I’m an academic who spends a lot of time studying sperm, and sperm’s collective behaviour. The twist? I’m a not a biologist, I’m a mathematician.

To many scientists, how tiny cells and organisms manage to swim is a fascinating topic in its own right. But it is also a crucial element of our understanding of how life comes about: How do cells assemble to form more complex structures? How does a sperm cell find the egg? How do algae stay in the sunlight? How do embryos differentiate between left and right? All these questions are related to the swimming mechanisms of cells. As it turns out, mathematics may indeed help address these fundamental biological questions. ‘Fluid dynamics’ is a discipline situated at the crossroads of physics, maths, and engineering, concerned with the motion of matter that can flow, like water, air, or honey. But it can also help to describe ‘swimming cells’ like sperm. On a fundamental level, sperm are really just moving objects immersed in a fluid.

Biologists have been interested in the details of swimming sperm for a long time. Human spermatozoa were discovered by Antonie van Leeuwenhoek, who was renowned for building the best microscopes in the 17th century. When analysing sperm under the microscope, he noticed one of their most striking features -- their swimming mechanism. Like other types of eukaryotic cells, sperm cells have a long, flexible tail-like appendage, called a flagellum, which they move in a wave-like, undulating manner to swim through surrounding fluid.

Implications of this swimming mechanism can also be understood in terms of physics. Neighbouring swimming cells that are flexing and waving near each other will interact. This is because each of them exerts a force on the fluid, and the fluid, in turn, exerts a force on its neighbour. Even if they individually move so as to swim in a straight line, this interaction can lead to net attraction, which means that they get closer and closer as they swim next to each other.

In the 1940-50s, Lord Victor Rothschild, a zoologist at the University of Cambridge and chairman of the Agricultural Research Council, tried to improve the process of artificial insemination in sheep and cows. A crucial step in artificial insemination is to select the most fertile semen samples. It was believed that more motile semen (the semen in which more sperm cells swim faster) was more fertile. This makes sense, based on the presumption that slow cells will probably not reach the ovum.

Although he agreed with the theory, Lord Rothschild thought that simply looking at sperm samples was subjective and open to vast inaccuracies. Lord Rothschild reasoned that changes in the local ordering and density could change how electrical currents flow through the sample, because the cells’ electrical properties differ from the fluids’. Hence, he came up with an intriguing idea to quantify the quality of a semen sample. By applying a voltage to a sample of ram or bull semen, he measured how its resistance (or impedance, to be precise) changed over time. In this way, he hoped to identify and measure the changes in spermatozoa’s collective motility, or “swirliness”. This was an important step towards making fertility scoring more quantitative.

Nowadays, the correlation between fertility and wave-like, swirly motion is backed up by empirical studies. Instead of electrical currents, digital image processing can be used to measure the collective wave formation in semen. But why are swirlier samples more fertile?

This is the question that has captured the attention of applied mathematicians. Compared to half a century ago, there are now more powerful computers and faster algorithms to help us solve certain computational problems that arise in fluid dynamics. They allow us to compute in detail how hundreds or even thousands of model sperm cells swim together -- something that was probably unthinkable at the time of Leeuwenhoek or even Victor Rothschild.

This gives rise to interesting opportunities from a biological perspective, as it allows us to switch off specific features of model spermatozoa, and observe how this affects the physics of swimming. In this way, we can try to better understand how the observed macroscopic phenomena correspond to microscopic behaviour or physical properties, such as the cell density, sperm cells’ length or the size of their heads. For example, in simulations we can choose the precise number of cells per fluid volume, we can choose how much they bend their flagella, or we can make them stubbornly swim in a straight line without being affected by chemicals or temperature changes. This helps in developing a more detailed picture of the underlying physical effects.

For example, through simulations we can show that sperm cells swimming in a thin film of fluid behave differently depending on their density. When sperm cells are packed more tightly, their interactions are stronger, leading to overall swirlier motion. This could, in turn, be connected to a samples’ fertility. So maybe it is merely the number of cells and not their individual motility that determines how fertile a sample is? Much more work in connecting different features of individual sperm cells to their mass motility is still needed. There's a lot left to discover, both on the biological and on the mathematical side of the problem!

It’s all very interesting, I hear you say, but why bother studying this?

Well, theoretical insights on collectively swimming sperm cells could be useful in guiding the development and control of micro-robots. In a few decades it might be possible to let swarms of small, partially biological, robotic devices, (kind of like medical micro-cyborgs) deposit drugs or cells within the human body with incredible accuracy and effectiveness. In order to make this a reality, we need to understand how these micro-robots should behave so that their interactions prevent them from forming swirls or clusters in the wrong spot within the human body.

So, through the combined efforts of mathematicians and biologists, you might hopefully soon be in a position where you can say “...not only did sperm start my life...they saved it too!”