Turing's last problem

Everything else on this site is geometry. A rule places points, a pattern comes out, and the essays are careful to say — repeatedly, because it is the easiest thing in the subject to forget — that reproducing a pattern is not explaining it. Douady and Couder got sunflower spirals out of magnetised ferrofluid droplets, which contain no biology at all.

That care has a cost. A collection that only ever says what a model does not show eventually owes an account of what a mechanism would look like, and this field is that account.

The obvious place to start is also the historically correct one. Turing’s paper The Chemical Basis of Morphogenesis appeared in 1952; the work he was doing when he died in 1954 was on phyllotaxis, it was left unfinished, and it was not published in any form until decades afterwards. It is the least-told part of a very well-told career.

The idea, which sounds impossible #

Two substances diffuse. Diffusion smooths things out — that is what it is for, and any intuition about it says that a system of diffusing chemicals started near uniform will stay near uniform.

Turing’s observation is that this is false when there are two of them with different diffusion rates and they react. Under conditions that can be written down, the uniform state is stable to everything except diffusion, and adding diffusion destabilises it into a pattern with a definite spacing.

The usual gloss is “short-range activation, long-range inhibition”: one substance promotes its own production and the other’s, the other suppresses it, and the inhibitor spreads faster. A spot that is slightly ahead builds itself up locally and suppresses its surroundings out to a distance set by the inhibitor’s range. What comes out is a set of peaks separated by roughly that distance.

The version used here is the Gierer–Meinhardt system, which is the one that has a steady state simple enough to write on a line:

$$a_t = D_a a'' + \frac{a^2}{h} - \mu_a a, \qquad h_t = D_h h'' + a^2 - \mu_h h$$

The uniform steady state is $a_0 = \mu_h/\mu_a$ and $h_0 = \mu_h/\mu_a^2$, which is worth checking by substitution because everything downstream is an expansion about it.

Predicting the answer before running it #

The Jacobian at that steady state is

$$\begin{pmatrix} \mu_a & -\mu_a^2 \\ 2\mu_h/\mu_a & -\mu_h \end{pmatrix}$$

Its trace is $\mu_a - \mu_h$ and its determinant $\mu_a\mu_h$. So with no diffusion, the uniform state is stable whenever the inhibitor decays faster than the activator, which it does in every parameter set used here.

Now put the system on a ring of circumference $L$. A perturbation of mode $n$ has wavenumber $k = 2\pi n/L$, and diffusion subtracts $D_a k^2$ and $D_h k^2$ from the diagonal. The growth rate of that mode is the larger eigenvalue of the modified matrix, and it is positive over a band of $n$.

This is the dispersion relation. It is a calculation on the equations, it takes no integration, and it says which pattern should appear.

At a circumference of 1.4 with the parameters used here, modes 4 to 15 grow and mode 8 grows fastest.

Then running it #

The prediction is worth nothing until the full nonlinear system is asked the same question and the answer is extracted rather than assumed.

So: 320 grid points around the ring, the uniform steady state plus 2% disorder, explicit stepping with a time step checked against the diffusion stability limit rather than chosen and hoped for, ninety thousand steps.

Then the peaks in the result are counted — by a routine that is handed an array of numbers and finds local maxima rising a stated fraction above the field’s own range. It is not told what mode was predicted. It is not told what mode was seeded, because none was: the starting disorder is a deterministic pseudo-random sequence with no structure in it at any wavelength.

The count is 8.

What agreement does and does not establish #

Prediction and count agree at $L = 1.4$, and at 0.7 and 1.3. At $L = 1$ the prediction is 5 and the count is 6; at 1.6 the prediction is 9 and the count is 8; at 1.9, 10 against 9; at 2.2, 12 against 11.

The disagreements are the informative part and they should not be smoothed away. The linear calculation describes what happens to an infinitesimal perturbation of a uniform state. The pattern that finally settles is a solution of the nonlinear system, and there is no theorem saying it has to be the linearly fastest mode — only that it will be one of the modes in the unstable band, which at $L = 2.2$ runs from 5 to 23.

So the honest statement is: the mechanism selects a spacing, and the spacing is what the dispersion relation predicts, to within the discreteness of fitting a whole number of peaks around a ring. Anything stronger would be a claim the arithmetic does not support.

That is still a real result. A model that produced a pattern with no relation to its own dispersion relation would be badly wrong, and a model whose peak count was insensitive to the parameters would not be selecting anything.

The selection is a distance, not a number #

The way to see that this is a spacing rather than a count is to vary the ring.

At circumferences of 0.7, 1.0, 1.3, 1.6, 1.9 and 2.2, the peak counts are 4, 6, 7, 8, 9 and 11 — climbing with the ring. The peaks per unit circumference stay between 4.74 and 6.00 across the whole range, a spread of a quarter against a threefold change in the ring.

That is the property the geometry needs from the chemistry, and it is worth naming precisely: the mechanism supplies a characteristic distance between elements, fixed by the diffusion constants and the reaction rates, and independent of how much room there is. How the elements are arranged given that distance is a separate question, and it is the question every other field on this site is about.

Why a ring, and not a disc #

The choice of geometry looks like a simplification and is partly a commitment, so it is worth separating the two.

A plant’s apex is a dome, and the region where new elements appear is an annulus around its edge — the part that is far enough from the centre to be competent and close enough to be young. Modelling that annulus as a one-dimensional ring is the standard reduction and it is a good one: the annulus is much longer than it is wide, so a pattern across its width would have nowhere to go.

What the reduction throws away is the radial direction, and with it every question about how a pattern at one moment relates to the pattern a moment later. That is not a small omission. It is precisely the information a spiral arrangement consists of, and reinstating it is the whole difficulty.

The commitment hidden in the choice is that pattern formation happens in that annulus rather than being inherited from elsewhere. Nothing here tests that; it is an assumption of the model, it is the assumption Turing made, and it is shared by the auxin account that replaced it.

The numerics, and how they could lie #

An explicit scheme integrating a stiff diffusion problem is one of the easier ways to produce a convincing picture of nothing.

If the time step exceeds the diffusion stability limit, the solution does not fail obviously. It develops a high-frequency oscillation that looks, at the resolution of a figure, like a fine regular pattern — precisely the sort of thing this model is supposed to produce. A peak counter run on it returns a large number with high contrast and the figure ships.

So the time step is not chosen. It is computed as a fraction of $\Delta x^2 / 2D_{\max}$ and then asserted to be inside the limit, so that a change to the grid or the diffusion constants that pushes it over stops the build rather than changing the answer.

The field is also required to stay finite at the end of the run. Gierer–Meinhardt has an activator term $a^2/h$ that will happily run away if the inhibitor is driven towards zero, and a run that blows up leaves NaN in every cell, which most downstream arithmetic propagates silently.

Neither check is interesting when it passes. Both exist because the failure mode they catch looks like success.

What has to be true for anything to happen #

The most important check on a pattern-forming model is that it can fail to form a pattern, and here the condition is classical and sharp.

Diffusion-driven instability requires the two substances to diffuse at different rates — specifically, the inhibitor must spread substantially faster than the activator. Set $D_h = D_a$ and the dispersion relation has no positive modes at all: every perturbation decays, and the system returns to uniform however long it is run.

The gate requires exactly that. With the diffusion constants equal, the count of unstable modes must be zero, and it is. An activator–inhibitor model that patterned regardless of its diffusion ratio would not be a Turing model; it would be a model with a pattern built into it.

This matters more than a formality because the ratio is the part of the theory that has been hardest to find in real tissue. The mathematics needs a substantial separation — here about eighty-fold — and for decades no one could point to a pair of molecules in a plant or animal with that relationship. It is the standard objection to reaction–diffusion accounts of biological pattern and it has never entirely gone away.

The counter has to be able to say nothing #

The peak count is extracted from the field, and like every other counter on this site it needs to be able to return an answer nobody wants.

It can. A field that is uniform to within floating-point noise has no local maxima rising above the threshold, and the routine refuses outright rather than reporting spurious peaks: with the range of the field at zero, there is nothing to take a fraction of, and the assertion fires.

This is what makes the equal-diffusion test meaningful. Without a counter that can return zero, “no pattern forms when the diffusion rates are equal” would have to be established from the dispersion relation alone — that is, from the linear theory, which is the thing the simulation is supposed to be checking rather than assuming.

The threshold itself is worth one sentence, since it is the only tunable quantity in the counting. A peak must exceed a quarter of the way from the field’s minimum to its maximum. That is a relative criterion, so it does not depend on the concentration scale, and it is loose enough that no result here changes if it is moved to a fifth or a third.

The parameters, and what they are not #

Four numbers appear: two diffusion constants and two decay rates. They are not fitted to anything.

They were chosen so that the ring of circumference 1 shows about eight peaks, which is a legible number of peaks to draw, and then left alone. No plant measurement enters, no attempt is made to match a real spacing, and none of the essays’ claims depend on the values.

What does depend on them is the shape of every result: the band of unstable modes, the fastest mode, the peaks per unit circumference. Change the diffusion ratio and all of these move together in the way the dispersion relation says they should, which is the only sense in which the model is being tested at all.

That is the appropriate ambition for a model of this kind, and stating it plainly is meant to head off the reading that these figures show a plant doing something. They show a mechanism doing something, at parameters chosen for legibility, in a geometry chosen for simplicity.

What the ring cannot do #

The ring selects a number of peaks and it does so robustly. There is a question it does not touch, and the next essay is about it.

A spiral arrangement is a history: element 1, then element 2 at some angle from it, then element 3 at the same angle again. The divergence angle is a relationship between successive elements, and it presupposes that there are successive elements.

A ring’s peaks do not arrive in a meaningful order. They emerge over a short window, and which of them happens to lead is set by the starting disorder — so the angle between the first two is a different number on every run, and measuring that is what settles the question.

What a stationary ring makes, in botanical terms, is a whorl: some number of elements at one level, arranged around. Which is a real pattern that real plants have. It is not the one this site is mostly about.

The objection that has followed it for seventy years #

It would be misleading to present this as the accepted account of how a plant places its leaves, because it is not, and the reasons are worth stating with the model rather than in a footnote.

The molecules were never found. A Turing mechanism needs two substances with a large separation of diffusion rates and the right reaction structure. Decades of looking has not produced a pair in the shoot apex that fits, and the account that did eventually get molecular support works a different way entirely — transport rather than diffusion.

The model is indifferent to too much. Almost any activator–inhibitor pair with the right diffusion ratio patterns, which is a strength mathematically and a weakness as evidence. A mechanism that would produce spots whatever the details is not strongly confirmed by the existence of spots.

Selecting a spacing is not the hard part. Several quite different mechanisms select spacings, and the thing that distinguishes phyllotaxis from ordinary spotted patterns is the arrangement, which no stationary ring supplies.

None of that makes the mathematics wrong, and none of it is a reason to leave the model out. It is the origin of the whole modern treatment, its dispersion argument is the one every successor uses, and what it cannot do turns out to be as informative as what it can.

What the essays around this one use it for #

Three things elsewhere on the site depend on this model doing what it does, and it is worth naming them so the figures are not read as a detour.

The next essay uses the emergence order to establish that a stationary ring supplies no divergence angle, which is the negative result the whole geometric half of the site implicitly relies on.

The transport account is compared against this one on exactly the quantities established here: whether a spacing is selected, whether the count scales with the domain, and whether either says anything about arrangement.

And the essay on what a mechanism would have to show uses the gap between “selects a spacing” and “produces a pattern” as its central example, because it is the clearest case on the site of a model doing something real and being credited with something else.

Where this leaves Turing #

The unfinished work does more than the ring, and it is worth recording what it contains, because the popular summary — “Turing worked on sunflowers” — undersells it in a specific direction.

He was not only doing chemistry. A substantial part of the manuscript is lattice geometry on a cylinder: parastichy pairs, the parameters that index them, the transitions between them. It is the same object this site’s cylinder field is built on, arrived at independently, and it sits alongside the morphogen work rather than being replaced by it.

That combination is the right instinct and it is the one this collection has ended up with by a much longer road. The chemistry supplies a spacing; the lattice supplies the arrangements that spacing admits; and neither on its own is an account of a plant.