# Cell jams and 3.81

### Traffic jams and cells

In applied mathematics,  quantitative techniques are used to analyze phenomena in the real world. There has been a wealth of applied mathematics used in areas like physics and engineering, and increasingly in the twenty-first century, we are witnessing applications of mathematics in biology and medicine.

I had a recent conversation with biomedical physicist Michael Kolios. Biomedical physics is very active at Ryerson, and Kolios is a leader in this field. He described some recent research which ties an otherwise unknown mathematical constant 3.81 to cell jamming. I delved deeper and was intrigued!

Cells may migrate from one location to another as part of their normal function. As two simple examples, think of how blood flows through our bodies, our how water moves from sweat glands to the surface of our skin. Cell mobility, however, underlies many other mechanisms in the body. Cells may move freely, or become jammed and immobilized. The jammed state is something like a solid, akin to cars piled up in a traffic jam, while the unjammed state resembles a fluid.

### The jam index

Phase transitions have a long history in physics and applied mathematics. In my field of discrete mathematics, they come up as percolation, which considers the emergence of connected components in various processes.  One of the most well-known examples of a phase transition is water freezing at zero degrees centigrade. Below zero Celsius, water forms a solid (aka ice), and above zero it is liquid. Cell jamming also experiences a phase transition.

A group of researchers studied lung cells and their mobility in asthmatic versus non-asthmatic patients. They discovered that in the asthmatic tissue, the cells jam less frequently or never jam at all. Further, they discovered that the property of jamming was based on a physical parameter called the jam index, which is defined as j = P/√ A , where P is the perimeter of the cell and A is its area. Jamming happens j < 3.81, and doesn’t occurs when j > 3.81.

For example, if the cell had a shape of a circle and has radius 1/2, then its perimeter is π and its area is π/4, so j is about 3.54 < 3.81 and so the cell wouldn’t jam. However, if we had a more irregularly shaped cell with lots of edges (see the figure below for an example), then a cell with area π/4 could have a much larger perimeter, and so the jam index is larger and it won’t jam.

Observe that the jam index is purely a physical parameter, depending only on the geometry of the cell involved. Here is a reference to the paper:

Park JA, Kim JH, Bi D, Mitchel JA, Qazvini NT, Tantisira K, Park CY, McGill M, Kim SH, Gweon B, et al. Unjamming and cell shape in the asthmatic airway epithelium. Nature Materials 2015; 14:10401048.

The featured image of this blog illustrates the phenomena of cell jamming when j is around 3.81 and is taken from this paper.

There are a couple of interesting observations to make from this discovery. The constant 3.81 was completely unknown before this line of research and came up naturally from physical observation. I’ve never seen this constant appear before in thirty years of studying mathematics. The discovery of the cell jam phase transition underscores not only the importance of genetic factors but physical/mechanistic factors within cells for the spread of certain diseases.

### Jamming cancer

What’s also provocative on the topic of cell jamming is its connection to cancer research. Cancerous cells may metastasize, spreading from one organ to another. Researchers discovered that the jam index and the parameter 3.81 are also at play with cancerous cells (just like the asthmatic ones described above).  If we could figure out a way for cancerous cells to jam, then that may potentially prevent metastasization of these cells. In this case, unlike with cars, cells jamming is actually a good thing.

There is a great article in Quanta magazine that goes detail about how the cell jamming index was discovered. I also include a few video from the paper cited above which actually shows cells jam or not.

I think there is low-hanging fruit here for applied mathematicians to add their insights. It would be terrific if mathematical arguments could actually explain why the cell jamming constant actually is 3.81. Perhaps there is a mathematical model for cell jamming where the constant pops out in a natural way.

Have medical physicists discovered a new mathematical constant?

Anthony Bonato