The beauty of mathematics applied to human biology

Keith Moffat, Emeritus Professor of Mathematical Physics at the University of Cambridge, talks about fluid mechanics.

Keith Moffatt

Emeritus Professor of Mathematical Physics

View Author ->

Keith Moffatt

Emeritus Professor of Mathematical Physics

02 Jul 2021

Key Points

Fluid mechanics has a very important part to play in biology. It’s something that we need to understand in our attempts to combat disease.
In the 1860s, Kelvin realised that if vortex lines in a fluid are knotted, they remain knotted for all time. In 2014, knotted vortices were realised for the first time in a laboratory.
Under the Navier–Stokes equations, these knotted structures can unknot. That leads to exciting work involving reconnection of vortex lines and associated jumps in vortex topology.

Fluid mechanics and biology

Photo by FlashMovie

Fluid mechanics has a very important part to play in biology. There’s a whole area described as biological fluid mechanics and physiological fluid mechanics, which is concerned with airflow in the lungs or flow of blood through the veins and arteries. It’s therefore very fundamental, and something that we need to understand in our attempts to combat disease.

Going down to the smaller scale, to the level of the human cell, viscose forces become of dominant importance. Inertia becomes totally negligible. Surface tension effects can also be very important. Some fascinating problems arise in that area. It’s an area that has developed under the title “microhydrodynamics” or even “nanohydrodynamics”, if one goes to extremely small scales. There’s been a lot of excitement in that area over the last 20 years or so.

Knotted vortices

Let me focus on the business of helicity. I should have mentioned that, in the dynamo context, the conclusion is that large-scale magnetic fields can arise out of small-scale turbulent activity. It’s a wonderful example of how order can emerge out of chaos.

This helicity invariant was already recognised by Kelvin in the 1860s, who realised that what we describe as vortex lines, or vortex tubes, in a fluid – if they are knotted, then they remain knotted for all time. That’s on the basis of the Euler equations. Kelvin got very excited about this result. It wasn’t until about seven years ago that knotted vortices were realised experimentally for the first time in a laboratory by a team in Chicago. This again led to great excitement in this area, the area of knotted vortices.

Knotted magnetic field lines

You can also have magnetic field lines that are knotted. The question that arises is whether you can ascribe an energy to a knot. You can, in the case of a magnetic field, because the magnetic field has an energy. So, a knotted magnetic flux tube has energy. Then the question is, suppose you allow that magnetic flux tube to relax to a minimum energy state. What will it look like? That has led to the concept of tight knots, just like a tight knot in a shoelace. A tight knot in a mathematical sense has an energy related to the particular knot that you’re dealing with.

There’s been a lot of work on this area in the last 20 years or so. It’s an area that I was involved in back in the 1990s, through realising that you could approach knot energy through this magnetic context. Why is it important in biology? Well, DNA strands can be knotted, and a knot in a DNA strand may be associated with genetic abnormalities. That is something that geneticists and knot theoreticians are interested in; and from the sidelines, I would say, fluid dynamicists. That was the origin of my interest in biological fluid mechanics.

The interaction between biology and mathematics

The idea of knotted structures in the biological context immediately stimulates activity among mathematicians, and particularly mathematicians who are experts in knot theory. There is a whole journal devoted to knot theory and its ramifications, and a number of the papers in this field are published in that journal. Equally, a number of papers are published in the Journal of Fluid Mechanics involving knotted vortex structures and the discovery experimental work trying to generate knotted vortex structures, which we believe are occurring all the time in turbulent flow. This is part of the interest, from a fluid dynamical point of view.

Under the Navier–Stokes equations, these knotted structures can unknot. When you have vortex lines that cross one another involving dissipation of energy, the knot topology changes. That, again, leads to a rather exciting area of work involving reconnection of vortex lines and associated jumps in vortex topology.

Topological jumps in soap films

Topological transition of a soap-film Möbius strip. Raymond E. Goldstein, H. Keith Moffatt, Adriana I. Pesci and Renzo L. Ricca.

A few years ago, my colleagues in Cambridge and I were involved in work on topological jumps in soap films. We succeeded in constructing a soap film in the form of a Möbius strip; that’s to say, a one-sided surface. This was a one-sided soap film bound by a wire – a rather curious structure. By slowly unfolding the boundary wire, we observed that the one-sided soap film jumped quite suddenly to a two-sided surface – like a disc, from a topological point of view. That was an interesting topological jump, and it happened quite rapidly. We resolved it by using a high-speed camera and achieved an understanding of just how that jump occurred.

This was rather fascinating because it involved interaction with pure mathematics. Pure mathematicians have been very much involved with what we describe as minimum area surfaces. Given a boundary, there is always a surface of minimum area with that boundary, and a great amount of activity on the mathematical side has been devoted to understanding the structure of such surfaces.

A connection to the human cell?

Surface tension and viscosity are the important physical effects that dominate behaviour down at the micro level of the human cell. Any little surface membranes that tend to form at that level will have a minimum area, subject to the boundary constraints as a result of the very strong effect of surface tension. This is one reason why it is important to analyse and understand such effects in the biological context. It’s been conjectured that one might have this sort of topological jump going from one-sided to two-sided surfaces in very fundamental developments at the cellular level – a rather exciting conjecture and an area in which mathematicians and biologists can interact fully.

A better understanding of cancer?

It’s quite possible that, in the course of time, our understanding of the behaviour at the cellular level will increase, even in the fluid dynamical idealisation, and that could potentially lead to better understanding of the proliferation of cancer cells, the growth of tumours. That is very much an aim of much biological research in this area. There’s also the question of remedial action – the targeting of such cells by controlling, for example, microscopic devices that are injected into the bloodstream and directed at areas of the system where the aggressive cells may be developing. That is an area where fluid dynamics is going to play an important part.

Mathematicians’ greatest motivations

Photo by Lorna Roberts

Applied mathematicians – I stress applied – are motivated by the need to understand natural processes that we observe around us, out in space, in the interior of the human body, or things like the flight of insects and birds.

Pure mathematicians are motivated more by the understanding of abstract structures that have their own internal beauty, rather like abstract art, and that may not be accessible to everyone. One does need a basic mathematical training to appreciate this sort of beauty. Yet, one of the miracles is that these beautiful structures often come, quite unexpectedly, to play a part in the understanding of natural systems.