Cell-Cell Commun.

Macroscopic PDE

Mesoscopic Langevin

Microscopic Master Eq.

Coarse-graining

Suitable Mathematical Languages for hierarchical System

Single Cell

Development

Scale

DNA

ProteinRNA

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

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

HeadTail

Time Time

Tail Head

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

Head Tail

Cell 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 mm / 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,

t1,t2: timescales of local (intra cellular) kinetics

v:inhibitor,

“Turing Pattern” never occurs.

Laplace Operator; From continuous into Discrete

• g changes the local behavior. (Oscillatory to Bistable)

Numerical Settings

t １ =0.58, t2=3.2×10, D=1.19×10-5, α =0.4, b=0.33, k=1.0, a=1.0, 　 b=1.0.

Fitzhugh-Nagumo type

Dx =0.01, Dt=0.001

Parameterization

Nagahara, Kageyama, KY, etc “Phys. Rev. E, 80, (2009).

g has spatial gradient: g(x)=0.21-0.20x

space

u,

v gu, v, g

Fgf gradient

Head

Tail

Propagation Failure

• Parameter γ is taken as follows:

t , x x t

3.0 2.5 x 4.6 10 4 tspace

tim

e

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?

min11

ca.5mm

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 scenario Turing 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 CellNumber Fluctuation Typical Number of Regulatory factors may be on

the order of 101.Temporal Fluctuation Breakdown of Detailed Balance, Occurrence of

Flow and Rhythm in State SpaceSpatial 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 = 57mm

elongateddensity

Free Energy

5mm

PRL, 76,3029(’96); JPCB, 101, 9396 (‘97).

110~n

Always Continuous

210~n

First-Order Phase Transition

Liquid 　　　 Solid

Cf: Ar-Cluster, H2O-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

l-ZAP II L-ZAP II

RNA Transcripts on Individual DNA in Micro-SphereTsuji, KY, JACS(2010).

+ Mg2+

Outer Environment

Metabolic Network

Loose PackingIntra-cellular Network

Genetic Regulators

Expression

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