Biology has two explanations for how a uniform tissue becomes patterned: Turing's reaction-diffusion, which sets its own spacing, and Wolpert's positional information, which reads position off a gradient. Growing the tissue and adding noise tell them apart.
The predator-prey equations of the previous essay make waves but no Turing spots: they lack a self-enhancing activator. This essay starts from a reaction that has one.
A fruit-fly embryo is at first one cell holding thousands of nuclei, and within three hours they carry stripes of gene expression that prefigure the segments. Alan Turing's 1952 explanation for such patterns was that two reacting, diffusing chemicals can make a uniform field unstable at a spacing set by the chemistry. Lewis Wolpert's 1969 explanation was positional information: a signal fades from one end of the tissue, and each cell reads its concentration as a coordinate.
In the Gray-Scott model a substrate U is fed in everywhere at rate \(f\), and V makes more of itself by consuming U, through \(U + 2V \to 3V\), and is removed at rate \(f + k\):
\[ \partial_t u = D_u \nabla^2 u - u v^2 + f(1-u), \qquad \partial_t v = D_v \nabla^2 v + u v^2 - (f+k)\,v. \]
V is the activator, and U inhibits by running out. U diffuses twice as fast as V (\(D_u = 0.21\), \(D_v = 0.105\) in grid units), so a spot's depleted zone reaches beyond it and keeps a second spot away, the activation with wider-range inhibition that Gierer and Meinhardt (1972) set out. Without V nothing makes V, so the empty state \(u = 1, v = 0\) is stable and the grid starts from seeds.
Figure 1. Gray-Scott reaction-diffusion on a 256×256 torus, colored by the concentration of V. Changing f or k reseeds the grid. Click or drag on the canvas to drop V seeds by hand; Perturb adds a few seeds without wiping the pattern.
A pair of V-rich uniform states exists only below the curve \(k = \sqrt{f}/2 - f\), and well below it the seeds tip the grid into one of them. Most patterns sit above it, where only the empty state exists: 28 of the 35 patterned tiles in the map, each a separate 32×32 run from the same four seeds.
Figure 2. The Gray-Scott parameter plane as a mosaic of 280 small simulations; click to send a point to Figure 1. Each tile shows V after 5,000 steps from identical seeds, so a dark tile means the seeds died out at those rates and a flat cream-to-red tile means the whole grid switched to the V-rich uniform state. The dashed curve is the exact bound \(k = \sqrt{f}/2 - f\) on V-rich uniform states. The named presets carry informal names (Pearson labelled his classes with Greek letters) and are points where V survives and patterns in a 64×64 run; the map redraws in whichever colormap Figure 1 is using.
Turing's argument starts from a uniform state that is stable on its own. Diffusion damps a ripple of wavenumber \(q\) at a rate \(D q^2\) in each species, so the ripple grows at
\[ \sigma(q) = \max \operatorname{Re}\,\operatorname{eig}\Big(J - q^2 \begin{pmatrix} D_u & 0 \\ 0 & D_v \end{pmatrix}\Big). \]
If the inhibitor diffuses enough faster than the activator, a band of ripples grows although uniform pushes die away, and the fastest one sets the wavelength. Gray-Scott's candidate is the V-rich state below the curve; at this page's ratio of 2 its band opens only in a strip under 0.001 wide in k, for f above about 0.048.
Figure 3. Left: what the V-rich uniform state does at each (f, k) for the chosen ratio, from the eigenvalues of \(J - q^2 D\) (dashed: the curve above which it does not exist); click to pick a state. Right: the growth rate of each ripple, as a curve from the Jacobian (with the grid's Laplacian, \(4\sin^2(q/2)\) in place of \(q^2\)) and as dots measured from the 256-cell ring itself, the slope of each Fourier mode's log amplitude over steps 50 to 350 of a run started from noise of size \(10^{-200}\) so that it stays linear. Below: the same ring from noise of size \(10^{-6}\), run until the pattern has saturated.
The dots sit on the curve across the unstable band, and the stripes that appear land close to the fastest wavelength: 8.0 cells against 7.45 at ratio 4, and 12.8 against 12.9 at ratio 2 inside the strip. Most patterns in Figures 1 and 2 sit above the curve, where there is no V-rich state to destabilize and spots must be seeded, yet they keep a spacing of their own.
Figure 4. The same chemistry on tori of side 32, 48, 64, 96 and 128 cells, each started from one seed per 400 cells (seed 7) and run for 10,000 steps, drawn at one scale. Below: the power in V's two-dimensional Fourier transform averaged over rings of equal wavenumber, per unit wavenumber, against wavelength. The readout gives each box's mean wavelength, the box side divided by the power-weighted mean ring radius.
A box that holds two or three waves forces a whole number of them, but from 64 cells up each preset keeps its wavelength within 0.6 of a cell, all between 12.7 and 14.5 cells. That spacing fixes no position, and an embryo needs its head at one end.
In Wolpert's model cells at one end make a morphogen that diffuses and decays, so at steady state (\(D\,C'' = k_{\text{deg}}\,C\)) it falls off with decay length \(\lambda = \sqrt{D/k_{\text{deg}}}\). Each cell picks one of three fates from two thresholds: Wolpert's French flag, blue, white and red. In the fruit fly, Bicoid falls off from the anterior pole with a decay length about a fifth of the egg, and hunchback switches on above a threshold. Embryos from mothers given extra copies of bicoid shift their head structures toward the tail (Driever and Nüsslein-Volhard, 1988), as a threshold reading of a stronger gradient predicts.
The first test is size. Unlike the seeded patterns of Figures 1 and 2, the Turing row below starts from a flat state that is genuinely unstable: Figure 3's default point at ratio 4, in one dimension, plus noise of size 0.01. The flags read the same thresholds off a gradient with a 40-cell decay length and off Wolpert's source-sink line, 1 to 0 at any length.
Figure 5. Top: three tissues of the chosen length. Turing peaks of V in purple; flag fates in blue, white and red. Bottom: where features sit as a fraction of tissue length, for every length. Each purple dot is a measured Gray-Scott peak (14 separate 1D runs of 40,000 steps from the flat state plus noise, seeded by length). The dashed lines are the source-sink flag boundaries, and the solid curves are the boundaries of the fixed-decay-length flag, from the exact steady state \(C = \cosh((L-x)/\lambda)/\cosh(L/\lambda)\) with a no-flux far end.
The Turing row keeps its spacing, 6.9 to 7.7 cells at every length, and adds peaks; the source-sink flag keeps proportions; the fixed-decay flag does neither. Body plans need proportion. In fly species with eggs of very different sizes, expression patterns sit at nearly the same fractional positions because the Bicoid decay length scales with the egg (Gregor and colleagues, 2005). Within one species the hunchback boundary scales although Bicoid varies (Houchmandzadeh, Wieschaus and Leibler, 2002), by a means still debated.
A fish grows while its pattern runs. A young angelfish (Pomacanthus) has three stripes, and Kondo and Asai (1995) saw new ones open between the old as it grew, keeping the spacing roughly constant. Figure 6 grows the row while it runs.
Figure 6. The Turing row of Figure 5 (f = 0.070, k = 0.060, ratio 4, no-flux ends) grown exponentially from 40 to 300 cells, starting flat plus seeded noise. Each time the growth rate times the length adds up to one cell, the profile is stretched onto one more cell, and both chemicals are diluted at the growth rate. Length runs down both panels on a log scale, which is time for exponential growth. Right: peaks (stretches with V above 0.3) on the growing row, against the fixed-length rows of Figure 5.
At \(10^{-5}\) per step peaks are added one or two at a time, as peaks split where the gaps have stretched most: with noise seed 1, 10 peaks at 73 cells, 20 at 162, 36 at 297, about 8.3 cells apart against 7.5 in the fixed rows. Crampin, Gaffney and Maini (1999) predicted that exponential growth can double the count at once over a range of rates; this setting shows no doubling at any rate from \(2 \times 10^{-6}\) to \(10^{-3}\) (noise seeds 1 to 3). Faster growth adds peaks irregularly and lags behind, and at \(1.5 \times 10^{-3}\) no pattern forms. New peaks come from old ones splitting, not from the gaps as in the fish.
The second test is noise. A cell that reads an exponential gradient with relative error \(\sigma\) misplaces a boundary by about \(\sigma\lambda\), 4 cells for a 10% error and a 40-cell decay length, and a source stronger by a factor \(a\) moves every boundary by \(\lambda \ln a\). A Turing pattern has no source to vary and its peaks rise steeply, so noisy reading barely moves their edges. Its count varies: the embryos below start from different noise and, since a band of spacings is stable, end with 26 to 29 peaks.
Figure 7. Ten embryos of 200 cells for each mechanism, the source at the left. The first slider scales each embryo's gradient by its own random factor; the second adds independent multiplicative noise to every cell's reading, of the morphogen in the top panel and of V in the bottom one (a cell counts as a peak if its reading of V exceeds 0.3). The readout below the panels is measured from the rows shown.
Positional information puts the pattern in the same place in every embryo, as precisely as the gradient allows. In Drosophila neighbouring nuclei, 1 to 2% of the egg length apart, see Bicoid levels about 10% apart, and Hunchback reads differences that small, near the limit set by the random arrival of Bicoid molecules (Gregor, Tank, Wieschaus and Bialek, 2007). Self-organization gives sharp edges without a calibrated source but fixes neither the number of repeats nor where the first falls.
Real tissues often use both. In the mouse limb a Turing-type network of Bmp, Sox9 and Wnt spaces the digits while Fgf and Hox gradients modulate the wavelength, so the digits fan out from a narrow base (Raspopovic and colleagues, 2014; Green and Sharpe, 2015).
The next essay makes the inhibitor slow and lets only the activator diffuse. The medium then fires when pushed past a threshold and carries waves that curl into spirals.