Morphogenesis

Paint a smear of chemical into a still dish and walk away. It does not fade. It splits, spots, branches, and settles into a pattern that looks designed, from one rule with no designer in it. This is how a leopard decides where its spots go.

The concept

Alan Turing's last great idea: two substances, one that activates and one that inhibits, diffusing at different speeds, can break a uniform state into a pattern all by themselves. The activator feeds itself but also feeds its own inhibitor; the inhibitor spreads faster and fences the activator into islands. Local self-amplification plus long-range suppression is the whole recipe for a spot.

The math

The Gray-Scott model is two coupled reaction-diffusion equations for concentrations $u$ and $v$:

$\frac{\partial u}{\partial t} = D_u \nabla^2 u - u v^2 + F(1 - u)$

$\frac{\partial v}{\partial t} = D_v \nabla^2 v + u v^2 - (F + k) v$

The $\nabla^2$ terms are diffusion; the $u v^2$ term is the reaction where $v$ consumes $u$ to make more of itself. $F$ is the feed rate, $k$ the kill rate, and the ratio $D_u / D_v$ sets the inhibitor's reach. Tiny moves in $(F, k)$ flip the dish between spots, stripes, mazes, and self-replicating blobs.

Why it stays strange

Nothing in the equations says "spot" or "stripe." The pattern is not stored anywhere; it is what the system relaxes into, and the same two numbers that grow a leopard's rosettes grow a maze a hair's width away in parameter space. Form here is not a blueprint being executed. It is an instability being expressed.

Further reading

• Turing (1952), "The Chemical Basis of Morphogenesis."
• Pearson (1993), "Complex Patterns in a Simple System" - the Gray-Scott zoo.

@article{turing1952,
  author  = {Turing, Alan M.},
  title   = {The Chemical Basis of Morphogenesis},
  journal = {Philosophical Transactions of the Royal Society B},
  volume  = {237},
  number  = {641},
  pages   = {37--72},
  year    = {1952}
}
@article{pearson1993,
  author  = {Pearson, John E.},
  title   = {Complex Patterns in a Simple System},
  journal = {Science},
  volume  = {261},
  number  = {5118},
  pages   = {189--192},
  year    = {1993}
}