- The Infinite Alleles Model
- The Charge Ladder Model
- The Infinite Sites Model
- Models of Sequence Evolution
The Infinite Alleles Model
In 1958, Motoo Kimura received a letter from James Crow which posed the following question:
Have you ever considered this problem? Suppose every mutant is to an entirely different allele (or at least is counted this way, so that the only homozygosity is homozygosity by descent). Under such a system with a finite population of size n what is the proportion of homozygous loci at equilibrium? Perhaps you have already solved this, but I am not sure. Some of Josh's work suggests that every mutant is distinguishable from every other one if a careful enough test is made; at least this is true for a large number.
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.
