Nambu and Living World: Symmetry Breaking and Pattern Selection in - - PowerPoint PPT Presentation

Nambu and Living World: Symmetry Breaking and Pattern Selection in Cellular Mosaic Formation

Noriaki OGAWA (小川軌明)

[ RIKEN QHP Lab. / iTHES ] in collaboration with: Tetsuo Hatsuda (初田哲男) [RIKEN Hatsuda QHP Lab. / iTHES] Atsushi Mochizuki (望月敦史) [RIKEN Mochizuki Theo. Bio. Lab. / iTHES] Masashi Tachikawa (立川正志) [RIKEN Mochizuki Theo. Bio. Lab. / iTHES]

2015 November 17

Osaka CTSR - Kavli IPMU - RIKEN iTHES International workshop “Nambu and Science Frontier”

Symmetry Breaking

cf.) Talks by Watanabe, Oda, Noumi

Cone Cellular Mosaic on Fish Retina

[Flamarique, Proc.B, 2012] [OIST Developmental Neurobiology (Masai) Unit, website] [T. Allison, website]

Noriaki OGAWA (RIKEN)

Vertebrate Eye & Retina

Retina R G B UV

[Figure: from Wikipedia]

Cone cells:

Noriaki OGAWA (RIKEN)

[OIST Developmental Neurobiology (Masai) Unit, website]

Mosaic of Cone Cells on Fish Retina

[T. Allison, website]

Noriaki OGAWA (RIKEN) Zebrafish type

[Flamarique, Proc.B, 2012]

Medaka type

p4mg c2mm

Conventional Model and SSB

2D Physical Modeling

• Cellular-level, Square-lattice Model (extended Potts model)

– 1 site is , or “double cone” – Binding energies between neighborhoods (binding proteins on the membranes) – Effective temperature for fluctuation

[Tohya-Mochizuki-Iwasa, 1999]

RG

R G B UV Noriaki OGAWA (RIKEN)

B

UV

Reproduction of Zebrafish-pattern

R G B UV

3 3 2 2

[TMI 1999] [Mochizuki 2002]

Noriaki OGAWA (RIKEN)

• r

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Metropolis Simulation

(stochastic replacements)

(Random Initial State)

Reproduction of Zebrafish-pattern

R G B UV

3 3 2 2

[TMI 1999] [Mochizuki 2002]

Metropolis Simulation

(stochastic replacements)

Noriaki OGAWA (RIKEN)

• r

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Reproduction of Zebrafish-pattern

R G B UV

3 3 2 2

[TMI 1999] [Mochizuki 2002]

Metropolis Simulation

(stochastic replacements)

Noriaki OGAWA (RIKEN)

• r

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

(Spontaneous Symmetry Breaking)

Selected Direction in Nature

Zebrafish “rotated Zebrafish pattern”

?

Marginal Central

Growth

Noriaki OGAWA (RIKEN)

Retina Growing Model and Analysis

Retinal Growth : Schematics

• D. L. Stenkamp et al. (1997)
• J. Comp. Neurol. 382: 272-284
• D. A. Cameron and S. S. Easter (1993)

Visual Neurosci. 10: 375-384

Noriaki OGAWA (RIKEN)

Retinal Growth : Modeling

“front-end”

“Pool” of cone cells

“new-layer”

Edge of retina

Noriaki OGAWA (RIKEN)

Markov-Chain of Layers

• Prob. distribution of 1 layer of cells

– 6 states for 1 cell: – Pi distribution for 6w states

w Noriaki OGAWA (RIKEN)

Layer-internal Inter-layer

Transition Matrix

Markov-chain

Transition Matrix around T=0

Fluctuation term from T>0 Noriaki OGAWA (RIKEN)

Wild-type Rotated

Multi-stability at T=0

?

Grow

Wild-type (1) Wild-type (2) Rotated Noriaki OGAWA (RIKEN)

Growth of “Rotated” at T>0 ?

Typical one-shot simulation:

( w = 16 , T=0.5) Noriaki OGAWA (RIKEN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Growth of “Rotated” at T>0 ?

Typical one-shot simulation:

( w = 16 , T=0.5) Noriaki OGAWA (RIKEN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Growth of “Rotated” at T>0 ?

Typical one-shot simulation:

Rotated zebrafish Wild zebrafish Dynamical transition process ( w = 16 , T=0.5) Noriaki OGAWA (RIKEN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Monte-Carlo results

• ▲

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10 20 30 40 50 n 0.2 0.4 0.6 0.8 1.0

I(n)

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10 20 30 40 50 n 0.2 0.4 0.6 0.8 1.0

I(n)

From rotated pattern From random configuration

T=0.4 T=0.5 T=0.65

Noriaki OGAWA (RIKEN)

“Agreement rates” along with growth

( w = 16)

Eigen-spectrum of Transition Matrix

Noriaki OGAWA (RIKEN)

1 0.2426 0.2426 0.2426 0.2426

0.00047 0.00045 0.00047 0.00045

• 0.948

1

• 1
• 0.225

0.225

• 0.948
• 0.504

0.504 1

• 1

0.910 1 1

• 1
• 1

0.865

• 1
• 1
• 1
• 1

0.8319 0.8076 0.8319 0.8076 0.864i 1

• 0.97 i
• 1

0.97 i

• 0.864i

1 0.97 i

• 1
• 0.97 i
• 0.863

0.024 0.024 0.024 0.024 1

• 0.971

1

• 0.971

0.321 1 1 1 1 0.0764 0.014 0.0764 0.014

…

(w = 4, T=0.5)

Wild-type Rotated

97%

Pattern Selection Mechanism from Toy-Model

Directional Assymmetry of Bindings

Zebrafish “rotated Zebrafish pattern”

?

R G B UV

3 3 2 2

3 3 3 3 3 3 3 2 2 2 2 2 2 2 3 2

Noriaki OGAWA (RIKEN)

2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3

Horizontal (intra-layer) bindings stronger Vertical (inter-layer) bindings stronger

Simplified Toy-Model

2-states model Pattern α: AAAAA... Pattern β: BBBBB...

Noriaki OGAWA (RIKEN)

A A A

VAA VAA UA

A

VAA UA UA UA

B B B

VBB VBB UB

B

VBB UB UB UB

A A

VAA UA UA

B B

VBB VBA UB UB UA VAA + UB VBB +

= W = >

UA + VBA UA + VAB

A A

VAA UA UA

B B

VBB UB UB VBA

Fluctuations

Simplified Toy-Model: fluctuations

2-states model Pattern α: AAAAA... Pattern β: BBBBB...

Noriaki OGAWA (RIKEN)

Fluctuations

Intra-layer Energy governs Stability

Noriaki OGAWA (RIKEN)

Fluctuations Stationary distribution

Eigen-spectrum

Larger layer-internal energy is favored.

Same Mechanism Working

Zebrafish “rotated Zebrafish pattern”

?

R G B UV

3 3 2 2

>

3 3 3 3 3 3 3 2 2 2 2 2 2 2

Noriaki OGAWA (RIKEN)

3 2

Summary

• Physical modeling of growing retina.

– Expressed by Markov-chain Dynamical System. – Seed for symmetry breaking

• Only “Wild-type” pattern can survive.

– “Dynamical selection” between two patterns with

same static energy.

– Agreement with observation in real animals.

Noriaki OGAWA (RIKEN)

Thanks!