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Pattern Formation

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

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  1. Pattern Formation and Diffusion Driven Instability

  2. 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

  3. 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.

  4. Activator-Inhibitor Reaction Kinetics:

  5. 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).

  6. Butterfly Patterns: (Murray 1989)

  7. Animal Coats: Murray (1981)

  8. Tail Geometry: • a)-c) are simulations • d) adult cheetah • e) adult jaguar • f) pre-natal genet • g) adult leopard

  9. Impossible Patterns:

  10. 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

  11. Primitive Streak Formation (Hensen’s Node)

  12. 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.

  13. Simulations:

  14. 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.

  15. Model simulation: Ermentrout (1986)

  16. Game of Life (cellular automata)

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