A tissue that grows forgets Lewis's law

A tissue that was never shaken took a golden head’s tiling and divided it cell by cell, holding it as a map because a division is a wall no set of points can make. It found Lewis’s law switching on once a tenth of the cells had divided, at a lower variance of side counts than any moved tissue reaches the law at. The commonest single division makes two half-sized fives and two full-sized sevens at once, so one wall carries a Lewis correlation in miniature. And the fit in the variance and Aboav’s $a$, which had placed every moved tissue on Lewis’s law, misplaced a divided one by fifteen times the noise.

That essay was clear about what it left out. Nothing grew. A division halved a cell and that was permanent, so the correlation it built could only accumulate. A real tissue’s cells grow between divisions and the two daughters of a division approach their parent’s size. Its closing question was what happens when they do: where on the plane of both laws a tissue in steady growth settles, whether it still switches Lewis’s law on at a low variance, and whether, once growth has evened out the areas, the two numbers that placed moved tissues begin to carry after all.

Two earlier rounds had deferred this, because a growing tiling needs its walls to move, and the first attempt to move them made the cells bend back on themselves until divisions refused. Giving the walls a tension, as a real tissue’s walls have, removes that difficulty. What it shows is simple to state. Growth does not modify Lewis’s law. It takes it away.

A tiling whose walls can move #

The tissue begins as every tissue here has begun, the same 900-organ golden head as a planar map, with each corner shared by the three cells that meet there. Each cell now has a target area, starting from the area it has, and the corners move by gradient descent on the standard energy of a vertex model: a cost for each cell’s departure from its target, and a line tension on every wall. The corners at the rim are held still. Between rounds of division, forty steps of that relaxation bring the tissue close to balance.

A round divides about one per cent of the head’s measured cells, by the shortest wall through each one’s centroid, Errera’s rule. Each daughter gets half its parent’s target. Then every target moves a stated share of the way towards the mean, and that share is what is being varied: nothing, three per cent, a tenth, three tenths, or all of it. A cycle of about a hundred rounds is a whole generation, so a share of three per cent a round restores about 95 per cent of what a cell lost by the time its next division is due. A share of one means every cell is back at the mean size at once, and only its position is left to tell it from any other.

Three seeds are read at each setting, at each of four line tensions, through nine stages from a twentieth of the cells divided to twice their number, by exactly the instrument used on moved and on divided tissues. The relaxation hardly ever refuses a division, once in the whole run. With no growth, the tissue’s readings are close to those the earlier essay found, with a Lewis slope of 0.163 at a tenth divided against its 0.164. So the walls’ movement has changed nothing in itself.

What growing looks like #

The picture shows it without any measurement. With no growth, cell sizes run in steps. There are large cells that have never divided, small ones that have, smaller ones that have done it twice, and the shaded and unshaded cells form obvious clusters by size. With a little growth, the steps blur. By the time the cells are regaining a tenth of their loss each round, sizes are close to even, and when they regain it all, every cell has about one area whatever its shape. What hasn’t changed in any of these pictures is the jumble of shapes. All four panels have plainly the same kind of disorder in their sides.

Lewis’s law, fading #

The law gives way in order. With no growth, the slope climbs throughout: 0.115 with a twentieth of the cells divided, 0.163 at a tenth, 0.255 at three tenths, 0.337 at three quarters and 0.416 when the head’s cells have doubled. That is the earlier result. It only goes up, because a halving is never undone.

Let cells regain 3 per cent of their loss each round. The slope reaches 0.137 at a tenth divided and 0.164 at three tenths, crossing the threshold on the way, and then it comes back down, to 0.140 at three quarters and 0.085 by the doubling, below the threshold again. So a tissue in this state passes through Lewis’s law and comes out of it. Regaining a tenth, it peaks at 0.098 and never reaches the threshold at all, ending at 0.030. Regaining three tenths, it starts at 0.047 and only falls. Regaining all of it, the slope never gets above 0.006.

There is a clean way to see why. A correlation between a cell’s area and its sides, on this tissue, is a correlation between how recently a cell divided and how many sides it has. A daughter is small and has few sides at the moment it is made. Growth makes the first half of that untrue over time, while the second half stays. So the slope reads a race between how fast divisions add small few-sided cells and how fast growth gives them back their size. Faster growth means that only the most recent divisions still count, and the fading slope after the peak is the tissue’s own older divisions being forgotten as more cells grow up.

The side counts do not notice #

Over the same range the variance of the side counts hardly moves with growth at all. With no growth it reads 0.43, 0.58, 1.01 and 1.52 at a twentieth, a tenth, three tenths and three quarters of the cells divided. Regaining all of its loss, it reads 0.42, 0.56, 0.88 and 1.32. Only at the doubling do they differ much, at 2.48 and 1.82, and part of that is a limit of the model that the end of the essay comes back to. Side counts come from the division’s geometry and from the topology of who touches whom, and they don’t care how large anything is. These are the sides as the map counts them, every wall included however short, and the qualification that matters is below: with all its size regained a tissue’s walls shrink further, and a person counting only the walls they could see would find rather less variance.

That makes a sharp point about the claim that Lewis’s law needs the sides to vary. It does need that. But it needs something more, and on moved tissues that something was present in every tissue read. A tissue fully regaining its size has a variance of 1.8 by the map’s count, and its Lewis slope is nothing. Counted by visible walls its variance is much lower, and it is still far above where moved tissues have their slope well on. So varying sides are necessary and they are nowhere near sufficient.

Aboav’s relation does not notice either #

Aboav’s relation behaves in the same way as the side counts. At a tenth of the cells divided, $a$ is 1.34 with no growth, 1.21 regaining 3 per cent, 1.18 regaining a tenth, 1.22 regaining three tenths and 1.18 regaining all of it. At three quarters, those are 0.79, 0.82, 0.91, 0.82 and 0.88. By the doubling, 0.60, 0.65, 0.69, 0.66 and 0.63. The curves lie on top of each other within the scatter of three seeds. Aboav’s relation is a statement about neighbours’ side counts, which involves no area, and it was already known to read something other than Lewis’s law.

So the two laws of cellular tissue finally come apart in the cleanest way so far. Moved tissues always had them tied together through how much the points had been moved. Division tied them in a different way, through the halving. Growth unties them. One keeps its value and the other goes to nothing.

A steady state, counted by what can be seen #

Asked where a tissue in steady growth settles, the map’s own count is the wrong place to look, because it counts walls that have shrunk to nothing. Counted by visible walls, those of at least a twentieth of a cell’s size, the variance gives the clearest answer in this essay. With no growth it never stops rising: 0.58 at a tenth divided, 1.36 at a half, 2.45 at one, 4.31 at two. With growth it stops. Regaining 3 per cent a round it reaches 1.03 at a half and then reads 1.09, 1.12, 1.11 and 1.11. Regaining a tenth, 0.88 then 0.93, 0.93, 0.93 and 0.97. Regaining all of it, 0.78 then 0.84, 0.86, 0.87 and 0.89.

So a tissue that grows has a steady state and a tissue that does not has none. Its visible variance settles at between about 0.9 and 1.1, set weakly by the rate. That is a real number that a tissue’s history does not move, and the second moment could be read for it on real epidermis, with nothing else than a count of sides. It is well above an ordered head’s 0.25 and well below a shaken random tissue’s, and it does not depend on how long the tissue has been dividing, which a report of a variance would otherwise have to state.

On the plane of both laws #

The first question was where a growing tissue settles on that plane. A tissue with no growth runs across the top, reaching a slope of 0.416 with $a$ at 0.60. A tissue regaining all its size runs along the floor of the plane, from an $a$ of 1.26 to one of 0.63, at a slope that never leaves the region next to nothing. Each intermediate rate traces a hump between them.

The bottom of the plane is a place that moved tissues reach only at small variance, near the ordered head, where the sides hardly vary. A growing tissue sits there with very varied sides, a variance around 1.8. Moved tissues at that variance have a slope of well over 0.2. So here is a further corner of the plane that was empty, the one with the sides varied and the area not following, reached by the same simple rule that made the earlier corners. Whatever the plane of two laws is, it does not have one natural path across it.

The second question was whether the law still switches on at a low variance. The share regained decides that. At 3 per cent a round, it does, at a tenth of the cells divided and a variance of 0.54, lower than any moved tissue’s 0.71. At a tenth a round, it does not switch on at all.

The two numbers do not carry here either #

The third question was whether evening out the areas might bring the moved tissues’ fit back into play. It does the opposite. That fit gives a slope for a tissue from its variance and $a$, and on growing tissues it predicts something near what a moved tissue of that variance would have, a sizeable slope. With a tenth of the cells divided and no growth it predicts 0.087 against a measured 0.163, the known under-prediction. With as many divisions as cells it predicts 0.164 for a tissue regaining all its size, against a measured 0.005, and 0.221 at the doubling against 0.006. That is about thirty times the measured slope and thirty-six times the noise. Rather than a tissue’s approaching the moved tissues as growth removes the halving’s mark, it moves away from them on the other side.

So the two numbers do not carry to a divided tissue, where they under-predict the slope, or to a grown one, where they over-predict it. A report of the variance and $a$ on a tissue of unknown history places its Lewis slope anywhere from nothing to more than the moved tissues reach. These are three disordered tissues, made by three routes, and they share no relation among the three numbers. Knowing a tissue’s route is not a detail to be added to a report. It is most of what the report would need to say.

What the walls contribute #

There is one more route for a slope to arise, which a tissue with no growth hides. A tissue regaining all its size has every target equal, so any correlation has to come from the mechanics. A many-sided cell has a longer perimeter on a given area, and its walls pull differently from those of a few-sided neighbour. With walls under a stated tension, cells with more sides are pulled a little larger. Across four tensions, from 0.02 to 0.2, the slope at the doubling rises from 0.002 to 0.025. So tension alone does make a Lewis correlation of a sort. It is small, a twentieth of what a tissue with no growth reads, even at the largest tension here, and it would need a good deal more to reach the threshold. How a law looks across a gradient made a related point about what a slope can be attributed to.

On the tissue that does not grow, tension changes little: 0.416, 0.416, 0.410 and 0.409. There the halving decides, and mechanics adds nothing visible to it.

What it says about a measured slope #

The result turns Lewis’s law from a property of tissue into something closer to a clock. A cucumber epidermis with a large Lewis slope is, on this account, one whose cells are slower to regain size than they are to divide: a tissue actively dividing, with cells that have not yet caught up. A tissue with cells that have had time to even out, or one that has stopped dividing, ought to read a much smaller slope with the same variance. Across a real organ, the slope might be measured at different ages and read as a history of how much division has happened lately.

That is a prediction the model makes and it is not a measurement of anything real. It also says what could not be concluded. Nothing here shows real tissues to behave this way, and a slope of 0.25, the value Lewis’s own numbers imply, sits above what this model reached at the regaining rates tried, other than with no growth at all. At a share of 3 per cent a round the peak is only 0.164. If real tissue’s cycle looks like these settings, its slope would be expected well below what is reported, and something else must be keeping cells’ sizes tied to their sides. A slower regaining rate might do it, or an orientation of division not read here. Either would be interesting to find out.

What this does not establish #

The model has no neighbour exchanges. Walls can only shrink, and in a real tissue a wall that shrinks to nothing is replaced by another across it. So by three quarters of the cells divided, between one and four per cent of the walls are under a twentieth of a cell’s size, and they still count as sides. Counting only the walls of a visible length, over every cell inside the rim cut, the variance at that stage is 1.86 with no growth and 0.84 with all of the loss regained. So a person looking at the tissue would see growth reducing the variance by half, where the map’s count sees a reduction of about a seventh, and every variance on the plane has this in it for a growing tissue. It does not rescue a moved tissue’s law: the moved tissues’ fit gives 0.074 at a visible variance of 0.84 with this tissue’s $a$, against a measured 0.005. And the slope falling to nothing does not come from it, since it falls from the first stages, well before the small walls appear. The growth law is a simple relaxation towards the mean rather than an exponential increase in each cell, and nothing changes the head’s overall size. Divisions land at random among the eligible cells, not on the large ones, and a division rule that picked large cells would keep sizes more even for a different reason. The tissue starts from one golden head.

Measurements that would overturn it #

A growing tissue, at a stated rate of regaining, whose slope kept climbing to the values of a tissue with no growth, would remove the claim that the law fades. A vertex model with neighbour exchanges whose slope behaved very differently would say that the shrinking walls were doing the work and not the growth. A tissue reaching a slope of 0.25 at a variance near 1.8 with its cell sizes nearly even would mean there is a route to Lewis’s law that has no memory of division in it.

Still open: a slower clock, and walls that can exchange #

Two things limit what is said here. Real cells may regain their size far more slowly than a few per cent per percent of the tissue dividing, and the rates that fall between no growth and 3 per cent a round are where the observed slope could lie: at what rate does a tissue in steady state settle at a slope of a quarter, if it does at all? And the model needs neighbour exchanges, so that a shrinking wall is replaced by one across it, as in every working vertex model of a real epithelium, because that would change the variance and perhaps $a$ at late stages. The next measurement adds the exchange at a stated wall length and reads the same tissues again, down to much slower rates of regaining, and asks for the steady state at each: its slope, its variance and its $a$, and whether the steady slope as a function of the rate gives a reading of how fast a tissue’s cells catch up, from a section with nothing but cell shapes in it.