Fig.

-RRB-

The

correlations

can

be

used

to

cluster

the

networks

into

distinct

superfamilies

-LRB-

-RRB-

The

TSPs

display

certain

conservation

relations

between

subgraph

types

For

example

networks

with

excess

triangle-shaped

triads

tend

to

have

a

deficit

of

V-shaped

triads

We

find

that

there

are

several

triad

conservation

rules

in

networks

that

conserve

the

degree

sequences

of

single

and

mutual

edges

-LRB-

SOM

Text

-RRB-

As

a

result

the

values

of

the

TSP

are

determined

by

only

seven

degrees

of

freedom

One

can

intuitively

interpret

these

conservation

laws

in

terms

of

reactions

that

convert

V-shaped

subgraphs

into

triangles

preserving

the

degrees

of

all

participating

nodes

Analysis

of

the

reactions

occurring

in

each

network

allows

a

compact

description

of

the

difference

between

networks

and

their

randomized

counterparts

-LRB-

SOM

Text

-RRB-

We

now

consider

nondirected

networks

in

which

edges

have

no

specified

direction

MARCH

VOL

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from

on

May

REPORTS

Fig.

The

subgraph

ratio

profile

-LRB-

SRP

-RRB-

for

various

nondirected

networks

The

networks

are

as

follows

-LRB-

-RRB-

-LRB-

i

-RRB-

The

electrical

power

grid

of

the

western

United

States

-LRB-

-RRB-

-LRB-

POWERGRID

N

E

-RRB-

and

a

geometric

model

with

similar

clustering

coefficient

-LRB-

GEO-MODELPG

N

E

-RRB-

-LRB-

ii

-RRB-

Networks

of

secondary-structure

elements

adjacency

for

several

large

proteins

-LSB-

structure

based

on

the

PDB

database

-LRB-

www

-RRB-

the

proteins

-LRB-

and

their

PDB

ID

-RRB-

were

an

immunoglobulin

-LRB-

N

E

-RRB-

a

serine

protease

inhibitor

-LRB-

N

E

-RRB-

and

an

oxidoreductase

-LRB-

N

E

-RRB-

-RSB-

and

a

geometric

model

with

similar

clustering

coefficient

-LRB-

GEO-MODEL-PS

N

E

-RRB-

-LRB-

iii

-RRB-

The

Internet

at

the

autonomous

system

level

-LRB-

-RRB-

-LRB-

AUTONOMOUS-SYSTEMS

to

N

E

-RRB-

-LRB-

iv

-RRB-

Networks

grown

according

to

the

preferential

attachment

BA

model

-LRB-

-RRB-

with

m

or

m

edges

per

new

node

-LRB-

BA

m

N

E

-RRB-

ality

Because

nondirected

networks

have

only

two

types

of

triads

-LRB-

V

and

triangle

-RRB-

we

analyzed

the

profile

of

the

six

types

of

nondirected

connected

tetrads

-LRB-

four-node

subgraphs

-RRB-

Unlike

triads

the

normalized

Z

scores

of

tetrads

show

a

significant

dependence

on

the

network

size

Therefore

instead

of

an

SP

based

on

Z

scores

we

use

the

abundance

of

each

subgraph

i

relative

to

random

networks

Nreali

i

<Nrandi >

<Nrandi >

Nreali

where

ensures

that

is

not

misleadingly

large

when

the

subgraph

appears

very

few

times

in

both

the

real

and

random

networks

-LRB-

here

-RRB-

The

subgraph

ratio

profile

-LRB-

SRP

-RRB-

is

the

vector

of

i

normalized

to

length

SRPi

i

-LRB-

2 1/2

i

-RRB-

A

nondirected

network

representing

the

electrical

power

grid

of

the

western

United

States

-LRB-

-RRB-

showed

an

SRP

with

overrepresentation

of

tetrads

to

-LRB-

Fig.

-RRB-

Nondirected

networks

of

protein

structure

in

which

nodes

are

secondary-structure

elements

-LRB-

helices

and

strands

-RRB-

and

two

nodes

are

connected

if

their

distance

is

smaller

than

have

similar

SRPs

with

overrepresented

tetrads

and

We

compared

these

networks

to

model

networks

in

which

connections

are

deter

mined

on

a

lattice

by

geometrical

proximity

In

these

geometrically

constrained

networks

the

nodes

are

arrayed

on

a

lattice

-LRB-

a

line

in

one

dimension

a

plane

in

two

dimensions

etc

-RRB-

Points

that

are

closer

than

a

distance

R

on

the

lattice

are

linked

by

an

edge

with

probability

p.

Points

at

a

distance

greater

than

R

are

unlinked

The

resulting

subgraph

distributions

of

these

networks

and

their

corresponding

randomized

versions

can

be

analytically

calculated

We

find

good

agreement

between

the

real-world

protein

structure

and

power-grid

SRPs

and

the

corresponding

geometrical

models

with

a

similar

number

of

nodes

edges

and

clustering

coefficient

-LRB-

Fig.

-RRB-

A

distinct

family

of

SRPs

was

found

for

the

Internet

at

the

level

of

nondirected

connections

between

autonomous

systems

-LRB-

AS

which

are

groups

of

computers

within

which

networking

is

handled

locally

but

between

which

data

flows

over

the

public

Internet

-RRB-

We

studied

the

structures

of

the

AS

network

sampled

at

different

time

points

from

to

-LRB-

-RRB-

The

SRP

of

the

AS

networks

was

similar

despite

their

different

sizes

We

find

that

the

SRP

of

these

networks

is

very

different

from

that

of

the

geometrically

constrained

superfamily

with

tetrads

to

underrepresented

and

tetrad

overrepresented

Finally

we

studied

the

preferential

attachment

model

of

Barabasi

and

Albert

-LRB-

BA

-RRB-

-LRB-

-RRB-

which

is

widely

used

to

study

network

evolution

In

the

BA

model

a

nondirected

network

is

grown

node

by

node

connecting

each

new

node

to

m

existing

ones

The

probability

of

connecting

to

an

existing

node

i

increases

with

the

number

of

edges

it

already

has

We

find

that

the

SRP

of

these

networks

-LRB-

Fig.

-RRB-

has

different

forms

for

m

m

and

high

m

-LRB-

-RRB-

This

is

because

not

all

tetrad

patterns

can

be

created

when

m

or

The

present

approach

can

thus

be

used

to

study

model

networks

-LRB-

-RRB-

and

allow

comparison

of

their

local

structure

to

that

of

real-world

systems

In

the

SOM

Text

we

also

show

the

SRP

of

tetrads

for

the

directed

networks

considered

above

We

find

that

generally

tetrad

profiles

of

related

networks

are

similar

However

networks

of

different

types

in

the

same

triad

superfamily

sometimes

show

distinct

tetrad

profiles

suggesting

that

higher

order

subgraph

profiles

can

help

refine

network

classification

The

present

approach

demonstrates

that

networks

of

the

same

type

share

not

only

network

motifs

but

also

characteristic

SPs

with

very

similar

proportions

of

motifs

and

antimotifs

significance

In

addition

we

find

several

superfamilies

of

networks

that

have

similar

SPs

even

though

they

describe

different

systems

of

vastly

different

sizes

What

do

the

superfamilies

mean

One

possibility

is

that

the

similarity

in

SP

is

accidental

and

that

distinct

evolutionary

histories

can

lead

to

a

similar

local

structure

SCIENCE

VOL

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REPORTS

Another

possibility

is

that

the

different

systems

in

a

superfamily

have

similar

key

circuit

elements

because

they

evolved

to

perform

similar

tasks

The

latter

possibility

leads

to

intriguing

hypotheses

that

connect

networks

from

different

disciplines

This

can

allow

for

a

better

understanding

of

a

given

network

based

on

results

from

other

networks

in

the

same

superfamily

It

would

be

interesting

to

use

the

present

approach

to

map

the

relation

between

the

function

and

the

local

structure

of

real-world

networks

References

and

Notes

Bioinformatics

in

press

-RRB-

was

used

to

generate

randomized

networks

and

to

calculate

the

subgraph

numbers

in

the

real

and

randomized

networks

The

randomized

networks

preserved

the

number

of

incoming

and

outgoing

edges

for

each

node

as

well

as

the

number

of

mutual

edges

for

each

node

-LRB-

a

mutual

edge

between

x

and

y

corresponds

to

and

-RRB-

Networks

and

their

sources

are

listed

in

table

Neural

synapses

of

C.

elegans

included

all

data

except

those

for

muscle

cells

Unlike

-LRB-

-RRB-

all

connections

were

used

not

only

those

with

synapses

Our

definition

of

the

TSP

can

be

sensitive

to

rare

events

of

a

mutual

edge

within

a

network

We

omitted

mutual

edges

in

networks

where

mutual

edges

make

up

of

the

total

number

of

edges

-LRB-

TRANSCYEAST

with

one

mutual

edge

and

with

two

-RRB-

D.

Thieffry

A.

M.

Huerta

E.

Perez-Rueda

J.

ColladoVides

Bioessays

-LRB-

-RRB-

T.

Ishii

K.

Yoshida

G.

Terai

Y.

Fujita

K.

Nakai

Nucleic

Acids

Res

-LRB-

-RRB-

T.

I.

Lee

et

al.

Science

-LRB-

-RRB-

N.

Rosenfeld

U.

Alon

J.

Mol

Biol

-LRB-

-RRB-

G.

Lahav

et

al.

Nature

Genet

-LRB-

-RRB-

M.

B.

Elowitz

S.

Leibler

Nature

-LRB-

-RRB-

J.

J.

Tyson

K.

C.

Chen

B.

Novak

Curr

Opin

Cell

Biol

-LRB-

-RRB-

E.

H.

Davidson

et

al.

Science

-LRB-

-RRB-

J.

P.

Eckmann

E.

Moses

Proc

Natl.

Acad

Sci

U.S.A.

-LRB-

-RRB-

M.

A.

J.

Van

Duijn

M.

Huisman

F.

N.

Stokman

F.

W.

Wasseur

E.

P.

H.

Zeggelink

J.

Math

Sociol

-LRB-

-RRB-

P.

Holland

S.

Leinhardt

D.

Heise

Eds.

in

Sociological

Methodology

-LRB-

Jossey-Bass

San

Francisco

-RRB-

pp.

D.

Cartwright

F.

Harary

Psychol

Rev.

-LRB-

-RRB-

R.

F.

Cancho

R.

V.

Sole

Proc

R.

Soc

London

Ser

B.

-LRB-

-RRB-

J.

Allen

Natural

Language

Understanding

-LRB-

Pearson

Addison

Wesley

ed

-RRB-

The

similarities

seen

in

the

TSPs

of

different

networks

can

be

further

appreciated

by

inspecting

the

distriNreali

bution

of

concentrations

of

subgraphs

Ci

-LRB-

Nreali

-RRB-

Even

though

they

show

similar

TSP

profiles

networks

of

different

types

in

the

same

superfamily

can

show

quite

different

concentration

distributions

-LRB-

SOM

Text

-RRB-

The

TSP

can

therefore

highlight

similarities

that

are

not

evident

from

merely

counting

subgraphs

J.

Berg

M.

Lassig

Phys

Rev.

Lett

-LRB-

-RRB-

K.

I.

Goh

E.

Oh

H.

Jeong

B.

Kahng

D.

Kim

Proc

Natl.

Acad

Sci

U.S.A.

-LRB-

-RRB-

We

thank

J.

P.

Eckmann

A.

L.

Barabasi

F.

N.

Stokman

M.

A.

J.

Van

Duijn

and

the

STKE

COSIN

GeNet

DBTBS

YPD

regulonDB

PDB

database

teams

for

valuable

data

sets

and

M.

E.

J.

Newmann

D.

Chklovskii

M.

Ronen

M.

Lapidot

and

I.

Yanai

for

discussions

We

thank

NIH

Israel

Science

Foundation

and

Minerva

for

support

R.M.

is

supported

by

a

doctoral

fellowship

from

the

Horowitz

Complexity

Science

Foundation

Supporting

Online

Material

SOM

Text

Figs

to

Tables

and

References

July

accepted

January

MARCH

VOL

SCIENCE

Downloaded

from

on

May

S.

H.

Strogatz

Nature

-LRB-

-RRB-

M.

Newman

SIAM

Rev.

-LRB-

-RRB-

A.

L.

Barabasi

R.

Albert

Science

-LRB-

-RRB-

D.

J.

Watts

S.

H.

Strogatz

Nature

-LRB-

-RRB-

S.

Maslov

K.

Sneppen

Science

-LRB-

-RRB-

S.

Shen-Orr

R.

Milo

S.

Mangan

U.

Alon

Nature

Genet

-LRB-

-RRB-

R.

Milo

et

al.

Science

-LRB-

-RRB-

N.

Rosenfeld

M.

B.

Elowitz

U.

Alon

J.

Mol

Biol

-LRB-

-RRB-

S.

Mangan

U.

Alon

Proc

Natl.

Acad

Sci

U.S.A.

-LRB-

-RRB-

S.

Mangan

A.

Zaslaver

U.

Alon

J.

Mol

Biol

-LRB-

-RRB-

The

algorithm

of

-LRB-

-RRB-

-LRB-

available

at

www

weizmann.ac.il

mcb/UriAlon

see

also

org/abs/cond-mat

and

N.

Kashtan

et

al.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

motif social biological complexity network


BIOINFORMATICS

APPLICATIONS

NOTE

Vol

no.

pages

DOI

MrBayes

Bayesian

phylogenetic

inference

under

mixed

models

Fredrik

Ronquist

and

John

P.

Huelsenbeck

Department

of

Systematic

Zoology

Evolutionary

Biology

Centre

Uppsala

University

Norbyv

Uppsala

Sweden

and

Section

of

Ecology

Behavior

and

Evolution

Division

of

Biological

Sciences

University

of

California

San

Diego

La

Jolla

CA

USA

Received

on

December

revised

on

February

accepted

on

February

ABSTRACT

Summary

MrBayes

performs

Bayesian

phylogenetic

analysis

combining

information

from

different

data

partitions

or

subsets

evolving

under

different

stochastic

evolutionary

models

This

allows

the

user

to

analyze

heterogeneous

data

sets

consisting

of

different

data

types

e.g.

morphological

nucleotide

and

protein

and

to

explore

a

wide

variety

of

structured

models

mixing

partition-unique

and

shared

parameters

The

program

employs

MPI

to

parallelize

Metropolis

coupling

on

Macintosh

or

UNIX

clusters

Availability

Contact

fredrik.ronquist@ebc.uu.se

Computational

complexity

has

long

been

a

major

obstacle

in

the

development

of

statistical

approaches

to

phylogenetic

inference

Even

moderate-sized

empirical

problems

have

posed

serious

challenges

to

computational

biologists

forcing

compromises

in

analytical

accuracy

However

the

recent

introduction

of

Bayesian

inference

and

Markov

chain

Monte

Carlo

-LRB-

MCMC

-RRB-

techniques

to

phylogenetics

has

changed

this

situation

Early

Bayesian

phylogenetics

papers

showed

that

Markov

chains

based

on

the

Metropolis

Hastings

algorithm

were

computationally

more

efficient

than

the

standard

Maximum

Likelihood

-LRB-

ML

-RRB-

bootstrapping

approach

-LRB-

Larget

and

Simon

-RRB-

It

is

now

known

that

problems

with

more

than

sequences

-LRB-

taxa

-RRB-

can

be

analyzed

successfully

with

moderate

computational

effort

using

Bayesian

inference

and

an

MCMC

convergence

acceleration

technique

known

as

Metropolis

coupling

-LRB-

Huelsenbeck

et

al.

-RRB-

Such

problems

are

set

on

tree

spaces

many

orders

of

magnitude

larger

than

those

amenable

to

ML

bootstrapping

The

increase

in

computational

efficiency

associated

with

the

Bayesian

MCMC

approach

makes

it

possible

to

analyze

more

complex

and

realistic

evolutionary

models

than

previously

Currently

an

important

but

commonly

To

whom

correspondence

should

be

addressed

invoked

constraint

on

model

complexity

is

the

assumption

of

data

homogeneity

Many

phylogenetic

data

sets

now

include

evidence

from

several

different

sources

morphology

and

molecules

amino

acid

and

nucleotide

data

or

sequences

from

the

mitochondrial

plastid

and

nuclear

genomes

However

the

available

software

commonly

forces

the

investigator

to

either

-LRB-

-RRB-

model

the

evolution

of

such

data

using

a

single

stochastic

model

-LRB-

-RRB-

analyze

the

different

data

partitions

or

subsets

separately

and

use

ad

hoc

methods

to

obtain

a

summary

result

or

-LRB-

-RRB-

resort

to

simple

search

algorithms

or

non-statistical

methods

of

these

alternatives

is

particularly

attractive

MrBayes

is

a

completely

rewritten

and

restructured

version

of

MrBayes

a

command-driven

program

for

Bayesian

phylogenetic

inference

-LRB-

Huelsenbeck

and

Ronquist

-RRB-

The

hallmark

of

the

new

program

is

a

powerful

framework

for

phylogenetic

inference

under

mixed

models

accommodating

data

heterogeneity

This

framework

will

help

the

user

to

specify

mixed

models

and

exploit

the

computational

efficiency

of

Bayesian

MCMC

analysis

in

dealing

with

composite

data

sets

Bayesian

phylogenetic

inference

is

based

on

Bayes

's

rule

Applied

to

the

phylogeny

problem

the

rule

can

be

expressed

as

follows

f

-LRB-

v

X

-RRB-

f

-LRB-

v

-RRB-

f

-LRB-

X

v

-RRB-

f

-LRB-

X

-RRB-

where

X

is

the

data

matrix

is

the

topology

of

the

tree

v

is

a

vector

of

branch

-LRB-

or

edge

-RRB-

lengths

on

the

tree

and

is

a

vector

of

substitution

model

parameters

The

distribution

f

-LRB-

v

-RRB-

is

referred

to

as

the

prior

and

specifies

the

prior

probability

of

different

parameter

values

f

-LRB-

X

v

-RRB-

is

the

likelihood

function

describing

the

probability

of

the

data

under

different

parameter

values

and

f

-LRB-

X

-RRB-

is

the

total

probability

of

the

data

summed

and

integrated

over

the

parameter

space

Bayesian

inference

is

based

on

the

so-called

posterior

distribution

f

-LRB-

v

X

-RRB-

Typically

it

is

not

possible

to

calculate

the

posterior

probability

distribution

analytically

instead

MCMC

tech

Bioinformatics

-LRB-

-RRB-

c

Oxford

University

Press

all

rights

reserved

MrBayes

Bayesian

phylogenetic

inference

under

mixed

models

niques

are

used

to

obtain

samples

from

it

MrBayes

uses

a

Metropolis

Hastings

sampler

and

updates

single

parameters

or

blocks

of

related

parameters

in

each

move

Assume

that

in

the

current

generation

the

Markov

chain

has

parameter

values

v

and

and

that

we

are

considering

a

change

in

to

the

new

value

picked

from

some

proposal

distribution

q

-LRB-

-RRB-

Then

we

accept

the

change

with

probability

r

min

f

-LRB-

-RRB-

f

-LRB-

X

v

-RRB-

q

-LRB-

-RRB-

f

-LRB-

-RRB-

f

-LRB-

X

v

-RRB-

q

-LRB-

-RRB-

Bayesian

inference

is

easily

extended

to

deal

with

heterogeneous

models

Assume

that

we

have

two

subsets

of

our

data

X

a

and

X

b

such

that

X

X

a

X

b

Assume

further

that

it

is

likely

that

these

data

subsets

evolved

on

the

same

phylogenetic

tree

but

according

to

entirely

different

substitution

models

with

parameters

a

and

b

respectively

such

that

a

b

Bayes

's

rule

now

becomes

f

-LRB-

v

a

b

X

-RRB-

f

-LRB-

v

a

b

-RRB-

f

-LRB-

X

v

a

b

-RRB-

f

-LRB-

X

-RRB-

-LSB-

f

-LRB-

v

a

b

-RRB-

f

-LRB-

X

a

v

a

-RRB-

f

-LRB-

X

b

v

b

-RRB-

-RSB-

-LSB-

f

-LRB-

X

-RRB-

-RSB-

During

a

MCMC

run

we

might

consider

a

change

of

the

substitution

model

parameters

affecting

one

of

the

subsets

for

instance

the

subset

X

a

Then

we

have

a

potential

change

from

a

to

a

and

the

jumping

kernel

of

the

Metropolis

Hastings

sampler

simplifies

to

r

min

f

-LRB-

a

-RRB-

f

-LRB-

X

a

v

a

-RRB-

q

-LRB-

a

a

-RRB-

f

-LRB-

a

-RRB-

f

-LRB-

X

a

v

a

-RRB-

q

-LRB-

a

a

-RRB-

Because

we

only

need

to

calculate

the

likelihood

ratio

-LRB-

the

second

ratio

in

the

product

-RRB-

for

the

affected

data

subsets

the

run

will

actually

be

faster

-LRB-

per

generation

-RRB-

under

a

mixed

than

under

a

homogeneous

model

Of

course

since

there

are

more

parameters

in

the

mixed

model

we

will

be

visiting

each

parameter

more

rarely

and

this

is

likely

to

lead

to

slower

mixing

and

a

need

to

run

the

chain

longer

before

an

adequate

sample

of

the

posterior

distribution

is

obtained

The

net

effect

is

dependent

on

the

particulars

of

the

analysis

but

there

is

no

obvious

reason

why

the

computational

complexity

would

necessarily

be

much

worse

under

a

parameter-rich

mixed

model

than

under

a

homogeneous

model

for

the

same

data

set

MrBayes

now

provides

the

tools

needed

to

examine

this

question

empirically

MrBayes

implements

a

wide

variety

of

stochastic

models

for

nucleotide

protein

restriction

site

and

morphological

-LRB-

standard

-RRB-

data

Single

doublet

-LRB-

for

stem

regions

-RRB-

and

codon

-LRB-

with

or

without

variation

in

the

nonsynonymous/synonymous

rate

ratio

across

sites

-RRB-

models

are

all

available

for

nucleotide

data

Protein

data

can

be

analyzed

using

a

range

of

fixed

or

variable

rate

matrices

Both

the

standard

and

restriction

site

models

can

include

correction

for

coding

biases

The

standard

model

appropriate

for

analysis

of

morphological

data

allows

up

to

ten

different

states

and

there

are

models

for

both

unordered

and

linearly

ordered

characters

Rate

variation

across

sites

can

be

accommodated

using

a

standard

gamma

or

for

nucleotide

and

protein

models

an

autocorrelated

gamma

distribution

For

nucleotide

and

protein

data

it

is

possible

to

allow

rate

variation

across

the

tree

using

a

variant

of

the

covarion/covariotide

model

in

which

sites

independently

switch

between

an

on

and

an

off

state

-LRB-

Huelsenbeck

et

al.

Huelsenbeck

and

Ronquist

-RRB-

Available

tree

models

include

unconstrained

standard

molecular

clock

birth-and-death

and

coalescent

models

MrBayes

reads

data

from

a

text

file

conforming

to

a

modified

NEXUS

format

allowing

mixed

data

sets

By

default

the

data

are

partitioned

according

to

data

type

The

user

can

further

subdivide

the

data

most

easily

by

specifying

character

sets

Once

the

data

set

has

been

partitioned

appropriately

the

user

can

set

the

model

structure

and

priors

of

each

data

subset

Individual

model

parameters

can

be

unlinked

or

linked

across

selected

data

subsets

The

currently

active

subsets

and

the

model

parameters

applying

to

these

subsets

can

be

listed

giving

the

user

a

means

of

checking

that

the

mixed

model

is

specified

correctly

In

addition

to

allowing

heterogeneity

across

data

subsets

in

overall

rate

and

in

substitution

model

parameters

MrBayes

also

allows

the

user

to

unlink

topology

and

branch

lengths

Different

data

subsets

can

thus

have

independent

branch

lengths

or

even

different

topologies

Correct

scaling

of

rate

parameters

is

important

in

mixed

models

MrBayes

scales

rates

such

that

branch

lengths

are

measured

in

the

expected

number

of

changes

per

site

For

instance

a

change

in

a

codon

model

is

counted

as

one

change

per

three

sites

Partition-specific

rate

multipliers

are

scaled

such

that

the

mean

rate

per

site

across

partitions

is

unity

MrBayes

provides

many

options

for

summarizing

and

diagnosing

the

results

of

an

MCMC

analysis

Simple

plots

of

overall

likelihood

and

individual

parameter

values

can

be

generated

to

determine

burn-in

and

examine

mixing

and

the

program

will

also

estimate

the

model

likelihood

used

in

Bayesian

model

testing

The

parameter

file

is

a

tab-delimited

text

file

which

can

be

imported

into

and

analyzed

with

most

standard

statistical

software

packages

The

program

will

summarize

trees

and

branch

lengths

in

the

form

of

consensus

trees

partition

tables

and

lists

of

trees

with

their

estimated

posterior

probability

Consensus

trees

are

written

with

both

branch

lengths

and

posterior

clade

probabilities

for

easy

graphical

representation

using

software

such

as

TreeView

-LRB-

Page

-RRB-

F.Ronquist

and

J.P.Huelsenbeck

By

default

MrBayes

uses

Metropolis

coupling

to

accelerate

convergence

of

the

Markov

chain

-LRB-

Huelsenbeck

and

Ronquist

-RRB-

Because

of

the

relatively

small

amount

of

information

communicated

among

the

multiple

chains

during

such

a

run

Metropolis

coupling

is

well

suited

to

parallel

implementations

in

which

chains

are

distributed

among

processors

Using

the

Message-Passing

Interface

-LRB-

MPI

-RRB-

this

type

of

parallelization

has

been

implemented

in

MrBayes

for

UNIX

and

Macintosh

clusters

With

large

data

sets

near

linear

speed-ups

can

be

achieved

using

this

approach

-LRB-

Altekar

et

al.

-RRB-

MrBayes

is

written

in

ANSI

C

and

is

available

free

of

charge

from

The

site

provides

precompiled

versions

for

the

MacOS

and

Windows

platforms

and

the

source

code

for

compilation

on

UNIX

machines

The

MPI-enabled

parallel

version

of

MrBayes

is

available

both

precompiled

for

Macintosh

OS

X

and

through

setting

the

relevant

compiler

switch

before

compilation

for

UNIX

The

parallel

Macintosh

version

requires

installation

of

POOCH

-LRB-

-RRB-

on

all

participating

machines

ACKNOWLEDGEMENTS

We

are

deeply

indebted

to

the

beta

testers

and

users

of

previous

versions

of

MrBayes

for

identifying

bugs

and

proposing

various

improvements

F.R.

was

supported

by

the

Swedish

Research

Council

grant

and

J.P.H.

by

NSF

grants

and

REFERENCES

Altekar

et

al.

-LRB-

-RRB-

Parallel

metropolis-coupled

Markov

chain

Monte

Carlo

for

Bayesian

phylogenetic

inference

Bioinformatics

in

press

Huelsenbeck

J.P

and

Ronquist

F.

-LRB-

-RRB-

MrBayes

Bayesian

inference

of

phylogeny

Bioinformatics

Huelsenbeck

J.P.

Ronquist

F.

Nielsen

R.

and

Bollback

J.P.

-LRB-

-RRB-

Bayesian

inference

of

phylogeny

and

its

impact

on

evolutionary

biology

Science

Larget

B.

and

Simon

D.

-LRB-

-RRB-

Markov

chain

Monte

Carlo

algorithms

for

the

Bayesian

analysis

of

phylogenetic

trees

Mol

Biol

Evol.

Page

R.D.M.

-LRB-

-RRB-

TreeView

an

application

to

display

phylogenetic

trees

on

personal

computers

Computer

Applications

in

the

Biosciences

O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O