The headline that Occupy Math first noticed told him that a new shape had been discovered — and as often happens, the journalists had focused on a simple and relatively minor point in the story. The new shape is called a *scutoid*, two adjacent scutoids are shown above, and the big story is that a mathematical model was used to explain a very clever solution to making your skin and the lining of your blood vessels stable and strong. These parts of your body are made of *epithelial cells* and they are packed to achieve high stability.

We start working our way up to describing the new shape. In geometry, a *prism* is a solid with two ends that are identical polygons — Newton used a triangular prism to split white light — joined by rectangular sides. A modification of the prism is the *frustum*:

This is what happens when you let one of the polygons at the ends be a different size from the other. To get a scutoid you chop off one corner of a frustum. Chopping off the corner creates a triangular face and — in the picture at the top of the page — lets you have a shape with a pentagon at one end and a hexagon at the other end. You can see the triangular face on the green scutoid at the top. That is a description of the shape of the scutoid.

Suppose you were using prisms to assemble a floor. Floor tiles are just short prisms after all. Then, as long as the polygons at the ends of the prisms tile the floor, this works well. In order to make a curved surface, like your skin or the lining of a blood vessel, we need to use frusta instead of prisms. The area on the inside of the curved shape is smaller than the area on the outside — so we need things with a smaller area on one end. It turns out that this is *not* what the cells in our body do. The chopped off triangles on the scutoids lets the top and bottom of cells that adopt a scutoid shape touch more neighbors, creating something stronger and more stable that a simple tiling with frusta. This is the discovery — our cells organize in tilings of scutoids. Here is the abstract from the scientist’s paper:

As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modeling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbors in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term “scutoid”. The detailed analysis of diverse tissues confirms that generation of apico-basal intercalations between cells is a common feature during morphogenesis. Using biophysical arguments, we propose that scutoids make possible the minimization of the tissue energy and stabilize three-dimensional packing. Hence, we conclude that scutoids are one of nature’s solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs.

**What was the mathematical model?**

A recent Occupy Math talked about using Voronoi tilings for artistic purposes. Once the biological team noticed that the cells had an odd shape, they used the three dimension version of Voronoi tiling to get an idea what the shapes were. A Voronoi tiling has to fill space — the only question is how well. Getting a solid Voronoi tiling that matched the behavior of the cells gave the team a digital model of the shapes of the cells. This yields a much cleaner version of the shapes that you can look at mathematically and with your computer, instead of peering through a microscope.

Occupy Math thinks that this sort of teamwork between math and biology is critical to progress in biology. This strong, efficient packing of cells will permit more effective work on issues from skin cancer to arteriosclerosis. These are both pathologies that occur in epithelial cells — knowing the arrangement of the cells may matter a lot. There may even be diseases resulting from something messing up the scutoid shape during development. Another key point is that mathematical discoveries arise from helping biologists explain their observations. This is a symbiotic relationship.

**Math and biology, a natural partnership!**

Occupy Math was hired as a *bioinformaticist*. In essence, his job is to dig biologists out from under piles of data. This job has two parts, helping people move their projects forward and figuring out when biology has dropped a mathematical ball. These errors happen in statistics, basic reasoning, models of biological phenomena, and algorithms. The first job is welcome, the second is often greeted with measured hostility. One of Occupy Math’s students has located, documented, and repaired a flaw in a widely used algorithm used for clustering. The results have been presented five times; each time the first question after the talk is a variation on “why are you wrong about this?”

Students often go into biology because they want to do science with a minimum of mathematics. This used to be an effective plan, but it isn’t any more. This week’s blog about scutoids and the geometry of tissues is one example, but there are thousands of papers in biology that use sophisticated math and thousands of papers that are *wrong* because they ignored the math. This is probably a substantial part of the replication crisis, where high-impact scientific papers have results that cannot be reproduced.

In the media, scientists are often portrayed as if they are Star Trek Vulcans — logical, almost always right, and working together with only polite, quickly resolved disagreements. Occupy Math’s experience in collaboration suggests a far messier situation, which actually makes sense. A mathematician walks into a field where he has no formal credentials and offers closely reasoned logic that is initially impenetrable about why hundreds of published papers are probably wrong. This is about as popular as a pig on the breakfast table. Add the general fear of math in our society and there is a big trust and communications gap. One of Occupy Math’s goals is to improve the situation and his contribution is to help when he can and to learn as much biology as he can.

If you’re a biologist, you may want to acquire a mathematical collaborator. Occupy Math can help. Are you interested in hearing about math and biology? Examples of places where biology has dropped the mathematical ball? Please let Occupy Math know in the comments.

I hope to see you here again,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics