パターン形成の数理モデルは統合できるか？阪大生命機能 近藤滋

Chemotactic modelNeural net model Mechano-chemical model

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反応拡散モデルで再現できる

しかし、同様のパターン形成能のあるモデルがたくさんある。

Q: パターン形成の本質は？

阪大生命機能 近藤滋

Experiment to show the RD mechanism

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v

Skin pattern change in tropical angelfish

Artificial pattern change in zebrafish

Interactions between two types of pigment cells are transferred by direct contact of the cells

The identified system should be mathematically homologous to the RD system

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x

DD

yDyxGt

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xDyxFt

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Act-Inh mechanism

RD wave pattern

inhibitor

activator

拡散

++

-- - - - -

?

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cell

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cellcell

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cellcell

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Diffusion rate in the growing real tissue???Activation rate of a gene in dividing cells???

In the real biological system, it is very difficult to identify the whole interaction network.

Detecting the kernel shape is easier.

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cell

cell

cell

cellcell

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cell

cell

cell

cell

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cell

Remove some cells

cell

cell

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cellcell

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cell

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cell

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Observe the effect in local and distant place

シグナル伝達の効果中心が、遠くにある場合

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cell

activation

inhibition

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Kernel based Turing model(KT model)

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𝑆𝑡𝑖𝑚 𝑝, 𝑞 = ඵ ሺ𝑢 𝑝 − 𝜉, 𝑞 − 𝜂 ∗ 𝐾𝑒𝑟𝑛𝑒𝑙2𝜉2 + 𝜂2)𝑑𝜉𝑑𝜂

𝑆ሺ𝑝, 𝑞) = ൞

0, 𝑆𝑡𝑖𝑚 𝑝, 𝑞 < 0

𝑆𝑡𝑖𝑚 𝑝, 𝑞 , 0 ≤ 𝑆𝑡𝑖𝑚 𝑝, 𝑞 < 𝑀𝑎𝑥𝑆

𝑀𝑎𝑥𝑆, 𝑀𝑎𝑥𝑆 < 𝑆𝑡𝑖𝑚 𝑝, 𝑞

𝜕𝑢

𝜕𝑡= 𝑆 − 𝑑𝑒𝑔 ∗ 𝑢

Periodic pattern formation by LALI condition(local activation + long inhibition)

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Periodic patternsby inverted LALI condition

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Wave length of the skin pattern sometimes differs drastically

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Nested patterns

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Resulting patterns are not related to the shape of the kernel

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Average value of the kernel

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いろいろなモデルを、KTモデルで統合できるか？

RD modelChemotactic modelNeural net model Mechano-chemical model

KTmodel pattern

複雑なネットワークは、計算できても理解できない例

In particular, the novel class of Type III networks has not been described before

and challenges models of short-range activation and long-range inhibition based

on differential diffusivity that have dominated the field of developmental and

theoretical biology for decades.

３因子のRD系で、任意のパラメータを入力、安定パターンを作るケースを抽出

crest 生命動態の理解と制御のための基盤技術の創出 18

W: inhibitor 拡散係数 0.01V： activator 拡散係数 0.1

LALIの逆なのに安定パターンができる

クイズ： 間違いを探せ！

問題

任意のネットワークをカーネルに置き換える方法は存在するか？

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カーネル