Spatial Dynamics and Turing Patterns

Spatial Dynamics and Self-Organizing Patterns

Why does a zebra have stripes while a leopard has spots? Why do sand dunes form regular rows? Why are brain neurons organized into columns? Alan Turing in 1952 proposed a mathematical mechanism: reaction-diffusion. The interaction of two chemicals with different rates of diffusion gives rise to stable spatial patterns.

Reaction-Diffusion Equations

General form: ∂u/∂t = D_u ∇²u + f(u, v) and ∂v/∂t = D_v ∇²v + g(u, v), where u and v are the concentrations of two morphogens (chemical "signals"), D_u, D_v are diffusion coefficients, f(u,v) and g(u,v) are the kinetics of the reactions.

Turing Mechanism: Activator-Inhibitor

Intuition: the activator (u) stimulates both itself and the production of inhibitor (v). The inhibitor suppresses the activator, but diffuses faster. Result: "spots" of activator surrounded by "seas" of inhibitor.

Conditions for Turing instability: (1) the homogeneous equilibrium is stable without diffusion; (2) D_v >> D_u (the inhibitor diffuses significantly faster); (3) the activator is self-enhancing (∂f/∂u > 0 at equilibrium).

Gierer-Meinhardt Model: ∂a/∂t = ρa²/h − μa + D_a ∇²a (activator a) and ∂h/∂t = ρa² − νh + D_h ∇²h (inhibitor h). With D_h >> D_a: patterns such as spots, stripes.

Pattern scale: characteristic length l ~ √(D_u/f_u) — depends on diffusion coefficient and kinetics. By changing D_u/D_v: you can obtain dots (small D_v/D_u) → stripes (medium) → labyrinths.

Real Biological Patterns

Animal coloration: Turing's model reproduces zebra stripes, leopard spots, shell spirals. Evidence: in the three-striped Danio fish, mutation changes the type of activator cells → transition from stripes to spots (Watanabe & Kondo, 2015).

Finger formation (digitization): proteins BMP (inhibitors) and Wnt/Sox9 (activators) organize the pattern of five fingers via Turing mechanism. Disturbances → polydactyly or syndactyly.

Neuronal columns: in the visual cortex of cats and primates—"ocular dominance columns" (alternating stripes responding to stimuli from the left/right eye). Turing mechanism in neural tissue.

Spatial Epidemiology

Reaction-diffusion SIR equations describe spatial propagation of infection:

∂S/∂t = −βSI/N + D_S ∇²S
∂I/∂t = βSI/N − γI + D_I ∇²I

Result: infection wave. Wave speed: c = 2√(D_I · β(1 − 1/R₀)). With D_I = 1 (diffusion), β=0.5, γ=0.1 (R₀=5): c = 2√(1·0.4) ≈ 1.26 units/day.

Real data: the plague from 1347–1353 spread ≈ 350 km/year (1 km/day)—corresponds to diffusion of rats and fleas along medieval trade routes.

Cellular Automata and Discrete Patterns

Conway’s “Game of Life” (1970): 2D grid of cells. A live cell with 2–3 live neighbors survives. With < 2 or > 3 — dies. A dead cell with 3 live neighbors—comes alive. Simple rules → incredible diversity: stable structures, oscillators, "gliders", self-replicating automata.

Wolfram’s rules (one-dimensional CA): 256 possible rules for 1D automata. Rule 110—Turing complete (proven by Matthew Cook, 2004). Wolfram: "Nature = computing cellular automaton."

Numerical example: desert dunes. Reaction-diffusion model of dune growth: sand is the "activator" (accumulates on irregularities), wind is the "transporter" (diffusion over large distances). Regular rows of dunes are obtained with spacing l ~ √(D_wind/α_deposition) ≈ 50–200 meters—matches observations in the Sahara and Namib deserts.

Exercise: Implement a 2D Turing model (Brusselator: f=A−(B+1)u+u²v, g=Bu−u²v). On a 100×100 grid: (1) Simulate 500 time steps (finite differences). With A=4.5, B=7.5, D_u=1, D_v=8: do you get spots or stripes? (2) Change the D_v/D_u ratio: at what value does the transition from spots to stripes occur? (3) Compare to real coloration: find three animal species whose coloration corresponds to different regimes of the Turing model.