Pattern Formation

andand

Diffusion Driven InstabilityDiffusion Driven Instability

Some Facts:

Alan Turing (1952) - first work on chemical Alan Turing (1952) - first work on chemical mechanisms for morphogenesismechanisms for morphogenesis

Reaction kinetics and diffusion alone can cause Reaction kinetics and diffusion alone can cause stable spatially nonhomogeneous chemical stable spatially nonhomogeneous chemical gradients.gradients.

Chemotaxis and haptotaxis can also induce similar Chemotaxis and haptotaxis can also induce similar patternspatterns

Diffusion Driven Instability

Some reacting species achieve stable equilibrium Some reacting species achieve stable equilibrium concentrations when well mixed (CSTR) but concentrations when well mixed (CSTR) but spontaneously form spatial patterns if reactants spontaneously form spatial patterns if reactants allowed to diffuse.allowed to diffuse.

Belousov-Zhabotinsky reaction (1951): oxidation Belousov-Zhabotinsky reaction (1951): oxidation of malonic acid in an acid medium by bromate of malonic acid in an acid medium by bromate ions and catalyzed by cerium.ions and catalyzed by cerium.

Diffusion driven instability is common in Diffusion driven instability is common in “activator-inhibitor” systems.“activator-inhibitor” systems.

Activator-Inhibitor Reaction Kinetics:

1 2

+

+

-

-

Activator-Inhibitor Reaction-Diffusion System: (2 species)

Two species is simplest caseTwo species is simplest case Diffusivities must be differentDiffusivities must be different Reaction-kinetics for concentrations u,v embodied Reaction-kinetics for concentrations u,v embodied

in functions f(u,v) and g(u,v). in functions f(u,v) and g(u,v).

Butterfly Patterns: (Murray 1989)

Animal Coats: Murray (1981)

Tail Geometry:

a)-c) are simulationsa)-c) are simulations d) adult cheetahd) adult cheetah e) adult jaguare) adult jaguar f) pre-natal genetf) pre-natal genet g) adult leopardg) adult leopard

Impossible Patterns:

Spemann Organizer (Agius et al - 2000)

Goosecoid gene activated by TGF-ß factors such as Activin, Goosecoid gene activated by TGF-ß factors such as Activin, Xnr1, Xnr2 and Xnr4.Xnr1, Xnr2 and Xnr4.

(eccentric) Nieuwkoop center produces higher amounts of TGF-(eccentric) Nieuwkoop center produces higher amounts of TGF-ß like factors that diffuse to the mesoderm.ß like factors that diffuse to the mesoderm.

Therefore, a gradient generated by a source region in the Therefore, a gradient generated by a source region in the endoderm induces gene activations in the mesoderm endoderm induces gene activations in the mesoderm

Primitive Streak Formation (Hensen’s Node)

Model: (Bull. Math. Biol. 62:501-526, 2000)

Uses chemotaxis to create pattern in streak cell Uses chemotaxis to create pattern in streak cell density n(x,t)density n(x,t)

Chemoattractant u(x,t) is either activin or cVgl.Chemoattractant u(x,t) is either activin or cVgl.

Simulations:

Neural Model for Shell Patterns:

Mantle edge secretes material interminttentlyMantle edge secretes material interminttently Neural stimulation S from surrounding regionsNeural stimulation S from surrounding regions Accumulation of inhibitory substance RAccumulation of inhibitory substance R Pigment P secreted only if mantle activity superthreshold.Pigment P secreted only if mantle activity superthreshold.

Model simulation: Ermentrout (1986)

Game of Life (cellular automata)