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>.
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Phys.
J.
I
France
(1995)
5
1011-1025
1995,
AUGUST
1011
PAGE
Classification
Physics
05.40+j
Abstracts
87.10+e
Self.Organized
Criticality
(*)
Vandewalle
N.
SUPRAS,
Institut
(Jleceived
10
and
1995,
received
Growths
Tree
(**)
Ausloos
M.
Physique BS, Sart Tilman,
de
March
Phylogenetic-Like
in
final
in
Université
form
de
Liège,
4000
ApriJ 1995, accepted
24
3
Liège, Belgium
May1995)
simple modèle stochastique
d'évolution
Darwinienne
engendrant des arbres
développé. Le modèle est basé sur un
de
branchement
tenant
processus
corrélations.
En présence de
corrélations
à courte
portée,
compte d'effets de compétitions et de
le
s'auto-organise dans un état critique
caractérisé
l'intermittence
d'explosions
processus
par
d'activité
de
tailles.
Sur une
échelle
pseudo-géologique, ce comportement
accord
toutes
est
en
les
caractéristiques
ponctualistes de
l'évolution
biologique.
Les
arbres
phylogénétiques
avec
auto-similaires.
simulés
La dynamique des régimes
transitoires
décroissance
sont
montre
une
eu
0+
loi de puissance du paramètre
d'ordre
caractérise
qui
critique
instable.
La portée
point
vers
un
génétique des
corrélations
espèces vivantes est un paramètre pertinent qui
et compétitions
entre
la classe
d'universalité
du
d'évolutiondétermine
Une portée infinie de ces
corrélations
processus
détruit
cependant le comportement critique auto-organisé. La dimension
fractale Di des arbres
l'infini lorsque k varie de 1 à l'infini.
L'exposant critique T de la
croît de 2,0 vers
distribution
des
avalanches
décroît à partir de 3/2 lorsque k augmente et
atteint
environ 1,2 pour k
10. Une
Résumé.
Un
phylogénétiques
est
=
relation
d'échelle
semble
moyen
montre
champ
de
Abstract.
effects
A
simple
developed.
is
le
que
des
comportement
différentes
engendré
processus
classes
plus complexe qu'un
bien
est
into
account.
model
stocllastic
Tlle
model
In
presence
based
is
of
on
of limite
Darwinistic
a
and
short
range
critical
a
the 0+
value
correlations
universality
however
like
trees
of
k
trie
=
10.
A
(*
e-mail
address:
(**)
e-mail
address:
©
Les
Editions
which
characterizes
between
living species
class
de
of
the
from
hyperscaling
is
decreases
relation
as
seems
1995
be
An
k
from
[email protected]
[email protected]
Physique
to
behaviour.
infinity
to
critical
found
process.
critical
2.0
size-distribution
unstable
an
evolution
self-organized
increases
avalanche
Une
théorie
de
processus
generating pllylogenetic-like
taking
competition-correlation
correlations, tlle process
self-organizes
of activity of ail
generated.
sizes
are
witllpunctuated
equilibrium features of
evolution
branclling
process
steady-state in wllich
intermittent
bursts
On a geological-like time scale, tllis
bellaviour
agrees
biological
evolution.
Tlle
simulated
pllylogenetic-like trees
show a power
law
decrease
dynarnics of the
transient
regimes
into
d'universalité-
décorrélé.
branchement
trees
lier le
to
relevant
found
to
the
order
genetic
parameter
self-similar.
be
parameter
fractal
dimension
The
towards
range k of competitionwhich
determines
trie
competition-correlation
infinite
The
of
The
state.
a
are
Di of the
destroys
phylogenetic-
range
from 1 to infinity.
Trie
critical
exponent T
goes
about 3/2 (for k = 1) and
reaches
about
1.2 for
relate
the
various
universality
classes.
Through
a
JOURNAL
1012
theory, we
branching
mean-field
uncorrelated
mention
that
the
PHYSIQUE
DE
evolution
process
I
is
N°8
much
complex
more
than
simple
a
process.
Introduction
1.
question of tue biological
fundamental
is certainly one of tue most
problems of
decade, some efforts bave been made in the application of
matuematical
concepts to tue biological evolution especially in systematics Iii. More recently,
also shown tuat
simulation
provides an interesting approacu to the problem
it was
computer
of evolution, thus opening new
research
fields in botu
statistical
physics and biophysics. Tue
models
correlations
dilferentiation
in DNA
[3], or
[2], cell
concern,
sequences
e-g-, long-range
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
number
between
species1 is associated a scalar fitness bj which is a random
and one,
zero
than a genetic code [Si. Trie quantity b, is supposed to be trie jitness (or a
of
rather
measure
trie barrier against mutation) of the species1; the higher trie fitness, trie more likely trie species
adapted to trie ecosystem and trie less likely the species mutates. At each "time step", trie
is
fitness bj is supposed to make an adaptative
species j having trie minimum
trie fitness
move:
number.
assumed
of short range
Because of trie
interactions
bj receives a new random
presence
between
trie
mutating species is supposed to affect trie fitnesses of its (2d) nearest
species,
a
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
The
cornes
puilosopuy.
and
science
biological
from
Since
evolution
tue last
arguments
[8].
and oversimplified but it follows trie fines of thought of
physicists develop complexity out of simplicity in contrast
with trie attempt to reduce complexity to simplicity [9].
steady-state for which
Trie BS model was found to self-organize into a critical
intermittent
of activity of ail sizes are generated. On a geological-like time scale, the
avalanches
avalanche
(quiescence)
found
be
separated
by
periods
of
stasis
much
longer
thon
trie duevents
to
are
of avalanches,
behaviour
wuich is
somewhat
simflar to the punctuated
features of tue
rations
biological evolution [loi. However, biological evolution is much more complex [loi. Nevertheless, trie idea of "critical self,organization" Ill] is of interest
non-linear
in
nature
since
il
biological
that
bave
organize
order
ail
length
structures
systems
into
to
processes
2]
appear
on
statistical physics by
scales [13,14]. Trie BS model is original and opens new investigations in
Actually, only a few extensions of the BS model bave been
realistic
models.
inventing
more
Trie
modern
statistical
is
restrictive
very
physics
where
trie
ils].
studied
Three
species
model
BS
is
Secondly,
latter
extinction
drawbacks
important
always kept constant:
an
extinction
consideration
has
is
multiple
is
a
on
trie BS
it is
associated
to
bave
emphasized.
to be
of
coevolution
the
strict
mutation
assumption, 1-e-, it
complex causes
which
strong
and
a
model
model
is
not
are
rather
than
First, the
a
model
number
of
species into another
Darwinistic
[16]. In fact,
specially related to a
not
of
N of
evolution.
a
one.
a
The
species
mutation
SOC
N°8
also
can
Thirdly,
[16].
event
not
several
mutate
PHYLOGENETIC
IN
well
taken
times
and
into
TREE
in
account
with
compete
its
GROWTH
trie BS
model
1013
trie
is
fact
that
species
one
or olfsprings.
branching process
updated species
own
developed the rules of trie BS model into a stochastic
[Iii
which allows for trie "neighbouring" species and trie mutated
with
other.
each
to
compete
ores
The advantages of this model are that, from very simple rules, ii) it generates phylogenetic-like
of real biological evolution, (iii) and it leads to a self-organized
trees, iii) it contains trie
essence
We
bave
critical
thus
behaviour
Here,
like
for
model.
trie BS
study trie elfects of trie genetic range of interactionwe
species on trie self-organized critical biological process.
entiated
competitions
Trie
number of new olfsprings appearing at each
mutation
biological considerations, various domains of physics
generalized model of tree-like
evolution is defined in trie next
rate, 1-e-, trie
these
Besides
The
of
processes
of trie
A
and
model
conclusion
model
2.
will be
trees
are
in
proposed and we also
is finally drawn in Section
discussed in Appendix.
is
Tree-Like
The
presented
principal
Model
Section
discuss
5.
In
3.
Section
4,
an
elfect
between
event, is also
concerned
are
section.
studied
here.
by this model.
Numerical
studies
mean-field
theory
elementary
origin of the self-organized
biological aspect and relevance
trie
The
dilfer-
branching
of trie
critical
process.
of the
present
Evolution
of
dilferentiated
species,
present model is that a mutation
event gives z
species and the z -1 other olfsprings. This mimics the apparition of z -1
This leads to a branching
and trie
species in a parent population of animais.
new
process
of tree-like
dilferent species are
formation
such that the genetically
located
structures
at trie
of tree
for phylogenetic trocs.
branches
Figure la presents a small tree which
extremities
as
from a single
labelled
bas been
mutation
"a".
Each
mutation
events
ancestor
grown by 6
differentiated
offsprings. Trie tree of Figure la could be
in Figure la bas given z
2
event
illustrated
On trie tree of Figure la, we
mapped into a phylogenetic-like troc
in Figure 16.
define trie
"distance"
number of segments
d~nn between two m and n species by trie minimum
trie m
As for trie BS model, a fitness
of trie tree
needed to
connect
to trie n species.
species
b, which is a random
number
between
and one is
associated
to each living species
at
zero
fitness value,
each
branch
extremity. At each "time" step, trie species j having trie
minimum
min(b,), mutates and gives z differentiated
Trie
1-e-, bj
parent species is one of
species.
fitness value.
these.
Each one
Trie branching points of trie tree in Figure
receives a random
understood
from trie mapping
la represent old configurations but Dot living species as cari be
The
1-e-, trie
idea of trie
parent
=
=
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
Trie
latter
thus
affected by trie
of trie species
bas
mutated.
j species
mutation
species
are
dermes
The
k
trie
genetic-hke
j but do not necessarily mutate
parameter
next.
range of the
between
differentiated
interactions
species.
Trie k
fllustrated
2
is
in Figure lc where a
mutation
3 and z
event
on
process
occurs
min(bi,
b7). Trie mutation of trie
labelled "4" in trie tree of Figure la, 1-e-, b4
trie species
differentiated
species "4" gives two
species with fitness, b[ and b[ (since z
2) and affects 3
distribution
other species labelled "5", "6" and "7", respectively, in Figure la. This gives a new
(b[,. ,b[) of fitnesses, where b[ bi, b[ b2, b[ b3 and other fitnesses (b[ to b[) being new
=
=
=
.,
=
=
=
random
either
(for
z
numbers.
One
should
note
that
gives rise to two offsprings (new
2) offspring (new "5").
=
in
"4"
=
so
doing,
and "5")
one
or
considers
evolves
that
trie
(new "4")
ancestor
and
(say
gives
a
"4"
single
JOURNAL
1014
DE
PHYSIQUE
hs
a)
I
N°8
b)
tg
ht
b2
b3 b4 bs
b6
b7
b6
a
6q
65
~~
~,
b'7
2
~3
,
c)
Fig.
a)
l.
mappmg
species
of
small
A
trie
labelled
tree
tree
of
4 in
a).
a)
a
by 6 mutation
events, each
phylogenetic-like tree; c) trie
grown
into
a
mutation
tree
of
giving
a)
after
z
=
offsprings; b)
2
trie
mutation
of
the
The
with an arbitrary
evolution
number of species arranged or Dot on a tree.
The
starts
described
for
mutation-competition
above
repeated
number
of
given
mutation
t
process
is
a
Thus, trie number of different species linearly increases with t with a z
The
1 rate.
events.
geological-like duration of a mutation of a species with fitness bi is assumed to be proportional
to
exp(Àbi)
like for
The only two
during the whole
not
to
predetermine
next
This
be
used
physics
the
BS
model.
parameters
evolution
at
each
are
the
integers
(or tree-growth)
mutation
k
which
event
and
process.
z
which
One
of the
z
are,
should
at
this
time, kept
also
note
that
differentiated
the
olfisprings
constant
model
has
a
does
chance
evolve.
rnodel
in
or
is not
the
only adapted
general study
computer
networks.
of
to
the
cornputational
branching
correlated
study of biological
processes
[Iii
as
evolution
well, like
but
in
could
nudear
SOC
N°8
Numerical
3.
Algorithms
stored
in
FALSE
Carlo
of such
arrays
nurnber
step,
PHYLOGENETIC
TREE
GROWTH
1015
Results
species
old
and
IN
the
are
branching processes
labelled by integer
do
not
major
cause
Trie
nurnbers.
computation
values
fitness
diiliculties.
species
these
of
Ail
are
living
simply
floating point numbers. Each species is represented by a logical TRUE or
old species (branching points), respectively.
For each
Montea living or
species having trie minimum fitness value bi (among the living species) gives z
of
for
living species which are immediately stored. The activity of trie species which bas given
turned off. Trie
Monte-Carlo
the z offsprings is then
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.
competition-correlations
between
the living species, one
In order to compute trie
needs to
of the
In
each
the
know the whole
doing,
associated
structure
at
tree.
growing
so
species
is
label is also stored in an array of integer
The
latter
numbers.
label of its parent.
This allows
for trie search of ail living species being at a given distance
from trie mutating species less than
random
the
number.
parameter k. Trie fitnesses of the latter species are updated by a new
One should
that the increasing number of species with
Monte-Carlo
time slows down trie
note
speed of trie algorithm for large trees. Trees of up to 50000 species have been computed here.
will present
the
numerical
In trie following paragraphs,
investigations of both physical
we
(via trie criticality) and geometrical (via the fractality) aspects developed by trie model for
values of the two integer
k and
various
parameters, 1-e-, trie range of competition-correlations
trie branching rate z.
new
STEADY-STATE.
FITNESS AT THE QUASI
Starting with dilferoF
configurations or ancestors, the stochastic
process is always round to self,organize
steady-state in which ail the fitnesses are
distributed
distribution
in a step-like
into a so-called
n(b) sharply vanishes below some limite
distribution
shown in Figure 2. Trie "steady-state"
as
critical
value bc and n(b) has a finite and
value above bc.
Such a "steady-state"
non-zero
THE
3.1.
DISTRIBUTION
arbitrary
ent
O.025
k=2
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Fig.
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events.
The
Three
distribution
dilferent
n(b,)
ranges
of
of
fitnesses
at
the
extreInities
correlation-competitions
have
of
trees
been
containing
(k
2,
used
JOURNAL DB PHYSIQUB L
=
t
4
T 3, MS,
=
20000
and
mutation
6).
AUGUST1993
41
JOURNAL
lo16
PHYSIQUE
DE
I
N°8
0.08
0.07
k"2
k=4
'~~ ~
.,
;
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0.05
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0.03
0.02
.,
,,
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O.O
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bmin
Fig.
t
The
3.
1 to
=
t
n(bm,n)
distribution
Three
20000.
=
of
different
fitnesses bm,n through
correlation-competitions
minimum
of
ranges
which
trie
has
tree
been
have
evolved
(k
used
=
from
2,
4
3
shows
and
6).
distribution
trie
illustrated
in
n(bm;n)
of the
values
various
been
of k and
constructed
One
should
out
turn
not
Figure
is
distribution
for
z
2 for
various
fitness
minimum
=
respective
Trie
2.
of k and
values
values
through
distributions
of up to 20000 living species.
that the "steady-state" is Dot
with
for
z
which
Figure
of
Figure
2.
=
trie
bas
tree
Figure
2 and
for
evolved
have
3
trees
emphasize
to take place through
fitness
values
reached,
which
are
long
as
less
thon
as
ail
a
do
mutations
critical
bc(k,z)
value
configurations. A true steady-state is of course
never
will still take place for species having a fitness close to bc and will
reached
mutation
since a
(see Section 3.2 below).
perturb the
distribution
further
One should note that a fraction of living species will net
participate in trie evolution
configurations. Indeed, those and only those species havbecause they will be screened by some
values strictly greater thon bc(k, z) and having in their neighbourhood (determined
ing fitness
by the parameter k) fitnesses ail greater than the threshold bc(k, z) cannot further evolve. Trie
independently
study of
such
model
The
works,
screening
does
of the
effect
is
determine
not
ecological
additional
extinction
account
choice
of the
related
not
initial
outside
trie
if such
screened
constraints
could
to
scope
of this
species
be
introduced
Figure 4 presents the measured values of bc(k, z) as a
branching parameter z. The bc(k, z) values decrease
from 1 to large k values.
These values are less than and
BS
model
which
is
little
a
bit
greater
than
2/3 [18].
asymptotically zero for k tending to infinity, 1e.,
ail living species in the phylogenetic-like tree.
3.2.
AVALANCHES.
avalanches
values
are
can
above
take
the
As for
place
in
threshold
the
extinct
BS
model, trie
when
function
further
be
stay olive.
or
model
in
order
are
The
evolution
process
of k and for
1/z
from
quite
to
zero
different
bc(k, z)
studied.
In
further
take
to
into
existence
k is
from
the bc
thresholds
of such b~
that at
m
bc(k, z)
values
various
when
seem
competition-correlations
[Si. Suppose
bc(k, z). The species having bi
the
the
in
wfll
mutation.
the
the
and
paper
become
sonne
trier
value
to
implies
"time"
mutates
of
reach
between
occur
thresholds
of
increased
ail
that
fitness
leading
N°8
SOC
PHYLOGENETIC
IN
GROWTH
TREE
1017
0.6
z=3
&
~ ~
~=~
a
à
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Ù.5
.
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.
~
*
»
.
~
~
O.O
2
0
6
4
8
10
12
16
14
k
Fig.
k
Tue
4.
and
to
for
species and
avalanche
an
values
[Si
tue
below
values
values
vanous
new
z
turesuold
or
local
to
fitness
rate
function
a
values
of
turesuold
as
of tue
competition-correlations
of
range
z.
fitness
new
activity as
bc(k, z). Tuis
burst
a
bc of tue
branching
of the
whicu
causally
a
connected
tuat
means
be less
can
the
bc(k,z).
tuan
of
sequence
evolution
process
defines
Tuis
activity witu fitness
self-organizes into
avalanches
of activity as for the BS model [Si.
intermittent
subsequent
avalanches
separated by long periods of quiescence having the longest
Two
are
geological-like durations m exp(Àbc generated by the evolution
The punctualist
aspect
process.
of tue
evolution
generated
by
tue
model
discussed
Appendix.
is
in
present
process
n(s) of tue avalanches bas been investigated. Here, tue "size" of an
Tue
size-distribution
avalanche is tue number of mutation
contained in a single avalanche.
An
avalanche of
events
s(z
-1)
of
bas
produced
beings.
duration
avalanche
linearily
related
size s
Thus, trie
new
an
is
of
this
the
avalanche
leading
the
model
equivalent
critical
size
in
to
present
to
exponent for
n(s) of avalanches is found to follow a power
both physical properties.
The size
distribution
of
state
a
law
behaviour
as
nls)
Il)
S~~
r~
finite
for ail
the
been
grown
n(s)
seen
BS
been
n(s)
large
s
for
6
[Si.
presents
various
known
are
=
Figure 5 which
branching
a
be
to
presents
with
from
deviations
finite-size
self-organizes
model
evolution
obtained
For
value
simulations
and
for ail
should
z
ail
10
usually
values,
indicate
trees
values
z
the
of
values
branching
of the
by simulating
mean-field
+cc
measured
the
values
parameters.
-
The
in
2
rate
z
in
log-log plot
a
The
2.
=
the
strict
power
law
effects
[19].
This
indicates
critical
[20]
into
a
has
tree
of
behaviour
that
steady-state
the
as
for
model
trie
k
for k
mutations.
our
(k, z)
is
illustrated
is
avalanches
values
of
This
z.
of
50000
process
Figure
k and
of k and
to
up
for
stochastic
the
values
distribution
size
T
made
T
rate
obtained
for
decreases
and
exact
a
z.
The
of 50000
for k
and
function
as
1, trie
=
a
asymptotic
critical
reaches
value.
interaction-range
of the
n(s)
size-distributions
mutation
critical
events
for
exponent
about
The
1.2
each
T
branching
is
process
for k
10.
continuous
=
of
parameter
avalanches
bave
couple of integer
close
Iii].
to
3/2
which
However, for
Further
extensive
of
exponents
variation
DE
JOURNAL
1018
PHYSIQUE
I
N°8
10°
i~l
1
~
0~3
i
s
Fig.
z
=
The
5.
The
2.
n(s)
distribution
size
tree
grown
was
up
to
t
avalanches
of
tree-like
trie
in
evolution
for k
process
2
=
and
50000.
=
2.0
z=~
z=3
z#4
.
~
&
"
6
~
~
l 2
l 0
0
2
6
4
8
10
12
k
Fig.
values
various
by
grown
of
50000
tue k and
witu
but
critical
The
6.
of trie
function
a
Each
z.
dot
of
range
the
represents
competition-correlations
results
of
the
simulation
k
and
of 10
for
trees
mutations.
z
TRANSIENT
critical
as
T
rate
parameters
does
not
seem
to
be
of tue
artefact
an
limited
quality of data,
intrinsic.
seems
3.3.
exponent
branching
the
REGIMES.
steady-state
tue
average
< b >
tue
system
reacues
We
starting from the
of
tue
fitnesses
critical
bave
aise
non-critical
observed
state
all living species
steady-state, < b >
over
as
how
of
a
a
trie
single
function
becomes
system
ancestor.
of the
evolves
We
have
simulation
towards
its
measured
time
t.
As
SOC
N°8
PHYLOGENETIC
IN
TREE
GROWTH
1019
où
jjj
~
~~
~
~
Î
'
»,~~m~
AA
a
£
k=6
m
.~
»
.
~ ~
»
~
,
~
l 0"2
*
,
.
a
,,~
a
,
AA
.
à
~
é
i Q"3
~
l 0~
0
100
1000
0000
t
Fig.
Evolution
7.
range
of tue
order
parameter
competition-correlations
of
< b
parameter"
"order
Tue
of tue
m
witu
m
time
t
for
z
and
2
=
for
values
various
Il
>=
bc)/2
+
puenomenon
evolution
present
12)
be
can
defined
[21]
as
(3)
m=bc-2<b>+1
since
in
a
shrinks
m
0+ in tue
to
semi-log plot for
different
One
cases.
lattices
(characterizing
should
One
above
a
that
as
seen
in
situations.
of k and for
the BS
trie
classical
note
to k
sensitive
that
note
order
Tuis
7.
7
=
shows
tue
evolution
Trie
2.
average
reached
with
be
to
presents
dimension
de
exponent
Figure
z
found
is
model
critical
second
trie
Figure
critical
values
steady-state
critical
should
hypercubic
various
Trie
trees.
tue
of
k.
exponential
an
<
>
of
was
law
power
a
behaviour
with
m
taken
for
time
t
1000
over
decay
ail
in
d-dimensional
4 [21]. Below de, a power
law
behaviour
phase transition [20]) was found for the BS model.
characterizing this power law decrease of m is weakly
weak dependence should be further
numerically aria=
lyzed.
GEOMETRICAL
3.4.
through
structure
different
two
distance
mean
largest
the
aspect
total
We
kinetic
trie
oF
because
numencally
1)t
it
of
the
expresses
found
that
the
in
< d
for ail limite
the
general
values
of k and
evolution
of
z.
Trie
dmax(t)
exponent
shows
also
>r-
trie
second
<
d >
a
evolution
of
the
and
species in the
differentiated
shows
power
a
law
tree.
behaviour
(4)
t"
was
found
scaling
law
v
a
The
most
length
characteristic
and
time
iii) trie evolution of
updated species. The first
length of the tree and the
aspect has some biological
species
ail
characteristic
tree.
of the
with
Monte-Carlo
and
the
between
evolution
the
trie
ancestor
common
contained
species
ii)
geometrical
single ancestor,
The
TREES.
Starting
interest.
ancestor
common
the
physical relationship
a
is also of
measured:
be
cari
PHYLOGENETIC-LIKE
THE
evolves
process
features
between trie
< d >
distance dmax between
expresses
number (z
relevance
PROPERTIES
which
to
be k and
behaviour
z
dependent.
Moreover,
JOURNAL
1020
PHYSIQUE
DE
I
N°8
S-ù
a
z=~
z=3
~
~_~
4.5
,
4.O
3.5
à
3.O
2.5
2.O
1.5
2
6
4
8
10
12
k
Fig.
8.
Tue
correlations
k
simulation
of 40
Di of trie phylogenetic
dimension
fractal
and
for
values
various
trees
by
grown
10000
scaling laws
self-similar
A
by
for ail
self-similar,
power
a
fl wuicu
exponent
an
Tuese
finite
tuai
values
fractal,
or
also
was
indicate
are
to
trie
function
a
Eacu
z.
range of competitiontue
results of tue
of tue
dot
represents
t~
là)
'~
strongly k- and z-dependent.
generated phylogenetic-like
tree
z.
property
This
bave trie
trees
to be
property
signature of criticality [20].
another
is
fractal-like
[22] of
Di is
dimension
indeed
characterized
law
nid)
relating trie
nid) from
as
trie
of k and
growing
rate
mutations.
dmax
witu
trees
branching
of tue
d~~~~
16)
r~
nid) of species to trie distance d away from a common
Integrating
ancestor.
dmax, one con find that biological (fl) and physical iv) evolution
exponents
Moreover, trie fractal-like
Di of trie phylogenetic-like trees is found
dimension
number
to
zero
sames.
be
D~
Figure
8
values
of k and
z.
The
2.0
large values
with
to
4.
tue
presents
fractal-like
increase
dimension
of the
Iv
m
i
/p
Di(k, z)
dimensions
fractal-like
the
i
m
ii)
genetic
phylogenetic
of tue
Df of the
trees
k of
range
found
is
for
trees
to
increase
various
limite
from
about
next
evolve
correlations.
Discussion
For k
giving
1, the
=
z
evolution
offsprings
process
or
process
is
and
uncorrelated.
A species has
process is
has a probability
1- bc to be stopped.
Galton-Watson
also
equivalent
to
process
iii].
the
invasion
Because
of
percolation
tue
a
choice
model
[23]
probability bc
This
of
is
the
on
a
a
to
classical
minimum
Bethe
branching
fitness, the
lattice.
In
this
SOC
N°8
case,
by
r
tue
=
and Df
k > 1,
described by
for bc
=
dassical
In the
step.
The
lattice.
then bc
From
TREE
1/z
exponents
and
critical
GROWTH
1021
Df
and
T
theoretically given
are
gradually
with
increase
k in
the
receive
a
branching
and
process
be
cannot
theories.
reaching asymptotically
For k
=
PHYLOGENETIC
2, respectively.
correlations
For
time
critical
is
process
3/2
IN
latter
+cc, ail species
model
becomes
the
case,
distribution
of the
numerical
results
fitnesses
is
always
flat
a
fitness
new
equivalent
the
to
value
Eden
distribution
at
model
between
each
mutation
[24]
on
and
zero
a
Bethe
one
since
0.
=
the
of
section,
previous
the
it
the
that
seems
model
generates
a
The
self-organizes into a
self-organizing evolution
process for all finite k and z values.
process
critical
steady-state and gives birth to self-similar phylogenetic-like trees. The possibility that
enuances tue idea tuat self-organized
tue model bas a great biological relevance (see Appendix)
natural
phenomena [13,14].
criticality is an archetype to describe
behaviour
be
understood
from the following
model, the self-organized critical
In our
can
One can imagine a large tree witu living species presenting a
simple mean-field
arguments.
flat
distribution
of fitness
values
between
and
The species having the
minimum
one.
zero
fitness value which is close to zero, is the one
selected for mutating and gives z offsprings
affects q other species which also receive
fitness
values.
This
fitness
with
mutation
new
new
One should
that ~ depends on k and on the local configuration of each
values.
mutating
note
species
here.
from
a
flat
1lin
above
On
ils].
extensions
+
contrary,
new
distribution,
the
great
z).
On
the
lime
is
it
tue
fixed
a
fitness
majority
b~
tuese
with
+ q
z
model
random
a
new
successive
over
BS
tue
in
cuosen
are
of
>t
<
average
parameter
values
species having fitnesses
of
accumulation
tue
Because
probably
values
are
brancuing events, an
fitness
mutation
previous
[Si and
number
generator
or
above
=
i
Ii<
j8)
+z)
~ >~
results, while species uaving fitness less tuan bc tend to disappear from trie tree. This leads
of fitness values with a discontinuity at bc. Trie branching
step-like
distribution
process
having bi < bc gives
reaches a critical
steady-state because trie mutation of only one species
statistically only one species ion the average) with a fitness below bc. In Section 3.2, we have
critical
unstable
and is
characterized
by intermittent
avalanches.
that this
state is
seen
1/(2d + 1) [21], 1-e-,
One should
that a
mean-field
theory of the BS model gives bc
note
neighbours).
number of affected species (mutating species and
the inverse of the total
nearest
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
of both and 1/bc
for z
mutation.
Figure 9 presents in a semi-log plot
2.
at each
measures
z
1/z value is exact because the model reduces to a usual
For decoupled species (k
1), the bc
mean-field
uncorrelated
branching process. Large deviations
between
the
theory and
are
seen
thus
to
a
=
=
=
=
for k
simulations
than
simulated
1.
»
< ~
>t
=
The
estimations
values.
This
shows
of < ~ >t in the
mean-field
that a true
evolution
process
simple
uncorrelated
branching process.
Further
investigating trie local configurations close to "blocked"
than
a
Moreover,
increase
the
mean
exponentially
number
of
affected
species by
are
less
complex
more
work should darify this problem by
species and mutating ones.
one
mutation
event
much
<
~
>t
is
found
to
like
e~(z)k
< ~ >
for k » 1. The
further
extensive
approximation
is
z-dependence
numerical
of j is
works,
outside
e-g-, for k
j~)
+~
the
=
scope
2,
j
is
of this
found
paper
to
be
and
0.367.
should
be
analyzed
in
JOURNAL
1022
PHYSIQUE
DE
N°8
I
iooo
l~i
.
1OÙ
Î/Î>~-z
°
.
.
.
.
o
.
o
o
.
10
,
.
o
o
,
o
~
o
.
o
i
.
~
o
0.1
0
k
Fig.
9.
the
range
Semi-log plot of the number
competition-correlations
k.
of Figure 4 are also suown.
values
b~
exponential
The
the
trees
This
Df
are
like
From
wuat
order
of k wuicu is
bave
we
parameter
pararneter
critical
a
above,
towards
behaviour
vamsuingly
Avalanches
value.
an
is
small
then
are
form is
effectively found
decrease
critical
towards
exponential
then
but
towards
1.e.,
as
function
a
derived
from
geometrical properties of
belong to the same
variable
For these, the
of k.
(or scale).
distance
of
measure
for < ~ >t.
irnplies that
Tuis
tue
tuat
rules
present
unstable
an
force
fluctuations
to
tue
steady-state.
critical
recognized to result frorn the tuning of tue
positive value and lying tuereafter exactly at
associated
of
tue
(Fig. 8).
understand
cari
0+,
3
z
which
trees
a
observed
Section
in
one
is
parameter
event
(l/b~)
>=
for k > 1, the
that
Phylogenetic-like
dimension
independent
latter
mutation
per
of < ~
k.
tue
discussed
to
m
self-organized
The
because
power
here since
a
fractal
unique
a
< ~ >
values
mean-field
indicates
parameter
of k
observed
function
a
have
dass
behave
is not
is
by the
species
affected
Tue
(9)
law of equation
determined
universality
suould
of
of
at all scales in tue
order
tuis
of tuis
response
recently proposed [25] as a conceptual framework
point. Tuis rnecuanism
was
for self-organized criticality.
Dissipation of perturbations (or avalanches) is the principal ingredient of time-dependent
Here, tue dissipative mediurn is represented by tue set of species.
tuerrnodynamics.
Tue
dissipation is strongly dependent on tue k-parameter.
For k
1 and k
2, a
mutation
affect the fitnesses of previous species or
avalanches
The
thus local
cannot
ancestors.
are
dissipations only. However, for k > 2 prior and posterior species con be affected and the
avalanches
branches
propagate along the tree. In tuis case, many
can
cari
grow on tue tree in
unstable
critical
=
a
single
avalanche
"delocalized"
We
have
in
seen
depends strongly
and
r
could
simple
be
trie
trees
but
they
on
for
each
cannot
is
should
10
by
a
shows
note
this
that
z-independent.
tuat tue universality
Section 3
both k and
related
Figure
rule.
One
event.
avalanches
z
unique
trie
One
parameters.
hyperscaling
(avalanche)
dass
=
wuicu
tue
evolution
tuat
tue
tuey
because
seem
r
to
relation:
versus
follow
"localized"
between
2
expect
exportent
Trie dots
couple of (k, z) parameters.
described by the usual
hyperscaling
be
to
could
relation
critical
at k
crossover
=
are
given by
trie
an
process
critical
fractal
and
belongs
exponents
a
Df
unique and
dimension
of
hyperscaling
relation
SOC
N°8
PHYLOGENETIC
IN
TREE
GROWTH
1023
.8
~
~"~
~
~'~
Î~I,
th.
.5
é~
.4
.3
.2
1-1
1.8
19
2.1
2
2.2
2.3
2.4
2.5
Df
Fig. 10.
possible
a
Tue
critical
uyperscabng
exponent
T
between
law
vs.
tue
fractal
dimension
Df(T -1)
for
usual
mean-field
of percolation
theories
hyperscaling
latter
relation
is
Di of tue
trees
showing
tue
existence
of
tuem.
=
(10)
1
[26] and self-organized
represented by trie
continuous
curve
critical
in
behaviour
Figure
[27]. Trie
10.
Conclusion
5.
evolution.
model for studying biological
Trie model
bave developed a relevant
generates
complex
which
competition-correlations
between difbranching process
takes into
account
a
self-organizes into a critical steady-state in which
ferentiated
beings. Trie evolution
process
(or
activity)
of
of ail sizes are generated.
avalanches
bursts
On a geological-like
intermittent
punctuated
equilibrium
of
discussed
theories
evolution
time scale, this is in
agreement with
as
competition-correlations
infinite
of
found
destroy
the
Appendix.
However,
in
to
an
range
is
self-organized critical (the punctuated equilibrium) behaviour.
Phylogenetic tree data
Moreover, the phylogenetic-like trees are found to be self-similar.
should be investigated through fractal
concepts.
The dynamics of the
transient
regimes have been also studied. They show a slowing down of
critical
critical self-organized
towards
unstable
The
behaviour
the order
state.
parameter
an
has been recognized here to be the result of the tuning of the order
parameter.
should be further
introduced
biological constraints
No need to say that
in the model.
more
infinite
of
correlations
For
that for the present model, an
instance, we have seen
range
(k - +cc) destroys the punctuated equilibrium behaviour. Long range correlations are howbiologically of interest.
Obviously, biological
correlations
certainly dilferently thon
oct
ever
through a random change of the fitness values. This should be taken into account in future
We
physics
statistical
It is
eter
and
dear
for
that
z
models
the
of
parameter
evolution.
dilferent
extensions, like a variable k
to considering
during evolution, and to exarnining other physical properties.
model
invites
param-
JOURNAL
1024
PHYSIQUE I
DE
N°8
Acknowledgments
This
and
D.
work is financially supported by the Belgium Fund for
Research
Formation
in Industry
Agriculture (FRIA, Brussels). The authors thanks Dr. D. Staulfer for his comments
and
Sornette for providing reference [25] before publication. They also thank SSTC (contract
SU/02/13)
and
(94-99/174).
ARC
Appendix
heavily pertain to paleontological observations.
punctuated equilibrium behaviour [10], 1-e-, rapid bursts
tg,
a
a
of activity separated by long periods of stasis,
established.
seems
Several
quantities could be investigated to
demonstrate
punctuated equilibrium. For examwhether
the
dilferentiated
ple, one can
examine
species
evolves.
Figure Il shows the
more
geological time evolution of dmax in the k
là has been fixed to 100.o). Each dot
2 case
the
of a species for the farthest
mutation
Each step
ancestors.
expresses
away species from its
height (or lump) corresponds to a burst of activity (avalanche) of the farthest away dilferenwhile the tread steps correspond to long periods
tiated species with respect to the ancestor,
dilferentiated
of quiescence for such
During these periods, avalanches may ocspecies.
more
less
dilferentiated
with
other
species
the
Que thus easily
understands
that dmax
tree.
on
cur
Two
of the
considerations
geological-like
On
model
time
scale
=
increases
in
propagate
limite
The
cific
Section
staircase
a
in trie
values
of k and
finite-size
3.3.
branches.
elfects.
In
trie
after
of trie
mutation
species the
farthest
equilibrium
punctuated
This
behaviour
evolution
recovered
is
must
for ail
z.
of
stages
earlier
since
manner
other
growth
trie
A
power
earliest
stages
the phylogenetic,like
characterized
trees
are
o+ was
decay of trie order
parameter
to
m
trie growth, trie
evolution
takes place through
of
law
of
by
shown
spein
fitnesses
1200
,1
1000
,
BOO
#
nÎ
600
400
200
~
O
.
1022
2 102Z
3 IOZZ
4 IOZZ
tg
evolution of the
Plot of trie
Fig. Il.
geological-Iike
the
living species
versus
competition-correlations
fixed to
of
was
maximum
time
tg. Trie
be 2.
distance
dmax
parameter
À
between
was
fixed
the
to
initial
be
species
100.0.
Trie
and
range
trie
k
SOC
N°8
which
could
be
steady-state
earth
on
lution
elfects
as
it is
species
new
when
illustrates
This
found
in
often
trie
finite-size
durations
the
fact
is
bave
characterized
models
r-
exp(Àb,)
occured
trie
at
always present in
analyzed per se and
to
critical
origin of life
evc-
Such
nature.
are
be
needed
trie
in
stages of biological
first
trie
elfects
should
time
longer than
much
to
seems
limite-size
that
1025
geological-like
the
elfect
bave
majority of physical
trie
GROWTH
stages of "life"
earliest
sonne
TREE
Thus,
than bc.
the
in
exp(Àbc). Trie
large geological
r-
[28].
larger
sometimes
several
obtain
PHYLOGENETIC
IN
neglected
not
case.
References
iii
Rammal
[2]
Ossadnik
M.
R.,
and
[3]
Kauffman
[4]
Stauffer
[si
Bak
Toulouse
and
M.A.,
Virasoro
Med.
Reu.
and
iii]
[12]
Glansdorff
(Wiley,
[13]
P.
New
P.,
Bak
K.
H-F-, Phys. Reu.
Chen
and
K., Phys. Reu.
aud Prigogiue I.,
York, 1971).
Thermodyuamics
K.
New
and
Lett.
(1993)
71
M.,
Creutz
Nature
59
Theory
(1989)
342
Gould
and
Lent.
of
Harris
[19]
Nature
332
(1988)
211.
387.
Stability
and
Fluctuations
780.
[16]
[18]
York, 1993).
Structure,
il?]
[15]
Simons
4087.
S-J-,
(1987)
E., Physica A 208 (1994) 336.
H-F-, Mak L. and Kwok P-K-, Physica A 215 (1995) 431.
Darwin C., The Origin of Species (John Murray, London, 1859)
[14]
765.
E 49
Wiesenfeld
P., Taug C. and
Bak
(1986)
J.
Flyvbjerg H., Sneppen
Chau
58
Mantegna R-N-, Peng C.-K.,
The
Bak P., Phys. Reu.
(1994)
4691.
[7]
Science
(1986)
M.D.,
1528.
251
[8] Raup
Anderson
P-W-, Phys. Today 44 (1991) 9.
[9]
[loi Gould S-J-, Pateobiology 3 (1977) ils; Eldregde N.
[6]
Phys.
S.,
Havlin
Origins of Order (Oxford University Press,
Phys. 24 (1994) 900.
Sneppen K., Phys. Reu. Lent. 71 (1993) 4083.
S-A-,
D., Braz.,
P.
G.
S-M-, Buldyrev S-V-, Goldberger A.L.,
Stanley H-E-, Biophys. J. 67 (1994) 64.
Bonabeau
Chau
T.E.,
Paczuski
de
M.
J.,
Boer
The
Theory
and
Bak
Branching
of
P., preprint (1995)
B., Flyvbjerg H.,
Derrida
(Spriuger, Berlin, 1963)
Processes
Jackson
A.D.
and
Wettig T., Phys.
Reu.
Lent.
73
(1994)
906.
[20]
Fisher
Fisher
[21]
[22]
M.E., Physics 3 (1967) 255;
M.E., Reu. Med. Phys. 46 (1974)
Ray T.S.
and
N., Phys. Reu.
Jan
B-B-,
Mandelbrot
The
[23]
[24]
Eden M., Symposium
York, 1958) p. 359; see
[25]
Sornette
Bunde
[27]
[28]
Feder
D.,
Johansen
Havhn
and
A.
Berlin, 1991)
p.
Tang C.
Bak
and
Volkenstein
597.
72
Geometry
Fractal
(1994)
of
4045.
Nature, (W.H.
Freeman
Information
on
Jullien
also:
and
A.
S.,
Fractals
Dornic
and
and
Co.,
New
York, 1982).
(1988) 2117.
Theory in Biology, H-P- Yockey, Ed. (Pergarnon Press, New
R. and Botet R., J. Phys. A 18 (1985) 2279.
I., J. Phys. I France 5 (1995) 325.
Disordered
Systems A. Bunde and S. Havlin, Eds. (Springer,
J., Aharony A. and J#ssang T., Phys. Reu.
Furuberg L.,
[26]
Lent.
Lent.
61
SI
P., Phys.
M.V., Physical
Reu.
Lent.
Approaches
60
to
(1988) 2347.
Biological
Evolution
(Springer, Berlin, 1994).