Клітинна автоматизація Тьюрінга: Морофоргенез

Diffusion causes pattern, not uniformity

Intuition says diffusion smooths things out — mix two chemicals and they blend into a uniform grey. Turing showed the opposite can happen: if a fast-diffusing inhibitor and a slow-diffusing activator react with each other, small random fluctuations get amplified into a stable, repeating pattern. This is diffusion-driven instability, and it is the leading mathematical model for how a uniform embryo develops spots, stripes and segments. A cellular automaton version discretises Turing's original system of coupled partial differential equations: space becomes a grid of cells, time advances in fixed ticks, and diffusion becomes an average over a cell's neighbours — the qualitative behaviour survives the simplification almost unchanged.

From PDE to grid: discretising the Laplacian

The continuous reaction-diffusion equation for a chemical concentration u is ∂u/∂t = D·∇²u + f(u,v). On a grid, the Laplacian ∇²u — which measures how much a cell's value differs from its surroundings — becomes a simple stencil:

∇²u[i,j] ≈ u[i-1,j] + u[i+1,j] + u[i,j-1] + u[i,j+1] − 4·u[i,j]

A cellular automaton keeps this update strictly local and synchronous —
every cell reads only its immediate neighbours from the previous grid,
and all cells update together, the same double-buffering discipline
used in Conway's Game of Life.

The three-step cellular automaton rule

Each cell holds two numbers: the concentration of an activator chemical A and an inhibitor chemical B. Every tick, every cell runs the same local rule: diffuse — blend each chemical toward the average of its neighbourhood, weighted by its own diffusion rate; react — activator is produced autocatalytically and consumed by the inhibitor, while the inhibitor is produced by the activator and decays on its own; clamp — bound both concentrations to a valid range so the automaton stays numerically stable indefinitely. The activator diffuses slowly and self-amplifies — the signal that says "become a spot here" — while the inhibitor diffuses quickly and suppresses the activator at long range: local activation, lateral inhibition, the general recipe behind every Turing pattern.

The Gray-Scott model

The Gray-Scott model is the most widely used concrete recipe. Two chemicals U (substrate) and V (activator) react as U + 2V → 3V — V autocatalytically converts U into more of itself — with a feed rate F replenishing U and a kill rate k removing V:

ΔU = D_u · ∇²U − U·V² + F·(1 − U)
ΔV = D_v · ∇²V + U·V² − (F + k)·V

Typical diffusion constants: D_u = 0.16, D_v = 0.08 — the
inhibitor diffuses roughly twice as fast as the substrate,
satisfying Turing's core requirement.

Spots, stripes and labyrinths: sweeping F and k

Almost the entire visual vocabulary of Turing patterns comes from sweeping just two parameters. Spots (Turing's "leopard" regime) appear around F ≈ 0.035, k ≈ 0.065 — isolated activator blobs surrounded by inhibitor. Stripes and mazes (zebra regime) appear around F ≈ 0.029, k ≈ 0.057 — connected ridges instead of isolated spots. Self-replicating spots (mitosis-like regime) appear around F ≈ 0.03, k ≈ 0.06, where spots split as they grow past a critical size. Narrow bands between the stable regimes produce chaotic, worm-like patterns that never settle. The boundary between "pattern forms" and "everything decays to a uniform grey" is itself a Turing bifurcation — the diffusion ratio D_v/D_u must exceed a critical threshold for the instability to occur at all.

Розташування цих закономірностей у біології

Смуги на рибці зебро виникають завдяки взаємодіям між пігментними клітинами (жовтими та меланопластичними), які слідують за моделі активатора-інгібітора, дуже схожій на модель Грей-Скотта. Візерунки хутра леопарда та жирафи показують розмір і відстань між точками, що корелює з прогнозованими довжинами тестування Тюрінга для відповідних швидкостей дифузії морфогенів шкіри. Утворення пальців у гілочках кінчиків тіла веретених істот передбачає передачу сигналів Sox9/BMP/Wnt, яка поводиться як реактивно-дифузійна система, періодично розташовуючи пальці. І зростаючий край раковини молюска є ефективною 1D системою Тюрінга, вихід якої записується як 2D закономірність у часі — тому пігментація раковин виглядає як вихід з саморозумного комп'ютера".

Frequently asked questions

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