A pump that works uphill

The Turing account of phyllotaxis has a long-standing problem, and it is not mathematical. The mathematics works. What has never turned up is the pair of molecules — an activator and an inhibitor with a large separation of diffusion rates and the right reaction structure — that the mathematics requires.

The account that did eventually acquire molecular support works differently, and the difference is worth stating up front because it inverts an intuition that most people carry without examining it.

There is one signalling molecule rather than two: auxin, the plant hormone that has been near the centre of developmental botany for a century. And it is not left to diffuse. Cells carry it across their membranes with a family of transporter proteins, the PINs, which are placed asymmetrically — concentrated on one face of the cell rather than distributed evenly — so that transport has a direction.

The direction is the surprise. Each cell polarises towards the neighbour that already has more auxin.

Why that is strange #

Every transport process one meets first moves things down a gradient. Diffusion does, heat conduction does, and the reason is thermodynamic: a gradient is a departure from equilibrium and unforced transport relaxes it.

Pumping up a gradient is not forbidden — cells spend energy on it constantly, and it is how every concentration difference across a membrane is maintained — but it produces a system with a different character. Instead of smoothing a disturbance, it amplifies one. A cell that is slightly ahead is pumped into by both of its neighbours, which makes it further ahead, which strengthens their polarisation towards it.

That is positive feedback with no inhibitor anywhere in it, and the immediate question is why the whole ring does not empty into a single cell.

The answer is that the feedback is competitive rather than absolute. A cell’s transporters are a finite resource split between its two faces in proportion to how attractive each neighbour looks, so a cell adjacent to two rising peaks divides its output between them rather than doubling it. Add production and decay, and the result is a set of peaks with a spacing, exactly as in the reaction–diffusion case.

The model, written out #

Auxin $a_i$ in cell $i$ on a ring changes by production, decay, diffusion, and polar transport:

$$\frac{da_i}{dt} = \rho - \mu a_i + D(a_{i-1} - 2a_i + a_{i+1}) + T\big(\text{in} - \text{out}\big)$$

where the transport terms use the polarisation

$$P_{i \to j} = \frac{a_j^{\,k}}{a_{i-1}^{\,k} + a_{i+1}^{\,k}}$$

— the share of cell $i$’s carriers facing neighbour $j$, larger for the richer neighbour, and summing to one across the two faces.

The exponent $k$ is the one knob that matters and it is deliberately not fitted. It controls how sharply a cell commits to its richer neighbour: at $k = 0$ the cell splits evenly and the model reduces to diffusion; as $k$ grows the commitment becomes all-or-nothing. Everything claimed here is about the shape of the outcome — that a spacing is selected — and not about a number matched to a plant.

Where the competition comes from #

The single line that keeps this model from collapsing deserves more than a clause, because it is the whole of the mechanism’s stability and it is easy to write down wrongly.

Each cell has a fixed amount of transporter. It divides it between its two faces according to how much auxin each neighbour has, and the two shares sum to one. So a cell’s total export is set by its own auxin content and by the transport rate, and the neighbours compete for a share of it rather than each drawing independently.

Get this wrong — let each face’s export depend on that neighbour’s concentration without normalising — and the model has no brake. Two adjacent rising peaks each pull at full strength, the cell between them empties, and then the peaks pull on each other’s neighbours until everything drains into a single maximum. The pattern has one peak, at whichever position the noise favoured, and the figure looks like a bug rather than like a result.

The normalisation is what makes the feedback local. A cell next to one peak commits nearly all its carriers to it; a cell between two commits half to each; a cell in a flat region splits evenly and exports nothing net. That is what fixes the spacing: a peak can drain its immediate neighbours completely, and cannot reach past them, because the cells beyond are committed elsewhere.

What comes out #

Forty-eight cells, started at the uniform value plus 6% disorder, integrated to a steady state.

Twelve peaks, at a contrast of 94% — that is, the difference between the highest and lowest cell is 94% of the highest, so this is not a ripple on a nearly uniform field but a set of well-separated maxima with near-empty cells between them.

The peaks are counted by the same routine used on the reaction–diffusion field: handed an array of numbers, told nothing about what produced them, and required to find local maxima rising a stated fraction above the field’s own range.

Twelve peaks over forty-eight cells is a spacing of four cells. That number is not in the equations anywhere; it is what the competition between transport, production and decay settles on.

The starting disorder, and why it is written down #

The initial condition is the uniform steady state times $1 + 0.06(u - \tfrac12)$ where $u$ comes from a small deterministic generator with an explicit seed.

Two things about that are deliberate. The disorder is small — six per cent, so nothing about the final spacing can be attributed to structure in the start. And it is reproducible, because a figure whose content changes between one deploy and the next cannot have an assertion written about it, and every figure on this site has assertions written about it.

The second point is a fleet convention rather than a scientific one, but it has a scientific consequence here that is worth noticing: it makes the seed a parameter, which makes “does the answer depend on the seed?” a question one can ask by changing a number. That is exactly the question the ring’s limitation turns on, and it would not be askable if the disorder came from an unseeded source.

What the two mechanisms share, and where they differ #

Put side by side, the two accounts agree on more than they disagree on.

Both select a spacing rather than a number. Give either more room and it produces more peaks at roughly the same density. That is the property the geometry needs and neither supplies anything else.

Both amplify a disturbance rather than responding to a pre-existing pattern. Neither is given a template; both start from something almost uniform and produce structure.

Both are indifferent to arrangement. The transport model on a stationary ring has exactly the limitation the reaction–diffusion model has: its peaks arise from the starting disorder, so there is no reproducible angle between them, and what it makes is a whorl.

Where they differ is in what can be said in advance. The reaction–diffusion system has a dispersion relation: a linear calculation on the equations predicts the fastest-growing mode before anything is integrated, and the prediction can be checked against the count. The transport model’s nonlinearity is in the polarisation, which is not a small perturbation of anything — at the uniform state every cell’s polarisation is exactly balanced, and the whole behaviour lives in how that balance breaks.

So the transport model has to be run to be known, which makes it a weaker instrument for the same job even though it is the better-supported biology.

What the exponent does #

The polarisation exponent is the only quantity in the model whose value changes the character of the answer rather than its scale, so it is worth saying what happens across its range.

At $k = 0$ every cell splits its carriers evenly whatever its neighbours have. The transport term becomes a constant times the difference between neighbours, which is a diffusion term with the wrong sign — or rather, with no sign preference at all — and the model reduces to production, decay and diffusion. No pattern forms.

At $k = 1$ the commitment is proportional. A cell with neighbours at 3 and 1 sends three quarters of its output to the richer. That is enough to pattern, weakly, with low contrast.

At $k = 2$, the value used here, the commitment goes as the square: neighbours at 3 and 1 split nine to one. The peaks separate cleanly and the intervening cells are drained to a few per cent of the peak value.

Above about $k = 4$ the commitment is effectively all-or-nothing and the peaks become single cells, which is a lattice artefact rather than a biological statement — the pattern’s wavelength has collapsed to the grid.

So the model has a working range rather than a value, and the figure sits in the middle of it. That is the appropriate ambition: the claim is that a mechanism of this shape selects a spacing, and a claim that survives a factor of four in its one sensitive parameter is a claim about the shape.

Why it won anyway #

The reason the transport account displaced the morphogen one is not that it is mathematically nicer. It is that its parts were found.

PIN proteins are real, their asymmetric placement in the membrane is visible under a microscope, and — the decisive observation — their orientation in the shoot apex points towards the sites where primordia are about to form. Mutants lacking them fail to make primordia at all. Applying auxin to a suppressed apex restores them at the point of application.

That is a different quality of evidence from “a system with these properties would produce this pattern”, and it is what the reaction–diffusion account never assembled. It is also a useful reminder about what mathematical models of biology can and cannot settle: two mechanisms with quite different molecular content produce the same spacing behaviour here, so the pattern does not distinguish them. Only the molecules did.

The polarisation arrows, and what they are showing #

The figure draws each cell’s polarisation as a short line towards its richer neighbour, and the resulting picture is worth reading carefully because it makes the mechanism visible in a way the concentrations alone do not.

Away from a peak the arrows all point the same way over a stretch of several cells — towards the nearest maximum. Between two peaks there is a cell where the direction flips, and that cell is the drained one: it is exporting in both directions and receiving from neither.

So the ring is divided into basins, each draining into one peak, with a watershed between. That is a structure the concentration profile shows only indirectly, and it is the structure that makes the spacing stable: a new peak cannot start in the middle of a basin, because every cell there is already committed to exporting towards an existing maximum.

It also makes a prediction about what should happen if a peak is removed. The basins on either side should extend into the vacated region and the watersheds should move, and after a while a new peak should form near the middle of the enlarged basin. That is a testable statement about the model, and the corresponding experiment on a real apex — ablating a young primordium and watching where the next one goes — is one of the classical results the transport account is built on.

What is being modelled and what is not #

Three simplifications, stated because each is load-bearing.

One dimension. The apex is a dome and the competent region is an annulus; treating it as a ring is the standard reduction and it discards the radial direction, which is where the ordering information lives.

One molecule, no growth. Real apices grow, and growth is what carries primordia out of the competent zone. A stationary ring has no such motion, which is the limitation that matters.

Instantaneous polarisation. In the model a cell’s carriers redistribute immediately in response to its neighbours’ concentrations. In a plant this involves protein trafficking and takes hours. Whether that delay matters is a question this model cannot ask, and delays in feedback systems are precisely the sort of thing that can change an answer.

None of these is hidden in the code and none of them is unusual — they are the standard reductions the published models make too. They are listed here because a figure showing twelve neat peaks invites the reading that something has been explained, and what has been shown is that a mechanism with a stated form produces a spacing with a measurable value.

What a stationary ring still cannot do #

It is worth closing the loop on the limitation, because it is easy to assume that a better-supported mechanism escapes it.

It does not. Run this model from six different starting perturbations and the peaks land in six different places, in an order set by the disorder, with no reproducible angle between the first two. That is the same result the reaction–diffusion ring gives, and it is for the same reason: the equations are rotationally symmetric, so nothing in them distinguishes any position on the ring, and the only thing that breaks the symmetry is the initial condition.

Better biology does not help with this. The limitation is geometric — a stationary one-dimensional domain has no notion of “the previous element” — and it would survive any amount of molecular detail.

What removes it, in the published transport models as much as here, is putting the ring on a growing apex so that primordia are carried out of the competent zone. Then each new peak forms in the space vacated by ones that have left, the arrangement inherits an order, and a divergence angle exists to be measured. That is the same ingredient the lattice essays call the rise.

The uphill pump elsewhere #

One last observation, which is why this mechanism is worth knowing beyond phyllotaxis.

Transport up a gradient with a competitive resource constraint is a general pattern-forming device. It shows up in vein formation in leaves, where the same auxin and the same carriers produce channels rather than spots because the geometry rewards a different solution. It shows up in root branching, which is the one place this site’s branching field and its mechanism field touch. The same equations with a different domain give a different pattern, which is a much more economical account of a plant than one mechanism per structure.

And it makes a prediction the diffusive account does not: interfering with the carriers should destroy the pattern while leaving the signalling molecule intact. That experiment is doable, it has been done, and it is the reason this is now the textbook account rather than the other one.

Which is the honest summary of this field’s position. Neither model explains phyllotaxis; both explain a spacing; one of them has its parts identified. Getting from a spacing to an arrangement is a separate problem, and it is a geometric one — a lattice on a cylinder, with a rise that a growing apex lowers.