Cells that fill the egg

We built a second embryo, on a different idea. Every cell owns a share of the egg and the shares fill it completely: no gaps between cells, no overlaps, and the boundary between two cells falls exactly where their claims meet. It runs one cell to a thousand in about a quarter of an hour, and the same program runs a nematode worm, whose embryos have been measured cell by cell.

Why fill the egg

Our first embryo gave each cell a membrane and pushed the membranes against each other. That is closer to the physics, but it is slow, and crowded cells could slide into one another. Here the rule is geometric: each cell has a centre and a weight, and it owns every point of the egg closer to its centre than to any other, once the weights are counted. Adjusting a cell's weight adjusts its volume, so cells can be told to take unequal shares, as the water bear's stem-line cells do.

What the two share is the developmental rules: the measured cycle lengths, the division axes, which cells move inside at gastrulation, and the epithelium. What changed is only how cells take up space.

The same program, two animals

Nothing in the engine is about water bears. A species is either a table of divisions, each with its time, its direction and the daughters' volumes, or a set of rules in code. The nematode Caenorhabditis elegans runs from a table, because its embryo has been measured: cell volumes, division axes, and the contact areas between cells, for seventeen embryos from four cells onwards.

That gives us something we cannot get from a water bear: a check against real measurements. At the four-cell stage, our predicted contact areas land within about 12% of the measured ones, close to the variation between embryos themselves. Later stages are harder, and the honest numbers are on the method page and in the graph.

Is a real embryo built this way?

The whole model rests on one assumption: that the wall between two cells is flat and square-on to the line joining their centres. It is made because it is convenient, so we set out to measure how well it holds in a real embryo — the worm, because worm embryos have been measured cell by cell and water bears have not.

You can check it without running anything. Hand the arithmetic nothing but the measured walls — no cell sizes, no nuclei, no lineage — and it works backwards to where the centres would have to be; rebuild the embryo from those and count how much comes back. Between four cells and three hundred and twenty, in two embryos, 92–100% of the real contacts come back. Shuffle the walls between cells first and almost none do.

What that is worth, honestly. Less than we first wrote here. There is a theorem, from 1987 and extended in 2008, which says that any tessellation whose cells are convex and which meets face to face is one of these diagrams. So the real question is not whether an embryo is one, but whether an embryo is convex and tidy — and we went and measured that too. It is, and more so than we expected: cells fill eighty-nine to ninety-six per cent of their own convex hull, and past about eighty cells they are as convex as a diagram we built this way deliberately, sometimes more — a real cell's rounded corners survive a voxel grid better than a mathematical one's sharp ones. The awkward junctions where five cells meet at a point are all but absent in both. So the theorem does most of the explaining. Our numbers confirm what it implies rather than discovering something it didn't.

We also found that the arithmetic itself was published in 2019, in a paper about designing contracts, of all things — the same equations, for a completely different purpose. What is ours is making it survive real measurements — a case that literature never took up at all — and giving the answer a floor and a chance level so the number means something.

Three measurements do stand on their own, because nobody had made them for an embryo: how convex its cells are, how flat its walls are — three per cent of their own width, and the same at fifteen cells as at three hundred — and how far what is left departs from the model.

Corrections, same day. This section has been rewritten twice. First we reported that the match weakens badly as the egg fills, and explained why; that weakening was our own doing — one number in the method was set a hundred times too strong and dragged the answer harder the more cells there were. Then we found the theorem and the 2019 paper, and had to give up the claim that we had asked a question nobody had asked. The wrong numbers, the sweep that caught the first mistake and the search that caught the second are all in the write-up. We would rather publish the corrections than quietly restate the result.

One caution worth keeping. The first version of the test asked only that each wall be square-on to its centre line, ignoring where along that line the wall sits. It gave beautiful numbers and was worthless: it bought them by sliding the centres fifteen micrometres through an embryo fifty micrometres long, and the embryo it rebuilt shared one contact in seven with the real one. A direction with no position is not a constraint.

How wrong is it allowed to be?

Our worm embryo gets the contact areas wrong by about 13% at four cells and about 50% by three hundred, and we spent a while treating that growth as the model's failing. Then we asked what the right answer even is. Two real embryos of the same species at the same stage do not have identical contact areas either, and whatever they differ by is a floor nothing can get under.

The public dataset holds seventeen embryos, and the tables needed to compare them with each other have been sitting in it since 2020 — it is the three-dimensional scans people download, not these. Comparing every pair on the contacts they share:

Our model, scored the same way against one embryo, gives 0.13 at four cells and 0.42 at a hundred and twelve. It is inside that spread at every stage. At four cells it is closer to the measured embryo than another embryo of the same species is.

So the growing error is not the model coming apart. It is the embryos themselves becoming less alike as they get more complicated, and the model following them down. Past this point, getting closer to one embryo would mean learning that embryo's particulars rather than how embryos are built.

Where you put the centres matters, and the field puts them in the wrong place

This one is simple and it survives everything above. Run the model twice over thirty-eight measured frames, changing nothing but where each cell's centre is put. Put it at the cell's measured centre of volume and the contact areas come out one to eleven points closer to the truth than if you put it at the cell's nucleus — at every stage, from four cells to three hundred and twenty, and it finds more of the real contacts too.

That matters because the nucleus is what lineage tracing gives you, so predicting which cells touch from nuclear positions is the standard move in this field. A nucleus is not at the middle of its cell, and in a model where the centre is the cell's handle on space, that offset spreads into every wall it touches.

An empty cell, for the gut

A lumen is a hole: a space with no cell in it. Here it is a region like any other, except that nothing is drawn in it. It takes up volume, it can grow, and it can divide, so it lengthens as a chain of beads along the gut instead of a single ball. The cells that share a face with it are its lining.

Signals across the cells

Cells talk across the faces they share, so the model carries signals on exactly that network: a substance is made in the cells a paper shows it in, passes between touching cells, and fades with distance. Colouring the embryo by the level shows the gradient the cells would experience. The sources come from published patterns of dpp, sog, the posterior Wnts and six3; the fade distance is a guess, and marked as one.

Limits

Cells here are polyhedra: flat faces, straight edges. Real cells round off where nothing presses them, so free surfaces come out too large. The model cannot show a gap, a loose cell or a cavity unless one is put there deliberately. The comma-shaped body is prescribed from the published stages, not grown. Limb buds, the pharynx and the ganglia are not in it yet.