Mathematics of Life (2011)

Chapter 19. The Sixth Revolution

Mathematics has played a central role in the physical sciences for hundreds of years. In 1623, in The Assayer, Galileo wrote:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.

His words were prophetic. By the seventeenth century, mathematics had become a major driving force behind dramatic advances in the physical sciences, and today mathematics and physics (along with astronomy, chemistry, engineering and related areas) have become inseparable.

Until fairly recently, however, mathematics played a much smaller role in the development of the biological sciences. One reason is the old joke about a farmer who hires some mathematicians to help him improve his milk yield. When they present him with their report, he opens it, only to read the opening sentence: ‘Consider a spherical cow.’ Galileo’s language of triangles and circles seems far removed from the organic forms of the living world. You don’t find a cow in Euclid.

This story is amusing, and holds a lesson for wannabee biomathematicians. But it also reveals a misunderstanding about mathematical models. They don’t have to be an exact representation of reality to be useful. In fact, making them less realistic generally makes them more useful, as long as they still provide useful insights. A model that is as complex as the process or thing it represents is likely to be too complex to be useful. A simple model is easier to work with. So a spherical cow is useless if you want it to give birth to a calf, but it might be a useful approximation if you’re wondering about the spread of some bovine skin disease.

A good model must, of course, be sufficiently realistic that it doesn’t leave out anything of vital importance. If you model a rabbit population using immortal rabbits, you will observe a population explosion that has little to do with reality. But even then, your model may capture how a small population grows before it hits environmental limits – so don’t dismiss it too readily. What counts is what the model predicts, not what it leaves out.

Part of the art of biomathematics is the selection of useful models. Another part is taking the biology seriously, and not missing out anything crucial. A third is to pay attention to the problems that biologists want to solve. But sometimes it is also necessary to take a step back, try out a new mathematical idea in a simple but unrealistic setting, and see where that leads. There is another old joke, about a drunk searching under a lamppost for his keys. ‘Did you drop them here?’ ‘No, but this is the only place where there’s enough light to look.’ It is not widely appreciated that the joke’s original context, in Computer Power and Human Reason by Joseph Weizenbaum, was an analogy with science. The point was that in science you have to search under the lamppost, or you’ll never find anything. Maybe, just maybe, you’ll find a torch, even if the keys are somewhere along the road in the gutter. Several of the topics in Mathematics of Life started out as wild oversimplifications, the best that could be done at the time, but eventually turned out to be really informative about biology. It’s important not to strangle a good idea at birth.

Looking back on the story of how biology started to embrace mathematics, one thing stands out: it was doing so long before anyone noticed. Mendel’s discoveries hinged on simple mathematical patterns in the numbers of plants with particular characters. Although the early development of the microscope was empirical, the mathematics of optics soon entered into the story, because you can’t develop really good microscopes without it. One of the clues to the structure of DNA was Chargaff’s rule, a striking but unexplained numerical relationship that couldn’t be coincidence. Bragg’s law for X-ray diffraction was also crucial, and much of what we know about the structure of biologically significant molecules depends on it. And although evolution did not acquire any mathematical expression until recently, Darwin was on the Beagle because, among other activities, the vessel was carrying out a chronometric survey – a mathematical technique for finding longitude.

My sixth revolution, then, is not revolutionary because no one ever used mathematics to solve a biological problem before. What is revolutionary is the breadth of the methods used, and the extent to which they are starting to set the agenda in some areas of biology. I doubt that mathematics will ever dominate biological thinking in the way it now does for physics, but its role is becoming essential. In the twenty-first century, biology makes use of mathematics in ways that no one would have dreamed of at the start of the twentieth. By the time we get to the twenty-second century, mathematics and biology will have changed each other out of all recognition, just as mathematics and physics did in the nineteenth and twentieth centuries.

In Darwin’s day, geology, not mathematics, was vital to the nascent theory of evolution. In the 1960s, chemistry became an essential foundation for cell biology. Then computer science joined in, with the advent of bioinformatics. Now physics and mathematics are entering the fray. And it’s not just biology that is changing in this way: so are all the other branches of science. Conventional borders in science are breaking down. You can no longer study biology as if the rest of science didn’t exist.

Instead of isolated clusters of scientists, obsessed with their own narrow speciality, today’s scientific frontiers increasingly require teams of people with diverse, complementary interests. Science is changing from a collection of villages to a worldwide community. And if the story of mathematical biology shows anything, it is that interconnected communities can achieve things that are impossible for their individual members.

Welcome to the global ecosystem of tomorrow’s science.