A growing organ is part of the rule

Every model in this collection has the same architecture, and it is so consistent that it has never been examined. There is a surface — a disc, a cylinder, a cone, a surface of revolution — and there is a rule that places elements on it. The surface is geometry and the rule is dynamics, and the two are cleanly separated: the surface says where things can be, the rule says where they go.

The separation is convenient and it is not quite true, and this essay is about the term it leaves out.

Where the surface enters #

Look at what the placement rule actually reads. To place primordium i it needs the positions of the primordia already present, and the distances between a candidate position and each of them. That is all. The surface enters only through those distances — through the metric — and once the metric is fixed the rule has no other access to the geometry at all.

The site’s essays on cones make this explicit and make good use of it. A cone is a cylinder whose rise falls; a disc is a cone of flare one half in element number; the local cylindrical metric is within half a per cent of a cone’s own on every hop of a four-hundred-node lattice. All of those are statements that one surface can be substituted for another because the rule only sees distances, and they are among the strongest results here.

But that argument has a consequence nobody drew. If the rule sees the surface only through the distances between elements, then any change in those distances is a change to the rule, whatever caused it. A surface that deforms between one placement and the next does not present the rule with a different boundary condition; it presents it with different neighbours.

What the models here assume, stated plainly #

Each of this site’s placement models makes an assumption about growth that has never been written down as one.

The Douady–Couder rule on a disc has elements drifting outward exponentially at a stated rate. That is growth, and it is uniform: every element moves radially by the same factor per step, so the pattern is carried outward rigidly and no element moves relative to its neighbours except through the drift the rule itself models.

The rising model on a cylinder has the rise declining, which is the apex narrowing or the plastochron shortening depending on which reading is taken. Again uniform: every internode is what it is, and nothing is displaced sideways.

The surface family generalises the shape and keeps the same assumption: growth carries elements along the meridian according to a stated law, and their azimuths are untouched.

In all three, growth is a rigid motion as far as the neighbour relations are concerned. Distances change in a way the rule knows about, because the rule computed them from the current positions, and no element ends up anywhere its neighbours did not expect.

That is a strong assumption about a shoot apical meristem, and it is not obviously true of one. An apex grows by cell division and cell expansion, both of which are spatially uneven; the region immediately around a forming primordium behaves differently from the region between primordia; and the primordia themselves are mechanically distinct from the tissue they sit in.

What the model does when the assumption is dropped #

The jostle kind of noise is the smallest possible way of dropping it. Every primordium is placed exactly where the rule says. Then it moves — by a small random amount, drawn once — and every later primordium computes against the moved position while the recorded position stays where the rule put it.

That is deliberately not a model of tissue mechanics. It has no correlation structure, no dependence on where on the apex a primordium sits, and no coupling to the primordia’s own stiffness. What it has is the essential property: the neighbours are not where they were put.

The finding is that this reaches the rule at a different point from anything the previous phase modelled. Because the displacement happens before the next primordium’s profile is computed, it can change which minimum that primordium goes to — one or two placements in a thousand, at amplitudes where the pattern is comfortable. Placement noise, which is the same size of displacement applied after the rule has chosen, changes none, ever.

What the transport model says about the same question #

The site’s other mechanism model is auxin transport on a ring of cells, and it is worth asking the same question of it, because the answer is different and the difference is instructive.

In that model each cell holds an auxin concentration, carriers on the cell walls move auxin towards whichever neighbour already has more, and a peak forms where the positive feedback runs away. The primordium is the peak. There is no placement rule and no argmin; the pattern is the steady state of a set of coupled ordinary differential equations.

Deform the tissue that model runs on and something quite different happens. Distances between cells enter through the transport coefficients — how much auxin crosses a wall per unit time — and stretching a wall changes that coefficient continuously. The peaks then move, smoothly, and there is no analogue of a basin change because there is no discrete choice being made anywhere. The system does not select among alternatives; it relaxes.

That is a real structural difference between the two mechanism models this site carries, and it is not one either of them was built to expose. A relaxation model is robust to substrate deformation in a way a selection model is not, because deforming the substrate perturbs a continuous solution rather than perturbing a comparison. Neither behaviour is obviously the one a plant has.

It also means the two models would respond differently to the same experiment. A perturbation applied to a growing apex — a physical prod, a local ablation, a treatment that softens a region of tissue — would move a relaxing pattern smoothly and could, occasionally, move a selecting pattern discontinuously. That is a distinguishing prediction, and it is the first one this site has that separates its two mechanism models rather than separating either from the geometry.

Why this is a mechanism question rather than a noise question #

It would be reasonable to file all of this under noise and leave it in the emergence field with the rest. It sits here instead, and the reason is what the result says about the class of mechanism the rule stands for.

This site is careful, everywhere, to say that its models are models of form and not explanations of development. A lattice that matches a sunflower does not show that the plant computes it. The mechanism is auxin transport, and the geometry was worked out long before anybody knew that; Douady and Couder got the same patterns out of magnetised ferrofluid droplets, which contain no biology at all.

The jostle result sharpens what that caveat is about. The placement rule is agnostic about what does the inhibiting — auxin depletion, a mechanical signal, a ferrofluid’s magnetic repulsion — because all it uses is a falling function of distance. But it is not agnostic about the substrate being rigid. Every one of those candidate mechanisms operates on a growing tissue, and a growing tissue supplies a channel into the rule that a dish of ferrofluid does not have.

So the analogy that the site’s own essays lean on has a seam in it. The droplets are a good physical analogue of the interaction and a poor one for the substrate: the dish does not deform, and the apex does. Whatever else is different between a plant and a ferrofluid experiment, this is a difference that enters the rule rather than sitting outside it.

What would have to be true for it to matter #

The honest position is that the effect measured here is small, and it is worth setting out what would make it large.

Amplitude. A jostle of a degree leaves a comfortable lattice. A jostle large enough to matter would have to displace primordia by a substantial fraction of the spacing between them, and whether real growth does that is a measurement nobody here has.

Correlation. The model displaces each primordium independently. Real tissue deformation is correlated over some length — neighbouring cells move together — and a correlated displacement field does something quite different from an independent one: it moves whole groups, which is much closer to what field noise does. The prediction that a jostle “behaves like field noise” would probably be more true, not less, with correlation added.

Timing. The model draws each displacement once, at placement. Real drift accumulates with age, so the neighbours a primordium computes against are displaced by amounts that depend on how long ago they formed — and the nearest neighbours, which dominate the profile, are the youngest and least displaced. That would reduce the effect, and it would introduce a dependence on the plastochron.

Each of those is a parameter, and each would need its own justification. What is here is the smallest version, which is enough to establish that the channel exists and where it enters, and not enough to say what a plant does with it.

A note on what “the surface grows” has meant so far #

It is worth going back over how the collection has used growth, because the word has done three jobs and only one of them is the one at issue here.

Growth as a clock. The rise falls as the plant develops, so a stem’s position on the Fibonacci ladder is a function of how far along it is. This is the sense in which the site’s rate essays use growth, and it is entirely a matter of the schedule: what changes is the parameter, not the geometry between existing elements.

Growth as a carriage. Elements drift outward on a disc, or upward on a stem, carrying the pattern with them. This is what makes a Vogel head a head rather than a ring, and it is the sense in which a cone’s transitions sit at node numbers rather than at lengths. Again rigid: the arrangement is transported, not reshaped.

Growth as deformation. The tissue changes shape between one placement and the next, so the arrangement itself is altered. This is the sense nothing here has ever modelled, and it is the one a jostle is the crudest possible version of.

Separating the three is useful because the first two are extremely well handled and the third was invisible partly because the word covered all of them. A model can be described as “including growth” while having no representation at all of the thing a developmental biologist would mean by it — and the description would be honest, because two of the three senses are there.

What it changes about the site’s other claims #

Not much, and the “not much” is worth being explicit about rather than implied.

The counting results are untouched: they are statements about point sets, and a point set is whatever it is.

The ladder results are untouched, because the ladder is a property of the geometry rather than of the rule, and a jostled pattern climbs it as any other does.

The branch results are untouched at these amplitudes, which is the previous phase’s headline and survives a third kind of noise: nothing here walks a stem from one ladder to another. What the phase adds is that the reason placement noise cannot do it is structural — it never changes a choice — and the reason the other two do not is that the rule repairs the changes they make faster than they accumulate.

The mechanism claims are the ones that move, and they move in the direction of being more carefully bounded. A sentence like the geometry does not depend on what does the inhibiting is true of the interaction and false of the substrate. That is one sentence’s worth of correction to four phases of hedging, and it is the sort of correction that only shows up when somebody puts a term into the model that was previously in the scenery.

What a measurement would have to catch #

Suppose somebody wanted to test whether an apex’s deformation reaches the placement rule. What would they measure?

Not the finished pattern. Everything in the previous two essays says the finished pattern is blind to it: the counts, the divergence, the scatter and the tolerance are the same whichever way the disturbance arrived.

Not the deformation alone either. Measuring how much an apex’s surface strains between plastochrons is a real and difficult experiment, and it would give an amplitude — which, on its own, says nothing, because the same amplitude produces the same pattern through three different channels.

What would settle it is a joint measurement: the deformation of the surface between one primordium’s initiation and the next, and the sequence of divergence angles that resulted. The model’s prediction is specific — a displacement of the neighbours leaves the recorded angles’ correlation intact, and a displacement of the primordium itself destroys it — so a pair of measurements on the same apex would discriminate where either alone cannot.

That is a harder experiment than anything else this collection has asked for, and it is worth writing down anyway. The site’s wrong field is largely a record of what each open question would cost, and this one costs live imaging of a developing apex with primordia tracked individually — which exists, is done, and is not usually reported with an angle sequence attached.

The general form #

A boundary condition that changes between steps is not a boundary condition. It is a term in the dynamics, and the way to tell is to ask what the rule reads: if the changing thing enters through something the rule computes, it is inside the model whatever the diagram says.

The test has a companion, and it is the one that would have caught this earlier: ask what the rule reads, and list it. This rule reads exactly one thing — the positions of the elements already placed — and once that list is written down, the question of whether growth is inside or outside the model answers itself, because growth changes positions. Four phases of essays described the architecture as a rule on a surface without anybody writing the list.

That test is worth carrying because the architecture it catches is extremely common. Almost every simulation of a pattern on a growing domain separates the domain from the pattern, updates the domain, and then runs the pattern rule on the updated domain — which is exactly right if the update is rigid and exactly wrong if it is not. The difference is invisible in the code, because both look like move the points, then place the next one, and it is visible in the results only through a statistic that asks which minimum was chosen.

It is worth ending on what this does not license. Nothing here shows that a real apex’s growth is irregular enough to matter, or that any pattern in any plant has ever been altered by it. What is shown is that the channel exists in the model, that it enters upstream of the choice, and that the site’s habit of treating the surface as furniture was a modelling decision rather than a description. Turning an unexamined assumption into a stated one is the entire content, and on a collection whose premise is state the claim, state the test it is the kind of content that should not need a result attached to be worth publishing.