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Reaction-Diffusion Systems and Turing patterns

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Reaction-Diffusion Systems and Turing patterns �

What is a reaction-diffusion system?

∂A∂t

= f (A,B) + DA∂ 2A∂x 2

∂B∂t

= g(A,B) + DB∂ 2B∂x 2

∂A∂t

= f (A,B)

∂B∂t

= g(A,B)

∂A∂t

= D∂2A∂x 2

Example: Shankenberg

f (A,B) = k1 − k2A + k3A2B

g(A,B) = k4 − k3A2B

S1⇔ AS2

k4 → BB + 2A k3 → 3A

Reaction Diffusion

A

B g=0

f=0

A

x

t=0

t>0

Units: [D]= L2/t

Alan Turing 1912-1954

•  Instrumental in early computer science (“Turing Machine”)

•  King’s College Cambridge, 1931-34; PhD Princeton, 1938

•  WWII: Bletchley Park, cryptography, deciphering German”enigma” code

•  1952: work on reaction-diffusion equations, pattern formation, and morphogenesis

A.M. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. R. Soc. London B237, pp.37-72, 1952

Turing (1952): �“The chemical basis of morphogenesis”

Activator

inhiBitor

+

+ -

-

Reaction (f,g) Diffusion

DA << DB

∂A∂t

= f (A,B) + DA∂ 2A∂x 2

∂B∂t

= g(A,B) + DB∂ 2B∂x 2

With “appropriate” reaction kinetics, an RD system can form spatial patterns due to the effect of diffusion.

A.M. Turing, (1952) Phil. Trans. R. Soc. London B237, pp.37-72, 1952

Ingredients required for “Turing” pattern formation

∂A∂t

= f (A,B) + DA∂ 2A∂x 2

∂B∂t

= g(A,B) + DB∂ 2B∂x 2

•  There is a stable spatially homogeneous steady state, f(A,B)=0, g(A,B)=0.

•  Range of activator << range of inhibitor

∂a∂t

= c11a + c12b + DA∂ 2a∂x 2

∂b∂t

= c21a + c22b + DB∂ 2b∂x 2

Close to that state: (Linearize eqns using Taylor Series for f, g)

Da

c11<<

Db

c22

“on center/off surround”

lateral inhibition: local excitation, long-range inhibition x

eiqx eσt

q

σ

Example: Shnakenberg RD system

f (A,B) = k1 − k2A + k3A2B

g(A,B) = k4 − k3A2B€

∂A∂t

= f (A,B) + DA∂ 2A∂x 2

∂B∂t

= g(A,B) + DB∂ 2B∂x 2

Starting close to the HSS, the system evolves a spatial pattern that persists with time.

A 0

time

200

Contributors to theories of pattern formation and chemotaxis:

Hans Meinhardt Lee Segel

Pattern formation in biology

Hans Meinhardt

http://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/home.html

Phyllotaxis (Wikipedia)

Reaction-Diffusion systems and animal coat patterns

J.D Murray, A pre-pattern formation mechanism for animal coat markings, J. Theor. Biol. 88: 161-199, 1981