A model in two acts : a commentary on ‘ A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population ’ by Timoko Ohta and Motoo Kimura

In the early 1970s, protein electrophoresis was the primary tool geneticists used to discover and measure allelic variation in natural populations. It was a relatively simple and inexpensive technique and, most importantly, it permitted the detection of multiple alleles regardless of polymorphism levels. This was a critical point because before the age of protein electrophoresis, segregating alleles were usually discovered only in cases where clearly discrete patterns of phenotypic variation were first observed. With protein electrophoresis a geneticist’s ability to identify multiple alleles did not depend on a prior indication of the presence of genetic variation (Hubby & Lewontin, 1966; Lewontin, 1974). The purpose of the short paper that Tomoko Ohta and her mentor Motoo Kimura published in Genetical Research in 1973 was to devise a mutation model that was explicitly appropriate for protein electrophoretic data and that would permit such data to be analysed with regard to questions on the relative roles of natural selection and genetic drift. As data on electrophoretic alleles began to accumulate, it was discovered that individuals were heterozygous, and many species were polymorphic, at a substantial fraction of the proteins that could be surveyed. These numbers on heterozygosity and polymorphism immediately began to feed a long-standing hunger, that had built up from decades of sophisticated modelling, for data on such topics as mutation rates, genetic load, the rate of neutral mutations, and the relative roles of natural selection and genetic drift in shaping levels and patterns of variation. Ohta and Kimura were the primary theoreticians of the neutral theory of molecular evolution and they had a very strong interest (as did most population geneticists of that age) in understanding how well the neutral theory explained the levels of polymorphism discovered by electrophoresis. The models they developed focused on amounts and patterns of genetic variation, and they tended to include explicitly a neutral mutation rate as well as assumptions about the nature of the mutation process. One prediction of the neutral theory was that the number of alleles in a population was expected to co-vary strongly with the effective population size. Earlier in 1964, Kimura and James Crow had developed the infinite alleles model, in which every mutation gives rise to a new allele (Kimura & Crow, 1964), and under this model the number of neutral alleles varies linearly with both effective population size and neutral mutation rate. Ohta and Kimura’s key idea in 1973 was a mutation model that explicitly gave rise to new allelic states in single steps that differed in net protein charge. Because four of the amino acids are normally charged at physiological pH, the surface of a soluble protein will carry a charge that affects its behaviour in gel electrophoresis, and mutations that raise or lower this charge will increase or decrease the rate of electrophoresis. In Ohta and Kimura’s model, which later came to be known as the ‘stepwise mutation model ’ (Kimura & Ohta, 1978) and also sometimes the ‘ ladder model ’, a protein may mutate to a different allelic state in +1 and x1 steps. Importantly, unlike the infinite sites model, it was possible under this model for two proteins to be identical in kind (i.e. have the same net charge), and not be identical by virtue of common descent from an ancestral gene of the same allelic state (i.e. identity by descent). Ohta and Kimura were a virtual dynamic duo of differential equations, and in this paper as in many others they took a diffusion equation approach. Their primary target was an expression for the effective number of alleles in a population, ne. Under the infinite sites model, Kimura and Crow had shown that ne=1+4Neu (Kimura & Crow, 1964). But under the stepwise model, Ohta and Kimura found that ne= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi