Reaction-Diffusion

🤖 Turing's 1952 Morphogenesis Paper

Alan Turing — best known for his work on computation — published "The Chemical Basis of Morphogenesis" in 1952, proposing that biological patterns form via diffusion-driven instability. Two substances called the activator and inhibitor react and diffuse at different rates. If the inhibitor diffuses much faster than the activator, a spatially homogeneous mixture becomes unstable and spontaneously forms patterns. Turing's prediction was vindicated in the 1990s when the predicted chemical waves were directly observed in the Belousov-Zhabotinsky reaction and in growing zebrafish fins.

💡 The Gray-Scott Equations

This simulator solves the Gray-Scott model — a two-chemical reaction-diffusion system:

∂u/∂t = D_u∇²u − uv² + f(1−u)
∂v/∂t = D_v∇²v + uv² − (f+k)v

u is the activator, v the inhibitor. The term uv² is the autocatalytic reaction. f (feed rate) replenishes activator; k (kill rate) removes inhibitor. D_u>D_v globally, but specific f-k values determine which patterns emerge.

🌈 Patterns in Nature

• Animal coats: Leopard spots, jaguar rosettes, zebra stripes, cheetah spots — all modelled by reaction-diffusion with different f-k values
• Marine biology: Seashell pigmentation patterns (Conus shells), coral branching morphology
• Developmental biology: Digit spacing in limb development; feather bud positioning in bird embryos
• Neuroscience: Cortical column formation and ocular dominance stripes
• Chemistry: Belousov-Zhabotinsky reaction spirals; CIMA reaction (chlorine dioxide-iodine-malonic acid)

📈 Exploring the f-k Phase Diagram

The f-k parameter space contains dozens of distinct pattern regimes. Key regions:

• Spots (mitosis): f≈0.028, k≈0.053 — spots replicate like cell division
• Stripes: f≈0.035, k≈0.065 — parallel running stripe patterns
• Coral: f≈0.055, k≈0.062 — branching dendritic growth
• Solitons: f≈0.025, k≈0.057 — isolated bullet-like moving spots
• Labyrinthine: f≈0.04, k≈0.058 — maze-like winding channels