Reaction-Diffusion & Pattern Formation (Turing)
Reaction-Diffusion & Pattern Formation (Turing)
In 1952 Alan Turing showed that a homogeneous steady state, stable to uniform perturbations, can become unstable when two interacting species have different diffusion rates. This diffusion-driven instability spontaneously breaks spatial symmetry, generating periodic patterns — spots, stripes, labyrinths — without an external template, offering a mathematical basis for embryonic pattern formation.
The canonical system has an activator $u$ (slow diffuser, self-activating) and an inhibitor $v$ (fast diffuser, $D_v\gg D_u$). The dispersion relation $\sigma(k^2)$ selects which spatial wavenumbers grow; the pattern wavelength is set by the peak.
Reaction-Diffusion System
$$\partial_t u=f(u,v)+D_u\nabla^2 u,\quad \partial_t v=g(u,v)+D_v\nabla^2 v.$$ Turing instability requires $D_v/D_u\gg1$ and specific Jacobian sign conditions.
Turing Instability Conditions
Jacobian $J=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ at the steady state: $\text{tr}(J)<0$, $\det(J)>0$ (stable without diffusion) but $aD_v+dD_u>2\sqrt{D_uD_v\det(J)}$ (unstable with diffusion).
Example 1
Schnakenberg model: $f=a-u+u^2v$, $g=b-u^2v$. Identify activator and inhibitor.
Solution: $u$ is the activator (self-activates via $u^2v$); $v$ is the inhibitor (consumed by $u^2$).
Example 2
How does increasing domain size affect Turing patterns?
Solution: Larger domains accommodate more wavelengths; spot/stripe count increases while individual feature wavelength stays roughly constant.
Practice
- Why must the inhibitor diffuse faster than the activator for Turing instability?
- Derive the dispersion relation $\sigma(k^2)$ for a linearised two-component RD system.
- Give two biological examples of putative Turing patterns.
- How does the Gray-Scott model differ from the Schnakenberg model?
Show Answer Key
1. If the activator diffused faster, local perturbations would spread both species equally, preventing pattern formation. The inhibitor must diffuse faster to create long-range inhibition with short-range activation (LALI), destabilizing the homogeneous steady state at finite wavelengths.
2. Linearize $\partial_t \mathbf{u} = D\nabla^2\mathbf{u} + \mathbf{f}(\mathbf{u})$ about the HSS. Fourier modes $\propto e^{\sigma t + ikx}$: $\sigma(k^2)$ are eigenvalues of $J - Dk^2I$ where $J$ is the Jacobian of $\mathbf{f}$. Turing instability: $\sigma > 0$ for some $k \neq 0$ while $\sigma < 0$ for $k=0$ (HSS stable without diffusion).
3. (1) Animal coat patterns (leopard spots, zebra stripes) modeled by Murray. (2) Digit formation in vertebrate limbs (BMP/Shh interactions). Also: seashell pigmentation patterns, bacterial colony patterns.
4. Schnakenberg: simple two-species (activator-substrate depletion) model, polynomial kinetics, well-understood analytically. Gray-Scott: feed/kill model with cubic autocatalysis, exhibits richer dynamics including self-replicating spots, labyrinthine patterns, and spatiotemporal chaos.