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Pattern Formation. and Diffusion Driven Instability. Some Facts:. Alan Turing (1952) - first work on chemical mechanisms for morphogenesis Reaction kinetics and diffusion alone can cause stable spatially nonhomogeneous chemical gradients.
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Pattern Formation and Diffusion Driven Instability
Some Facts: • 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 • 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-Diffusion System: (2 species) • 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).
Tail Geometry: • a)-c) are simulations • d) adult cheetah • e) adult jaguar • f) pre-natal genet • g) adult leopard
Spemann Organizer (Agius et al - 2000) • 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
Model: (Bull. Math. Biol. 62:501-526, 2000) • Uses chemotaxis to create pattern in streak cell density n(x,t) • Chemoattractant u(x,t) is either activin or cVgl.
Neural Model for Shell Patterns: • Mantle edge secretes material interminttently • Neural stimulation S from surrounding regions • Accumulation of inhibitory substance R • Pigment P secreted only if mantle activity superthreshold.