• Coupled Dynamical Systems

(Coupled map, chaotic itinerancy,…)

magic number 7+-2

　　　　　　 dynamical systems inspired by biology

• Cell Model with chemical networks

(universal statistical properties, evolution)

selection of dyn.systems through evolution biology study based on dyn.sys, statistical phys.

• Coupled Cell Dynamics

cell differentiation and development

biology study based on coupled dyn.sys.With growth

？ Q: entanglement versus module:

　 Is 　 there some limit in degrees of freedom so that the system behaves ‘’nicely ‘’?

background of this question;

(1) Magic number 7 ＋－　２ in psychology

--- number limit of chunk for a short-term memory, (e.g., phone number) Muller

(2) Use of modularity in biological systems

degrees of freedom are often clustered

??is there some critical number of degrees of freedom beyond which the ‘memory’ (separation into attractors) by dynamical system is unstable??

feed-forward network, globally coupled map

underlying universal logic?

Example 1:Feed-forward thresholdNetwork

M elements each layer

Neural network;Gene network with thresholdDynamics

Effect of entanglement?

Each element +1,-1 if β ＞１Attracted to +1,-1 dep. on input

input

Ishihara and KK 200 ５　 PRL

Random coupling

regarded as tim

e

Distribution of output x at downflow

Well separated

Plot of x_1 vs x_2 for all layers

Folding and Stretching; chaos

M=6 M=8

• Outputs are clearly separated to +1, -1 for most inputs for most networks (coupling ε ）

　　　　　　　　 for M ≦ ６

･ For M ≧ ７、 outputs can take intermediate values. Chaotic Dynamics is observed along the layer

(stretching and folding dynamics are observed commonly)

･　 This critical number ~ ７　 is rather universal

The average fraction of x (with some bin), where orbits visited (i.e. x such that P(x)>0)

Model with zero threshold

Model with distributed threshold

The ratio of networks that　 show chaotic behaviorPlotted as a function of M

Computed from Lyapunov exp.

Why?

consider just 2-layer dynamics by first

fixing input x2,x3,..xM and study the map

from the input x1 to the output x1

　 σ １＞０　 σ ２＜０

　 σ ３＞０

　 ν １　 ν ２　 ν ３

Let us renumber ν 　 as orderingν １＜ ν ２＜…Then, if 　 σ １、 σ ２、… changes itsSign alternately folding Process like logistic map if satisfied at all i=1,2,,,M-1 、　 then folding at every region

x(inp)

X(out)

• probability to satisfy such sign changes

----- 2^{-M} • By change of input values x_2,…x_M

there can be a variety of （ M-1)! for such ordering for ν(using original numbering ν)

If >1 then, the folding occurs at every region at least for somewhere in the layer chaos dominates cover the phase space

‘’If Combinatorial Variety wins exponential growth,

then the clear separation by threshold dynamics

is not possible!”

Crossover occurs at M =6

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 clusteringsDue to the symmetry there are

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

Another example: GCM

a

ε

• At some parameter region (partially ordered phase) 　 Milnor attractors occupy large portion of basin 　　　　 (KK,PRL97 、 PhysicaD98)

• 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 the orbit

out of the attractor, while it has a finite measure of basin ( orbits from many initial conditions are attracted to it)

Milnor attractor network chaotic itinerancy

The fraction of basin(i.e. initial values) forMilnor attractors,Plotted as a function ofLogistic map parameter

Note! Fraction is almost1 for some region

Result for N=10,50,100…. a

1

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

DependenceOn theNumber ofElements N

(accumulation over 1.55<a<1.72)

（ｋｋ、 PRE,200 ２ )

• Why?　　 Conjecture 　 by 　 combinatorial explosion of basin boundaries

Simple separation x(i)>x* or x(i)<x*; one can separate 2 ^N attractors by N planes.

In this case the distance between attractor and the basin boundary does not change with N ---- Order of (N-1)!

　 The boundary makes combinatorial explosion

• The number of basin boundary planes has combinatorial explosion, as factorial wins over exponential ( around N ≒ ７） . Then, the basin boundary is ‘crowded’ in the phase space. Thus often attractors touch with basin boundaries

dominance of Milnor attractors here (complete symmetry is unnecessary)These two examples show that a separation of attractors

cannot work well when combinatorial variety wins over exponential increase of the phase space.

This explains psychological magic number 7 + －　２　？

If elements more than 7 are entangled, clear separation behavior is difficult

A solution could be use of modules with each <7 elements

Constructive Biology Project : to understand basic biological features

theme experiment theory question

replicating

system

in vitro

replication with

enzymatic reaction

minority

control

origin of

heredity;

evolvability

cell

system

replicating cell

with internal

reactions

universal

statistics in

reaction dynamics

condition for

recursive

growth

cell differentiation.

development

differentiation

of E Coil

by interaction

emergence of

differentiation

rule from dynamics

irreversibility

robustness

Spontaneous

adaptationArtificial gene network

Adaptive attractor selection by noise

Robust adaptation without signalling

evolutionRelevance of

phenotypic fluctuation and dynamics

Genetic assimilation of phenotype fluct.an

d dynamics

geno-pheno type

relationship

ERATO Project Complex Systems Biology (2004 -2010, wiith Tetsuya Yomo (experimentalist)

How recursive production of a cell is sustained ？　　 each 　 cell 　 complex 　 reaction 　 network

with 　 diversity 　 of 　 chemicals;The 　 number 　 of 　 molecules 　 of 　 each 　

species 　 not 　 so 　 large

●●● ●●●●●●●●●●●

●●●● ●●●●●●●●

●●● ●●●●●●●●●●●

●●●● ●●●●●●●●

●●●● ●●●●●●●●

●●● ●●●●●●●● ●●

？Fluctuations

Universal characteristics as reproducingDynamical system

Ｔｏｙ Cell Model with Catalytic Reaction Network

‘Crude but a cell model that reproduces’

（ resource）

reaction

catalyze

cell

medium

diffusion

ｋ ｋ species of chemicals species of chemicals 、、 XXoo……XX ｋ－１ｋ－１ number ---nnumber ---n ００ 、 、 nn １１ … … nn ｋ－１ｋ－１

some chemicals are some chemicals are penetrablepenetrable

through the membrane with the through the membrane with the diffusion coefficient Ddiffusion coefficient D

resource chemicals are thus transforresource chemicals are thus transformed into impenetrable chemicals, leadimed into impenetrable chemicals, leading to the growth inng to the growth in

Ｎ＝ Σni, when it exceeds N when it exceeds Nmaxmax

the cell divides into twothe cell divides into two

random catalytic reaction networkrandom catalytic reaction network

with the path rate pwith the path rate p

for the reaction for the reaction 　　ＸＸ ii ＋Ｘ＋Ｘ jj －＞Ｘ－＞Ｘ kk

+X+Xjj

modelmodel

C.Furusawa & KK 、 PRL2003

・・・ K >>1 species

dX1/dt X0X4; rate equation;∝Stochastic model here

(Cf. KK&Yomo 94,97)

In continuum description, the following rate eqn., but we mostly use stochastic simulation

Michalel-Menten kinetics, higher order catalysis same results

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

0.00

0.05

0.10

0.15

0.20

0.001 0.01 0.10

0.2

0.4

0.6

0.8

1

grow

th sp

eed

(a.u

.)

simila

rity

H

diffusion coefficient D

growth speedsimilarity

D = Dc

※similarity is defined from inner products of composition vectors between mother and daughter cells

・・

・・

D

Dc

No Growth

(only resources)

Growth

Zipf’s Law is oberved at D = Dc

nnii 　　 (number(number 　　 ofof 　　 moleculesmolecules ））

rankrank

Furusawa &KK,2003,PRL

Average number of each chemical ∝ 　 1/(its rank)

(distribution 　 of 　 x ： ρ （ｘ）∝ｘ　）-2

number rankX1 300 5X2 8000 1X3 5000 2X4 700 4X5 2000 3…….. (for example)

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03

ratio

of m

RNA

rank

Human Liver

Human Kidney

Human Heart

Human Colorectal Cancer

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03 1e+04

ratio

of m

RNA

rank

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03 1e+04

ratio

of m

RNA

rank

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03ra

tio of

mRN

Arank

α= -1

Confirmed by gene expression data

Mouse ES cell

C. elegans

Mouse Fibroblast Cell

Yeast

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03 1e+04

ratio

of m

RNA

rank

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03ra

tio o

f mRN

Arank

1e-05

1e-04

1e-03

1e-02

1e+00 1e+01 1e+02 1e+03 1e+04

ratio

of m

RNA

rank

1e-04

1e-03

1e-02

1e-01

1e+00 1e+01 1e+02 1e+03

ratio

of m

RNA

rank

α= -1

Formation of cascade catalytic reactionFormation of cascade catalytic reaction

Rank of nRank of ni i

Catalyze chemicals of higher rank mainly

1

10

100

1000

10000

100000

1 10 100 1000 10000 100000

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

１１２２

３３４４

１：１： minority moleculesminority molecules

２：２： catalyzed by 1, synthesized by resourcecatalyzed by 1, synthesized by resource

３：３： catalyzed by 2catalyzed by 2

　　　　　　　　　　　　　　：　　　　　　　　　　　　　　：

★★

nini 　　

With conservation law,The exponent -1 is explained

Mean-field type (self-consistent) calc.)

Evolution of Network to satisfy Zipf’s law? Yes Critical D value depends on connectivity in the network; mutation of network + selection approaches Zipf’s law

Furusawa,kk

Later, the connectivity in the network approaches “scale-free” network through evolution.statistical properties; embedded into network structure

Zipf’s law holds, irrespective of network structure, but

Furusawa 、 KK, submitted

initial

evolved

４2

evolutionary embedding of dynamics into network

Dynamics (abundance) first, structure (equation for dynamics) later

probability for a path to chemical with abundances x is selected; q(x) transformation of abundance distrb. to connectivity distrib.

So far average quantity of all components;

Next question: fluctuation by cells: distribution of each Ni by cells

Each color givesdifferent chemical species

LOG SCALE

Furusawa,..KK,Biophysics2005

Log normal distribution !

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

Experiment; protein abundances measured by fluorescence

Log-normalDistributionConfirmed experimentally

Furusawa,Kashiwagi,,Yomo,KK

+flow-cytometry

Also studied in GFP synthesis in liposome

☆☆Heuristic explanation of log-normal distributionHeuristic explanation of log-normal distributionConsider the case that a component X is catalyzed by Consider the case that a component X is catalyzed by other component A, and replicate; the number --other component A, and replicate; the number --NNXX 、、 NNAA

d Nd NXX /dt = N /dt = NXX N NAA

thenthen

d log( Nd log( NXX )/dt = N )/dt = NAA

IfIf 、 、 NNA A fluctuates around its mean < NNAA ＞ , with fluct. η（ｔ） d log( Nd log( NXX )/dt = )/dt = ＜ NNAA ＞ ＋ ＋ η （ｔ）

log( Nlog( NXX ) shows Brownian motion ) shows Brownian motion NNX X log-normal distributionlog-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

Significance of fluctuations phenotypic fluctuation ∝ 　 evolution speed (experime

nt, cell model) extending fluctuation dissipation theorem in physics (cf)recall in standard evolutionary genetics, only the distribution of gene is discussed, by assuming

unique phenotype from a given genotype--in terms of dynamical systems genetic fluctuation (mutation): change in equation of dy

namical systems, parameter,Phenotype: variable according to the dynamical system

s (with fluctuations)Relationship! From evolutionary stability Sato et al., (PNAS, 2003), KK&Furusawa JTB2005

Ｃｅｌｌ　ｄｉｆｆｅｒｅｎｔｉａｔｉｏｎ：　　 Isologous Diversification: (KK,Yomo1997)

internal dynamics and interaction : development phenotype

instability

distinct phenotypes

interaction-induced

Example: chemical reaction network

specialize in the use of some path

1 2( , ,.., )m

km

dxf x x x

dt

Universal characteristics as Coupledreproducing dynamical systems Assuming oscillatory reaction dynamics (gene expression) Study of coupled dynamical systems (globally coupled map) etc., +GROWTH differentiation??

GROWTH

Dilution by the increaseOf cell volume

Catalytic reaction dynamics (or gene expression)

Diffusion to/from media; resource included

synchronous division: no differentiation

Instability of homogeneous statethrough cell-cell interaction

formation of discrete types with different chemical compositions: stabilize each other

recursive production

Assuming oscillatory dynamics as a single cell

Assuming chaotic dynamics

Hierarchical differentiation from ‘stem cell’;by taking initially dynamics with instability (e.g., chaotic)

(higher order catalysis) 　　　　　 Furusawa&KK 98 BMathBiol

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

Irreversible loss of omnipotency characterized by the decrease in chemical diversity and phenotype variation

Generated Rule of Differentiation (example) A1 A A2 A3 S B

(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 A if #(A) then pA ―― global info. is embedded into internal cell statesSTABILITY

(4) Differentiation of cell ensemble (tissue) ――multiple stable distrib. { Ni }

Cf hematopoietic system

　　 Stem cell= 　 chaotic 　 Milnor 　attractor 　 hypothesis?

As long as the stem cell state is stable, it reproduces itself

With the increase of the number, it becomes unstable differentiate to other types

If the number of differentiated cells increases

then the stability of the stem cell is recovered,

and it reproduces itself

Stem cell stays at a marginally stable state

*recall that gene network mutually connected,

with more than 7(?) elements, easily expected

Universality?　 checked a huge number of networks; only some fraction of them show chaotic dynamics & differentiation

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

Such networks are selected Furusawa,KK 2001PRL

Explained:

Robustness in development under large fluctuation in 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 process characterized by the loss of phenotypic variation

• Irreversibility: Loss of multipotency

• diversity in chemicals

• stability of the cellular state

• growth speed

Prediction:gene expression of stem cells 　 diverse, weakregulation of proliferation/differentiation 　 dep. on cell distributionIf #(stem) proliferation 　 decreased 　 externally 　　　 then 　 proliferation 　 increases

• Summary:(1) Magic number 7?: crossover in (globally coup

led) dynamical systems at that number, combinatorial-exponential crossover (general?)(2) Cell model with reaction network; evolution to a ‘critical state’; universal statistical

laws on gene expression: log-normal distrib. evolution --- phenotype fluctuation (3) Coupled cell model: robust hierarchical differentiation from stem cell Collaborators: Ishihara(magic7) Furusawa (the

ory),Yomo and Kashiwagi (expriment), See http://chaos.c.u-tokyo.ac.jp for most papers