Darwin 200: Biology meets Mathematics
12 May 2009
James Marshall of the Department of Computer Science explains how mathematical modelling has both confirmed and improved Darwin’s ideas on evolution.
Biology and evolutionary theory have moved on a long way since Darwin’s time, both in the depth of our under-standing and in the techniques available to us. Darwin formulated his theory without knowledge of how traits are inherited – he assumed that inheritance occurred by a blending of parents’ characteristics, despite Gregor Mendel publishing evidence for the discrete, genetic nature of inheritance only six years after publication of The Origin of Species. When Mendel’s results were rediscovered and interpreted by the scientific community, it took mathematical modellers such as Ronald Fisher, Sewall Wright and J. B. S. Haldane, working in the first half of the 20th century, to reconcile genetics and evolutionary theory in what is now known as the ‘synthetic view of evolution’, or ‘modern synthesis’. Darwin himself was no mathematician, arriving at his startling conclusions solely through observation of the natural world and verbal reasoning. However, his ideas find natural expression as mathematical models of how gene frequencies change in populations over time, and Darwin’s theory was vindicated and strengthened by these earliest modelling efforts.
Today, Darwin’s ideas remain as current and exciting as they did on first being revealed to the world and many of the mathematical frameworks developed by Fisher, Wright and Haldane are also still in widespread use. What has changed is that many of today’s biologists are often much more comfortable than Darwin was with presenting purely theoretical results, so that they or others might subsequently look for confirmatory evidence. For those working in the field of theoretical biology, Darwin’s work still represents a treasure trove of ideas to be modelled, investigated and refined. Mathematical models are now used extensively to address the problems that Darwin himself was concerned with.
On the Origin of Species
Darwin’s magnum opus is, as its name suggests, concerned with the problem of how new species arise. One possibility is that populations of a single species can become geographically isolated from each other and, over time, genetic change due to natural selection and other causes will lead them to diverge, so eventually they can be recognised as distinct. This is precisely the mode of speciation famously illustrated by the various finch species that Darwin collected from the different islands of the Galapagos. But is this the only means by which new species can arise? Or can species also arise in response to natural selection alone, as Darwin originally thought, without the need for isolation? It is hard to address such a question empirically. John Maynard Smith was probably the first to treat this question mathematically, in 1966, and over 40 years later the modelling is still going strong. (The answer is ‘yes’ under certain circumstances, by the way.)
Any good book should have some sex in it and Darwin’s The Descent of Man, and Selection in Relation to Sex was no exception. Darwin was interested in how differences between males and females of a species could evolve. The classic example is the peacock’s tail; peahens are rather drab, unassuming birds, yet the peacock is resplendent with brightly coloured, very long feathers. Furthermore, these feathers actually interfere with a peacock’s flight, so why should it have them? Darwin suspected that the answer to such questions was due to female choice and, indeed, mathematical modelling supports his intuition; males must vie with each other to win over the females and one way of doing this is by having more conspicuous and costly symbols of virility. Thus a peacock’s tail and an expensive Saville Row suit have quite a lot in common.
The evolution of co-operative behaviour
The problem of why individuals should co-operate with, or behave altruistically towards, each other has long exercised evolutionary theorists. If natural selection acts on individuals, why should they do anything detrimental to their own fitness in order to help others? Verbal reasoning can let us down badly here and for a long time it was supposed that altruism could spread because it was beneficial to the group as a whole. Darwin himself teetered on the edge of a rare logical error when he wrote, in The Descent of Man, that an altruistic primeval man ‘might thus do far more good to his tribe than by begetting offspring with a tendency to inherit his own high character’. In 1964 Bill Hamilton showed how altruism could be favoured if the degree of relatedness between donor and recipient exceeds the ratio of the costs and benefits of the act. This formalises Haldane’s famous response when asked if he would risk death to save a drowning brother: ‘No, but I would to save two brothers, or eight cousins,’ he said. Haldane recognised that on average we share one half of our genes with our siblings. Interestingly, it has recently been shown that Hamilton’s rule can also be thought of in terms of selection acting on groups – so perhaps Darwin wasn’t so far off the mark after all.