Cell-Cell Commun. Macroscopic PDE Mesoscopic Langevin Microscopic Master Eq. Coarse-graining Suitable Mathematical Languages for hierarchical System Single Cell Development Scale DNA Protein RNA Real-World Modeling on Hierarchical Dynamics of Living Matter K. Yoshikawa, Kyoto Univ. 吉川研一（京大・理・物 理）

Current Hypothesis on Morphogenesis: Turing Pattern (1952) Reaction Diffusion System u:activator, v ： inhibitor Necessary Condition

Y. Saga, H. Takeda Nature Reviews 2, 835, (2001). Somitogenesis

Traveling Waves Lead to Stationary Periodic Structure Masamizu, Kageyama, KY, et al., PNAS 2006 Spatio temporal Pattern Phase difference among the single cells Traveling waves of Gene expression (Hes1 / Hes7) move across the presomitic mesoderm (PSM). Scheme Head Tail Time TailHead

Y. Saga, H. Takeda Nature Reviews 2, 835, (2001). HeadTail Cell Level of Fgf Cell-Cell Coupling with Activator Spatial Gradient of Morphogen

・ Total number of lateral train of the cells: ca.50 R ～ 10 m / each cell ・ Diffusion “inside” a cell is quite fast. Total Length of the System(PSM) The system is regarded as the connection of discrete cells

f, g: local kinetics of u, v Model with Activator Coupling u:activator,      : timescales of local (intra cellular) kinetics v:inhibitor, “ Turing Pattern ” never occurs. Laplace Operator; From continuous into Discrete

  changes the local behavior. (Oscillatory to Bistable) Numerical Settings  １ =0.58,   ×10, D=1.19×10 -5, α =0.4, =0.33, =1.0, a=1.0, b=1.0. Fitzhugh-Nagumo type  x =0.01,  t=0.001 Parameterization Nagahara, Kageyama, KY, etc “Phys. Rev. E, 80, (2009).

 has spatial gradient: (x)= x space u, v  u, v,  Fgf gradient Head Tail Propagation Failure

Parameter γ is taken as follows: space time Emergence of Stationary Pattern on a Growing Embryo Assumption of linear growth

・ Discrete medium: ・ Continuous medium: stationary pattern may appear. (Propagation failure in Bistable medium) is small enough, i.e. If, Why does propagation fail? ca.5m ca.1mm/min

Through Galilei Transform, the growth of the embryo is equivalent to the directional flow. cf; M. Kaern et. al., Faraday Discuss 120, 295, (2002). Mathematical Equivalence between the growth and directional flow

Discrete medium Stationary but vividly dependent on past history and boundary. Continuous medium Stationary pattern independent on past history and boundary. Present scenarioTuring scenario de Kepper, et al., Science 2009 Difference from the Turing model

Schibler, Current Opinion in Biology, 2005 E. S. Maywood et al, Current Biology, 2006 Circadian Clock: Regulator of daily rhythm “Internal Body Clock”: SCN (suprachiasmatic nucleus) Activator-Inhibitor Cell-Cell coupling with Activator Ma and K. Yoshikawa. Phys. Rev. E (2009).

Current Hypothesis: Switching &Rhythm are Caused by Chemical Network Hypothesis of Jacob & Monod Who manages the expression of 20,000 genes?

Noise in a Single Cell Number Fluctuation Typical Number of Regulatory factors may be on the order of Temporal Fluctuation Breakdown of Detailed Balance, Occurrence of Flow and Rhythm in State Space Spatial Fluctuation Inhomogenity within Intracellular Space Is there any other scinario to cause the reduction of freedom in complicated noisy system?

JCP,102,6595(’95); PRL, 76,3029(’96); JPCB, 101, 9396 (‘97). On/Off Switching of Conformation on a Giant DNA Demonstrated by Fluorescence Microscopic Observation. T4 DNA 166kbp; L = 57m

elongated density Free Energy 5m5m PRL, 76,3029(’96); JPCB, 101, 9396 (‘97).

Always Continuous First-Order Phase Transition Liquid Solid Cf:Ar-Cluster, H 2 O-cluster N=100 corresponds to DNA with 30kbp, where Kuhn length is 100nm (300bp)

Luckel, Tsumoto, et al., Biophys. Chem.(2003). Compaction of giant DNA completely inhibits the transcriptional activity -ZAP II L-ZAP II

RNA Transcripts on Individual DNA in Micro-Sphere Tsuji, KY, JACS(2010). + Mg 2+

Outer Environment Metabolic Network Loose Packing Intra-cellular Network Genetic Regulators E xpression Transcription Nuclear Environment Folder of genes Tight Packing Unfolded DNA Network: Fluctuating pairs of key-locks Environment: Robust owe to large number Life is a hybrid system between network and environment.

Lot of Thanks Thank to all of the colleagues at Kyoto Univ. and collaborators: Especially, Dr. Nagahara Dr. Ma Prof. Kageyama Dr. Marcel Hoerning Welcome to Kyoto 謝謝