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Pattern Formation
and
Diffusion Driven Instability
Some Facts:
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Alan Turing (1952) - first work on chemical
mechanisms for morphogenesis
Reaction kinetics and diffusion alone can cause
stable spatially nonhomogeneous chemical
gradients.
Chemotaxis and haptotaxis can also induce similar
patterns
Diffusion Driven Instability
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Some reacting species achieve stable equilibrium
concentrations when well mixed (CSTR) but
spontaneously form spatial patterns if reactants
allowed to diffuse.
Belousov-Zhabotinsky reaction (1951): oxidation
of malonic acid in an acid medium by bromate
ions and catalyzed by cerium.
Diffusion driven instability is common in
“activator-inhibitor” systems.
Activator-Inhibitor Reaction Kinetics:
Activator-Inhibitor Reaction-Diffusion
System: (2 species)
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Two species is simplest case
Diffusivities must be different
Reaction-kinetics for concentrations u,v embodied
in functions f(u,v) and g(u,v).
Butterfly Patterns: (Murray 1989)
Animal Coats: Murray (1981)
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Tail Geometry:
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a)-c) are simulations
d) adult cheetah
e) adult jaguar
f) pre-natal genet
g) adult leopard
Impossible Patterns:
Spemann Organizer (Agius et al - 2000)
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Goosecoid gene activated by TGF-ß factors such as
Activin, Xnr1, Xnr2 and Xnr4.
(eccentric) Nieuwkoop center produces higher amounts of
TGF-ß like factors that diffuse to the mesoderm.
Therefore, a gradient generated by a source region in the
endoderm induces gene activations in the mesoderm
Primitive Streak Formation (Hensen’s Node)
Model: (Bull. Math. Biol. 62:501-526, 2000)
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Uses chemotaxis to create pattern in streak cell
density n(x,t)
Chemoattractant u(x,t) is either activin or cVgl.
Simulations:
Neural Model for Shell Patterns:
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Mantle edge secretes material interminttently
Neural stimulation S from surrounding regions
Accumulation of inhibitory substance R
Pigment P secreted only if mantle activity superthreshold.
Model simulation: Ermentrout (1986)
Game of Life (cellular automata)