Coupled Dynamical Systems (Coupled map, chaotic itinerancy,…)  magic number 7+-2

1. 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

2. ？Q: entanglement versus module: Isthere some limit in degrees of freedom so that the system behaves ‘’nicely ‘’? background of this question; • 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?

3. Example 1: Feed-forward threshold Network M elements each layer Random coupling input regarded as time Neural network; Gene network with threshold Dynamics Effect of entanglement? Each element  +1,-1 if β＞１ Attracted to +1,-1 dep. on input Ishihara and KK 200５　PRL

4. Distribution of output x at downflow Well separated

5. Plot of x_1 vs x_2 for all layers

6. Folding and Stretching;  chaos

7. M=6 M=8

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

9. 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

10. The ratio of networks that show chaotic behavior Plotted as a function of M Computed from Lyapunov exp.

11. 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 Let us renumber νas ordering ν１＜ν２＜… Then, ifσ１、σ２、…changes its Sign alternately  folding Process like logistic map if satisfied at all i=1,2,,,M-1、 then folding at every region σ２＜０ σ１＞０ X(out) σ３＞０ x(inp) ν１　ν２　ν３

12. 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

13. Another example: GCM ε 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)

14. 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

15. The fraction of basin (i.e. initial values) for Milnor attractors, Plotted as a function of Logistic map parameter Note! Fraction is almost 1 for some region Result for N=10,50,100 …. 1 a

16. The Milnor attractors become dominant around N ＞~（５－８） Dependence On the Number of Elements N (accumulation over 1.55<a<1.72) （ｋｋ、PRE,200２)

17. Why? Conjecturebycombinatorial 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

18. 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

19. Constructive Biology Project : to understand basic biological features ERATO Project Complex Systems Biology (2004 -2010, wiith Tetsuya Yomo (experimentalist)

20. How recursive production of a cell is sustained？ eachcellcomplexreactionnetwork withdiversityofchemicals; Thenumberofmoleculesofeachspeciesnotsolarge ●●● ●●●● ●●●● ●●● ●●● ●●●● ●●●● ●●● ●●●● ●●●● ●●●● ？ ●●●● ●●●● ●●●● ●●● ●●●● ●●●● ●● ●●●● ●●●● ●●●● Universal characteristics as reproducing Dynamical system Fluctuations

21. Ｔｏｙ Cell Model with Catalytic Reaction Network ‘Crude but a cell model that reproduces’ C.Furusawa & KK、PRL2003 model (Cf. KK&Yomo 94,97) ｋ species of chemicals 、Xo…Xｋ－１number ---n０ 、n１… nｋ－１ （resource） cell random catalytic reaction network with the path rate p for the reaction Ｘi＋Ｘj－＞Ｘk+Xj reaction some chemicals are penetrable through the membrane with the diffusion coefficient D catalyze diffusion resource chemicals are thus transformed into impenetrable chemicals, leading to the growth in Ｎ＝Σni, when it exceeds Nmax the cell divides into two ・・・ K >>1 species medium dX1/dt ∝ X0X4; rate equation; Stochastic model here

22. In continuum description, the following rate eqn., but we mostly use stochastic simulation Michalel-Menten kinetics, higher order catalysis  same results

23. ☆Growth speed and fidelity in replication are maximum at Dc Dc D Growth No Growth (only resources) ※similarity is defined from inner products of composition vectors between mother and daughter cells D = Dc ・ ・

24. Furusawa &KK,2003,PRL Zipf’s Law is oberved at D = Dc ni(numberofmolecules） 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 (distributionofx：ρ（ｘ）∝ｘ　）

25. Confirmed by gene expression data Human Liver Mouse ES cell Human Heart Mouse Fibroblast Cell α= -1 α= -1 Human Kidney Human Colorectal Cancer C. elegans Yeast

26. Catalyze chemicals of higher rank mainly Rank of ni 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 ★ ４ ３ ni ２ １ １：minority molecules ２：catalyzed by 1, synthesized by resource ３：catalyzed by 2 　　　　　　　　　　　　　　： With conservation law, The exponent -1 is explained Mean-field type (self-consistent) calc.)

27. 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

28. 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 embedding of dynamics into network ４ 2 evolved probability for a path to chemical with abundances x is selected; q(x)  transformation of abundance distrb. to connectivity distrib. initial Furusawa、KK, submitted

29. Furusawa,.. KK, Biophysics2005 So far average quantity of all components; Next question: fluctuation by cells: distribution of each Ni by cells e.g. cell1 X1 10000 cell2 8000 cell3 15000 cell4 20000 ….. histograrm Log normal distribution ! Each color gives different chemical species LOG SCALE

30. Experiment; protein abundances measured by fluorescence +flow-cytometry Log-normal Distribution Confirmed experimentally Furusawa,Kashiwagi,,Yomo,KK Also studied in GFP synthesis in liposome

31. ☆Heuristic explanation of log-normal distribution Consider the case that a component X is catalyzed by other component A, and replicate; the number --NX、NA d NX /dt = NX NA then d 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 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

32. Significance of fluctuations phenotypic fluctuation ∝evolution speed (experiment, 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 dynamical systems, parameter, Phenotype: variable according to the dynamical systems (with fluctuations) Relationship! From evolutionary stability Sato et al., (PNAS, 2003), KK&Furusawa JTB2005

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

34. Dilution by the increase Of cell volume Catalytic reaction dynamics (or gene expression) Diffusion to/from media; resource included

35. synchronous division: no differentiation Instability of homogeneous statethrough cell-cell interaction Assuming oscillatory dynamics as a single cell recursive production formation of discrete types with different chemical compositions: stabilize each other

36. Assuming chaotic dynamics

37. Hierarchical differentiation from ‘stem cell’;by taking initially dynamics with instability (e.g., chaotic)(higher order catalysis) Furusawa&KK 98 BMathBiol

38. 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

39. 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 states • STABILITY • (4) Differentiation of cell ensemble (tissue) • ――multiple stable distrib. { Ni } Cf hematopoietic system

40. Stem cell=chaoticMilnor attractorhypothesis? 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

41. 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

42. 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

43. Irreversibility: Loss of multipotency • diversity in chemicals • stability of the cellular state • growth speed Prediction: gene expression of stem cellsdiverse, weak regulation of proliferation/differentiation dep. on cell distribution If #(stem) proliferationdecreasedexternally thenproliferationincreases

44. Summary: • Magic number 7?: crossover in (globally coupled) 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 (theory),Yomo and Kashiwagi (expriment), See http://chaos.c.u-tokyo.ac.jp for most papers