An experiment a needle could run

Eight instalments of this collection have ended with the same outstanding item: a survey of real plants that would settle what the machinery says is settleable. It has been specified three times, each time more carefully, and each time it has got harder to justify — hundreds of specimens, a protractor good to a fraction of a degree, a counting radius that has to be stated, and a set of exclusions that threw away most of what a herbarium holds.

The intervention is a different kind of ask, and it is worth setting out properly, because for the first time the specification is short.

What is being asked for #

One apex, accessible enough to work on under a dissecting microscope. A way to remove a single primordium without disturbing its neighbours — the standard tools are a fine needle or a laser. And a record of which organs were made in which order, which is not an extra requirement: it is visible on the apex, since the youngest primordium is the smallest and nearest the tip and the order runs outwards.

The measurement is: did the next primordium appear where the undisturbed sequence says it would, or somewhere else?

That is the whole of it. No angles need to be measured to a fraction of a degree, no counting radius has to be agreed, and no specimen has to be photographed flat.

The effect sizes, against the noise a plant has #

The comparison that decides whether an experiment is worth running is between the predicted effect and the variation the measurement already has.

A real apex’s divergence angles scatter. This site has taken half a degree as the working figure throughout and has measured what a stem looks like at a range of amplitudes either side of it; the amplitude at which a lattice stops being one is between one and two degrees, depending on how the disturbance is delivered.

Against that, the intervention’s smallest predicted signal is 2.6°, at the very last organ of the front, and its largest is 168°. Twelve of the thirteen offsets inside the front predict a displacement of sixteen degrees or more. The ratio of signal to the plant’s own scatter runs from five to three hundred.

For comparison, this site’s best observational readout — recovering the parastichy pair from a list of divergence angles — needs about nine hundred organs on one stem and a protractor good to about four tenths of a degree before its second comb clears the sampling band.

It is worth being careful about what “signal to noise” means for a yes-or-no measurement, because the arithmetic is not the arithmetic of a mean. The question at each offset is whether the next primordium’s position is drawn from the undisturbed distribution or from a displaced one. The undisturbed distribution has a spread of about half a degree; the displaced one is centred between 2.6° and 168° away. At the smallest offset that is five standard deviations, so a single ablation at that offset separates the two hypotheses about as well as twenty-five specimens separate a difference of one standard deviation in the survey. At every other offset inside the front it is thirty standard deviations or more, which is a distinction no amount of within-plant variation is going to blur.

What is not five standard deviations is the plant-to-plant variation in the undisturbed position itself, which nobody has measured — because until now nobody had a reason to want it. That is the one number this experiment needs that this collection cannot supply, and it is measured by the controls: the ablations past the boundary, where the model predicts nothing happens, give the undisturbed spread directly.

How many cuts #

The boundary is what is being measured, so the cuts have to bracket it.

Sixteen ablations — one at each offset from one to sixteen places back — would give the whole step at one rise on one plant, with three offsets past the predicted boundary as the control. Each is a separate apex or a separate plant, since the first ablation changes the pattern that the second would be measured against.

That is the real cost, and it should be said plainly: sixteen apices, not sixteen cuts on one plant. With repeats to establish that the boundary is where it looks, call it three plants per offset and forty-eight apices for one rise.

There is a cheaper design worth naming, and it trades statistics for assumptions. If the boundary is all that is wanted — the count, not the shape of the step — then the offsets far inside the front and far outside it are already known and need not be spent on. Six ablations bracketing the predicted boundary, at n − 2 through n + 3, would locate it if the prediction is roughly right, and would be uninformative if it is badly wrong. The full sixteen is what to do first, on one plant, precisely because it does not assume the answer.

Forty-eight is a greenhouse and a term. The survey this collection has been asking for is hundreds of specimens across a genus, and it needs them to be comparable in ways that herbarium sheets mostly are not.

The four ways it can come out #

It is worth writing these down before the experiment rather than after, because each of them says something and the temptation afterwards is to have expected whichever one happened.

Every offset inside the front moves the next organ and no offset outside it does. The placement rule’s prediction, and the transported-error account has nothing to say: it predicts no displacement at any offset. This would be the first evidence on this site that separates a plant computing its pattern from a plant merely having one, and it would come with a spirals count as a by-product, since the boundary is the larger parastichy number.

No offset moves the next organ. The rule is refuted as a description of that apex. That is a real possibility and it is not a remote one: it is what happens if primordium sites are laid down by something the existing organs do not influence — a pre-patterned field, or a genetically timed sequence — and it is the account this collection’s kinematic control was built to represent.

Every offset moves it, including offsets well behind the front. The neighbourhood the plant is placing against is wider than the rule’s, which is a measurement of the interaction range rather than a refutation. This site has already found that the range is not identifiable from a finished pattern; the ablation would identify it directly, which would be the most valuable outcome of the four.

The next primordium appears in the vacancy for some offsets and merely leans towards it for others. The apex has a freedom the model does not — the primordium’s radial position, or the timing of the next one — and the size of the lean would be a measurement of how much.

Which plant #

The experiment needs a rise, because the prediction is a function of it: the boundary is the larger parastichy number and that number is roughly 0.9/√h. A species whose apex sits on the 5/8 rung predicts a boundary at eight, and one on the 8/13 rung predicts thirteen. Choosing a species is therefore choosing which prediction is being tested, and the choice should be made before the cutting and written down.

The coarser rungs are the better first target, and for a reason that is not about botany. At the 3/5 rung the front is five organs wide, so the whole step is eight ablations rather than sixteen, and the displacement past the boundary is under 1.4° against spacings of 65° — a control that is easy to read. At the fine end the ablations are harder to perform, the organs are more crowded, and the last offset inside the front moves the next organ by only a few degrees.

What the coarse end costs is the strength of the count: a boundary at five is a weaker signature than a boundary at thirteen, because five is a number a lot of things could produce. Doing it at two rungs on two species, and getting five and then eight, is the version of the experiment that is hard to explain any other way.

What a real apex adds that the model does not have #

Three confounds, and the reason to state them here is that two of them are testable within the same experiment.

The wound. Removing a primordium leaves damaged tissue, and damaged tissue is not the same as absent tissue. If a wound response — callus, altered mechanics, a burst of signalling — is what moves the next primordium, then the displacement would appear at every offset rather than only inside the front, because a wound is a wound wherever it is made. So the confound has its own control, and it is the offsets past the boundary. A displacement at fourteen and fifteen places back is evidence of wound response; the absence of one is evidence that what moved the primordium was the vacancy.

Regrowth. The neighbours of the removed organ go on growing, and on an apex the surface expands; a hole may close before the next primordium is placed. That would shrink the displacement without abolishing it, and it predicts a specific signature — the effect falls off with the plastochron, so a species with a slow plastochron would show less than one with a fast one. This is not controlled by anything in the experiment as described and would need a second species.

The plastochron itself. In the model the next organ arrives on schedule whatever has been done to the apex. On a plant the removal might delay it, or bring it forward, and a shifted plastochron changes where the organ would have gone even with no vacancy. The only defence is to record the timing as well as the position, which costs nothing and turns a confound into a measurement.

The control cuts do three jobs, which is why they are worth their apices #

The ablations past the predicted boundary are the most expensive part of the design relative to what the model expects of them, since the model expects nothing. They are also the part that carries the most information, and it is worth counting how much, because the three jobs are usually discussed as though they were one.

They measure the undisturbed spread. The prediction’s signal-to-noise arithmetic uses half a degree for a plant’s own divergence scatter, and that figure comes from this collection’s stems rather than from any apex. A cut at fifteen places back is a plant that has been operated on and whose next primordium the model says goes where it was always going: the spread of those positions across replicates is the denominator every other offset is judged against, measured on the same species, the same apices and the same observer.

They separate the vacancy from the wound. A wound response is indifferent to where the wound is. A vacancy is not. Two cuts outside the front turn that difference into an observation instead of an assumption.

And they locate the boundary’s outer edge. The inner edge is fixed by the last offset that moves the next organ, and a boundary needs both. Without cuts past it, the strongest available statement is that the effect persists to at least thirteen places back, which is compatible with a front of thirteen and with a front of forty.

Three quantities from one set of cuts, none of which the interesting offsets can supply. A design that drops them to save apices has not saved a third of the experiment; it has kept the third that cannot be interpreted alone.

Blind the reading, because the prediction is a shape #

One more thing costs nothing and protects everything. The predicted displacement is not flat inside the front — 168° at four places back, 29° at five, 2.6° at thirteen — so an observer who knows which offset was cut knows roughly what to expect, and the measurement being made is a judgement about where a small primordium sits on a crowded apex.

The defence is the site’s own habit applied to a person rather than to a counter: the machinery here reads patterns without being told what rule made them, and the same discipline is available for free at the microscope. Photograph each apex before the cut and after, index the offsets from the first photograph, and have the positions read from the second by someone who does not know which offset each apex belongs to. The order of the front is recoverable from the picture, so nothing is lost.

That turns the experiment’s four possible outcomes into four that an unblinded reading could not have produced, and it is the difference between an intervention that settles something and one that will be argued about.

What it costs to get wrong #

A specification is worth as much as its failure modes are worth, so here are the three ways this experiment could be run and produce nothing.

Ablating the wrong organ. The prediction is indexed by how many places back, and on an apex that is a judgement about the order of production. The youngest primordium is the smallest and nearest the tip, and the order runs outwards — but near the tip the size difference between consecutive primordia is small, and a mistake of one place is a mistake of one row of the answer. Since the displacement varies wildly between adjacent offsets — 168° at four places back and 29° at five — a mis-indexed ablation does not produce a wrong number so much as an uninterpretable one. The defence is to photograph the apex before the cut and index from the photograph.

Reading the answer too late. The prediction is about the next primordium. Two plastochrons later the pattern has begun to respond to its own response, and what is being measured is a mixture of the first displacement and the cascade that follows it. Both are interesting; only the first is predicted here.

Not recording the offsets past the boundary. These are the control, and the temptation is to skip them because the model says nothing happens there. If nobody cuts at fourteen and fifteen places back, then a displacement at eight is consistent with the rule and with a wound response, and the experiment has cost a term and settled nothing. Of everything in this specification, this is the item most likely to be dropped and the one that carries the most weight.

What it would not settle #

It would not confirm the golden angle. Nothing here is about which angle a plant settles on; the whole experiment happens at whatever angle the plant already has. That is a feature — the site’s oldest result is that the angle is an output rather than a constant, and an intervention that assumed a value would be assuming the conclusion — but it means a successful ablation experiment leaves the divergence question exactly where it was.

It would not tell a placement rule from any other local process. The retraction of the comb is not undone by this. A plant whose primordium sites are decided by mechanical stress, by auxin depletion, or by any other mechanism that responds to what is present, would displace the next primordium too. What the ablation separates is processes that respond to the neighbourhood from processes that do not, which is a real and large division and is not the same as identifying the rule.

And it would not travel far from the species it was done on. One genus, one apex geometry, one plastochron. The survey the site has been asking for is expensive precisely because breadth is what it buys, and the intervention buys depth instead. They are complements, and if only one is ever done it should be this one, because it is the one whose result cannot be predicted from what is already known.