Pattern Formation and Diffusion Driven Instability

Some Facts: Alan Turing (1952) - first work on chemical mechanisms for morphogenesis Alan Turing (1952) - first work on chemical mechanisms for morphogenesis Reaction kinetics and diffusion alone can cause stable spatially nonhomogeneous chemical gradients. Reaction kinetics and diffusion alone can cause stable spatially nonhomogeneous chemical gradients. Chemotaxis and haptotaxis can also induce similar patterns 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. 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. 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. Diffusion driven instability is common in “activator-inhibitor” systems.

Activator-Inhibitor Reaction Kinetics:

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).

Butterfly Patterns: (Murray 1989)

Animal Coats: Murray (1981)

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

Impossible Patterns:

Spemann Organizer (Agius et al ) 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: , 2000) 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: 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)