Reaction-Diffusion Models of theFormation of Animal Pigmentation Patterns

AASU Mathematics and Computer Science ColloquiumSeptember 17, 2003

Eastern Box Turtle

Ruffed Lemur

Cheetah

CorroboreeTree Frog

Leopard Frog

And then there are the less exotic …

BeltedGalloway

Eastern Box Turtle

Mojo(Felinus Maximus)

Alan Turing

Turing, 1952e Chemical Basis of Morphogenesis

(Phil. Trans. Roy. Soc. London)

Diffusion-driven instability

Under appropriate conditions, a spatially homogeneous equilibrium of a chemical reaction can be stable in the absence of diffusion and unstable in the presence of diffusion.

Such a reaction is capable of exhibiting spatiallyinhomogeneous equilibria, i.e., patterns.

Diffusion-driven instability might explain some of the complex dynamics of nature.

Diffusion

One dimensional model [ u = uo(t,ox) ]

%†¨ = d %≈«€ ¨o™ for 0 < x < 1 %≈¨ = 0 at x = 0 and x = 1 uo(0,ox) = uºo(x) for 0 ? x ? 1

diffusion equation:

boundary condition:

initial data:

Two dimensional model [ u = uo(t,ox,oy) ]

%†¨ = dooãou for (x,oy) l G = (0,1)o!o(0,1) %ñ¨ = 0 for (x,oy) l $G uo(0,ox,oy) = uºo(x,oy) for (x,oy) l Gouo$G

Notation: ãou = %≈«€ ¨o™ + %¥«€ ¨o™ (Laplacian of u) %ñ¨ = Ùuo…on (normal derivative on $G)

Every solution approaches a constant steady state u*as toáo"; namely, u*= avgí uºoo.

In fact, the spatial average is independent of t, and …

all spatial variation is smoothed out over time.

So it seems that, since diffusion is a smoothing process, it should also be a stabilizing process …

?

Reaction

Reaction RatesA + 2oB + 3oC á product(s)

&† [A] = ™⁄oø &† [B] = £⁄oø &† [C]

Law of Mass Actione rate of an elementary reaction is proportional to the product of the reactant concentrations.

A + B → C

&†Å = &†ı = -kooaob, &†Ç = okooaob

2A + B ⇆ C

™⁄oø &†Å = &†ı = -kƒoa €ob + k®oc &†Ç = koƒoa €ob - k®oc

Reaction and Diffusion(Two components)

%†¨ = d¡ ãou + fo(u,v) %†◊ = d™ ãov + go(u,v)

where f and go come from the chemical kinetics.

Diffusion-driven Instability:It is possible for both of the following to be true:

(1) If d¡ =ood™oo = 0o, then (u,v) always approaches a constant equilibrium as toáo".

(2) ere exist diffusion coefficients d¡ `ood™oo such that (u,v) approaches a nonconstant equilibrium as toáo".

An Activator-Inhibitor Reaction(Gierer-Meinhardt)

A → 0B → 0

2A + D → 2A + B2A + C → 3A + C

B + C ⇆ E

Assume that [D] and [E ] remain constant.

&†´ = kƒoboc - k®oe = 0 ì coo„oobo —o ⁄

&†Å = k¡oa €obo —o ⁄ - k™oa &†ı = k£oa €o - k¢ob o.

Equilibrium at (1,1) if k¡ = k™o and k£ = k¢o.

0.5 1 1.5 2 2.5 3

1

2

3

4

0.5 1 1.5 2 2.5 3

1

2

3

4

0.5 1 1.5 2 2.5 3

1

2

3

4

k¡oo=ook™ =o1k£oo=ook¢ >o1

k¡oo=ook™ =o1k£oo=ook¢ =o1

k¡oo=ook™ <oo1k£oo=ook¢ = 1

e dynamics without diffusion

Activator-Inhibitor Reaction-Diffusion System

Variation

%†Å = d¡ooãa + k¡oa €o/b - k™oa

%†ı = d™ooãb + k£oa € - k¢ob (plus boundary conditions and initial data)

%†Å = d¡ooãa + k¡oa €o/((1o+ok∞oa €o)o(k§o+ob)) - k™oa

%†ı = d™ooãb + k£oa € - k¢ob (plus boundary conditions and initial data)

d¡ < d™o is necessary for pattern formation to occur.

(e inhibitor must diffuse faster than the activator.)

1 2 3 4 5 6 7 8 9 10 11 12 13

Simulations created withMathematica

Periodic boundary conditions (think torus)Initial distributions with random variations

1–10 : Activator-inhibitor model 11–13 : Activator-substrate model1–3 : Initial distribution over the entire domain4–7 : Initial distribution confined to a “spinal strip”8–9 : Initial distribution confined to a diagonal band (barber pole)10 : Initial distribution confined to a spot11–13 : Initial distribution confined to a “spinal strip”

Shameless plug for my own work(in a previous life) :

Q1: If diffusion can cause instability, can it also cause finite-time blow-up?

Q2: Does conservation of mass in a reaction-diffusion system imply (pointwise) boundedness?

Under certain conditions satisfied by most “ordinary” systems, NO and YES. (Hollis, Martin, & Pierre; Morgan; Hollis & Morgan)

In general, YES and NO. (Pierre, et al.)

References

1988 Scientific American article by Murray: “How the Leopard Gets its Spots”