1. Introduction
In 1952, Alan Turing published "The Chemical Basis of Morphogenesis", a paper proposing a mathematical mechanism to explain how a uniform embryo can develop differentiated spatial structures. His model, based on two interacting chemical substances, is the origin of Turing pattern theory.
2. Reaction-Diffusion Equations
The Turing system is described by two coupled partial differential equations for concentrations u (activator) and v (inhibitor):
The key is that D_v >> D_u: the inhibitor diffuses much faster than the activator. This diffusion asymmetry is the necessary condition for Turing instability.
3. Turing Instability
Turing instability is a counterintuitive phenomenon: a uniform state that is stable without diffusion becomes unstable when diffusion is added. The activator self-amplifies locally, but the inhibitor, diffusing faster, suppresses this amplification at a distance. The result is a periodic spatial structure whose wavelength is determined by reaction rates and diffusion coefficients.
4. Spots, Stripes and Spirals
Depending on system parameters, different patterns emerge. Spots appear when the activator is strongly localized. Stripes appear when both substances have comparable diffusion lengths. Spirals appear in excitable systems — a variant of reaction-diffusion systems where cells can be excited, refractory or at rest — as in the Belousov-Zhabotinsky reaction.
5. Biological Applications
- Mammal coat patterns: leopard spots, zebra stripes (Murray models)
- Mollusk shells: pigment patterns generated by reaction-diffusion along the mantle
- Finger development: regular spacing of digits in the limb bud
- Cardiac fibrillation: spiral waves in excitable cardiac muscle
6. Conclusion
Reaction-diffusion systems show how chemistry and physics can spontaneously generate complex biological patterns. Turing spirals are stable dynamic solutions that appear in excitable systems, and their presence in cardiac fibrillation makes them an active medical research subject.