Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

380

A

500 1000 1500 2000

Arabidopsis

B

500 1000 1500 2000

Arabidopsis

c

O

..

··~·

···

-

··

·

···

-·

·

o 500 1000 lS00 2000 2500

Arabidopsis

o

15001

'

ff)

.

::I

"3 I

§'1000t

CL

'

SOO~'"

o

E

F

500 1000 1500 2000

Oryza

SOD

1000

1S00

2000 2500

Oryza

2500r ... -

••

_-_

..

.

....

. .

2000'

E ,

.E

1500+

g'

i

en

1000·

500

..

j

o

..

"

...

.

......

.....

....

...

.

.......

,

...

. '

o 500 1000 lS00 2000 2500

Populus

Figure

2.

Quantile·quantile

plots

of

the

variance

of

hexanucleotides

in

promoters.

Pair·

wise

comparison

of

the

variance

of

hexanucleotide

motifs

in

the

regulatory

promoter

se·

quences

of

different

species

by

using

QQ·plots.

The

variances

of

all 4096

hexanucleotides

were

assessed from

the

respective

frequency

distribution

curves

for Arabidopsis, Populus,

Sorghum

and

Oryza

promoters

[·1000

bp

sequence

upstream

of

the

annotated

start

of

the

gene].

Shown

are

the

plots

for all six different

QQ-combinations

(A

F)

between

the

four

species.

The

bisecting

line is

integrated

and

indicates

a

linear

steepness

of 1, i.e.

the

variances

of

the

motifs

between

the

two

species

are

of

the

same

value.

larly,

the

QQ-plots of

the

Arabidopsis

and

Sorghum hexanucleotide vari-

ances (Fig. 2C) were also proximal

to

the

bisecting line,

but

with

slightly

381

higher

variances

in

Sorghum. Here again,

the

majority

of

difference was

found in

the

quantiles between 0.90

and

0.96.

This

can

not

readily

be

ex-

plained

by

e.g. different sizes

of

the

promoter

dataset

-

both

species

had

a

similar

number

of

annotated

promoters.

However,

it

might

be

possible

that

the

Sorghum genome

data

or

the

dataset

contains

a

certain

degree

of

re-

dundancy

and,

thus,

there

were

more

highly conserved motifs

with

a

higher

variance found

that

lead

to

higher quantiles. Therefore,

further

research

on

the

nature

of

these

elements

and

on

the

Sorghum

dataset

needs

to

be

done

to

clarify

this

matter.

The

comparison

ofthe

hexanucleotides for Populus

and

Oryza (Fig. 2D)

displays overall similarities

and

trends

to

the

Populus

and

Arabidopsis com-

parison

(Fig. 2A). Hence,

the

Populus

promoter

dataset

contains

unique

features

that

account

for

these

differences.

This

observation

can

not

be

ex-

plained

by

differences in

the

sizes

of

the

dataset

as

the

number

of

promoters

contains in

the

Oryza

and

Populus

dataset

are

of a similar

magnitude,

while

Arabidopsis is larger. Nevertheless,

the

same

trends

were observed for all

comparisons

with

the

Populus

promoters.

Figure

2E

displays

the

compar-

ison

of

hexanucleotide variances

of

Sorghum

and

Oryza. Interestingly,

the

quantiles

of

Sorghum were

much

higher

than

those

of

Oryza. As

has

been

noted

previously,

an

explanation

would

be

that

the

underlying

promoter

re-

gions in

Sorghum have a higher

amount

of high-consensus hexanucleotides

-

probably

hinting

to

a higher

number

of

regulatory

active sequences

or

to

redundance

within

the

dataset.

In

Figure

2F, Sorghum is

compared

to

the

Populus

dataset.

While

Populus displayed

the

lowest values

in

its

quan-

tiles,

the

Sorghum

dataset

contains

the

highest values

in

its quantiles

of

our

organisms

and,

hence,

the

resulting

plot

has

a very high steepness

due

to

the

high

amount

of high-consensus hexanucleotides (those

with

specific

positional bias)

in

the

Sorghum

promoters.

While

we consider

the

variance in hexanucleotides a

measure

for over-

all

promoter

architecture

and

motif

composition, we

expected

that

the

variances

of

CREs

would provide us

with

information

on

conserved core-

promoters

and

retained

CRE

function

in

gene regulation.

Figure

3 gives

QQ-plots

of

the

CRE-motives

for

the

four

plant

species

under

investigation.

For

comparative

reasons, we also included

the

data

of

the

hexanucleotide

variances [in grey] from

the

in

the

CRE-plots.

This

will pro-

vide us

with

the

information,

whether

the

variances

of

the

CREs

differ from

the

general

promoter

architecture.

Since

the

CRE-motives

harbor

known

regulatory

information, we could

expect

from

them

to

have high variances

in

the

promoter

regions

and,

thus,

harbor

a

higher

order

of

information.

382

A

B

c

o

2000 4000

Arabidopsis

6000

2000 4000 6000

ArabidopsiS

2000 4000 6000

Arabidopsis

o

2000 4000 6000

Oryza

E

o

2000 4000 6000

Oryza

F

o

:2000

4000 6000

POpulus

Figure

3.

Quantile-quantile

plots

of

the

variance

of

known

cis-regulatory

ele-

ments

(CREs)

in

promoters.

Pairwise

comparison

of

the

variance

of

known

cis-regulatory

elements

(CREs)

in

the

regulatory

promoter

sequences

of

different

species

by

using

QQ-plots.

The

variances

of

426

known

CREs

from

the

PlaCE

(http://www.dna.affrc.go.jp/PLACE/)

database

were

assessed

from

the

respective

fre-

quency

distribution

curves

for

Arabidopsis,

Populus,

Sorghum

and

Oryza

promoters

[-1000

bp

sequence

upstream

of

the

annotated

start

of

the

gene].

Shown

are

the

plots

for

all

six

different

QQ-combinations

(A

-

F)

between

the

four

species.

QQ-plots

for

the

variances

of

CRE-motives

are

displayed

in

black; for

comparison,

the

QQ-plots

for

the

variance

of

the

hexanucleotides

shown

figure 2 [in grey]

have

been

incorporated

into

this

figure.

The

solid

line

indicates

the

bisector

of

an

angle

and

indicates

a

linear

steepness

of

1, i.e.

the

variances

of

the

motifs

between

the

two

species

are

of

the

same

value.

383

In

Figure

3A

the

quantiles

of

Populus were

plotted

against

Ambidop-

sis.

Similar

to

the

hexanucleotide comparison,

the

quantiles

of

Populus

were

much

lower

than

those

of

A mbidopsis.

On

the

one

hand

this

find-

ing could

be

explained

by

the

on

average smaller sizes of

promoters

with

higher

information

content

in

Ambidopsis.

On

the

other

hand,

the

Pop-

ulus

dataset

is

of

a

poor

quality

due

to

mis-annotation

or

problems

to

curate

the

genome information.

Next,

the

quantiles

of

the

CRE

variances

of

Oryza were

compared

to

Ambidopsis (Fig. 3B).

It

was obvious

that

the

variances were

of

approximately

of

the

same

values for

the

promoters

of

both

species

and,

thus,

close

to

the

bisecting line. Here, also

the

steepness

of

the

QQ-plot

for

the

CRE

variances was similar

to

that

of

the

QQ-plot

of

the

hexanucleotides, which

points

towards

the

fact,

that

the

amount

of

relative

information

between

two species is similar for hexanucleotides

and

CRE-motifs.

Figure

3C

shows

the

QQ-plot

of the

CRE

variances

of

Sorghum

and

Ambidopsis.

The

quantiles

of

the

variances were

very

similar for

both

species, which is indicative

of

similar frequency

distribution

curves for

the

CREs

in

the

promoters

of

the

two species. Especially for

the

lower quantiles

the

graph

follows

the

dissecting line. Hence, one

can

conclude

that

the

motif

composition in

the

promoters

of

the

two species

and

their

architectures

are

highly similar in general. However,

this

similarity was highest for

the

known

functional

CREs.

Only

for

the

quantiles

0.97 - 0.99,

the

variances diverged.

The

simplest

explanation

for

this

observation

was

the

presence

of

longer

CRE-consensi

or

naturally

rare

motifs, which

occur

slightly

more

in one

genome

than

the

other

and,

thus,

have

exorbitantly

high

variance over a

very low

mean

frequency.

The

QQ-plots

of

the

CRE

variances

of

Populus

against

the

Oryza

(Fig. 3D)

or

Sorghum (Fig. 3F)

datasets

revealed a very low

information

content

for Populus. Again,

the

values

in

the

quantiles of Populus were

the

smallest in

the

whole

dataset

and,

therefore, Populus also displayed a

low

CRE

consensus.

The

same

trend

was seen for

the

Sorghum

promot-

ers.

In

Figure

3E,

the

quantiles

of

CRE

variance in Sorghum were

plotted

against

the

Oryza

dataset;

these

quantiles

displayed similarities for

both

species

and,

thus,

the

distribution

of

the

known

functional elements

in

the

promoters

likely follows a similar

trend

in frequency. Since

the

quantiles

of Sorghum have also

been

similar

to

that

of

Ambidopsis (Fig. 3C), we

could conclude

that

CRE

distribution

in

the

promoters

might

indeed

be

conserved between

the

three

species. Moreover,

these

similarities could

be

detected

with

QQ-plots

of

the

variance

as

a

measure

irrespective

of

the

384

A

B c

Figure

4.

Distance

matrix

tree

graphs

for

evolutionary

distance

of

the

species

and

Euclidean

distance

of

the

DNA-motifs.

Tree

graphs

display

the

relative

relatedness

of

the

organisms

on

the

basis

of

their

protein

sequence

similarities

(A)

or

the

similarities

between

the

DNA-motif

variances

of

all

hexanucleotides

(B)

and

known

CREs

(C).

Sim-

ilarities

and

dissimilarities

between

the

tree

topologies

in

A - C

are

highlighted

by

lines

that

interconnect

the

four

organisms

between

the

graphs.

different

dataset

sizes

or

DNA-base composition

(GC/AT-content).

Our

presumption

was

that

information

content

within

promoters

is re-

tained

as positional disequilibria

in

promoter

sequences

when

observed

at

a

genomic scale. As a simple,

but

effective measurement, we

took

the

variance

of

DNA-motifs frequency-distribution

to

indicate

the

degree

of

information

a

motif

carries,

and

compared

these

variances between species

to

capture

global similarities in

promoter

architecture. To make conclusions

on

these

interspecies comparisons, we established distance matrices for

protein

se-

quence similarity

and

for

the

variance

of

hexanucleotides

or

CREs;

these

results were displayed as

tree

graphs

(Fig. 4).

The

last

common ancestor

of

dicot (Arabidopsis

and

Populus)

and

monocot (Sorghum

and

Oryza) species

lived

at

approx. 140 Mio years before present

10.

The

phylogenetic

tree

in

Figure

4A

captures

this

evolutionary

time

and

serves as

our

reference.

The

Oryza

dataset

appears

to

be

closer

Arabidopsis

than

to

its

evolutionarly close relative Sorghum when looking

at

the

hexanucletide

variances.

One possible

explanation

for

this

similarity in hexanucleotide

composition is

that

both

species have relatively small, compressed genome

sizes

11.

As a consequence,

the

intergenic space, i.e.

the

DNA-region be-

tween

the

genes, is relatively small,

and

as such, more information

must

be

packed

into

shorter

promoter

regions

3.

The

Arabidopsis thaliana genome

is one

of

the

smallest eukaryote genome 4

and,

thus,

regulatory

promoter

sequence

must

be

short

and

enriched for motifs

with

a

high

order

of

infor-

mation.

This

notion

might

be

true

for Oryza as well, as

its

genome size is

one

of

the

smallest

amongst

other

Poaceae crop plants.

The

CRE

variance is shown

in

Figure

4C. Here,

the

overall

tree

topology

follows

our

reference

tree

(Fig. 4A),

with

the

exception

of

the

Populus

385

dataset,

which could likely

be

considered as

an

outlier.

The

reasons for

the

aberrant

Populus set

might

be

manifold,

but

most

likely

originate

from a

low

quality

in

the

annotation

of

the

gene

starts.

4.

Conclusions

We have shown

that

promoter

architecture

is

strongly

dependent

on

the

sizes of intergenic regions.

The

observation

that

the

CRE

variances for

Arabidopsis, Oryza

and

Sorghum were highly similar

supports

the

idea

that

the

changes

in

protein

sequence

or

in cis-regulatory

DNA

over

time

might

be

proportional.

Thus,

known functional elements display

the

same

evo-

lutionary

concepts

as

are

considered for genome

or

protein

evolution.

To

further

support

and

refine

our

findings, well

annotated

genome

information

of

more

species is needed

and

has

to

be

integrated

in

our

analysis.

The

sequence

information

of species

with

different genome sizes

and

larger evo-

lutionary

distances

would

be

of

special

interest.

Moreover,

the

analysis

of

the

CRE

variances is

hampered

by

too

little

information

on

transcription

factor

(TF)-CRE

relationships. Hence,

our

approach

is well

worth

to

be

repeated

with

updated

information

on

both,

TF-CREs

and

a larger

number

of

promoter

datasets.

Acknowledgements

We

thank

Jochen

Supper

for his

constant

input

and

critical discussions.

This

work was

supported

by

the

DFG

(HA2146/11-1).

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Co. (pp.

387-4

01)

ENTROPY

TYPE

COMPLEXITIES

IN

QUANTUM

DYN

AMICAL

PROCESSES

NOBORU WATANABE

Department

of

Information Sciences,

Tokyo University

of

Science,

Noda

City, Chiba 278-8510, Japan

E-mail: watanabe@is.noda.tus.ac.jp

In

a

study

of

several

systems,

we

are

int

eres

ted

to

examin

e (1)

the

dynamics

of

state

change

and

(2)

th

e

comp

lexity

of

states

.

Ohya

introdu

ced

in

1991 (ref.[21])

a

general

idea

so-called a

Information

Dynamics

(ID),

which

const

itute

s a

theory

under

a

frame

of

the

ID

by

sy

nthesizin

g

the

formaliti

es

of

the

investigati

ons

of

(1)

and

(2).

There

are

two

kind

of

complexities

in ID.

One

is a

complexity

of

state

describing

system

itself

and

anoth

er is a

transmitted

complexity

betw

een

two

systems.

Entropies

of

classical

and

quantum

systems

are

the

example

of

these

complexiti

es.

In

order

to

treat

a flow

of

dynamics

proce

ss,

dynamical

entropies

were

introduced

in

not

only

classical

but

also

quantum

systems

.

The

main

purpose

of

this

paper

is

to

compare

with

these

mean

entropies

and

the

complexiti

es

in

rD,

we calc

ulate

these

mean

entropies

for

some

simple

models

to

discuss

the

complexity

of

information

transmission

for

OOK

and

PSK

modulations.

1.

Introduction

In

several complicated systems,

it

is

important

to

study

(1)

the

dynamics

of

state

change

and

(2)

the

complexity of

states

of

systems.

Information

Dynamics (ID)

introduced

by

Ohya

is a general

idea

constructing

a

theory

by synthesizing

the

research schemes

of

(1)

and

(2)

under

a frame

of

the

ID.

In

ID,

there

are

two

type

of complexities,

that

is, (a) a complexity

of

state

describing

system

itself

and

(b) a

transmitted

complexity between

two systems. Entropies

of

classical

and

quantum

information

theory

are

the

example

of

the

complexities

of

(a)

and

(b).

In

quantum

information theory,

Ohya

introduced

a

compound

state

and

defined

Ohya

mutual

entropy

18

based

on

quantum

relative

entropy

of

Umegaki 31

in

1983, he

extended

it

19

to

general

quantum

systems

by using

the

relative

entropy

of

Araki

5

and

Uhlmann

32.

Based

on

the

quantum

mutual

entropy,

he

quantum

capacity

is discussed in

24,25,28.

One

can

discuss

the

coding theorems by means

of

387

388

the

mean

entropy

and

the

mean

mutual

entropy

defined

by

the

dynamical

entropy.

The

KS

entropy

14

was

introduced

in classical

systems.

Several

quantum

dynamical

entropy

were

studied

by

Emch

10

,

Connes-Stormer

9,

Connes, Narnhoffer

and

Thirring

8,

Park

29,

.Alicki

and

Fannes

4,

Hudetz

12.

Ohya

21,

Voiculescu

33,

Accardi,

Ohya

and

Watanabe

2,

3,

Kossakowski,

Ohya

and

Watanabe

15,34,

Ohya

and

Petz

23

and

Choda

7,

Bennati

6,

Muraki

and

Ohya

16

and

so on.

In

this

paper,

we briefly

explain

the

mean

entropy

and

the

mean

mutual

entropy

defined

by

Ohya

21.

We

calculate

these

mean

entropies

for some

simple

models

to

discuss

the

complexity

of

information

transmission

for

subsets

of

the

initial

state

space.

2.

Quantum

Channels

The

concept

of

channel

has

been

played

an

important

role in

th

e progress

of

the

quantum

communication

theory. Here we briefly review

the

notion

of

the

quantum

channels.

Let

Hk

(k

=

1,2)

be

complex

separable

Hilbert

spaces. We

denote

the

set

of

all

bounded

linear

operators

on

Hk

by

B(Hd

(k =

1,2)

and

we

express

the

set

of all

density

operators

on

Hk

by

6(Hk)

(k =

1,2).

Let

(B(H

k

),6(H

k

))

(k

=

1,2)

be

input

(k

= 1)

and

output

(k = 2)

quantum

systems,

respectively.

(1) A

mapping

from

6(Hd

to

6(H2)

is called a

quantum

channel

A*.

(2) A * is called a linear

channel

if A * satisfies

the

affine

property

such

as

A*(2:k

AkPk)

= 2:k

Ak

A

*

(Pk)

for

any

Pk

E

6(Hl)

and

any

non-

negative

number

Ak

E [0,1]

with

2:k

Ak

=

1.

For

the

quantum

channel

A *,

the

dual

map

A

of

A * is defined

by

trA*(p)B = trpA(B),

Vp

E

6(H

1

),

VB

E B(H2)'

(3) A * is called a

completely

positive

(CP)

channel

if A * is

linear

channel

and

its

dual

map

A :

B(H

2

)

--+

B(H

1

)

of

A * holds

(

x,t

AiA(A;Aj)AjX);::::: 0

(Vx

E HI)

t,J=1

for

any

n E

N,

any

{Ad

c

B(H

2

)

and

any

{Ai}

C

B(Hd·

Almost

all physical

transform

of

states

can

be

denoted

by

the

CP

chan-

nels

18,19,13,23,27.

389

2.1.

Noisy

optical

channel

Now we explain

the

noisy optical channel such as

an

example

of

the

quan-

tum

communication

channels

19,26.

Let

Kl

and

K2

be

complex separable Hilbert spaces

of

noise

and

loss

systems, respectively. For

an

input

state

p in

6(Hd

and

a noise

state

~

E 6

(Kd,

we defined

in

26

a

mapping

IT*

by

IT*

(p

@~)

==

V

(p

@

~)

V*,

which is called a generalized

beam

splitting, where V is a linear

mapping

from

HI

@

Kl

to

H

2

@K

2

given by

nl+ml

V(lnl;

@

Iml;)

= L

Cj",m"lj;

@ Inl +

ml

-

j;

j=O

for

the

nl,ml,j,(nl+ml-j)

photon

number

state

vectors

Inl;

E

HI,

Iml;

E K

l

, Ij; E

H2,

Inl +

ml

-

j;

E K

2

,

and

K . I , ,

"(

_

')'

Cn1,m1

=

''"'(-It"+

j

-

r

ynl·ml·]·

nl

+ml

].

J

~

r!(nl -

j)!(j

-

r)!(ml

- j + r)!

r=L

xaml-j+2r

(-73)

nl

+j-2r

.

(1)

a

and

{3

are

complex numbers satisfying lal

2

+

1{31

2

=

l.

K

and

L

are

constants

given by K =

min{nl,j},

L =

max{ml

-

j,

O}.

For

the

coherent

input

state

p =

Ie;

(el @

Ih;;

(h;I

E 6

(Hl@Kd,

the

output

state

of

IT*

is

obtained

by

IT*

(Ie; (el @

Ih;;

(h;I)

=

lae

+

{3h;;

(ae

+

{3h;1

@

1-73e

+

(ih;)

\

-73e

+

(ih;I·

IT*

with

the

vacuum

noise

state

~o

=

10;

(01

is called

the

beam

splitter

ITo

given by

ITo

(Ie; (el

@~O)

=

lae;

(ael

@

1-73e)

\-73

e

l

for

the

coherent

input

state

p @

~o

=

Ie;

(el @

10;

(01

E 6

(Hl@Kd·

ITo

was

described by means of

the

lifting

Eo

from 6 (H)

to

6

(H@K)

in

the

sense

of

Accardi

and

Ohya

1 as follows

Eo

(Ie; (el) =

lae;

(ael

@

l{3e)

({3el·

Based

on

the

liftings,

the

beam

splitting

was

studied

by Accardi -

Ohya

and

Fichtner

- Freudenberg - Libsher

11.

Noisy

quantum

channel A *

introduced

in

26

with

a fixed noise

state

~

E 6

(Kd

was defined

by

A*(p)

==

trJC

2

IT*(p

@~)

=

trJC2

V (p

@~)

V*.

(2)

Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

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