Where the noise gets in

The sweep that asked whether noise can move a pattern between branches produced an answer and, underneath it, an asymmetry that is sharper than the answer.

One run in a hundred and sixty escaped its branch. That run belongs to one of the two kinds of noise, and the split is not close: fifty-six intact runs of field noise, one of which changed branch; thirty-four intact runs of placement noise, none of which did, at any amplitude that left a lattice standing.

Two kinds of noise, the same rule underneath, the same seed, the same rate, ensembles that overlap almost exactly in the quantity a botanist could measure — and a different set of possible outcomes. The difference is where in the rule the disturbance enters.

The rule has two halves and they are not symmetric #

The placement rule does two things in sequence. It computes an energy profile around the circumference — the repulsion a new node would feel at each azimuth, from the nodes already placed — and it takes the minimum of that profile.

Field noise perturbs the profile. The rule still takes a minimum, faithfully; the profile it takes the minimum of is not quite the true one.

Placement noise perturbs the outcome. The rule finds the true minimum and the node is then put down a little to one side of it.

Those sound like the same disturbance seen from two angles, and in the quantity most easily measured they are: both widen the scatter of the divergence angle, continuously and by comparable amounts. The rest of what they do is not the same at all.

The energy profile around the circle is not a smooth bowl with one bottom. It has a local minimum in each gap between the existing nodes near the growing tip — half a dozen of them at a moderate rise, more as the rise falls — and the rule is choosing between those gaps. Which gap the node lands in is which nodes it will be a neighbour of.

Perturbing the profile can move the winner from one gap to another, because the gaps are competing on a quantity that has been disturbed. Perturbing the position after the fact cannot: it displaces the node within the gap it already won. A large enough displacement carries it into the neighbouring gap, but by then the displacement is comparable with the gap width, which is the amplitude at which the lattice is being destroyed anyway.

So the two disturbances differ in kind, not only in magnitude. One can change a discrete choice; the other can only blur a continuous one.

What a branch is, restated so the asymmetry is obvious #

A branch is not a value. It is a neighbour graph.

The Lucas branch at 7/11 means that node i is closest to nodes i ± 7 and i ± 11, and that this has been true, node by node, for the whole stem below the point being counted. The Fibonacci branch at 8/13 is a different assignment of the same nodes to different neighbours. There is no continuum between them — a node’s nearest neighbour is a whole number of places away, and 7 does not shade into 8.

Changing branch therefore means re-laying the neighbour graph over a run of consecutive nodes. One node placed into the wrong gap does not do it: the next node is placed among the nodes that are actually there, and the great majority of them are still where the old branch put them, so the anomaly is surrounded and the pattern continues as before. What is needed is a run of nodes choosing the same new gap consistently enough for the new arrangement to become the majority.

Field noise can start such a run, because it acts on the choice. Placement noise cannot, because after the choice there is nothing left to choose.

The competition between gaps, with numbers #

It helps to see the size of the quantities the two disturbances act on, because the asymmetry is not a matter of degree and the numbers say why.

Take a stem at a rise of 0.02, which is where these runs spend most of their length. The circumference is 1, so the nodes near the growing tip sit roughly a tenth of a turn apart in azimuth; the energy profile around the circle has a local minimum in each of the gaps between them, and there are of the order of ten such minima competing.

The winning gap beats its nearest rival by a margin that is small compared with the depth of either — the rule is choosing between arrangements that are all nearly as good, which is precisely why the golden angle is the answer at all. Field noise with an amplitude of 0.015 of the barrier is therefore not a small perturbation of the competition: it is a perturbation comparable with the gap between the best and second-best options, which is what it takes to change the winner.

Now do the same arithmetic for placement noise. To move a node from its own gap into the neighbouring one, the displacement has to be about half the gap width — a twentieth of a turn, which is eighteen degrees. The amplitude at which the lattice comes apart is a degree and a half. The displacement needed to change which gap a node is in is more than ten times the displacement that destroys the lattice, so there is no amplitude at which placement noise reassigns a node’s neighbours while the pattern still exists.

That is the whole asymmetry, and it is not a subtle balance. It is two quantities separated by an order of magnitude.

Where a real apex’s noise would enter #

The two kinds here are model constructs, and the useful question is which of them a plant’s actual disturbances resemble. The mechanism half of this collection has the vocabulary for it, and the answer is genuinely mixed.

A morphogen concentration that is uneven around the meristem disturbs the choice. The modern account of phyllotaxis is auxin transport: carriers polarise towards the neighbouring cell with the most auxin, a maximum forms, and a primordium is specified there. An unevenness in the auxin field, or in the density of carriers around the ring, changes where the maximum forms. That is field noise: the rule takes the true maximum of a field that is not quite the ideal one.

A primordium displaced after it is specified disturbs the position. Once a founder cell group is committed, the tissue keeps growing, and the visible organ ends up wherever the mechanics of the surrounding cells put it. That is placement noise, and it is the part a ruler on a finished stem is best placed to measure — which is exactly the part that turns out not to matter for what the pattern does next.

And the positions of the existing primordia are themselves uncertain, which is a third thing. The rule computes the energy profile from where its neighbours are. If those are displaced, the profile is perturbed — so a disturbance that is placement noise on one node arrives as field noise for the next one. That is not built here and it is the most realistic of the three; the prediction the account above makes is that it behaves like field noise, because it enters the profile rather than the outcome.

The proportions among these are not known and would be hard to measure. What the sweep establishes is that the proportions are what a plant’s capacity to change branch depends on, and that no measurement of the finished pattern’s scatter recovers them.

A single escape is not a lot to build on #

One run out of fifty-six is thin evidence for a mechanism, and it should be said plainly that the mechanistic story above is doing more work than that one run can support on its own. What supports it is the other side: zero out of thirty-four, at every amplitude up to the one that destroys the pattern, with the same seed and rate and a scatter distribution that overlaps the field runs almost exactly.

The asymmetry is what is measured. The account of why is an interpretation, and it makes two predictions that this collection can be held to later.

A field disturbance with a shorter correlation length should be less able to change a branch, not more. The perturbation used here is smooth around the circle — eight Fourier modes — so it displaces whole gaps relative to each other. A perturbation varying on a scale much finer than the gap spacing would push the minimum around inside a gap without changing which gap wins, and should behave like placement noise however large its amplitude.

A disturbance to the positions of the existing nodes, rather than of the new one, should behave like field noise. If a node’s neighbours are where the rule thinks they are but slightly displaced, the profile it computes is a perturbed profile, and the gaps’ competition is disturbed. That is a third kind of noise, it is arguably the most realistic of the three, and it is not built.

Neither is tested here. They are recorded because the interpretation is only worth having if it is the kind of thing that could be found wrong.

Why this makes “how noisy is an apex” the wrong question #

There is a natural thing to want from a model like this: a single number for the apex’s noisiness, measured on a plant, that could then be fed in. This says there is no such number.

Two apices with identical divergence scatter — the one quantity a measurement could plausibly recover — do not have the same set of futures, because the scatter records the size of the disturbance and not its point of entry. The distinction is upstream of anything visible in the finished pattern: it is the difference between a primordium that appeared in the wrong place and a primordium that appeared in the right place for a field that was wrong.

The realistic sources are not obviously on one side either. A morphogen concentration that is uneven around the meristem disturbs the choice. A primordium displaced by the mechanics of the tissue after it is specified disturbs the position. A cell file that divides unevenly does something between the two. A plant has all three, in unknown proportion, and the proportion is what the branch outcome depends on.

The consequence for this collection is a limit stated rather than a result. The measurement of the previous essay — a lattice tolerates about two degrees of scatter — is real and comparable. The inference from that scatter to what an apex can do is not available, and would not become available with a better measurement of the scatter.

What would distinguish them on a plant #

If the distinction decides what a pattern can do and is invisible in the pattern’s scatter, it is worth asking what would make it visible. Three things would, in principle, and each is harder than the last.

Watch the apex rather than the finished stem. The distinction is about the moment of specification, so an observation that catches primordia as they are specified — rather than the organs they become — separates the two directly. If the specified positions are regular and the mature organs are scattered, the noise is downstream of the choice and cannot change branches. If the specified positions are themselves scattered, it is upstream and can. This is the cleanest test and it needs live imaging of a meristem, which is a real technique and not one this collection can apply.

Look at what the scatter is correlated with. Placement noise displaces each node independently of its neighbours, so successive divergence angles should be uncorrelated: a node pushed clockwise makes the angle to it larger and the angle from it smaller, producing a negative correlation at lag one and nothing beyond. Field noise moves the whole profile, so neighbouring nodes are disturbed together and the correlation should extend further. That is a statistic computable from a single stem’s angle sequence, and it does not need a microscope — only the angles in order, which almost no published count records.

Or look for the thing that only one of them can do. A plant that changes branch part way up — counted at 4/7 low down and 8/13 higher, on one axis, without a transition of the ladder between them — is evidence of a disturbance to the choice, because nothing downstream of the choice can produce it. Such reports exist in the literature as oddities; whether they survive being counted carefully is exactly the sort of thing the survey this phase specifies would settle.

The second of those is worth dwelling on, because it is the one this collection could act on with no new machinery. The angle sequence is already produced by every run of the model, and the correlation structure of it is a prediction of each noise kind. What is missing is the plant’s sequence, and what stands in the way is that a published count reports a pair rather than a series of angles.

What it does settle #

Two things, both of which are worth having.

The branch threshold measured in the previous phase is not undermined by noise. The objection was that fluctuations would knock a pattern off a metastable branch regardless of rate. The kind of noise most easily imagined — imprecise placement — is now measured to be entirely incapable of it, across ninety runs, at every amplitude short of destruction. The kind that can do it manages once in fifty-six, at the edge of destruction. Neither is a rate-independent route onto the Fibonacci branch.

And the model has a place where realism would matter. If a phase later wants to make this rule more like a plant, the productive direction is not a better amplitude. It is the third kind of noise — disturbance to the positions of the existing nodes, which is what tissue mechanics actually supplies — and the prediction to test is that it behaves like field noise rather than like placement noise, because it enters the profile rather than the outcome.

That is the shape of the finding. Where a disturbance enters decides what it can do, the point of entry is not visible in the pattern it leaves, and the most measurable quantity is the one that cannot distinguish them.

A note on how nearly this was missed #

The sweep was built to answer a different question, and the asymmetry was almost thrown away before anybody looked at it.

The two noise kinds were written because an amplitude means whatever its implementation makes it mean, and a result holding for one kind and not the other is a result about code. That is a defensive reason: the second kind was there to be redundant, and the expected outcome was that both would say the same thing, which would have licensed quoting either.

They did say the same thing about the question being asked — neither provides a rate-independent route onto the other branch — and that was written up first, with the two kinds pooled into one sweep of a hundred and sixty runs. Pooled, the escape is one run in a hundred and sixty and the two columns are indistinguishable.

Unpooled, one column has an escape and the other has none, and the difference between them turns out to have an account, an order-of-magnitude argument behind it, and two predictions attached. The redundancy that was there for defence was carrying a result.

The general form of that is worth keeping: a control built to make a finding safe is a measurement in its own right, and pooling it back into the thing it was controlling for throws that measurement away. This collection’s habit is to state a claim and give it a test it could fail; the corollary, learned here, is to look at what the controls said before averaging over them.