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The chemical basis of morphogenesis
Severin Bang
Paper by A. M. Turing about pattern formation
29 November 2016
Concept Model Turing-Analysis Pattern Formation Example Take home message
Motivation
Periodic pattern in nature,despite homogeneous stage inembryo genesis.
Too many possibilities to beencoded in genes.
Question: How? Why?
source: wikimedia.org
Severin Bang The chemical basis of morphogenesis 29 November 2016 1 / 25
Concept Model Turing-Analysis Pattern Formation Example Take home message
Idea
This Paper: Pattern due to diffusion and reaction.
Different morphogenes
Inhibitory and excitatory
Unstable equilibrium
Diffusion, production and destruction
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Illustration
Forest in California.
Random fire outbreaks.
Fast “diffusing”firefighters.
Burned area as pattern.
Inhibitor must be faster.
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Setting
Two morphogenes c1,2(x ,y , t), change in concentration due to:
Reaction: f (c1,c2), g(c1,c2).Diffusion: D1,2∇2c1,2.
⇒ Reaction-diffusion systems:
c1 = f (c1,c2) +D1∇2c1
c2 = g(c1,c2) +D2∇2c2
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Notation
u = γf (u,v) + ∇2u
v = γg(u,v) +d∇2v
γ = L2/D1T scaling factor
L spatial scale
T time scale
d = D2/D1
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Properties
To be a Turing-Mechanism it must hold, that:
(I) There is a (homogeneous and stationary) positive solution (u0,v0)for f (u0,v0) = g(u0,v0) = 0
(II) Stable if no diffusion.
(III) Instable under diffusion.
The clue: diffusion creates patterns!
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (I)
Determine solution for f (u0,v0) = g(u0,v0) = 0
Does a positive solution exists? Fine! next!
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (II)
No diffusion: u = γf (u,v), v = γg(u,v).Consider small perturbation around stable state (u0,v0):
~w =
(u−u0v −v0
)A =
(fu fvgu gv
)∣∣∣∣∣u0,v0
⇒ w = γAw
Linear system: w ∝ eλ t
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (II)
Solution: λ1,2 for which det(γA−λ1) = 0
λ1,2 =1
2γ
[(fu +gv )±
√(fu +gv )2−4(fugv − fvgu)
]For Re(λ1,2) < 0:
fu +gv = Tr(A) < 0 , fugv − fvgu = det(A) > 0
Condition fulfilled? Fine! next!
Severin Bang The chemical basis of morphogenesis 29 November 2016 9 / 25
Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (III)
Bringing back the diffusion:
w = γAw +D∇2w , D =
(1 00 d
)Separation ansatz:
w(r , t) = ∑k
ckeλk tWk(r)
⇒ λ (k)Wk = γAWk +D∇2Wk
ck from start state w(r ,0).With ∇2W (r) +k2W (r) = 0 time independent solution of spatialeigenvalue problem:
⇒ λ (k)WK = γAWk −Dk2Wk
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (III)
Again: λ1,2 from det(λ1− γA+Dk2) = 0:
0 = λ2 + λ
[k2(1 +d)− γ(fu +gv )
]+h(k2)
h(k2) = dk4− γ(dfu +gv )k2 + γ|A|
Instable when Re(λ (k)) > 0, that means that either:
[k2(1 +d)− γ(fu +gv )] < 0h(k2) < 0
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (III)
[k2(1 +d)− γ(fu +gv )] < 0:
k2(1 +d) > 0∀k 6= 0, (fu +gv ) < 0⇒ not possible
h(k2) = dk4− γ(dfu +gv )k2 + γ|A|< 0it’s necessary, that:
dfu +gv > 0
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (III)
dfu +gv > 0, but from first condition: fu +gv < 0.
Implications:
d 6= 1
fu > 0 because activator activates itself ⇒ gv < 0
fu,gv different signs.
d > 1 ⇒ D2 > D1: Inhibitor diffuses faster.
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition (III)
dfu +gv < 0 by itself not sufficient. With:
hmin = γ2
[|A|− (dfu +gv )2
4d
], k2min = γ
(dfu +gv )
2d
it holds, that h(k2) < 0 for:
(dfu +gv )2
4d> |A|
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Pattern Formation
Critical for:
|A|= (dc fu +gv )2
4dc
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Concept Model Turing-Analysis Pattern Formation Example Take home message
k2c = γdc fu +gv
2dc= γ
√|A|dc
For d > dc :roots at k1,k2, instable fork ∈ [k1,k2]
For d = dc :root at kc , instable for kc
k /∈ [k1,k2]:no pattern!
source: J.D. Murray, Mathematical Biology II: Spatial Modelsand Biomedical Applications
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Recap
Two component reaction–diffusion system.
Three conditions restrict the solution set:
(I) Positive solution for g(u0,v0) = f (u0,v0) = 0(II) fu +gv < 0, fugv − fvgu > 0
(III) dfu +gv > 0, (dfu +gv )2 > 4d |A|
Linear approximation.
No instabilities for small and large k .
Waves with k ∈ [k1,k2] create patterns.
Diffusion creates patterns!
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Example
Explicit example for f ,g :
f (u,v) = a−u+u2v
g(u,v) = b−u2v
u = γ(a−u+u2v) +uxx
v = γ(b−u2v) +dvxx
With a,b parameters describing the reaction.
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition I:
0 = a−u+u2v
0 = b−u2v
⇒ u0 = a+b, v0 =b
(a+b)2, b > 0, a+b > 0
At the stationary state:
fu =b−a
a+b, fv = (a+b)2 > 0
gu =−2b
a+b, gv =−(a+b)2 < 0
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Condition II:
fu +gv < 0 ⇒ (a+b)3 > b−a
fugv − fvgu > 0 ⇒ (a+b)2 > 0
Condition III:
dfu +gv > 0⇒ d(b−a) > (a+b)3
(dfu +gv )2−4d(fugv − fvgu) > 0⇒[d(b−a)− (a+b)3
]2> 4d(a+b)4
The conditions on f (u,v),g(u,v) set the Turing-Space.
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Concept Model Turing-Analysis Pattern Formation Example Take home message
source: J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical ApplicationsNote: constant surface was used in model, shown size only to illustrate the behavior.
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Concept Model Turing-Analysis Pattern Formation Example Take home message
source: J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications
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Concept Model Turing-Analysis Pattern Formation Example Take home message
source: J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications
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Concept Model Turing-Analysis Pattern Formation Example Take home message
source: J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications
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Concept Model Turing-Analysis Pattern Formation Example Take home message
Take home message
One mechanism for pattern formation.
Reaction Diffusion system with conditions:
(I) Homogeneous state, which is:(II) Stable if D1 = D2 = 0
(III) Instable if D1,2 6= 0d ≥ dc > 1: short range activation, long range inhibition.
Spotted animals can have striped tails, not vice versa!
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