• First Talk
• General Introduction to Complex Systems
Biology• Recursive Production of Cells• （A few Basic Problems in Origin of Life）• Development--Cell Differentiation• Coupled Dynamical Systems• Summary----------------------------------------------------------2nd; phenotype evolution; robustness,
evolovability,….

Complex Systems Biology

cf. Life as Complicated System: （current trend)
Enumeration of molecules, processes
detailed models describing the life process
Life as Complex System:
Understand Universal features at a System withmutual dependence between parts and whole
Simplistic Physicists’ Approach
Strategy:
1) Dynamical Systems ++＆Statistical Physics ++Æ Catch consistency between micro-macro levels2) Constructive Approach: (Exp & Theory)` construct simple system to catch universal features'
as simple as possible

Ecosystem
Phenotypic Plasticity vs Symbiosis
Or Ecological diversification

Multicelluarity
Evolutionary relationship on

Consistency between
Multicelluar development
and cell reprodcution

Cell

Genotype

Phenotype

Stochsatic dynamics

Adaptation asa result of consistency
between cell growth andgene expression dynamics

Molecule

Robustness and Fluctuation

Gene regulatio
network

Consisteny between Cell reproduction
and molecule replication

Catalytic reaction network

Consistency between different levels(1)Cell reproduction vs molecule replication (03-)(2)Reproduction of multicellular organism vs of cells(97(3)Adaptation vs Reproduction(06-)(4)Genetic change vs Phenotypic Fluctuation (03-Constructive Biology Project
theme

experiment

theoryquestion

replicating
systemin vitro
replication withenzymatic reaction

minority
controlorigin ofheredity;
evolvability

replicating cellwith internal
reactions

universal
statistics inreaction dynamics

condition forrecursive
growthcell differentiation.
development

differentiationof E Coilby interaction

emergence ofdifferentiation
rule from dynamics

irreversibility
robustness

Spontaneous
adaptation

Artificial gene network

Adaptive attractor
selection by noise

Ubiquitous ability inadaptation

evolution

Laboratory evolution
using bactreia

Fluctuation-response
relationship

Robustness,
evolvability

cellsystemComplex Systems Biology Project (JST,ERATO; KK,Yomo,…)Replicating artificial cell (experiment)(ÅÆtheory; fluctuation, minority control)translation

replication

division

RNA polymerase
RNA polymerase geneRNA（Yomo‘s group)

By 菅原ら(東大総合分化）(Sugawara’s group)

Tranlation in liposome
Continouos division of liposomes
RNAreplication in liposome

How is recursive production of a cell sustained？
each cell complex reaction networkwith diversity of chemicals;
The number of molecules of each species
not so large

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？

Fluctuations
Naiive Physicist View

Ｔｏｙ Cell Model with Catalytic Reaction Network‘Crude but whole cell model’
C.Furusawa & KK、PRL2003
ｋ species of chemicals 、Xo…Xｋ－１
number ---n０ 、n１ … nｋ－１

⚦⢩

model

(Cf. KK&Yomo 94,97):（nutrient）

random catalytic reaction networkwith the path rate pfor the reaction
Ｘi＋Ｘj－＞Ｘk+Xj
some chemicals are penetrable
through the membrane with the
diffusion coefficient D
resource chemicals are thustransformed into impenetrable
chemicals, leading to the growth in
Ｎ＝Σni, when it exceeds Nmax
the cell divides into twocellreaction

:

:
:

:
:
diffusion

catalyze:・・・

K >>1 species

mediumdX1/dt ∝ X0X4; rate equation;
Stochastic model here

☆Simulation procedure
１：Pick up randomly 2 molecules at each time step, if thepair reactions, change the substratre molecule into
productt (with the probability of the reaction rate)otherwise leave as it is
2 With a certain rate per time step (（≈１／Ｄ）, exchange amolecule of inside in the cell by that in anenviromenment. If the molecule is impermeable, it stays
３：If the total number of molecules N goes beyodn Nmax
cells are divided into two, eahc of which consists ofmolecules chosen randomlyIn continuum description, the following rate eqn.,
but we mostly use stochastic simulation• Cf:• Use of ODE (+ fluctuation)versus stochastic model with discreteness of molecules• Basically same if the number of molecules N therein issufficiently large.
If N is small (in comparison with the number of species
K), then several molecule species can be zero atsome time, then there appears a qualitatively distinct
behavior
N>>K we are used in physics
N ~K or N<<K needs different view
• --- Discreteness Induced Transition --(Togashi,KK, PRL 2002, Awazu, KK PRE 2007)

☆Growth speed and fidelity in replication
are maximum at Dc
Dc
1

growth speed
similarity

Growth0.15(only nutrients)
0.6

0.10
0.4
0.05
D = Dc

0.00
0.001

D
No Growth

0.8

0.01
0.1
diffusion coefficient D

0.2

similarity H

growth speed (a.u.)

0.20※similarity is defined from innerproducts of composition vectors
between mother and daughter
cells
・

0
・

Furusawa &KK,2003,PRL
Zipf’s Law is oberved at D = Dc
ni (number of molecules）

rank

number rank
X1 300 5
X2 8000 1
X3 5000 2
X4 700
4
X5 2000 3…….. (for example)

Average number of each chemical ∝ 1/(its rank)-2(distribution of x：ρ（ｘ）∝ｘ ）Confirmed by gene expression data
Human Liver

1e-01

α= -1

1e-02

1e-03

ratio of mRNA

ratio of mRNA

1e-01

1e-04

ratio of mRNA

1e-01

1e+01

1e+02
rank

Human Kidney

1e-02

1e-03

1e-03

1e-04

1e+01

1e+02
rank

1e+03

1e+04

Human Colorectal Cancer
1e-01

1e-02

1e-03

1e-04

1e-04
1e+00

1e-02

1e-05
1e+00

1e+03

ratio of mRNA

1e-05
1e+00

Human Heart

1e+01

1e+02
rank

1e+03

1e+00

1e+01

1e+02
rank

1e+03

1e+04

Mouse ES cell

Mouse Fibroblast Cell
1e-01

1e-02

ratio of mRNA

ratio of mRNA

1e-01

α= -1

1e-03

1e-04

1e-02

1e-03

1e-04
1e-05
1e+00

1e+01

1e+02
rank

1e+03

1e+04

1e+00

C. elegans

1e+01

1e+02
rank

1e+03

Yeast
1e-01

ratio of mRNA

ratio of mRNA

1e-02

1e-03

1e-04

1e-05
1e+00

1e+01

1e+02
rank

1e+03

1e+04

1e-02

1e-03

1e-04
1e+00

Later confirmed by several other groups

1e+01

1e+02
rank

1e+03

Formation of cascade catalytic reaction★ 栄養から直接生成される成分の数を多くなるようパラメータを変えて、階層構造が判りやすくした例
100000ni

Catalyzechemicals ofhigher rank
mainly

４

10000

３

1000

２

100

１

10

Rank of ni

1
1

10

100

1000

10000

100000

１：minority molecules

With conservation law,
The exponent -1 is explained

２：catalyzed by 1, synthesized by resource
３：catalyzed by 2：
Mean-field theory in phase transition (self-consistent)
calc.)

• Remarks:（０）Universality(1) Evolution to the critical state (with Zipf law) is
confirmed numerically(2) Evolution to scale-free network appears lateras embedding of power-law abundances into
network (Furusawa,KK, PRE 2006)（3） Self-organization to critical state, if
transport of ‘nutrition chemicals’ is catalyzed bysome chemicals （no need for choice of D)(instead of simple diffusion) (Furusawa,KK,2007)

Evolution of Network to satisfy Zipf’s law?

Yes

Critical D value depends on connectivity in the network;
mutation of network + selection Æ approaches Zipf’s law

Furusawa

Zipf’s law holds, irrespective of network structure, but
Later, the connectivity in the network approaches“scale-free” network through evolution.
statistical properties; embedded into network structure
Dynamics (abundance) first, structure (equation for dynamics) later
evolutionary embeddingof dynamics into network
４
2

probability for a path tochemical with abundances x
is selected; q(x) Ætransformation of abundance
distrb. to connectivity distrib.
Furusawa、KK, PRE, 2006

evolvedinitialRelationship between Zipf’s law (abundance) and scale-free
network (structure): ???
(1) Abundance x: density ρ（ｘ） ∝ x^-2(2) ｘ； a path to molecule species with abundances x have more
influence on growth speed: the simplest casevariation of growth speed to a path going out of a chemical with
abundance x -> is x times higher ;the evolution speed of a path from a chemical with x is effectively
amplified by ｘ: in general accelerated by some function ｑ(x),say x^{α｝（３） as the path number is larger, there are some better networks.
Then the distribution of paths k by transformation ｑ（ｘ） Æ ｋ(4) Distribution of k ; P(k) ∝ (dx/dk) ρ（ｘ）
if q(x) =x, then P(k) k^{-2}, if α=1/2 then k^{-3}
NOTE abundance dynamics first, topology of network (scale-free
network) is later embedded accordingly

☆Distribution of paths in reaction network

Furusawa

頻度

反応パスの数Fluctuation of each chemical
Abundance;
Æ long-tail to abundant size

P(ni)

e.g.
cell1 X1 10000
cell2
8000
cell3
15000
cell4
20000
…..
histograrm

3

2.5

Frequency of ni

2
1.5
1

ni （abundances）

0.5
0

500

1000

1500

2000

2500

3000

ni (abundances）

So far average quantity of all components;
Next question: fluctuation by cells:
distribution of each Ni by cells
Log normal distribution !

Furusawa,..
KK,
Biophysics2005
e.g.
cell1 X1 10000
cell2
8000
cell3
15000
cell4
20000
…..
histograrm

Each color
gives
different
chemicalspecies

LOG SCALE

☆Heuristic explanation of log-normal distribution
Consider the case that a component X is catalyzed byother component A, and replicate; the number --NX、NA
d NX /dt = NX NA
thend log( NX )/dt = NA
If、 NA fluctuates around its mean < NA＞, with fluct. η（ｔ）
d log( NX )/dt = ＜NA＞ ＋ η（ｔ）
log( NX ) shows Brownian motion Æ NX log-normal distribution
too, simplified, since no direct self-replication exists here
But with cascade catalytic reactions, fluctuations are

successively multiplied, (cf addition in central limit
theorem.);Hence after logarithm, central limit th. applied☆Heuristic explanation of log-normal distribution☆Cascade leads to multiplicative propagation ofnoise (at critical region)ｄNx/dt=Ny Nｚ
with cascade catalytic reactions, fluctuations are
successively multiplied,(cf addition in central limit theorem.);Hence after logarithm, central limit th. applied☆Cascade leads to multiplicative propagation ofnoise (at critical region)
Ａ

Ｂ

Ｃ

Ｄ

Ｅ

Propagation of fluctuation, feedback toitself, leading to log-normal distribution tail.
Cf. If parallel,Ａ

Cf??
weight – log-normalheight -- normal

Fluctuations come in parallel:
Usual central limit theorem is valid;normal distribution.

Experiment; protein abundances measured by fluorescence+flow-cytometryLog-normal
Distribution
Confirmed
experimentally

Furusawa,Kashiwagi,,Yomo,KK

Also studied in GFP synthesis in liposome

Statistics in gene expression in the present cell

IPTG
Ptrc

rfp

tetR

Doxycycline (Dox)
Ptet

gfp

Chromosome in E. coli

Relative frequency (a.u.)

lacI

P

Relative frequency (a.u.)(b)log10[GFP/FSC (a.u.)]

GFP/FSC (a.u.)

Log-normal like distribution at each Doxycycline concentration

Tsuru,Ichinose,Kashiwagi,KK, Yomo (in prep.)Cf. Recent studies on fluctuations:
log-normal or not?
Cell Growth has to be seriously considered
In theory and experiment.
X just stochastic gene expression
X condition suppressing cell growth
Condition for steady growth state --- should be
carefully prepared in experiment
Size itself is lognormal: either selection by some
size, or normalized by size
Analysis for all gene expressions in yeast （Bar－Even etal., 2006)

Growth Fluctuation induces log-normal-type distrb.
Figure 1

Fluctuations in a Cell; Cell Volume Growth effect
A

B

Concentration

Stochastic gene
expression that are
current concern of many

Concentration

Consequence ofCell volume growthfluctuation tha we areinterested

Tsuru,Ichinose,Kashiwagi,Ying,KK,Yomo

Growth fluctuation can lead to Log-tailed
phenotypic fluctuation• protein concentration x• dx/dt=f(x)-(μ+η）ｘ
dilution term by cell volume growth
μ －－ growth rate
η －－ fluctuation (noise)
multiplicative noise Æ log-tailed distribution(exp； Tsuru ｅｔａｌ）
Growth rate μ is a result of an ensemble ofgene expression μ(x1,x2,x3,…) -(consistency)?

Replicating artificial cell (experiment)(ÅÆtheory; fluctuation, minority control)translation

replication

division

RNA polymerase
RNA polymerase geneRNA（Yomo‘s group)

By 菅原ら(東大総合分化）(Sugawara’s group)

Tranlation in liposome
Continouos division of liposomes
RNAreplication in liposome• A Lesson:
Necessity of mutual dependency to form
self-consistent reproduction system
Liposome growthÅÆ synthesis of proteinsDNAso far, not consistentÆ
too large fluctuations(not necessarily log-normal)

10um
10um

0 .1

0 .1

0 .0 8Frequency

0 .0 8

y
c
n
e
u
q
e
r
F

0 .0 60 .0 40 .0 20
0 .0 0 1

0 .0 60 .0 40 .0 20 .0 1

0 .1

1

10

100

0
0 .0 0 1

0 .0 1

0 .1

1

g
o
l
S
F
g
o
l
P
F
G

GFPlog-FSlog

10

GFP synthesis in liposome and in bacteria

100• Basic question related with Protocells (origin-of-life)-- origin of genetic information (genetic take over?)(Minority control, kk,Yomo JTB 2002,kk PRE2003,,,)-- dynamics versus algorithmic: discreteness induces
sequential procedure -- Discreteness Induced
Transition, Togashi,kk; PRL2002, Awazu,kk 2007+)-- ‘jamming’ problem in reaction dynamics?
(Awazu,kk; PRE 2007 +)-- how non-equlibrium condition is sustained?
(Awazu,kk; PRL 2004, + arXiv2008)

Question :All molecules have such large fluctuation?
Important ones are ‘protected’? (DNA?)• Q: Origin of heredity?
• * some molecules in a cell are regarded as‘’important’’, and control the behavior of celle.g., differentiation in roles between DNA and protein,
Minority Control hypothesis (KK & Yomo, 2002)
in a replicating system composing of mutually catalytic
molecules, minority molecules play therole of heredity-carrier
Condition for heredity
preservation
controllability

X and Y mutually catalyze
the synthesis of each other;
Y is synthesized muchslower than X molecules.
Rate equation may leadto (active) Y moleculeof the concentr. < 1/N
A few Y molecules are necessary
to continue reproduction

divsionSelected are ‘rare’ stateswith a few Y molucules
Active Y molecules;(i) Preserved well,(ii)Control the behavior
Carrier of heredity

N molecules

N molecules• Hypothesis based on Minority Control
minority ( preservation+ control) Æevolution ofmachinery of faithful transmission of minority
molecule. Æ
more chemicals are synthesized with it
Package life-critical info. Into minority molecule.
From compositional information to sequence
information**Evolvablity: ‘’mutation’’ of minority moleculeÆ large influence
Genetic Takeover(from loose reproduction to tight replication)

Mutual Catalytic Reaction network
Stochastic collision

Awazu,KK,PRE2007

C

H
D

F
D

G

A

G
C

F

G

K
H
B

E
C

J

C
A

K

E

F
A

Total number of molecules N
Number of molecules species M

A

I

I

J

F
B
I

E
If not N >>M, the number of someRandom catalytic network
species can be 0 Æ discreteness

Reaction Rate RR =reaction evens per time
total number of molecules

（Species： M=100、 path： K=12）

An example in network

Plot of N(i) for species i same network, with different N
N(i) / N

N=800

N=12

N(i) / N

time

time
species i

species i

Steady state + small fluct. Intermittent switch among states

Average composition differs
drastically（）<N(i)/N>

0.02
0.01
0

N=12

N=800 （~ rate eqn）

0.04
<N(i)/N>

0.02
i
0

50

100

0

i
0

50

100

Jamming in reaction by crowding often suppresses theprocess, especially when complex reaction is put
ARS
Amino acids
tRNAs

EF-Tu-GTP

MTF
f

50S
ribosomalsubunit

EF-Tu-GTP

IF2

30S
ribosomal IF1
IF3
subunitMatsuura
et alYomo’s
group

EF-Ts

GDP
EF-G-GDP

IF3

IF2
IF1

EF-Ts
EF-Tu

GTP

EF-Tu-GDP

RF1/2
EF-G-GTP
RF3 EF-G
RRF

RF1/2

RF1/2
RF3 EF-G
RRF

Self-replicating RNA
AMP
β

ATP

ATP

MK
2 ADP
+
2 Creatine
phosphate
CK
2 Creatine

β subunit

β

EF-Tu
EF-Ts
S1

β
β
Qβ replicase

PPi

GTP

GDP

PPase
NDK
2Pi

ADP

ATP

CK
Creatine Creatine
phosphate

More than 5000 reaction steps
run with 144 components ofbiological molecules for the invitro self-replication.

How is non-equilibrium condition sustained?
Our tentative answer: a large class of Catalytic network
exhibits ‘glassy’ (slow,)relaxation with bottlenecks
Å negative correlation between substrate and catalysts
General in catalytic networks Æ ‘Chemical Net Glass’(Awazu-kk)
with spatial pattern -- further hindered --good news
Reinforcement？
nonequilibrium condition
ÅÆ Structure formation in network and in spaceÆ CompartmentalizationÆreproduction of molecules & of compartment
Reproduction of Non-equilibrium Condition Itself？？
But there appear reaction bottlenecks (bad news) +successive switch --- may underlie ‘slow dynamics’ inpresent cells?

Proposal of ‘Chemical Reaction Net Glass’’With A.Awazu
Simple catalytic reaction network

Ei distributed : stnd deviation ε
RELEVANT parameter βεEquilibrium distribution exp(-βEi) : (Ådetailed balance)
Relaxation process: Initial : β=0(high temp) for all
species, i.e., equal probability for all chemical species.

Two salient features inrelaxation analogous to‘glass’(1) Log-t slow relaxation(rather than exponential)(2) Existence of plateaus

Why Log-t slow relaxation?
due to energy distribution, the relaxation time
( kinetic coefficients exp(-βＥ）) distribute extensively
C(t)~
by

Æ log(t)• Why plateaus?
• Local equilibrium within cluster
x3 X2 X6
• Equilibrated with other clusters
X3
X8 X1
X2
is suppressed by deficiency of
X7 X5
X1 X4
catalytic molecules
X0
X7
X2,X8 small:• negative-correlation with
Larger in the molecules X5
In the cluster
abundances versus catalysts
Say catalysts X2,X7
Are suppressed• ΔX ↑ Æ ΔJout ↓ ？
General in catalytic networks Æ ‘Chemical Net Glass’Consolidation (Reinforcement)？
nonequilibrium condition
ÅÆ Structure formation in network and in spaceÆ CompartmentalizationÆreproduction of molecules & of compartment
Æ increase inhomegeneity in space
Reproduction of Non-equilibrium Condition Itself
But reaction can be easily stopped (bad news?)
Bottlenecks in reactions + successive switch --may underlie ‘slow dynamics’ in present cells?
True in irreversible reactions/ open systems?
Æ Study irreversible Catalytic Reaction Network

Multicelluarity:Ｑｕｅｓｔｉｏｎ ｏｎ
Ｃｅｌｌ ｄｉｆｆｅｒｅｎｔｉａｔｉｏｎ：

Insight by Conrad Waddington
Waddington’s
Canalization
Cell types as
Attractors?
How genes guide
this process?

Multicelluarity:Ｑｕｅｓｔｉｏｎ ｏｎ
Ｃｅｌｌ ｄｉｆｆｅｒｅｎｔｉａｔｉｏｎ：
1 diversification of cell types
2 Ｌｏｓｓ ｏｆ Ｐｌｕｒipotency(plasticity)(‘time’s arrow’?)
3 Robustness in cell types andTheir distribution

dxm= fm ( x1, x2 ,.., xk )dt

GROWTH

Coupled Dynamical Systems
with growth in dimension

Proposal of Isologous Diversification:(KK,Yomo,Furusawa1997-)(Oscillatrory) Gene expression dynamics* cell-cell interaction + Reproduction of a cell
Growth as a multicellular organism
Æ irreversible and robust developmental processIsologous Diversification:internal dynamics and interaction : development
instability
distinct phenotypes
interaction-induceddx m= f m ( x1 , x 2 ,.., x k )dt

Example: chemical reaction network
specialize in the use of some path
Coupled Dynamical Systems
Æ development
Internal chemical reaction dynamics
and interaction and cell divisionphenotypesynchronous division:
no differentiation

Instability of homogeneous statethrough cell-cell interaction

Assuming oscillatory dynamics as a single cellrecursive production

formation of discrete types
with different chemicalcompositions:stabilize each other• (1) Synchronous oscillations of identical units
Up to some threshold number of units, all of them oscillate
synchronously, and their states are identical.}• (2) Differentiation of the phases of oscillations of internal
states. When the number of units exceeds the threshold, theylose identical and coherent dynamics. Although the state ofunits are different at an instance, averaged behaviors overperiods are essentially the same. Only the phase of oscillations
differs by units.
• (3) Differentiation of the amplitudes of internal states. At thisstage, the states are different even after taking the temporal
average over periods. It follows that the behavior of states(e.g., composition of chemicals, cycles of oscillations, andsoon) are differentiated.
• (4) Transfer of the differentiated state to the offspring byreproduction. This ``memory" is made possible through thetransfer of initial conditions (e.g., of chemicals) during the
reproduction ( e.g., cell division).
• (5) Hierarchy of organized groups. This stage is the result of
successive differentiation with time. Thus, the total system
consists of units of diverse behaviors, which forms acooperative society.

ÆWith the increase of the number

Distinct types are formed through instability in ‘developmental
dynamics’ and interaction (both types are necessary)Single cell dynamics --- bifurcationinteraction term works as bifurcation parameter
Self-consistent choice of bifurcation parameterBifurcation parameter is given by interaction –self-consistent stateFor

fixed v ---Bifurcation

Self-consistent choice ofBifurcation parameter

A.Nakajima,KK(2007)

Robustness of developmental process
both states of each cell type and number
distribution of each cell type(1) against molecular fluctuations;(a few % fluctuations, ( ~ 100-1000 molecules))(2) against macroscopic damage;i.e., type A and type B, determined
but if type A is eliminated, then B dedifferentiates
and initial A-B cell ensemble is recovered(since A,B is stabilized each other)Differentiation of E Coli

Measurement by fluoresecent
proteins
Character of bacteria differentiate in a crowded condition(Kashiwagi, Yomo,…)Hierarchical differentiation from‘stem cell’;by taking initiallydynamics with instability (e.g.,chaotic)(higher order catalysis)

Furusawa & KK

Hierarchical differentiation from ‘stem cell’;by taking initially dynamics with instability (e.g., chaotic)(higher order catalysis)
Furusawa&KK

Hierarchical differentiationfrom ‘stem cell’;by taking
initially dynamics withinstability (e.g., chaotic)(higher order catalysis)
pS
pA

pB

probability depends on # distrib. of cell types
with prob. pA for S Æ A
if #(A) decreases then pA increases:

STABILITY

Generated Rule of Differentiation (example)
A1
A2
A3

A
S
B

Cf hematopoietic system(1)hierarchical differentiation:stem cell system(2) Stochastic Branching:stochastic model proposed in hematopoietic system(3) probability depends on # distrib. of cell types
with prob. pA for S Æ A
if #(A)
then pA―― global info. is embedded into internal cell statesÆSTABILITY(4) Differentiation of cell ensemble (tissue)――multiple stable distrib. { Ni }

Explained:

Robustness in development under large fluctuationin molecule numbers

Recall: (signal) molecules of few number -- relevant;Loss of potency from totipotent cell (ES),to multipotent stem cell, and to determination
Irreversibility in cell differentiation processcharacterized by the loss of phenotypic variation• Loss of pluripotency is characterized by
Decrease in the degrees of expressed genes(chemical diversity)
Decrease in cell-cell variation
Decrease in temporal variation ingene expression ( loss of chaos)
To recover pluripotency
increase the degrees of freedom(# of excpressed genes)
prediction confirmed by iPS (Yamanaka)
To confirm the theory
Measure gene expression dynamics(oscillatory gene expression andits change through differentiation)
partially observed by Sui Huang’s (Nature 2008)Universality?
checked a huge number of networks; only some fraction ofthem show chaotic dynamics & differentiation

Cells with such networks
with differentiation
higher growth speed as
an ensemble

Such networks are selectedMechanism: approach to Milnor attractor?
(that touches with basin boundary)
As long as the stem cell state is stable, it
reproduces itselfÆWith the increase in the cell number, theattractor touches with its basin Æ differentiate
to other types
ÆIf the number of differentiated cells increases
then the stability of the stem cell is recovered,and it reproduces itself

Underlying Mechanism in Dynamical Systems

Ｋａｎｅｋｏ，１９８９－

Globally coupled map (no spatial structure)

Cf

Coupled map latticeÆ space-time chaos

Cf. synchronized state is stable if
Synchronization of all elements with chaos is possible

Clustering
Example 1
3-clusters, with each synchronized oscillatios
Differentiation of behavior from identicalelements and identical interaction

Cluster of synchrnoized elemens+ non-synchronized elements

DesycnhronizedOnset of chaos

Chaotic Itinerancy
High-dimensional
High-dimensional
Almost-3 cluster
disordereddisordered
Almost-2 cluster

Collaborators
Chikara Furusawa
experiment
Tetsuya Yomo
Saburo Tsuru
Akiko Kashiwagi
Most papers (biology,
Dynamical systems)
Available at
http://chaos.c.u-tokyo.ac.jp
ERATO Complex Systems Biology Project(2006,August)• Why?
Conjecture by combinatorial explosion of basin boundaries
Simple separation x(i)>x* or x(i)<x*; one can separate 2 ^Nattractors by N planes.
In this case the distance between attractor and the basinboundary does not change with N ---- Order of (N-1)!
The boundary makes combinatorial explosion

ε

GCM

a
Cluster: group of elements such that x(i)=x(j);Number of elements in each cluster; N1,N2,…,Nk•at some parameter region many attractors with different clusterings
Due to the symmetry there are

attractors of the same clusterings -combinatorially many increase with the order of (N-1)!
or so(KK,PRL89)

Milnor attractor( i.e., Attractor in the sense of Milnor minus
usual attractor with asymptotic stability);
attractor and its basin boundary touches,i.e., any small perturbation from it can kick theorbit out of the attractor, while it has a finite
measure of basin ( orbits from many initialconditions are attracted to it)
Observed; Milnor attractors large portion of basin
for the partially ordered phase in GCM (kk,97,98)

The fraction of basin(i.e. initial values) for
Milnor attractors,
Plotted as a function of
Logistic map parameter1

Note! Fraction is almost
1 for some region
Result for N=10,50,100….

aThe Milnor attractors become dominant around N＞~（7－８）

N=3, almost 0
5, few cases
7,8,9,.. dominant

The Milnor attractors become dominant around N ＞~（５－８）

Dependence
On theNumber of
Elements N(accumulationover
1.55<a<1.72)（ｋｋ、PRE,200２)• Why?
Conjecture by combinatorial explosion

Chemical Gradient for Positional Information is generated
cell differentiation ÅÆ graidient for pattern
図９．５

Consolidation to Patterns