Mathematical Challenges from Molecular Evolution

The first documented revolution in biology took place last century when Charles Darwin’s centennial book on the origin of species was published 1859 in England [60]. The only contribution to nineteenth century evolutionary biology, which is worth mentioning from the point of view of mathematics, however, was Gregor Mendel’s statistical analysis of plant crossing experiments [6, 61]. Biology was not yet ripe to accept the usefulness of mathematical models and, indeed, Mendel’s work did not reach the attention of evolutionary biologists. As a matter of fact his studies had to be rediscovered at the beginning of the twentieth century. The revival of controlled cross-breeding led to genetics, a biological discipline in its own rights, but the geneticists were at odds with the selectionists. It took almost half a century before Darwin’s principle of variation and selection and Mendel’s laws of inheritance were united in form of the synthetic theory [45, 59]. Populations genetics, nevertheless, had been developed already in the thirties by the three scholars Ronald Fisher, John Haldane, and Sewall Wright. Thus, the unification of genetics and natural selection has first been successful in terms of a mathematical model. Ronald Fisher was a mathematician and apart from his works in theoretical biology he made also important contributions to differential equations, probability theory and stochastic processes. Most relevant for this chapter, however, is Fisher’s selection equation which refers to sexual reproduction in large populations, random mating, and frequent recombination.