Download Turing Pattern (1952)

Real-World Modeling on Hierarchical
Dynamics of Living Matter
K. Yoshikawa, Kyoto Univ.
吉川研一（京大・理・物理）
Suitable Mathematical Languages for hierarchical System
Scale
Protein
RNA
Microscopic Master Eq.
DNA
Single Cell
Coarse-graining
Mesoscopic Langevin
Cell-Cell Commun.
Development
Macroscopic PDE
Current Hypothesis on Morphogenesis:
Turing Pattern (1952)
Reaction Diffusion System
Necessary Condition
u:activator,
v：inhibitor
Somitogenesis
Traveling Waves Lead to
Stationary Periodic Structure
Scheme
Spatio temporal Pattern
Tail
Head
Tail
Head
Traveling
waves of Gene expression
(Hes1 / Hes7) move across the
presomitic mesoderm (PSM).
Masamizu, Kageyama, KY, et al., PNAS 2006
Time
Time
Phase difference
among the single cells
Cell-Cell Coupling with Activator
Cell
Cell
Spatial Gradient of Morphogen
Level
of Fgf
Head
Tail
Y. Saga, H. Takeda Nature Reviews 2, 835, (2001).
The system is regarded as the
connection of discrete cells
・Total number of lateral train of the cells: ca.50
R ～ 10 mm / each cell
Total Length of the System(PSM)
・Diffusion “inside” a cell is quite fast.
Model with Activator Coupling
u:activator,
v:inhibitor,
f, g: local kinetics of u, v
t1,t2: timescales of local
(intra cellular) kinetics
“Turing Pattern” never occurs.
Laplace Operator; From continuous into Discrete
Parameterization
Fitzhugh-Nagumo type
• g changes the local behavior.
(Oscillatory to Bistable)

Numerical Settings
t１=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.
Nagahara, Kageyama, KY, etc “Phys. Rev. E, 80, (2009).
Dx =0.01, Dt=0.001
Fgf gradient
Propagation Failure
Tail
Head

g
u, v
u, v, g
space
g has spatial gradient:
g(x)=0.21-0.20x
Emergence of Stationary Pattern on
a Growing Embryo
• Parameter γ is taken
as follows:
t,x
x t
4
3.0 2.5 x 4.6 10 t
space
time
Assumption of linear growth
Why does propagation fail?
(Propagation failure in Bistable medium)
・Continuous medium:
・Discrete medium:
If
i.e.
is small enough,
,
stationary pattern may appear.
t1  1min
ca.1mm/min
ca.5mm
Mathematical Equivalence between
the growth and directional flow
• Through Galilei Transform, the growth of the
embryo is equivalent to the directional flow.
cf; M. Kaern et. al., Faraday Discuss 120, 295, (2002).
Difference from the Turing model
Present scenario
• Discrete medium
• Stationary but vividly
dependent on past
history and boundary.
Turing scenario


Continuous medium
Stationary pattern
independent on past
history and boundary.
de Kepper, et al., Science 2009
Circadian Clock: Regulator of daily rhythm
Ma and K. Yoshikawa. Phys. Rev. E (2009).
“Internal Body Clock”: SCN
(suprachiasmatic nucleus)
E. S. Maywood et al , Current Biology, 2006
Activator-Inhibitor
Cell-Cell coupling
with Activator
Schibler, Current Opinion in Biology, 2005
Current Hypothesis: Switching &Rhythm are
Caused by Chemical Network
Who manages the expression of 20,000 genes?
Hypothesis of Jacob & Monod
Noise in a Single Cell
Number 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 Space
Spatial Fluctuation
Inhomogenity within Intracellular Space
Is there any other scinario to cause the reduction
of freedom in complicated noisy system?
On/Off Switching of
Conformation on a Giant
DNA Demonstrated by
Fluorescence Microscopic
Observation.
T4 DNA
166kbp; L = 57mm
JCP,102,6595(’95); PRL, 76,3029(’96); JPCB, 101, 9396 (‘97).
Free Energy
density
elongated
5mm
PRL, 76,3029(’96); JPCB, 101, 9396 (‘97).
1
n ~ 10
2
Liquid
Solid
n ~ 10
Always
Continuous
First-Order Phase
Transition
Cf: Ar-Cluster, H2O-cluster
N=100 corresponds to DNA with 30kbp,
where Kuhn length is 100nm (300bp)
Compaction of giant DNA completely
inhibits the transcriptional activity
l-ZAP
II
L-ZAP
II
Luckel, Tsumoto, et al., Biophys. Chem.(2003).
RNA Transcripts on Individual DNA in Micro-Sphere
Tsuji, KY, JACS(2010).
+ Mg2+
Network: Fluctuating pairs of key-locks
Environment: Robust owe to large number
Metabolic Network
Expression
Genetic Regulators
Unfolded DNA
Transcription
Intra-cellular
Network
Nuclear
Environment
Loose Packing
Tight Packing
Outer
Environment
Folder of genes
Life is a hybrid system between
network and environment.
Lot of Thanks
• Thank to all of the
colleagues at Kyoto Univ.
and collaborators:
謝謝
Especially,
Dr. Nagahara
Dr. Ma
Prof. Kageyama
Dr. Marcel Hoerning
Welcome to
Kyoto