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Reaction–diffusion

Pattern formation

Reaction–diffusion: spots and stripes from two chemicals

Nothing in this dish knows what a spot or a stripe is. Two chemicals spread out and react, and the pattern is what is left when the spreading and the reacting come to terms. Drag on the dish to add chemical, pick a regime, or click anywhere on the map of (F, k) to try your own.

The model

The dish holds two chemicals, U and V. Everywhere, fresh U is fed in at a rate F, and V is drained away at a rate F + k. Where they meet, one U and two Vs make three Vs: V is autocatalytic, it makes more of itself. And both diffuse, but U spreads twice as fast as V.

∂u/∂t = Dᵤ∇²u − uv² + F(1 − u)
∂v/∂t = Dᵥ∇²v + uv² − (F + k)v

That is the Gray–Scott model, introduced by Peter Gray and Stephen Scott in the 1980s to describe a chemical reactor. Here it runs on a grid that wraps at the edges, with Dᵤ = 1, Dᵥ = 0.5 and a time step of 1, using a nine-point stencil for the Laplacian ∇².

Why patterns appear

A blob of V grows, because V makes more V. But it feeds on U, and U near the blob runs short. U diffuses in from further away faster than V can spread out, so the blob can keep itself going while starving the ground around it. Activation close by, inhibition further out: that combination is what Alan Turing identified in 1952 as a way for chemistry alone to break the symmetry of a uniform tissue, in a paper titled The Chemical Basis of Morphogenesis.

Which pattern wins depends on F and k. In 1993 John Pearson ran the model across that plane and catalogued a dozen distinct regimes — spots, stripes, labyrinths, spots that divide, waves — many of them in a sliver near one curve. The map in the controls panel draws that curve: to its right the only steady state is an empty dish, so anything you paint there fades away.

Things to try

  • Pick Mitosis, press Clear, and paint a single dot. Watch it grow, pinch and split; then watch the daughters do the same.
  • In Mazes, wipe a gap across the labyrinth. The stripes heal, but rarely back into the same shape.
  • Step F up or down a little at a time from a named regime. Some patterns survive large changes; others switch suddenly from spots to stripes.
  • Click the map just left of the curve, then just right of it. On the right, even a large painted blob dies away.

What it is and is not a model of

Turing-type mechanisms are now well supported for some biological patterns: the spacing of digits in the mouse limb, the stripes of zebrafish skin (driven by interacting pigment cells rather than diffusing molecules), and the regular arrangement of hair follicles and feather buds. The Gray–Scott model itself is not a model of any of those. It is the simplest well-studied system that shows the mechanism at work, which is why it is the one to play with.

References