The Surprising Link between Maths and our Immune System!

By Kate Huckstep, 2019 Alumni

Take a moment to transport yourself back in time:

It’s the year 2008, and you still use your DVD player. You’re sitting comfortably in front of the TV, waiting for family movie night to begin. Nobody has touched the remote for a while, so the screensaver has been activated. It’s the DVD logo, and it begins to bounce around the screen.

That memorable DVD-video logo: Picture by Joe Lewis via Flickr

Now, if you’re anything like me, you might just find yourself captured by this simple animation; watching with such anticipation, hoping that the logo might just bounce in a way that sends it perfectly into one of the corners. (Side note: you can relive the glory days at

It turns out that this actually a good example of a classic maths problem, known as the “narrow escape problem”. This problem essentially refers to a hypothetical scenario where a particle is trapped in a given space, and moves around randomly, making collisions and changing direction based on these collisions. In this hypothetical scenario, there is also a small “hole” in the “walls” of the space, known as an “escape gap”. There are a bunch of equations that exist which allow you to calculate how long, on average, it would take the particle to hit that escape gap and escape the space.

This is all well and good, you might be thinking, but what actual applications might these equations have? Well, recently, a massive collaborative study by mathematicians, biologists, and immunologists from Australia, the UK, and Sweden has revealed that the equations from the narrow escape problem actually help explain how T-cells work in our body!

What are T-cells?

T-cells are immune cells, and they essentially act as our body’s last line of defence against a pathogen. So once a pathogen has made it past your skin barrier, into your body… the role of the T-cell is to first identify these invaders, and then decide to attack it and destroy it. Up until now, we haven’t actually understood how this response was triggered. And why T-cells, for the most part, don’t attack the other healthy cells in the body.

What this study found was that the equations used in the narrow escape problem actually translate really well to this biological scenario, and play a key role in determining whether an immune response is triggered.

Why is this?

In terms of cell surface and appearance, T-cells aren’t your run-of-the-mill cell. Most cells in the body have a relatively smooth surface, but T cells are covered in a bunch of bumps and protrusions. Some of these protrusions are known as T-Cell Receptors, or TCRs.

The T-Cell: a strangely shaped cell! Picture by Polygon Medical Animation via Flickr

What scientists have found is that when a pathogen is in contact with one of these TCRs for long enough, this is what triggers the immune response.

Now it’s important to understand that the surface area of these TCRs is tiny. Like, we’re talking 1000x smaller than a human hair tiny. This means that the chance of one of these TCRs being in contact with a normal healthy thing in the body is pretty dang low.

When there is something foreign in the body, however, there is a much greater chance of it making contact with a TCR for the required amount of time to trigger an attack. And this phenomenon can be very accurately modelled using the equations from the narrow escape problem!

This understanding of T-cells and what triggers them to attack has some super important implications in furthering our understanding of auto-immune diseases, where T-cells actually go rogue and start attacking normal healthy tissue.

It just goes to show: the overlap between different fields of STEMM should never be underestimated, and some amazing things can be learned through collaborative studies like this one!


If you liked this article, Kate also runs a podcast with their brother called Curiosity Killed the Rat. Check them out on Instagram and in Apple Podcasts and Spotify.