In a letter dated July 24, 1959, Kimura
gave his answer by first considering the case of neutral alleles, alleles
with no influence from selective forces. Under these conditions, Kimura
found that the probability (F) that an individual is homozygous at a
locus is

F = 1/(4Neu+1),

where Ne is the effective population number
and u is the mutation rate per gene per generation. Crow was greatly
impressed by the simplicity of Kimura's solution and when Kimura returned
to the University of Wisconsin for two years starting in 1961, they
returned to the problem. The result was their 1964 publication, "The
Number of Alleles That Can Be Maintained in a Finite Population."

Kimura and Crow's infinite alleles model
had two key assumptions: (1) it assumed that there was a large enough
number of alleles such that any change was a change to a new allele,
and (2) it assumed that mutations can have a range of effects from drastic
to neutral. Kimura and Crow explicitly noted that they did not want
to argue for the plausibility of neutral alleles, but they did think
it was likely that such alleles could exist.

Kimura and Crow examined some of the population
consequences of three different allele systems; namely, "(1) A system
of selectively neutral isoalleles whose frequency in the population
is determined by the mutation rate and by random drift. (2) A system
of mutually heterotic alleles. (3) A mixture of heterotic and harmful
mutants." In other words, in each of the three cases being studied,
every mutation produced a new allele which was neutral, heterotic (overdominant),
or either heterotic or harmful depending on the case at hand. The results
would be systems or sets of only neutral alleles, only heterotic alleles,
or a mixture of heterotic and harmful alleles.

In the neutral case, Kimura and Crow showed
that the effective number of alleles maintained in a population of effective
size Ne and mutation rate u is

n = 1/F = 4Neu+1.

In this situation, if 4Ne << 1/u,
then F approaches 1 and "almost all the genes in a population at a given
locus will be descended from a single mutant." Conversely, if 4Ne >>
1/u, then many alleles will be maintained per locus. In this scenario,
as the effective population size (Ne) increases more individuals should
be heterozygous. In fact, this scenario provides an estimate of the
maximum number of alleles that can be maintained for a given effective
population size.

In the case of heterotic alleles and systems
of mixed heterotic and harmful alleles, Kimura and Crow constructed
an equilibrium model that allowed them to calculate the proportion of
homozygous loci, the effective number of alleles, and the segregational
load. A segregational load occurs when the most fit genotype is the
heterozygote and Mendelian segregation insures that in each generation
inferior homozygous combinations will be formed. The segregation load
is the decrease in the fitness of the population that occurs as the
result of the formation of the less fit homozygotes. As the number of
heterozygote superior loci increases so does the segregation load. What
Kimura and Crow's calculations showed, given their admittedly unrealistic
assumptions, was that "corresponding to a given value of s, Ne, and
u there is a certain [segregation] load required to maintain the alleles
in the population," where s is the selection coefficient, Ne is the
effective population size, and u is the mutation rate.

Kimura and Crow admit that their calculations
do not put a severe limit on the number of segregating loci, but they
do cast doubt on Bruce Wallace's 1958 assertion that the average __Drosophila__
individual from his study is heterozygous for 50% or more of all its
loci. Kimura and Crow's calculations of the minimum segregational load
associated with heterozygous loci in __Drosophila__ lead them to
the opposite conclusion; namely, that "it is more likely that the typical
__Drosophila__ is homozygous for the majority of its genes, though
the segregating minority may still be hundreds of loci." The absolute
number of segregating polymorphisms could still be quite large,according
to Kimura and Crow, since "in large populations, the possibility of
many very nearly neutral, highly mutable multiple isoalleles cannot
be ruled out, although there is no experimental evidence for the existence
of such systems." Since neutral and near neutral alleles create no segregation
load, there could be a large number of polymorphisms and a tolerable
segregation load if many of the alleles were neutral or nearly neutral.
In 1983, Kimura stated that he thought the evidence for neutral alleles
in nature came two years later with the large amounts of variation revealed
by the electrophoretic surveys done by Harris, Hubby, and Lewontin.

So, Kimura and Crow state that they do
not want to argue for the plausibility of systems of neutral isoalleles,
but neither do they want to rule them out. The question is then whether
Kimura and Crow wanted to suggest the neutral case as a possible situation
in nature or whether they were simply using it as a simplifying or tractable
mathematical case. Evidence points, I think, to the use of neutral alleles
as a mathematically tractable case. The neutral case is used to work
out the basic mathematical model which is then applied to more complicated
and more "plausible" cases of alleles which are either selected for
or against. It is important to note that the paper's argument is intended
to cast doubt on Wallace's assertion of the amount of heterozygosity
in __Drosophila__. The shift to the advocacy of the neutral theory
and the existence of neutral alleles then involves realizing and advocating
the fact that the simplest mathematical case may in fact hold in nature.
With the advent of the neutral theory, the mathematical treatment of
the neutral case first presented in 1964 became much more important
than the argument against Wallace, so much so that it now seems to overshadow
Kimura and Crow's main conclusion against large numbers of polymorphisms.

__References:__

Motoo Kimura and James Crow, "The Number of Alleles that Can Be
Maintained in a Finite Population," Genetics 49 (1964), pp. 725-738.