Self-Organized Criticality in Phylogenetic-Like Tree Growths

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Self-Organized Criticality in Phylogenetic-Like Tree Growths

N. Vandewalle, M. Ausloos

To cite this version:

N. Vandewalle, M. Ausloos. Self-Organized Criticality in Phylogenetic-Like Tree Growths. Journal de

Physique I, EDP Sciences, 1995, 5 (8), pp.1011-1025. �10.1051/jp1:1995180�. �jpa-00247112�

Classification Physics Abstracts

05.40+j 87.10+e

Self.Organized Criticality in Phylogenetic-Like Tree Growths

N. Vandewalle

(*)

and M. Ausloos

(**)

SUPRAS, Institut de Physique BS, Sart Tilman, Université de Liège, 4000 Liège, Belgium

(Jleceived 10 March 1995, received in final form 24 ApriJ 1995, accepted 3 May1995)

Résumé. Un simple modèle stochastique d'évolution Darwinienne engendrant des arbres

phylogénétiques est développé. Le modèle est basé _{sur un} processus de branchement tenant compte d'effets de compétitions et de corrélations. En présence de corrélations à courte portée, le _processus s'auto-organise ^dans _un ^état critique caractérisé _par l'intermittence d'explosions

d'activité de toutes tailles. Sur

une échelle pseudo-géologique, _ce comportement est _en accord

avec les caractéristiques ponctualistes de l'évolution biologique. Les arbres phylogénétiques

simulés sont auto-similaires. La dynamique des régimes transitoires montre _une décroissance _eu loi de puissance du paramètre d'ordre _vers 0+ qui caractérise _un point critique instable. La portée génétique des corrélations et compétitions entre espèces vivantes est _un paramètre pertinent qui

détermine la classe d'universalité du _processus d'évolution- Une portée infinie de _ces corrélations détruit cependant le comportement critique auto-organisé. La dimension fractale Di des arbres croît de 2,0 _vers l'infini lorsque k _varie de 1 à l'infini. L'exposant critique _T de la distribution des

avalanches décroît à partir de

3/2

lorsque k augmente et atteint environ 1,2 _pour k

= 10. Une

relation d'échelle semble lier le comportement des différentes classes d'universalité- Une théorie de champ _moyen montre que le processus engendré est bien plus complexe qu'un _processus de

branchement décorrélé.

Abstract. A simple stocllastic model of Darwinistic evolution generating pllylogenetic-like

trees _is developed. Tlle model is based _{on a} branclling _process taking competition-correlation

effects into account. In presence of limite and short range correlations, tlle _process self-organizes

into _a critical steady-state in wllich intermittent bursts of activity of ail _{sizes are} generated.

On _a geological-like time scale, tllis bellaviour _agrees witllpunctuated equilibrium features of biological evolution. Tlle simulated pllylogenetic-like trees _are found to be self-similar. The

dynarnics of the transient regimes show _a power law decrease of the order parameter towards the 0+ value which characterizes _an unstable critical state. The genetic _range k of competition-

correlations between living species _is found to be _a relevant parameter ^{which determines trie} universality class of the evolution _process. An infinite competition-correlation _range destroys however trie self-organized critical behaviour. The fractal dimension Di of the phylogenetic-

like trees _increases from 2.0 to infinity _as k goes from 1 to infinity. Trie critical exponent _T

of avalanche size-distribution decreases from about

3/2 (for

k

= 1) and reaches about 1.2 for k = 10. A hyperscaling relation _seems to relate the _various universality classes. Through _a

(* e-mail address: [email protected] (**) e-mail address: [email protected]

© Les Editions de Physique 1995

1012 JOURNAL DE PHYSIQUE I N°8

mean-field theory, _we mention that the evolution _process

is much _more complex than

a simple

uncorrelated branching _process.

1. Introduction

The

question

of tue

biological

evolution is

certainly

_one of tue most fundamental

problems

of science and

puilosopuy.

Since tue last

decade,

_some efforts bave been made in the

application

of matuematical concepts to tue

biological

evolution

especially

in systematics

Iii.

More

recently,

it _was also shown tuat computer ^simulation

provides

_an

interesting approacu

_to the

problem

of

evolution,

thus

opening

_new ^{research fields} in botu statistical

physics

and

biophysics.

Tue models _{concern, e-g-,}

long-range

correlations in DNA _sequences [2], ^cell dilferentiation [3], or

aging problems

_[4].

Recently,

Bak and

Sneppen (BS)

bave introduced _a

simple

model in order to

reproduce

tue

complex

features of trie real

biological

evolution

[5,6].

^{Trie BS model}

(aise

called trie

punctuated equilibrium model)

considers _an ecosystem with _a constant number N of

interacting species arranged

_on a d-dimensional

hypercubic

lattice with

periodic boundary

conditions. At each

species1

is associated _a scalar fitness bj which is _a random number between _zero and _one, rather than _a

genetic

code [Si. ^Trie

quantity

b, is

supposed

_to be trie

jitness (or

_{a measure} of trie barrier

against mutation)

of the

species1;

the

higher

trie

fitness,

trie _more

likely

trie species

is

adapted

to trie ecosystem ^{and trie less}

likely

the

species

mutates. At each "time

step",

trie

species j having

trie minimum fitness bj ^is

supposed

to make _an

adaptative

_move: trie fitness

bj

receives a new random number. Because of trie assumed _presence of short _range interactions

between trie species, a

mutating species

is

supposed

to affect trie fitnesses of its

(2d)

_nearest

neighbours

which _are then also

updated.

A

geological-time

scale

tg

^{has also been defined} _[Si

such that trie

geological

duration of _a mutation

lextinction through

_a fitness b; is assumed to be

proportional

to

exp(Àb,)

where is _some

positive

real

geological

_or

biological

parameter

which should be

large

_[7]. Trie

assumption

behind this

exponential

form of mutation durations

cornes from

biological

arguments _[8].

Trie BS model is _very restrictive and

oversimplified

but it follows trie fines of

thought

^of

modern statistical

physics

where trie

physicists develop complexity

out of

simplicity

in contrast with trie attempt to reduce

complexity

to

simplicity

_[9].

Trie BS model _was found to

self-organize

into a critical

steady-state

for which intermittent avalanches of activity of ail sizes _are

generated.

On _a

geological-like

time

scale,

the avalanche

events _are found to be

separated by periods

of stasis

(quiescence)

much

longer

thon trie du-

rations of

avalanches,

behaviour wuich is somewhat simflar to the

punctuated

features of tue

biological

evolution

[loi. However, biological

evolution is much _more

complex [loi.

Nev-

ertheless,

trie idea of "critical

self,organization" Ill]

is of interest _since in _nature non-linear processes

organize biological

systems into structures

il

_2] that _{appear to} bave order _on ail

length

scales

[13,14].

^{Trie BS model} is

original

and _{opens new}

investigations

_in statistical

physics by

inventing more realistic models.

Actually, only

_a few extensions of the BS model bave been

studied

ils].

Three

important

drawbacks _on trie BS model bave to be

emphasized. First,

the number N of species is

always kept

constant: it is _a model of coevolution rather than _a model of evolution.

Secondly,

_an extinction is associated _to the strict mutation of _a

species

into another _one. The latter consideration is _a strong

assumption,

_1-e-, it is not Darwinistic [16]. ^In

fact,

_a

species

extinction has

multiple

and

complex

_causes which _are not

specially

related to _a mutation

event [16].

Thirdly,

not well taken into account in trie BS model is trie fact that _one

species

can also mutate several times and compete with its _own

updated species

_or

olfsprings.

We bave thus

developed

the rules of trie BS model into _a stochastic

branching

_process

[Iii

which allows for trie

"neighbouring" species

and trie mutated _{ores to} compete ^with each other.

The

advantages

of this model _are

that,

from _very

simple rules, ii)

it generates

phylogenetic-like

trees,

iii)

it contains trie _essence of real

biological evolution, (iii)

and it leads to _a

self-organized

critical behaviour like for trie BS model.

Here,

_we

study

trie elfects of trie

genetic

_range of interaction-

competitions

between dilfer- entiated

species

_on trie

self-organized

critical

biological

_process. Trie elfect of trie

branching

rate, 1-e-, trie number of _new

olfsprings appearing

at each mutation event, is also studied here.

Besides these

biological considerations,

various domains of

physics

_are concerned

by

this model.

The

generalized

model of tree-like evolution is defined in trie next section. Numerical studies of processes and trees _are

presented

in Section 3. In Section 4, _an

elementary

mean-field

theory

of trie model is

proposed

and _we also discuss trie

origin

of the

self-organized

critical _process.

A conclusion is

finally

drawn in Section 5. The

biological

aspect and relevance of the present model will be discussed in

Appendix.

2. The Tree-Like Model of Evolution

The

principal

idea of trie present model _is that _a mutation event

gives

_z dilferentiated

species,

1-e-, trie parent

species

and the _z -1 other

olfsprings.

This mimics the

apparition

of _z -1

new

species

in _a parent

population

of animais. This leads to _a

branching

_process and trie formation of tree-like _structures such that the

genetically

dilferent

species

_are located _at trie extremities of tree branches _as for

phylogenetic

trocs.

Figure

la presents a small tree which bas been grown

by

6 mutation events from _a

single

ancestor labelled "a". Each mutation

event in

Figure

la bas

given

_z

= 2 differentiated

offsprings.

Trie _tree of

Figure

la could be

mapped

into _a

phylogenetic-like

troc illustrated in

Figure

^{16. On} trie tree of

Figure

la, _we define trie "distance" d~nn ^between two _m and _n

species by

trie minimum number of

segments

of trie tree needed to connect trie _{m species} to trie _n species. As for trie BS

model,

_a fitness b, which is _a random number between _zero and _{one is} associated to each

living species

at

each branch extremity. ^At each "time" step, trie species

j having

trie minimum fitness value, 1-e-, bj =

min(b,),

mutates and

gives

z differentiated _species. Trie parent

species

is _one of

these. Each _one receives _a random fitness value. Trie

branching points

of trie tree in

Figure

la represent ^old

configurations

but _Dot

living species

_{as cari} be understood from trie

mapping

shown in

Figure

16.

Furthermore,

trie fitness bi of ail trie

species1

which _are

separated by

_an

arbitrary

distance

d~

^from trie

mutating species j

less far _away than _a parameter

k,

_are also

updated

with _{a new} random number

by

_a kind of

competition-correlation

effect in the

branching

_process after trie

j species

bas mutated. Trie latter _{species are} thus affected

by

trie mutation of trie

species j

but do not

necessarily

mutate next. The parameter ^{k dermes trie}

genetic-hke

_range ^{of the}

interactions between differentiated species.

Trie k

= 3 and _z

= 2 _process is fllustrated in

Figure

lc where _a mutation event _{occurs on}

trie species labelled "4" _in trie tree of

Figure

la, 1-e-, b4 ₌

min(bi,

.,

b7).

Trie mutation of trie

species

"4"

gives

two differentiated

species

with

fitness, b[

and

b[ (since

_z

=

2)

and affects 3 other species labelled "5", "6" and "7",

respectively,

in

Figure

la. This

gives

_{a new} distribution

(b[,. ,b[)

of

fitnesses,

where b[ =

bi, b[

= b2, b[ = b3 and other fitnesses

(b[

to

b[) being

new

random numbers. One should note that in _so

doing,

_one considers that trie ancestor

(say

"4"

either

gives

rise to two

offsprings (new

"4" and

"5")

_or evolves

(new "4")

and

gives

a

single (for

_z

=

2) offspring (new "5").

1014 JOURNAL DE PHYSIQUE I N°8

a) ^hs b)

tg

ht b2 b3 b4 bs b6 b7 b6

a

6q 65

~~

~, ^b'7

2

,

~3

c)

_a

Fig. l.

a)

A small tree grown by 6 mutation events, each mutation _giving

z = 2 offsprings;

b)

mappmg of trie tree of

a)

into _a phylogenetic-like tree; c) trie tree of

a)

after trie mutation of the

species labelled 4 in

a).

The evolution starts with _an

arbitrary

number of

species arranged

_or Dot _{on a} tree. The

mutation-competition

_process described above _is

repeated

for _a

given

number t of mutation events.

Thus,

trie number of different _species

linearly

increases with t with _{a z} 1 rate. The

geological-like

^duration of _a mutation of _a

species

with fitness bi is assumed to be

proportional

to

exp(Àbi)

like for the BS model.

The

only

two parameters are the

integers

k and _z which _are, at this time,

kept

_constant

during

the whole evolution

(or tree-growth)

_process. One should also note that the model does not

predetermine

at each mutation event which of the _z differentiated

olfisprings

has _a chance to next evolve.

This rnodel is not

only adapted

_to the

cornputational study

of

biological

evolution but could be used in the

general study

of correlated

branching

processes

[Iii

_as

well,

like in nudear

physics

_or computer networks.

3. Numerical Results

Algorithms

of such

branching

_processes do _{not cause}

major

computation diiliculties. Ail

living

and old species are labelled

by integer

nurnbers. Trie fitness values of these

species

_are

simply

stored in arrays of

floating point

numbers. Each

species

_is

represented by

_a

logical

^TRUE _or

FALSE nurnber for _a

living

_or old

species (branching points), respectively.

For each Monte- Carlo step, ^the

species having

trie minimum fitness value bi

(among

the

living species) gives

z new

living

species ^which are

immediately

stored. The

activity

of trie species which bas

given

the _z

offsprings

is then turned off. Trie Monte-Carlo time t is

updated

to _a t + i

integer

^value

while the

geological-like

time

tg

is

updated

to trie

tg

+

exp(Àbi)

real value. Trie _new

mutating species

with trie smallest bi is then looked for

through

trie _new

forger

set of fitness values.

In order _to compute trie

competition-correlations

between the

living species,

_one needs to know the whole structure of the growing tree. In _so

doing,

at each species is associated the label of its parent. ^{The latter label is also stored in} an array of

integer

numbers. This allows for trie search of ail

living species being

at _a

given

distance from trie

mutating species

less than the parameter k. Trie fitnesses of the latter

species

_are

updated by

_{a new} random number.

One should note that the

increasing

number of

species

with Monte-Carlo time slows down trie

speed

of trie

algorithm

^for

large

trees. Trees of _up to 50000

species

have been

computed

here.

In trie

following paragraphs,

_we will present the numerical

investigations

of both

physical (via

trie

criticality)

and

geometrical (via

the

fractality)

aspects

developed by

trie model for various values of the two integer parameters, _{1-e-, trie range} of

competition-correlations

k and trie

branching

rate _z.

3.1. THE DISTRIBUTION _oF FITNESS _{AT THE}

QUASI

STEADY-STATE.

Starting

with dilfer-

ent

arbitrary configurations

_or ancestors, ^{the stochastic} process is

always round

to

self,organize

into _a so-called

steady-state

_in which ail the fitnesses _are distributed in _a

step-like

distribution

as shown in

Figure

2. Trie

"steady-state"

distribution

n(b) sharply

vanishes below _some limite critical value bc ^and

n(b)

has _a finite and _non-zero value above bc. Such _a

"steady-state"

O.025

k=2

o.o20 ---k=4

".."-.-"-k=6

, ' t ~'

0.015 ,

j i

Î[

i'

'il'i,/(i'Jii

'i'>

'~ j' j'" ~p,1,, _' 1....'j 1 'j

J~ ₁₁ ~.j~[l'i _{o" i} i>,i~"","1~ ^'" _'; ?o -iii

~fl' il fi

.,

? "' j Q "' r

o-o10

j

'

j '

Ù-Ù

b

Fig. 2. The distribution

n(b,)

of fitnesses at the extreInities of trees containing t = 20000 mutation

events. Three dilferent ranges of correlation-competitions have been used (k

= 2, 4 and 6).

JOURNAL DBPHYSIQUBL T3, MS, AUGUST1993 41

lo16 JOURNAL DE PHYSIQUE I N°8

0.08

0.07

., k"2

0.06 ^; k=4

_

0.05

'~~ ~

(

i ~.

~ 0.04 _',1

É 'S.

0.03

'/;',

0.02 _.,

,,

O.Ol

0.00

O.O 0.2 0.4 O.fi 0.8 1-ù

bmin

Fig. 3. The distribution n(bm,n) of minimum fitnesses bm,n through which trie tree has evolved from t = 1 to t ₌ 20000. Three different ranges of correlation-competitions have been used (k

= 2, 4 and

6).

distribution is illustrated in

Figure

2 for various values of k and for _z

= 2.

Figure

3 shows trie distribution

n(bm;n)

^{of the} minimum fitness values

through

which trie tree bas evolved for

various values of k and for _z

= 2. Trie respective distributions of

Figure

2 and

Figure

3 have been constructed with trees of _{up to} 20000

living

species.

One should

emphasize

^that the

"steady-state"

is Dot

reached,

_as

long

_as ^ail mutations do not turn out to take

place through

fitness values which _are less thon _a critical value

bc(k,z) independently

of the choice of the initial

configurations.

A true

steady-state

is of _{course never} reached since _a mutation will still take

place

for

species having

_a ^fitness close to bc and will

perturb

the distribution

(see

Section 3.2

below).

One should _note that _a fraction of

living species

will net further

participate

in trie evolution because

they

will be screened

by

_some

configurations. Indeed,

those and

only

those species ^hav- ing fitness values

strictly

greater ^thon

bc(k, z)

and

having

in their

neighbourhood (determined by

the parameter

k)

fitnesses ail greater than the threshold

bc(k, z)

cannot further evolve. Trie

study

of such

screening

effect is outside trie scope of this _paper and wfll be further studied.

The model does not determine if such screened

species

become extinct _or stay olive. In further works, additional

ecological

constraints could be introduced in the model in order to take into

account extinction not related to mutation.

Figure

4 presents the measured values of

bc(k, z)

_{as a} function of k and for various values of the

branching

parameter _z. The

bc(k, z)

values decrease from

1/z

to _zero when k is increased from 1 to

large

k values. These values _are less than and _are

quite

^different from the bc value of the BS model which is _a little bit greater ^than

2/3

_[18]. The

bc(k, z)

thresholds _seem to reach

asymptotically

_zero for k

tending

to

infinity,

1e., when

competition-correlations

_occur between ail

living species

_in the

phylogenetic-like

tree.

3.2. AVALANCHES. As for the BS model, trie existence of such b~ thresholds

implies

that avalanches _can take

place

_in the evolution process [Si.

Suppose

that _{at sonne} "time" ail fitness values _are above the threshold

bc(k, z).

The species

having

bi _m

bc(k, z)

trier mutates

leading

0.6

Ù.5 ^~ ~

& z=3

~ ~

~=~

a .

à

^°'~

a .

»

a .

0.2 "

, a .

»

. ~ ~

O.I . "

~ .

» ~ .

* » ~ .

~

O.O

0 2 4 6 8 10 12 14 16

k

Fig. 4. Tue turesuold values bc of tue fitness _{as a} function of tue _range of competition-correlations

k and for _vanous values of the branching rate _z.

to z new

species

and to local _new fitness values whicu _can be less tuan

bc(k,z).

Tuis defines

an avalanche _{[Si or a} burst of

activity

_{as a}

causally

connected _sequence of

activity

witu fitness values below tue turesuold

bc(k, z).

Tuis _means tuat the evolution process

self-organizes

^into

a state of intermittent avalanches of

activity

_as for the BS model [Si.

Two

subsequent

avalanches _are

separated by long periods

^of quiescence

having

the

longest geological-like

durations _m

exp(Àbc generated by

the evolution process. The

punctualist

aspect of tue evolution _process

generated by

tue present ^{model is discussed in}

Appendix.

Tue size-distribution

n(s)

of tue avalanches bas been

investigated. Here,

tue "size" of _an avalanche is tue number of mutation events contained in _a

single

avalanche. An avalanche of size _s bas

produced s(z -1)

_new

beings. Thus,

trie duration of _an avalanche _is

linearily

related to the size of this avalanche

leading

in the present model to

equivalent

critical exponent ^for

both

physical properties.

The size distribution

n(s)

of avalanches is found to follow _{a power} law behaviour _as

nls)

r~

S~~

Il)

for ail finite values of k and _z. This is illustrated in

Figure

5 which presents ⁱⁿ a

log-log plot

the size distribution

n(s)

of avalanches for k ₌ 2 with _a

branching

rate z = 2. The tree has

been grown up ^to 50000 mutations. The deviations from the strict power law behaviour of

n(s)

_seen for

large

_s values _are known to be finite-size effects [19]. ^{This indicates that the} stochastic process of _our evolution model

self-organizes

into _a critical [20]

steady-state

_as for the BS model [Si.

Figure

6 presents ^the measured values of _{T as a} function of the

interaction-range

parameter k and for various values of the

branching

rate z. The size-distributions

n(s)

of avalanches bave

been obtained

by simulating

10 trees made of 50000 mutation events for each

couple

of

integer (k, z)

parameters. ^{For ail} z values and for k

= 1, trie critical exponent _T is close to

3/2

which is trie mean-field value

usually

obtained for _a critical

branching

_process

Iii]. However,

for k _- +cc and for ail _z values, _T decreases and reaches about 1.2 for k

= 10. Further extensive simulations should indicate the exact

asymptotic

value. The continuous variation of exponents

1018 JOURNAL DE PHYSIQUE I N°8

10°

i~l

1

~ 0~3

i

s

Fig. 5. The

size distribution

n(s)

of avalanches

in trie tree-like evolution process for k

= 2 and

z = 2. The _{tree was grown up to} t ₌ 50000.

2.0

. z=~

~

& z=3

" z#4

6

~

~

l 2

l 0

0 2 4 6 8 10 12

k

Fig. 6. The critical exponent _{T as a} function of trie range of competition-correlations k and for

various values of the branching rate _z. Each dot represents the results of the simulation of 10 _trees

grown by 50000 mutations.

witu tue k and _z parameters ^does not seem to be _an artefact of tue limited

quality

of data,

but _seems intrinsic.

3.3. TRANSIENT REGIMES. We bave aise observed how trie system evolves towards its

critical

steady-state starting

from the non-critical state of _a

single

ancestor. We have measured tue average < b _> of fitnesses _over all

living species

as a function of the simulation time t. As tue system ^{reacues tue critical}

steady-state,

< b > becomes

où

~

~

j _j j

~~ ^~

Î

'

»,~~m~ ^m k=6

.~

a AA

~ ^{. »}

~ »

~ ,

£

l0"2 ^~ *

, .

a

,,~

aAA , à .

~

é

i Q"3 ~

l0~

0 100 1000 0000

t

Fig. 7. Evolution of tue order parameter _m witu time t for _z

= 2 and for various values of tue range of competition-correlations k.

< b _>=

Il

₊

bc)/2

12)

Tue "order

parameter"

_m of tue present ^evolution

puenomenon

_can be defined [21] as

m=bc-2<b>+1

(3)

since _m shrinks to 0+ in tue critical situations.

Figure

7 shows tue evolution of _m with time t in _a

semi-log plot

^for various values of k and for _z

= 2. Trie _average < > was taken _over 1000 different trees. Trie critical

steady-state

is found to be reached with _{a power} law

decay

in ail

cases. One should note that the BS model presents an

exponential

behaviour for d-dimensional

hypercubic

lattices above trie critical dimension de ₌ 4 [21]. ^Below

de,

_{a power} law behaviour

(characterizing

_a classical second order

phase

transition [20]) was found for the BS model.

One should note that trie exponent

characterizing

this power law decrease of _m is

weakly

sensitive to k _{as seen} in

Figure

7. Tuis weak

dependence

^{should be further}

numerically

_aria-

lyzed.

3.4. GEOMETRICAL PROPERTIES _oF THE PHYLOGENETIC-LIKE TREES. The

geometrical

structure

through

which trie process evolves is also of interest.

Starting

with _a

single

ancestor, two different kinetic features _cari be measured:

ii)

trie Monte-Carlo time evolution of the

mean distance < d > between trie _common ancestor and ail species ^and

iii)

trie evolution of the

largest

distance dmax between the _common ancestor and trie

updated species.

The first

aspect expresses a

physical relationship

between the characteristic

length

of the tree and the total number

(z 1)t

of _species contained in the tree. The second aspect ^has some

biological

relevance because it expresses the evolution of the most differentiated species in the tree.

We

numencally

found that the characteristic

length

< d _> shows _a power law behaviour

< d _>r- t"

(4)

for ail limite values of k and _z. Trie exponent v was found _to be k and _z

dependent. Moreover,

the

general

evolution of

dmax(t)

shows also a

scaling

law behaviour

1020 JOURNAL DE PHYSIQUE I N°8

S-ù

4.5

, z=~

a z=3

4.O

~ ~_~

à

3.5 3.O

2.5

2.O

1.5

2 4 6 8 10 12

k

Fig. 8. Tue fractal dimension Di of trie phylogenetic trees _{as a} function of tue _range of competition- correlations k and for various values of tue

branching

rate _z. Eacu dot represents ^{tue results} of tue simulation of 40 trees grown by 10000 mutations.

dmax '~

t~ là)

witu _an exponent

fl

wuicu _was also

strongly

^k- and

z-dependent.

Tuese

scaling

laws indicate tuai trie

generated phylogenetic-like

trees bave trie property to be self-similar for ail finite values of k and _z. This property ^is another

signature

of

criticality

_[20].

A

self-similar,

_or

fractal,

_growing tree [22] ^of fractal-like dimension Di is indeed characterized

by

_{a power} law

nid)

r~

d~~~~

₁₆₎

relating

trie number

nid)

of

species

to trie distance d _away from _{a common ancestor.}

Integrating nid)

from _zero to

dmax,

_one con find that

biological (fl)

and

physical iv)

evolution exponents

are trie _sames.

Moreover,

trie fractal-like dimension

Di

of trie

phylogenetic-like

trees is found

to be

D~ _m i

Iv

m i

/p ii)

Figure

8 presents ^tue fractal-like dimensions

Di(k, z)

of tue

phylogenetic

trees for various limite values of k and _z. The fractal-like dimension

Df

of the _{trees is} found _{to increase} from about 2.0 _to

large

values with the increase of the

genetic

_range k of correlations.

4. Discussion

For k

= 1, the evolution _process is uncorrelated. A species ^has a

probability

bc to next evolve

giving

_z

offsprings

and has _a

probability

1- bc to be

stopped.

This _{is a} classical

branching

process or Galton-Watson _process

iii].

Because of tue choice of the minimum

fitness,

the process is also

equivalent

to the invasion

percolation

model [23] on a Bethe lattice. In this

case, tue process is critical for bc ₌

1/z

and critical exponents T and Df _are

theoretically given by

_r

=

3/2

and

Df

= 2,

respectively.

For k _> 1, correlations

gradually

increase with k in the

branching

process and cannot be described

by

dassical theories.

For k

reaching asymptotically

+cc, ail

species

receive _{a new} fitness value at each mutation time step. ^{In the latter} case, the model becomes

equivalent

to the Eden model [24] on a Bethe

lattice. The distribution of the fitnesses is

always

_a flat distribution between _zero and _one since then bc ₌ 0.

From the numerical results of the previous section, ^it _seems that the model generates a

self-organizing

evolution process for all finite k and _z values. The _process

self-organizes

into _a critical

steady-state

and

gives

birth to self-similar

phylogenetic-like

trees. The

possibility

that tue model bas _a great

biological

relevance

(see Appendix)

enuances tue idea tuat

self-organized

criticality

is _an

archetype

to describe natural

phenomena [13,14].

In _our

model,

the

self-organized

critical behaviour _can be understood from the

following simple

mean-field arguments. ^One can

imagine

_a

large

tree witu

living species presenting

_a flat distribution of fitness values between _zero and _one. The

species having

the minimum

fitness value which is close _{to zero, is} the _one selected for

mutating

^and

gives

z

offsprings

with _new fitness values. This mutation affects _q other

species

which also receive _new fitness values. One should note that _~

depends

_on k and _on the local

configuration

of each

mutating

species ^{here. On tue} contrary, it is _a fixed parameter in tue BS model _[Si and

previous

extensions

ils].

Because tue _new fitness values _are cuosen with _a random number

generator

from _a flat

distribution,

the great

majority

of tuese _z + q new fitness values _are

probably

above

1lin

+

z).

On the lime average ^< >t over successive mutation _or

brancuing

events, an

accumulation of

species having

fitnesses above

b~ = i

Ii<

_{~ >~}

+z) j8)

thus

results,

while

species uaving

fitness less tuan bc tend to

disappear

from trie _tree. This leads to a

step-like

distribution of fitness values with _a

discontinuity

at bc. Trie

branching

_process

reaches _a critical

steady-state

because trie mutation of

only

_one

species having

bi < bc

gives statistically only

_one

species ion

the

average)

with _a fitness below bc. In Section

3.2,

_we have

seen that this critical state is unstable and is characterized

by

intermittent avalanches.

One should note that _a mean-field

theory

of the BS model

gives

bc =

1/(2d

+

1)

[21], 1-e-, the inverse of the total number of affected

species (mutating species

and nearest

neighbours).

Equation (8) gives

the mean-field value of the BS model for _z

= 1 and < ~ >t= 2d. The form of

equation (8)

_was also

numerically

verified

by measuring

^{the number of aflected}

species

< ~ >t at each mutation.

Figure

9 presents ⁱⁿ a

semi-log plot

_measures of both and

1/bc

_z for _z

= 2.

For

decoupled species (k

=

1),

the bc =

1/z

value is exact because the model reduces to _a usual uncorrelated

branching

_process.

Large

deviations _{are seen} between the mean-field

theory

and simulations for k _» 1. The estimations of _{< ~ >t} in the mean-field approximation _are less than simulated < ~ >t values. This shows that _a true evolution _process is much _more

complex

than _a

simple

uncorrelated

branching

_process. Further work should

darify

this

problem by investigating

trie local

configurations

close to "blocked"

species

and

mutating

_ones.

Moreover, the _mean number of affected species

by

_one mutation event < ~ >t is found to increase

exponentially

like

< ~ ^> +~

e~(z)k

j~)

for k » 1. The

z-dependence

of _{j is} outside the _scope of this _paper and should be

analyzed

in further extensive numerical

works,

_e-g-, for k

= 2, _{j is} found to be 0.367.

1022 JOURNAL DE PHYSIQUE I N°8

iooo

.

l~i

1OÙ ^°

Î/Î>~-z

_.

. .

. o

. o

. o

10 .

, _o

, o

~ o

o .

i o

. ~

o

0.1

0 k

Fig. 9. Semi-log plot of the number of affected species < ~ > per mutation event _{as a} function of the _range of competition-correlations k. Tue mean-field values of < ~ ^>=

(l/b~)

_z derived from tue b~ values of Figure 4 _are also suown.

The

exponential

law of

equation (9)

indicates that for k > 1, the

geometrical properties

of

the trees _are determined

by

the parameter ^k.

Phylogenetic-like

trees which

belong

to the _same

universality

dass have _a unique ^{fractal dimension}

independent

of k. For

these,

the variable suould behave like _a power of k because tue latter parameter is _{a measure} of distance

(or scale).

This is not observed here since _an

exponential

form is observed for _{< ~ >t.} Tuis

irnplies

^that

Df

is _a function of k wuicu is

effectively

found in Section 3

(Fig. 8).

From wuat _we bave discussed above, _{one cari} understand tuat tue present ^{rules force} tue order parameter m to decrease towards

0+,

_1.e., towards _an unstable critical

steady-state.

The

self-organized

critical behaviour is then

recognized

to result frorn the

tuning

of tue order pararneter ^towards a

vamsuingly

small but

positive

value and

lying

^tuereafter

exactly

at tuis critical value. Avalanches _are then associated to fluctuations _at all scales _in tue _response of _tuis unstable critical

point.

Tuis rnecuanism _was

recently proposed

_{[25] as a}

conceptual

framework for

self-organized criticality.

Dissipation

of

perturbations (or avalanches)

is the

principal ingredient

of

time-dependent tuerrnodynamics. Here,

tue

dissipative

mediurn _is

represented by

tue set of species. Tue

dissipation

_is

strongly dependent

_on tue

k-parameter.

For k

= 1 and k ₌ 2, a mutation

cannot affect the fitnesses of

previous species

_or ancestors. The avalanches _are thus local

dissipations only. However,

for k _> 2 prior and

posterior species

_con be affected and the avalanches _can propagate

along

the tree. In tuis case, many branches _cari grow on tue tree in

a

single

avalanche _event. One should note that this _crossover at k

= 2 between "localized" and

"delocalized" avalanches _is

z-independent.

We have _seen in Section 3 tuat tue

universality

^dass to wuicu tue evolution _process

belongs depends strongly

_on both k and _z parameters. One could expect ^{tuat tue critical} exponents Df and _r could be related

by

a unique

hyperscaling

relation because

tuey

_are

given by

a unique and

simple

rule.

Figure

10 shows trie

(avalanche)

critical exportent _r versus trie fractal dimension of trie trees for each couple of

(k, z)

parameters. ^{Trie dots} seem to follow _an

hyperscaling

relation

but

they

cannot be described

by

the usual

hyperscaling

relation:

.8

~

~"~

~'~

~ _Î~I,

th.

.5 é~

.4

.3

.2

1-1

1.8 19 2 2.1 2.2 2.3 2.4 2.5

Df

Fig. 10. Tue critical exponent T vs. tue fractal dimension Di of tue trees showing tue existence of

a possible uyperscabng law between tuem.

Df(T -1)

= 1

(10)

usual for mean-field theories of

percolation

_[26] and

self-organized

critical behaviour

[27].

^Trie

latter

hyperscaling

^relation is

represented by

trie continuous _curve in

Figure

10.

5. Conclusion

We bave

developed

_a relevant model for

studying biological

evolution. Trie model generates

a

complex branching

process which takes into account

competition-correlations

between dif- ferentiated

beings.

Trie evolution _process

self-organizes

into a critical

steady-state

_in which

intermittent avalanches

(or

bursts of

activity)

of ail sizes _are

generated.

On _a

geological-like

time

scale,

this is _in agreement with

punctuated equilibrium

theories of evolution _as discussed in

Appendix. However,

_an infinite _range of

competition-correlations

_is found to

destroy

the

self-organized

critical

(the punctuated equilibrium)

behaviour.

Moreover,

the

phylogenetic-like

trees _are found _to be self-similar.

Phylogenetic

tree data should be

investigated through

fractal concepts.

The

dynamics

of the transient

regimes

have been also studied.

They

show _a

slowing

down of the order parameter ^towards an unstable critical state. The critical

self-organized

behaviour has been

recognized

here to be the result of the

tuning

of the order parameter.

No need to say that _more

biological

constraints should be further introduced in the model.

For

instance,

_we have _seen that for the present

model,

_an infinite _range of correlations

(k

_-

+cc) destroys

the

punctuated equilibrium

^behaviour.

Long

_range correlations _are how-

ever

biologically

of interest.

Obviously, biological

correlations oct

certainly dilferently

thon

through

_a random

change

of the fitness values. This should be taken into account in future statistical

physics

models of evolution.

It is dear that the model invites to

considering

dilferent

extensions,

like _a variable k _param- eter and

for

_z parameter

during evolution,

and to

exarnining

other

physical properties.