Genetics Laboratory, University of Wisconsin, 425 G Henry Mall, Madison, WI 53706-1574, USA

Abstract

In 1959 Ernst Mayr challenged the relevance of mathematical models to evolutionary studies and was answered by JBS Haldane in a witty and convincing essay. Fifty years on, I conclude that the importance of mathematics has in fact increased and will continue to do so.

In 1959 Ernst Mayr (Figure

Ernst Mayr (1904–2005)

Ernst Mayr (1904–2005). Photograph reproduced with permission from the Archives of the Ernst Mayr Library of the Museum of Comparative Zoology, Harvard University.

**(a)** R A Fisher (1890–1962), **(b)** Sewall Wright (1889–1988) Photograph reproduced with permission from The Capitol Times, and **(c)** J B S Haldane (1892–1964)

**(a)** R A Fisher (1890–1962), **(b)** Sewall Wright (1889–1988) Photograph reproduced with permission from The Capitol Times, and **(c)** J B S Haldane (1892–1964).

Mayr was, however, criticizing textbook simplifications, rather than the actual work of the three pioneers. Far from treating gene frequency changes as analogous to the consequence of beans jostling at random in a bag, both Fisher and Wright considered gene interactions in detail. Fisher (Figure

Who was to answer Mayr's criticism? Fisher was already dead, and in any case preferred attack to defense, and Wright was too gentle – though admittedly not always when Mayr was involved: returning from Italy where he had received the prestigious Balzan Prize in 1984, Wright told me that the value of the prize was considerably diminished when he discovered that Mayr had won it the year before. In the event, however, it was Haldane (Figure

But the larger question remains: what indeed has been the contribution of mathematical theory to evolution? Mathematics is not central to evolution in the way it has been in theoretical physics. Solid advances have been made without using mathematics, much being due to Mayr himself

For example, the idea that polymorphisms become stabilized in populations because heterozygotes are at an advantage is now found in elementary textbooks, but Fisher was the first to formulate it. Loss of heterozygosity with inbreeding is also textbook knowledge, but it was not clear until Wright developed the theory and invented a simple algorithm for quantifying it. Similarly, the idea that the impact of mutation on the population depends on the mutation rate rather than the magnitude of the mutant effect is now taken for granted, but that was not known until Haldane showed it mathematically. One final example is the inheritance of the ABO blood groups, which was in doubt from the time of their discovery at the turn of the twentieth century until Bernstein's mathematical population analysis in 1924

Ironically, Mayr himself unwittingly provided an especially compelling argument for mathematical analysis. His theory of "genetic revolutions" assumed that from a well integrated population, genetic drift in a small founder offshoot will sometimes produce a population with a new set of genotypes integrated in a new way. Intuitively, a small founder population seemed a particularly unlikely place to find a new favorable gene combination, and this was indeed shown to be the case in a very detailed mathematical analysis by Barton and Charlesworth

Recent mathematical work has gone well beyond that of the three pioneers. Partly this is due to skilled mathematicians entering the field and bringing new techniques with them; especially noteworthy are stochastic processes. Second, and perhaps more important, is the extensive use of computers. Often you can use a computer to get by without deep mathematical knowledge. An additional influence is the explosive growth of molecular data, which lend themselves to mathematical treatment. In the first half of the twentieth century, population genetics and evolution had a beautiful theory, but there were very limited opportunities to apply it. Now the situation is reversed. Molecular data accumulate too fast to be assimilated.

What are some of the newer developments in evolution that are owed to mathematical theory? Here are a few.

Neutral theory, molecular clocks and selective sweep

One striking result in the post-Mayr period was Motoo Kimura's neutral theory, independently developed in 1968 by him and by Jack King and Thomas Jukes

One contribution of the neutral theory has been to provide a rationale for a molecular clock. Essentially, all our estimates of evolution rates depend on the assumption that the molecular changes used in constructing the clock are mutation-driven. The near constancy of average mutation rates permits reasonably accurate time estimates. Fortunately, enough of the DNA does not have an obvious function and can reasonably be supposed to be evolving by neutral kinetics, or near enough so that the neutral theory can be used in practice. And the experimenter can choose genomic regions most likely to behave in a neutral manner.

A second important attribute of the neutral theory is that it supplies a natural null hypothesis for the study of selection. And yet another outgrowth of the neutral theory is the view that much of the molecular polymorphism in natural populations is effectively neutral. This is especially useful now that variation in the frequencies of single-nucleotide polymorphisms (SNPs) is easily observed.

The various measures that are used to quantify genetic variability are outgrowths of population genetics theory. One striking result of such theory is the realization that all of the worldwide human population is descended from Africa, and moreover from a small area within Africa. The evidence for this striking conclusion is that molecular variance is greater in African peoples than elsewhere. The molecular clock can be used as one measure of the time taken during various human migrations and, of course,

Another outgrowth of population thinking is the 'selective sweep'. A new favorable allele arises by mutation, spreads through the population and becomes fixed at a rate that is determined mainly by how favorable it is. A consequence of this fixation is that neutral or weakly selected alleles linked to the locus are swept along with it. Because of this, there is a region on either side of the selected locus that is deficient in genetic variability. Such regions of reduced variability are footprints of a selective sweep in the past and, remarkably, provide evidence for events that occurred long ago and which can no longer be observed. Although the basic idea is simple and requires no mathematics, an assessment of how much the variability is reduced and the linkage distance over which the reduced variability occurs depend on mathematical theory.

Mathematics and the computation of family relationships

An area of biology in which mathematics, and especially computers, have become absolutely essential is systematics, Ernst Mayr's own field. Formerly, assessing species relationships and building phylogenetic trees based mainly on morphological differences was a matter of intuition and judgment. Systematists often disagreed, sometimes violently. Then came the DNA revolution. A mammalian DNA sequence supplies billions of bits of information, thus for the first time providing an opportunity for a procedure independent of personal judgments

One striking example from such studies, which came as a complete surprise to classical systematists, is the close relationship of the elephant to the shrew. Another example is in primates. For many decades the relationship of chimpanzee, gorilla and man has been uncertain. Molecular analysis of DNA sequences, using the newly developed theory, has shown that our closest relatives are chimpanzees. Furthermore, that we and the chimpanzees are 99% identical at the DNA level came as a surprise to many. Equally surprisingly, we share some 90% of our DNA with mice, rabbits, dogs, horses and elephants. Yet this is no surprise to those acquainted with the neutral theory. These numbers are fully consistent with expectations based on mutation rates and the times involved. Finally, there is now help available in the form of computer programs that can work out phylogenies and display the information graphically (see

Coalescence and speciation

Finally, there has been a major theoretical advance, coalescent theory

Until recently, mathematical theory had contributed little to the study of speciation. Mayr emphasized allopatric speciation and the prevailing model, due to Dobzhansky and Muller

Looking to the future

I have given only a few examples of the part that mathematical theory has played in evolution studies. There are many more, but these, I hope, constitute a convincing sample of the importance of mathematics in population genetics and evolution. I do not intend to imply that all evolutionary study need be mathematical and theory-driven. Much exciting evolutionary biology is done in the Mayr non-mathematical tradition

The rise of molecular methods has led to an increase in the importance of mathematics in population genetics and evolution. The abundance of data that require mathematical analysis has greatly increased. At the time of Mayr's challenge, evolution had a beautiful theory but very few opportunities to apply it. Now the situation is reversed: data appear faster than existing theory can deal with them. That mathematics will play an increasingly important evolutionary role in the near future seems clear.

Envoi

I think these examples show not only that mathematical theory is helpful, but that it is often essential. I don't know what Ernst would say today. He might have had a change of mind, but I doubt it. Knowing how much he enjoyed arguing, I suspect he would be quite critical of much that I have written. Unfortunately, although he lived to be 100, he was not immortal and died in 2005. Were he still alive, I would surely hear from him and whatever his opinions, he would not keep them to himself. He would have enjoyed an argument, preferably over a glass of sherry. And so would I.

Acknowledgements

I am indebted to Bret Payseur for reading the manuscript and offering some very useful suggestions.