A gradient keeps Lewis's slope and takes its share

Lewis’s law wants disorder measured the law on the tissue it was quoted for and found it holding on a random set of points, with a slope of about a fifth of the mean cell area per side, and failing on a golden head. Lewis’s law needs the sides to vary walked from one to the other, and reported a second number alongside the slope: the share of the variation in cell area that a cell’s number of sides accounts for. It reached 41 per cent on one tissue, against 31 on a random set. That share is the number that says whether the law describes a cell or only describes an average of cells, and in most reports of the law it is not given, or is given as a correlation that readers treat as the strength of the law.

Every one of those tissues had cells of about one size from one side of the field to the other. A real tissue need not. An epidermis has zones of expansion, a meristem’s cells are smallest at its centre and grow as they leave it, and a head’s radial law sets a gradient of cell size whenever it departs from a square root. A published Lewis plot pools its cells over one field of view and normalises each by that field’s mean, and that mean is then wrong for nearly every cell in it. The expected damage is unclear in advance, and the two numbers need not go together, so both are measured here.

A tissue with a gradient in it #

The way to build such a tissue is to use the law already in hand. Points are placed at random, with radii drawn so that the density matches a Vogel head with radius growing as the organ’s index to the power $p$. At a half the density is uniform. Above a half the cells get larger outwards, in proportion to their distance raised to $(2p-1)/p$. The tissue is otherwise exactly the random set the earlier essays used, and is read by the same steps: a Voronoi tiling, cells inside 86 per cent of the radius, and each area taken against the field’s mean.

The picture shows how little a gradient of this kind looks like an interesting change. At 0.8 the shaded cells lie outside and the unshaded inside, as an effect of position alone. The cells keep the irregular shapes that a random tissue always has. Nothing about the side counts looks different, and the correlation between a cell’s side count and its radius across the head is $-0.010$. The sides do not know about the gradient.

A second tissue goes alongside the first, to test whether any of this belongs to randomness: a golden head with the same radial exponent, every organ moved by half a spacing. That is well past where Lewis’s law switches on, and it has a different slope and a different share, so it provides an independent check. Eight seeds of 1,500 organs are read for each tissue, at each of six exponents from a half to 0.9.

The plot does not move #

The first result is simple to state and it is what makes the rest possible. The classes of cell have the same mean areas with a steep gradient as with none. Cells with five sides have a mean area of 0.77 of the field’s mean with no gradient and 0.79 at an exponent of 0.9. For six sides it is 1.00 and 0.99, for seven it is 1.22 and 1.22, for eight it is 1.46 and 1.46. Only the rarest classes differ, at three sides or nine and with few cells in them.

So a reader who was shown the two plots with their labels removed could not tell which tissue had its cells growing by several times across the field. Since the law as usually drawn is exactly this plot, of class means, the law as usually drawn cannot see a gradient at all.

The reason is the absence of any link between the side count and the radius. Each class is sampled evenly over the gradient, its mean being taken over small inner cells and large outer ones alike, and that mean is the same whichever class it is. The gradient lifts every class’s mean by the same factor and the normalisation divides that factor back out.

A slope that holds #

Fitted to the classes in the usual way, the slope agrees. On the random tissue it is 0.223, 0.219, 0.219, 0.220, 0.219 and 0.218 at exponents from a half to 0.9. The moved head reads 0.186, 0.187, 0.187, 0.189, 0.184 and 0.182. Those are eight seeds a point, and the scatter between seeds is around 0.006, so across a gradient that spreads cell areas from nearly nothing up to a coefficient of variation of a half, the slope has not moved by as much as its own noise.

This is a strong statement in the law’s favour, and it is worth more than it first seems. It means the slope can be compared between fields of view of different size, taken from tissues with different gradients, or taken from a tissue at different stages of expansion, without knowing those things. There are few measurements in the study of tissue about which that can be said. And as far as the slope is concerned, an argument can be made that nobody needs to worry about gradients at all.

A share that does not #

The share tells a very different story. On the random tissue the sides explain 33 per cent of the variation in area with no gradient, 31 at an exponent of 0.55, 28 at 0.6, 23 at 0.7, 19 at 0.8 and 15 at 0.9. On the moved head they go from 41 per cent to 39, 34, 25, 18 and 14. A law that explained two fifths of what there was to explain now explains one seventh, in the same tissue, with the same slope, because of something the sides have nothing to do with.

Anyone reading the share as the strength of the law would conclude the law had collapsed. And the moved head, which began a clear eight points above the random tissue, ends level with it at about 14 per cent. A share in a gradient has stopped comparing tissues and started measuring the gradient.

The arithmetic of the fall #

Unlike most of what has been measured about the law, this fall can be predicted. Take a cell’s area as a local factor times a trend. The local factor is what the cell would have had with no gradient, with a variance $V$ and a share $E$ explained by the sides. The trend is what the closed form for a radial law gives, with a variance $T$ that follows from the exponent alone. Sides are uncorrelated with radius, so the part of the variance the sides explain is unchanged in size, which is $E \times V$. The total it is a share of has grown, since two independent factors multiply: it is now $(1+V)(1+T) - 1$. So the share becomes

$$E' = \frac{E\,V}{(1+V)(1+T) - 1}.$$

Every input to that comes from the flat tissue and the exponent. The lines in the figure are that formula. On the random tissue it gives 33, 31, 29, 23, 18 and 15 per cent, against the measured 33, 31, 28, 23, 19 and 15. On the moved head it gives 41, 39, 33, 24, 17 and 13 against 41, 39, 34, 25, 18 and 14. Those agree to about a percentage point everywhere, with nothing fitted.

That has a consequence worth stating in a different direction. If $V$ and $T$ are known, a measured share under a gradient can be converted to the share the same tissue would have had without one, or the other way about. A share is not, then, a property of a tissue alone but of a tissue and a field of view, and the conversion between the two is a line of algebra.

How much a field can carry #

The formula also gives a scale for when any of this matters, which is a question for anyone reading an actual report. A random tissue’s local variance is about 0.28. A gradient begins to take a noticeable share when its own variance $T$ is a substantial fraction of that: at $T = 0.03$, a coefficient of variation of about 0.17 across the field and an exponent of 0.6, the share has dropped by about a seventh, from 33 per cent to 28. At $T = 0.09$, a coefficient of variation of 0.3, it has dropped by nearly a third, to 23. A field in which cells at one edge are about twice the area of those at the other is plausibly already in that range.

For a tissue that is more ordered than a random set the effect is stronger, since its local variance $V$ is smaller and a given gradient takes a larger part of the total. That is why the moved head, which starts above the random tissue, falls further. It is a warning about the comparison a reader would most want to make: two tissues reported at different shares might differ in order, in the size of their field, in the gradient across it, or in some combination, and the share by itself says nothing about which.

Getting the share back #

If the damage comes from normalising by the wrong mean, then the repair is to normalise by a better one. Divide the field into bands of equal numbers of cells along the gradient and read each cell against the mean of its own band. On the random tissue at an exponent of 0.9, one band gives the pooled share of 15 per cent; two give 20, three 22, six 26, twelve 29, and twenty-four 30, against 33 with no gradient at all. On the moved head at the same exponent the same bands give 14, 21, 25, 30, 34 and 37, against 41.

Twelve bands of 1,500 cells gets most of the way back, about a hundred and twenty measured cells a band, and the slope is untouched throughout: 0.217 to 0.222 at every setting on the random tissue. So a gradient costs nothing that cannot be recovered, provided the gradient is known. It also shows the correction is not free: the share creeps back slowly, since a band still has a little gradient inside it, and by twenty-four bands each band holds few enough cells for its mean to be noisy, and the share has begun to level off below its true value.

For an actual tissue this requires knowing the direction of the gradient, which a person looking at an image usually would. A radial gradient in a head wants rings. A longitudinal one in an epidermis wants strips across the axis. The question to ask of any such report is whether a cell’s area was normalised by the mean of its neighbourhood or by the mean of the field, since the share comes out quite differently for the two.

An ordered head cannot be switched on #

The arithmetic above has a limiting case that should be checked rather than assumed: a tissue with no sides to vary. An exact golden head under the same gradients reads a slope of 0.006, 0.002, 0.022, 0.008, 0.009 and $-0.004$. Here the cell areas vary enormously from one part of the head to another, and the slope stays at nothing, because nearly every cell has six sides, and the fives and sevens it does have sit on rings with one of each bound together. So a gradient by itself cannot manufacture Lewis’s law, which is the risk one might have worried about first.

That result has one rough edge. The ordered head’s slope is not stable under the banded reading: with a few bands on a steep gradient the local slope has reached 0.09 and even 0.21, on classes of a few dozen fives and sevens from whichever rings fall in a band. The rings of a head with a steep gradient are widely separated in radius and they hold their defects in one small part of the field, so a band that contains a ring has only those defect cells to fit a line through. A slope through a pair of classes each drawn from one narrow radius on a gradient is a statement about that radius. Banding is a repair for a tissue with many cells of every class, and it is not a safe thing to do to an ordered one.

What it says about the literature’s practice #

Lewis measured cucumber epidermis in 1928 and the numbers he reported have been repeated ever since as a property of tissue. This essay does not bear on what he found. It bears on how a later reader should treat a correlation drawn from a section. The slope carries the law, and it can be compared across studies. The explained share carries the law and the section’s own geometry mixed together, and it should either be reported with the band structure it was computed in, or be converted using the variance of the trend. And a tissue that seems to obey Lewis’s law weakly, in a section that takes in a zone of expansion, might not be obeying it weakly at all.

That has a direct effect on the earlier reading of how a tissue’s two numbers relate, which used the slope and never the share, and so is safe. It is also why a fit that placed a tissue on both laws from the side counts alone, without measuring an area, was reasonable in principle. It reports the part of the law that a gradient leaves alone.

What this does not establish #

Every gradient here is radial, smooth and a pure power law. A gradient with a step in it, or a tissue with two distinct zones, might do something different, in particular if side counts differ between the zones, which would break the one assumption the arithmetic depends on. A real tissue that divides as it grows could well have a correlation between cell size and side count that is not local at all. In that case a gradient would not leave the slope alone, and that case is not measured. The tissues here have 1,500 organs and a share read from fewer cells would be noisier. And the random tissue is a null model rather than a tissue, which the essay that first measured the law here was careful to say.

Measurements that would withdraw this #

A gradient across which the fitted slope moved by more than the scatter between seeds, on a tissue whose sides do not follow radius. A share that departed from the formula by more than a few points at moderate gradient. A correlation between side count and radius in tissues built this way, which would remove the argument for why the slope holds. A band-normalised share that did not rise towards the flat tissue’s value. Any of those would mean the account here was incomplete, and the first would mean it was wrong.

Still open: a gradient that the sides do follow #

The whole result depends on sides ignoring position, and that is a property of how these tissues are built rather than of tissue. In a growing tissue, young cells near a meristem have just divided and they are both small and few-sided, and a divided tissue’s few-sided cells are its small ones by construction. A tissue that divides more often at one end has side counts that vary along its gradient. The next measurement builds such a tissue, with division concentrated at one end of the field at a stated rate, and asks whether the pooled slope now moves, how much of what it reads is the law and how much is the zone, and whether reading by bands separates them. If it does not, then a published slope from a section across a zone of division is not the number that this essay says it can be compared by.