reaction-diffusion systems and turing patternskeshet/mcb2012/slideslek/rdturing_ppt.pdf · what is...
TRANSCRIPT
What is a reaction-diffusion system?
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∂A∂t
= f (A,B) + DA∂ 2A∂x 2
∂B∂t
= g(A,B) + DB∂ 2B∂x 2
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∂A∂t
= f (A,B)
∂B∂t
= g(A,B)
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∂A∂t
= D∂2A∂x 2
Example: Shankenberg
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f (A,B) = k1 − k2A + k3A2B
g(A,B) = k4 − k3A2B
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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
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∂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
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∂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
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∂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)
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Da
c11<<
Db
c22
“on center/off surround”
lateral inhibition: local excitation, long-range inhibition x
eiqx eσt
q
σ
Example: Shnakenberg RD system
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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
Pattern formation in biology
Hans Meinhardt
http://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/home.html
Phyllotaxis (Wikipedia)