Kimura and Crow’s 1964 article is justly regarded as one of the foundational articles of evolutionary molecular genetics. It is based on what they called the “infinite alleles” model (now called the “infinitely many alleles” model). This model was motivated by the recognition that a gene is a sequence of perhaps several thousand nucleotides, implying that an astronomically large number of alleles (equal to nucleotide sequences) is possible at any locus. In the later sections of their article Kimura and Crow discuss aspects of this model when selection exists. However, the theory for this case is still unresolved and here only the first section, which in any event has been by far the most influential section, is discussed. In that section only selectively neutral alleles (“isoalleles”) were considered.

Properties of the selectively neutral infinitely many alleles model were introduced by Malécot (1948). The essentially infinite number of possible alleles at a locus leads, in this model, to the assumption that a mutation creates an allele of an entirely novel type. Malécot’s analysis assumed a simple Wright–Fisher evolutionary model (Fisher 1922; Wright 1931) in which the genes in any offspring generation are assumed to be chosen at random and independently from the genes in the parental generation (binomial sampling). He showed that, in a diploid population of fixed size *N* and with mutation rate *u*, the probability that two genes taken at random from the 2*N* genes in any (stationary) generation are of the same allelic type is very close to (4*Nu* + 1)^{−1}. This may be taken as a measure of the genetic diversity in the population at that locus.

Kimura and Crow were no doubt motivated by the fact that information about the extent of genetic variation in natural populations was becoming available in the 1960s, so that Malécot’s result was relevant for a theoretical analysis of the reasons for this variation. They made the straightforward generalization of Malécot’s formula to (4*N*_{e}*u* + 1)^{−1}, where *N*_{e} is the effective population size. [There are at least four concepts of effective population size (Ewens 2000); Kimura and Crow used the “inbreeding” effective population size.] From this they defined the “effective number of alleles” as *n* = 4*N*_{e}*u* + 1, which would be the mean number of alleles present in the population at any time if all alleles had the same frequency. However, in the infinitely many alleles model the alleles present at any one time usually have widely varying frequencies, and as a result the mean number of alleles present is much larger than *n*. For example, the “effective number of alleles” *n* is 4 when *N*_{e} = 250,000 and *u* = 4 × 10^{−6}, whereas the mean number of alleles present in the population is about 42 (Ewens 1964). Despite the title of their article, Kimura and Crow did not give a formula for the mean number of alleles present and claimed that it should be close to the effective number, which the above example shows is not the case. As a result the effective number of alleles concept never gained traction.

Malécot obtained his equation (4*Nu* + 1)^{−1} by considering a retrospective analysis, looking backward in time (as opposed to the traditional evolutionary prospective analysis, looking forward in time). The retrospective approach has dominated population genetics theory for the last few decades. For example, Kimura’s (1968) neutral theory and the investigation of the properties of a sample to test for neutrality (Ewens 1972; Watterson 1974, 1978) were directly influenced by Kimura and Crow’s article. In another direction, Kimura (1971), also a foundational article of evolutionary molecular genetics, explicitly took into account the gene as a sequence of nucleotides. Watterson (1975) found many important properties of this model, again focusing on a retrospective analysis by considering the ancestry of a sample of genes. This form of analysis culminated in Kingman’s (1982) concept of the coalescent, in which the entire ancestry of a sample of genes, or all the genes in a population, is traced back to a common ancestor, leading to a revolution in population genetics theory. All of these developments, and many others, can be traced back to Kimura and Crow’s foundational article.

## Footnotes

*Communicating editor: C. Gelling***ORIGINAL CITATION**The Number of Alleles That Can Be Maintained in a Finite Population

Motoo Kimura and James F. Crow

*GENETICS*April 1964**49:**725–738Photo of Motoo Kimura (left) and James Crow (right) is courtesy of PPGBM Museum of Genetics, Federal University of Rio Grande do Sul, Brazil.

- Copyright © 2016 by the Genetics Society of America