I'm a D'Arcy Wentworth Thompson groupie, a common condition in developmental biologists. I've lost track of how many times I've read On Growth and Form (more often than I have Darwin's Origin, anyway), and I particularly like his chapters V and VI, where he discusses mechanical properties, like surface tension, that can determine the shapes of cells and tissues. I've also got Boys' classic Soap Bubbles and Isenberg's The Science of Soap Films and Soap Bubbles on my bookshelf—my developmental biology students will be getting a lab late this semester in which they try to model cell shapes seen in histological sections with soap bubbles. It sounds silly, but there's some serious math and biology behind behind soap bubbles and cell and developmental biology.

Now I also get to assign them a contemporary paper in the subject, a beautiful paper by Hayashi and Carthew in which they push soap bubbles around to model cell configurations in the Drosophila eye. See? I told you this was some serious stuff…it got published in Nature, after all.

The math and physics of simple interactions between bubbles have been worked out, and they follow fairly simple rules. They form boundaries to minimize the amount of exposed surface area, and they form junctions of three bubbles in which all the membranes meet at 120° angles. This can be seen in the drawing of three soap bubbles in (a), below.

Pattern formation in different tissues. a, The configurationof three soap bubbles. Theinterfaces(a,b,c) are orientedat 120°-angles from each other. b, Packing symmetry in an eight-cell mollusc embryo. c, d, Schematic Drosophila ommatidium at 35% of pupal life. Cross-section view at the level of the adherens junction(c), and the side view is equatorial to the midplane(d). Cone cells (eqc, equatorial; pc, posterior; plc, polar; ac, anterior) are surrounded by two primary pigment cells (pp) plus secondary pigment (sp), tertiary pigment (tp) cells and bristles (b). The cone cells sit over a cluster of photoreceptor cells (R). e, A retinas tained with cobalt sulphide. f, An ommatidium with an arrow marking the junctional interface between cone cells. g, A cluster of four soap bubbles, with an arrow marking the junctional interface.

The same thing is seen in many cells, shown in the pattern of cells in a dividing mollusc embryo in (b); the four topmost cells cluster together in a pattern that is exactly like the cluster of four soap bubbles in (g). This rule of 3 cells in contact with 120° planes always holds true, as long as the membranes have equal adhesivity (always true in these soap bubbles, but not always the case in cells). Hayashi and Carthew are studying a specific set of cells, the visual receptors of the Drosophila eye, which form a lovely 'neurocrystalline lattice' containing a predictable arrangement of cells. The compound eye of the insect is made of repeating, identical cartridges, called ommatidia and shown in (e), which are organized in a hexagonal lattice. Within each ommatidium are an array of cells diagrammed in (c) and photographed in (f). The only ones that matter here are the four receptor cone cells, ac, pc, plc, and eqc, which are wrapped up into a small cluster by two primary pigment cells, pp. Look at the four cone cells in (f): they follow the rule, just like the molluscan blastomeres in (b) or the soap bubbles in (g)!

Those images show how the normal, highly reproducible configuration forms. The ommatidia do this over and over again, never varying in the wild-type fly. The authors also have a mutant, called Rough eye (Roi), that disrupts the number of cone cells; instead of just four in that one plane, they can have one to six cones instead. In the figure below, you can see an arrangement of soap bubbles on the left, a diagram in the middle, and a photograph of the arrangement of cone cells in a Roi mutant—they're the same. Simple mechanical rules seem to be dictating the precise organization of the cells.

I said that these simple rules apply when the adhesivity between the cells or bubbles are equal. They aren't always equal in cells. The cone cells are all similar in their membrane properties and are all expressing the same cell surface molecules, so they stick together. The primary pigment (pp) cells surrounding them have different adhesivities, though, so the behavior is different. Look at diagram (i) below: when three cells butting up against one another have identical adhesivities, they form the usual 120° angles. When two cells stick more tightly to one another than to a third, though, they maximize their surface area with one another (the yellow line), and you see 180° angles instead.

In this experiment, they've identified one of the cell-surface molecules that make the cone cells adhere more tightly to one another than to the surrounding pp cells: it's DN-cadherin. Because normal cone cells have DN-cadherin, they cluster together and form those 180° boundaries with the pp cells. If the investigators knock out DN-cadherin, the cone cells then have similar adhesive properties to the pp cells, and they shift to establish boundaries closer to 120°.

To make the experiment even niftier, though, they've made mosaics: some of the cells lack DN-cadherin, and others have it. The wild-type cells that can make DN-cadherin are stained pink in figures a-h, while the mutant cells are black and lack DN-cadherin, and all of the cell membranes are green.

DN-cadherin is required for cone cell patterning. DN-cadherin mutant cells are marked by the absence of LacZ (purple) and visualized with b-catenin (green). a, b, A mosaic retina. c-h, Mosaic ommatidia, showing patterns when zero (c), one (d), two (e), three (f) and four(g, h) cone cells are mutant. i, Schematic of three interfaces that meet at a point. Left, adhesion is equivalent among the interfaces. Right, one interface (yellow) has greater adhesion than the other interfaces. j, k, Three-way interfaces between a primary pigment, a polar- and posterior- cone cell are highlighted. Primary pigment interfaces meet (arrowhead) at an angle close to 180° when cells are wildtype (j), and close to 120° when cells are mutant (k). The cone-cone interface (arrow) is shorter in the mutant. l, A mosaic ommatidium with an interface between wild-typecells (yellow), and an interface between a mutant and wild-type cell (white).

You can find clusters of cells that are all pink, or wild-type, as in (c), and the central cone cells are clustered and meet the pp cells at sharp 90° and 180° angles. In (g), all of the cells lack DN-cadherin, and now the boundaries between cone and pp cells try to form 120° angles.

D'Arcy Thompson would have loved this stuff. What it is showing is that form is regulated in part by the geometry of cell-surface interactions and the mechanical properties of adhesion. It's not just passive physical interactions, though; cells also modulate how they adhere to their neighbors by expressing different cell surface molecules. It's genes and environment working together to generate form, both indispensable to one another.

By the way, here's the really important information: the soap bubble recipe. They used ordinary dishwashing detergent, diluted 1:7 in glycerol.

Hayashi T, Carthew RW (2004) Surface mechanics mediate pattern formation in the developing retina. Nature 431:647-652.