Mathematical Modelling: Biology by the numbers

Before 2020, the phrase “mathematical modelling” may have made your eyes glaze over. But as COVID-19 escalated across the globe, we’ve watched mathematics combine with epidemiology to make predictions, guide government decisions, and inform the public about the state of the pandemic.

To gain insight into epidemiological modelling and its incredible influence on policy, Royal Institution lead scientist Alan Duffy recently headed a Cosmos briefing on epidemiology in which he talked with Professor Jodie McVernon from Doherty Epidemiology and Professor Tony Blakely from the Melbourne School of Population and Global Health.

But the usefulness of tackling biological problems with the framework of maths extends far beyond viruses – as Matthew Simpson, a mathematician at Queensland University of Technology (QUT), knows well.

Simpson collaborates with experimental biologists to study melanomas: a serious form of skin cancer that develops from pigment-producing cells.

“Here in Queensland, melanoma biology is always a hot topic – lots of white people living in the tropics,” he says with a chuckle.

His collaborators grow small populations of cells in the lab to mimic how a tumour behaves in living tissue, observing how it takes in nutrients, diffuses waste, and generally how the cells grow, live and die.

“We use mathematical models to help our experimentalists understand what they see in the lab,” Simpson explains.

“This is of high interest to clinicians, because cancer drugs are often targeted at interrupting the cell cycle.”

This quantitative approach is potentially revolutionary.

“One of the challenges in cell biology is that very often people are studying molecular-level information…but that’s just one very small part of the story,” Simpson says.

Studying such specific pathways doesn’t necessarily explain how populations of cells behave in and respond to complicated environments.

“Mathematical modelling is useful because it tries to tie together all sorts of different observations in space and in time to make a more meaningful prediction,” he explains.

“If we want to study a problem in engineering – to build a bridge or reservoir – there are basic principles and laws of physics that people adopt. One of the exciting things about mathematical biology is that fundamental principles are still being discovered.”

Matthew Simpson

Because here’s the thing: unlike physics or engineering, biology isn’t built on a firm foundation of mathematical principles. In effect, the field has no fundamental laws.

This is what drew Simpson – originally trained as an environmental engineer – to biology.

“When I first learned about mathematical biology, I was astounded at what we didn’t know – really, really basic questions,” he says.

“If we want to study a problem in engineering – to build a bridge or reservoir – there are basic principles and laws of physics that people adopt. One of the exciting things about mathematical biology is that fundamental principles are still being discovered.”

By providing big-picture rigour, maths can help biologists make more informed decisions in their research – for example, about how to design their experiments. This is vital, according to Simpson, because biology is facing a reproducibility crisis –an experiment conducted in one lab often cannot be repeated with the same results in another lab, even with the same set-up.

Other areas of physical sciences don’t have the same problem, likely because experimental design is based on mathematics and data science – but in biology, that’s not the case.

“My expectation is that once biological experimentation and mathematical modelling come closer together, that might be part of the answer of solving that reproducibility crisis,” Simpson says.

But this isn’t a straightforward task. Biology is messy – evolution is a tinkerer, not a designer, and so biological systems are full of complexities and variations. This makes it challenging to untangle the correct mathematical questions to ask.

In part, this challenge stems from the fact that mathematical biology is still establishing itself as a field. Early research was sparse: one of the first contributions came from Swiss mathematician and physicist Daniel Bernoulli, who – in a 1760 article in Mercure de France – suggested a differential equation model to understand the effect of cowpox inoculation on the spread of smallpox, and therefore quantify the advantages of vaccines.

As the 20th century dawned, research began probing the underlying mathematics of the natural world, including the classic Lotka-Volterra equations in the 1920s, which are a pair of differential equations that model the interactions between predator and prey populations, predicting how they influence each other’s numbers over time.

Mid-century, things began to pick up: a famous 1952 paper by British mathematician Alan Turing explored the origin of morphogenesis – that is, how development occurs – by looking at how stripes arise on animals. In the same year, Nobel-prize-winning work was published that quantified nerve conduction and excitation, and later in the 1980s and ’90s maths was instrumental in modelling how HIV spread – both within the cells of an individual’s body and through populations.

In the last few decades the field has exploded to encompass thousands of active researchers.

“One of the things that we struggle with is having a common language. An applied mathematician talking to a physicist is often a very simple conversation – but an applied mathematician talking to an epidemiologist or cell biologist or field ecologist is actually something quite different.”

Matthew Simpson

Still, forging partnerships between mathematicians and biologists seems to be a matter of chance and circumstance.

“Some people working in life sciences are very keen to work with theoreticians, and other people are less keen,” Simpson says.

“I don’t know that there’s any recipe to make this happen. It’s a result of hard work and usually lots of door-knocking or emails.”

Simpson’s current collaborator previously worked with an astrophysicist to understand the spatial dynamics of cells, and so saw the value of a theoretical framework.

But the surprising commonalities between seemingly separate disciplines are tempered by dissimilarities – such as the fact that mathematicians and experimental biologists are trained very differently, resulting in cultural and linguistic barriers between them. This, ironically, is largely because biology lacks many of the fundamental laws and basic principles inherent in the physical sciences; for a mathematician, physicist or engineer, it feels like being set adrift in a strange new world.

“One of the things that we struggle with is having a common language,” Simpson explains. “An applied mathematician talking to a physicist is often a very simple conversation – but an applied mathematician talking to an epidemiologist or cell biologist or field ecologist is actually something quite different.

“It’s a matter of finding common ground on which to tackle problems.”

This common ground may skew towards biological or mathematical language, but one thing is certain: a mathematician must be able to clearly understand the biological problem at hand before being able to successfully formulate a meaningful calculation or model. Most models require simplification, but they must retain the biological system’s essential features.

Now, to add to the difficulties, the whole field is rapidly changing.

“The big question facing the field is what we do with data,” says Simpson, who is now working at the interface of data science and modelling.

As techniques to collect biological data becomes more refined, information is flowing in fast – spanning from genomic data to satellite data. This not only influences the questions biologists are asking, but the questions they’re even capable of asking.

“As more data becomes available, we are updating our mathematical models all the time,” Simpson says – a stark contrast to other fields, such as physics, where many fundamental models and laws are largely unchanged by new data.

“It’s no longer enough to collect data and it’s probably no longer enough to have a mathematical model,” he adds.

“What we need to do is to develop models in light of data as it becomes available, so we can make real-time decisions.”

In Simpson’s team, these decisions help inform and optimise experiments. But for mathematicians working in COVID-19 modelling, processing the most recent available data is crucial to inform rapid decisions, such as whether to go into shutdown – thus affecting millions of people and potentially saving lives.

“What I’ve been trying to do is take the field of mathematical biology and make it more integrated with data,” Simpson says.

“I’m not sure that’s ever going to end. There’s an ocean of activities for us to continue with.”

The Royal Institution of Australia has an Education resource based on this article. You can access it here.