The chemical basis of morphogenesis - uni …jeti.uni-freiburg.de/.../Presentation_Severin_Final.pdfSeverin Bang The chemical basis of morphogenesis 29 November 2016 21 / 25 ConceptModelTuring-AnalysisPattern

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


Idea

This Paper: Pattern due to diffusion and reaction.

Different morphogenes

Inhibitory and excitatory

Unstable equilibrium

Diffusion, production and destruction



Illustration

Forest in California.

Random fire outbreaks.

Fast “diffusing”firefighters.

Burned area as pattern.

Inhibitor must be faster.



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



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



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!



Condition (I)

Determine solution for f (u0,v0) = g(u0,v0) = 0

Does a positive solution exists? Fine! next!



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



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!



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



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



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



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.



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|



Pattern Formation

Critical for:

|A|= (dc fu +gv )2

4dc



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



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!



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.



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



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.



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.



source: J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications









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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The chemical basis of morphogenesis - uni …jeti.uni-freiburg.de/.../Presentation_Severin_Final.pdfSeverin Bang The chemical basis of morphogenesis 29 November 2016 21 / 25 ConceptModelTuring-AnalysisPattern