Infinite Allele Model Research Articles

THE STEPWISE MUTATION MODEL: AN EXPERIMENTAL EVALUATION UTILIZING HEMOGLOBIN VARIANTS

The stepwise mutation model of Ohta and Kimura (1973) was proposed to explain patterns of genetic variability revealed by means of electrophoresis. The assumption that electrophoretic mobility was principally determined by unit changes in net molecular charge has been criticized by Johnson (1974, 1977). This assumption has been tested directly using hemoglobin. Twenty-seven human hemoglobin variants with known amino acid substitutions, and 26 nonhuman hemoglobins with known sequences were studied by starch gel electrophoresis. Of these hemoglobins, 60 to 70% had electrophoretic mobilities that could be predicted solely on the basis of net charge calculated from the amino acid composition alone, ignoring tertiary structure. Only four hemoglobins showed a mobility that was clearly different from an expected mobility calculated using only the net charge of the molecule. For the remaining 30% of hemoglobins studied, mobility was determined by a combination of net charge and other unidentified components, probably reflecting changes in ionization of some amino acid residues as a result of small alterations in tertiary structure due to the amino acid substitution in the variant. For the nonhuman hemoglobins, the deviation of a sample from its expected mobility increased with increasing amino acid divergence from human hemoglobin A.-It is concluded that the net electrostatic charge of a molecule is the principal determinant of electrophoretic mobility under the conditions studied. However, because of the significant deviation from strict stepwise mobility detected for 30 to 40% of the variants studied, it is further concluded that the infinite-allele model of Kimura and Crow (1964) or a "mixed model" such as that proposed by Li (1976) may be more appropriate than the stepwise mutation model for the analysis of much of the available electrophoretic data from natural populations.

Weak limit for the stationary distribution of the infinite-alleles model

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models. In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

Weak limit for the stationary distribution of the infinite-alleles model

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

Exact sampling distributions from the infinite neutral alleles model

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.A study is made of the joint distribution of the allele frequencies in two samples taken timetapart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at timetapart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2tapart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

Exact sampling distributions from the infinite neutral alleles model

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.A study is made of the joint distribution of the allele frequencies in two samples taken timetapart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at timetapart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2tapart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

A transition density expansion for a multi-allele diffusion model

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele typesAi→Ajare assumed to beuj(≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates.The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity.A density for the time to fixation with zero mutation is found for theKallele, and infinite alleles model.

A transition density expansion for a multi-allele diffusion model

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele types A i → A j are assumed to be u j (≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates. The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity. A density for the time to fixation with zero mutation is found for the K allele, and infinite alleles model.

Mutation rates, population sizes and amounts of electrophoretic variation of enzyme loci in natural populations.

A method is presented for estimating relative mutation rates or relative effective population sizes, under the hypothesis of adaptively neutral allelic variation. This method was applied to seven surveys of electrophoretic variation. It was observed that electrophoretic mutation rates so obtained follow the gamma distribution and, in Drosophila, are positively correlated with the molecular weights of the enzymes subunits. The variance in mutation rate is larger under the step-wise model of electrophoretic mutation than under the infinite-alleles model. Rates for the most variable loci may exceed rates for less variable loci by a factor of 500. For completely invariant loci, this factor may be as high as 4 X 10(4), an observation suggesting that these loci are subject to purifying selection. In contrast to mutation rates, effective population sizes may vary at the most by a factor of ten. These results support the hypothesis that differences in the amount of electrophoretic variability among polymorphic loci may reflect differences in the rate by which electrophoretically detectable variation is generated in population.

Variance of genetic distance and correlation of heterozygosity between populations under the pressure of stepwise mutation

Using the stepwise mutation model of Ohta and Kimura (1973), formulas are developed for the correlation of heterozygosity and the variance of genetic distance between two finite populations. Studied in detail is the case where the sizes of the two descendant populations are equal to that of the ancestral population and the mutation rate is the same for all loci. Numerical computations are carried out by using the present formulas and those of Li and Nei (1975 Genet. Res. 25) for the infinite-allele model. The results are as follows: The correlation of heterozygosity decreases with time faster for the stepwise mutation model than for the infinite-allele model. However, the relationships between the correlation of heterozygosity and the normalized genetic identity for the two models are very similar, if the average heterozygosities of the two populations are around 0.20 or less. On the other hand, the variance of genetic distance for the stepwise mutation model may become considerably smaller than that for the infinite-allele model, if the average heterozygosities of the two populations are larger than 0.05. The ratio of the standard deviation to the mean is, however, very large for the stepwise mutation model as well as the infinite-allele model.

Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field

A rare allelic type is modeled as a field of individuals diffusing independently in d-dimensional space (d = 1,2,...), in which individuals are replaced by random numbers of offspring at a constant rate. In an infinite-allele model with selectively identical alleles, the offspring distribution would have mean 1 - u, in which u is the mutation rate; otherwise mean (1 - u)w, in which w is the relative fitness of the allele. Let p(x) be the probability that some descendent of an individual initially at zero diffuses unilaterally as far as x(d = 1), or else the probability that some descendent of an individual at x diffuses within a > 0 of the origin (d >/= 2). A nonlinear differential equation is found for p(x) that is solvable for d = 1 and related to Emden's equation for d >/= 2. For p(x) >/= 10(-4) and u </= 10(-5), genetic drift is more important than mutation in the behavior of p(x) (d = 1). If u = 0 and w = 1, p(x) approximately C/x(2) as x --> infinity for d </= 3. As a mathematical application, it is shown that if the initial distribution is uniform Poisson, a bounded open set K is visited by individuals in the field at arbitrarily large times if d >/= 2 but not if d = 1.

Is intragenic recombination a factor in the maintenance of genetic variation in natural populations?

INTRAGENIC recombination between two different existing alleles in a population can create new alleles. The role of this process in maintaining variation in a natural population has been investigated1 by assuming that a gene consists of two sites, each of which can mutate to an infinite number of unique ‘alleles’ (the infinite allele model of Kimura and Crow2). Using this model it was shown that if the product of the population size, N, and the mutation rate, μ, is ⩾1, and the recombination rate, r, is the same order of magnitude as the mutation rate, then intragenic recombination significantly increases the number of alleles maintained in a finite population. Moreover, this is not equivalent to an increase in the mutation rate as the variance of homozygosity and the variance of the number of alleles are larger. This implies that the sampling theory of neutral alleles developed by Ewens3 does not apply to the situation where intragenic recombination is important in maintaining variation4. Application of these results to the study of intragenic recombination in natural populations would require a large number of independent samples in order to estimate the variance of homozygosity and the variance of the number of alleles. However, data from natural populations usually consist of the number of alleles and their frequencies in a single sample. Therefore, for the effect of intragenic recombination to be observable, there must be a detectable change in the distribution of the allele frequencies in a sample which contains k alleles. We present here the results of Monte Carlo simulation which show that intragenic recombination in the equilibrium population increases the frequency of the most common allele, increases the number of alleles which occur only once in the sample, and increases the homozygosity.

The theory of genetic distance and evolution of human races.

1. Theoretical works on Nei's genetic distance and its extensions are discussed. New formulae for the sampling variances of genetic distance estimates are presented. Formulae for the genetic identity of genes at the electrophoretic level when the mutation rate varies from locus to locus are also presented. 2. Empirical data suggests that the rate of gene substitution or mutation rate per locus varies considerably among protein loci, and if this factor is taken into account, the rate of decline of genetic identity (I) is no longer constant but decreases with evolutionary time. Using both the infiniteallele model and the stepwise mutation model, the numerical relationship betweenI and evolutionary time is presented. This relationship may be used for estimating the time after divergence between populations. The value of genetic distance or genetic identity is also affected considerably by the bottleneck effect. The bottleneck effect generally accelerates the increase of genetic distance with time, and the effect remains for a long time after the population size returns to the original level. A method for correcting for this effect is presented. 3. Application of the theory of genetic distance to data on protein polymorphism in man indicates that the genetic variation between the three major races, Caucasoid, Negroid, and Mongoloid, is much smaller than the variation within them, despite the fact that there is a conspicuous difference in some morphological characters such as pigmentation, facial structure, and hair texture. It is proposed that the differentiation of these morphological characters was brought about by relatively strong natural selection through a small number of gene substitutions, whereas general protein loci are subject to little or very weak selection. Analysis of blood group gene frequency data gives essentially the same result as those from protein loci, though they are likely to have been affected by nonrandom sampling of the loci. It is also shown that at the protein level the racial differences in man correspond to those between local races in other organisms. 4. Rough estimates of the number of codon differences between an individual of man and his various relatives are presented. It seems that the mean number of codon differences between man and chimpanzee is about 10 times larger than that between second degree relatives in Caucasians or Japanese, but about 1/19 of that between man and horse. 5. Genetic distance estimates suggest that among the three major races of man the first divergence occurred about 120,000 years ago between Negroid and a group of Caucasoid and Mongoloid and then the latter group split into Caucasoid and Mongoloid around 60,000 years ago. It is also shown that the genetic identity between man and chimpanzee corresponds to a divergence time of 4–6 million years if the assumption of constant rate of amino acid substitution is correct. 6. Methods of constructing a phylogenetic tree from genetic distance estimates are discussed. For constructing the topology of a tree, Fitch and Margoliash's method is quite efficient. For estimating branch lengths, however, Nei's method of averaging distances seems to be better. 7. A phylogenetic tree for twelve races of man is constructed by using gene frequency data for 11 protein and 11 blood group loci. This tree roughly agrees with what we expect intuitively from the morphological characters and the historical record of these races.

The dynamics of neutral mutation

A variety of mathematical models have been proposed, over the years since the pioneering work of Fisher and Wright, for the evolution of gene frequencies in large populations under the pressure of selection and mutation. It is broadly true to say that deterministic models are adequate, at least to a first approximation, when selective differences are large compared with the reciprocal of the effective population size. When selection is weaker than this, genetic drift is sufficiently obtrusive to make stochastic models essential. Such models are typically much more difficult to analyse than deterministic ones, and detailed studies have usually been confined to the very special situation of statistical equilibrium. But of course no biological system is really in equilibrium for very long, and moreover the relaxation of nearly neutral systems is usually slow; hence the need for dynamical stochastic models and for their analysis outside a state of equilibrium. The usual way of attacking this problem is by diffusion approximations, and the theory of such processes is surveyed. Questions of existence, uniqueness and adequacy of approximation are well understood, but much less is known about methods of deriving explicit quantitative results or qualitative insight. A new approach is suggested which is particularly useful for studying a locus at which many different alleles are possible. It leads, for example, to a dynamical picture (in terms of a diffusing Poisson process) for the neutral infinite-alleles model of Kimura &amp; Crow.

Stepwise mutation model and distribution of allelic frequencies in a finite population.

A mathematical theory is developed that enables us to derive a formula for the equilibrium distribution of allelic frequencies in a finite population when selectively neutral alleles are produced in stepwise fashion (stepwise mutation model). It is shown that the stepwise mutation model has a remarkable property that distinguishes it from the conventional infinite allele model (Kimura-Crow model): as the population size increases indefinitely while the product of the effective population size and the mutation rate is kept at a fixed value, the mean number of different alleles contained in the population rapidly reaches a plateau which is not much larger than the effective number of alleles (reciprocal of homozygosity).

THE EFFECT OF INTRAGENIC RECOMBINATION ON THE NUMBER OF ALLELES IN A FINITE POPULATION

A two-site infinite allele model is constructed to study the effect of intragenic recombination on the number of neutral alleles and the distribution of their frequencies in a finite population. The results of theory and Monte Carlo simulation of the two-site model demonstrate that intragenic recombination significantly increases the mean and variance of the number of alleles when the rates of mutation and recombination are as large as the reciprocal of the population size. Data from natural populations indicate that this may be a significant process in generating variation and determining its distribution.

Simulation results with stepwise mutation model and their interpretations.

Monte Carlo simulations are performed to compare the predictions based on the two presently used theoretical models for studying genetic variations in natural populations, the infinite allele model and the stepwise mutation model. Distribution of heterozygosity is noticed to be similar under these models until the product of population size and mutation rate is large. It is seen that electromorphs with high population frequency usually contain older allels (at the codon level) than an electromorph of low population frequency. The interpretations of these results in explaining the allelic variations at electrophoretic level is also discussed.

BOTTLENECK EFFECTS ON AVERAGE HETEROZYGOSITY AND GENETIC DISTANCE WITH THE STEPWISE MUTATION MODEL.

When a population goes through a small bottleneck, the genetic variability of the population is expected to decline rapidly, but as soon as population size becomes large, it starts to increase owing to new mutations (Wright, 1931; Mayr, 1954). Recently, Nei et al. (1975) studied this problem mathematically and showed that the pattern of change in genetic variability largely depends on the size of bottleneck, rate of population growth, and mutation rate. They provided a general formula for computing the average heterozygosity per locus in a population of changing size. This study is based on the assumption that new mutations occurring in a population are always different from the preexisting ones (Kimura and Crow's (1964) infinite allele model). This assumption seems to be roughly correct if allelic differences are studied at the nucleotide or codon level. In practice, however, the genetic variability of natural populations is usually studied by electrophoretic mobility of proteins. Electrophoretic mobility of a protein is determined mainly by the net charge of the protein, and a positive or negative change in the net charge due to an amino acid substitution may be canceled by the second opposite charge change. Therefore, strictly speaking, the infinite allele model is not appropriate for the study of protein variation detected by electrophoresis. In view of this circumstance, Ohta and Kimura (1973) introduced the so-called stepwise mutation model. In this model each allele is represented as one of the infinite sequence of allelic states and mutation is assumed to produce a one-step change in either the positive or the negative direction. They derived a formula for average heterozygosity in an equilibrium population. Later, Wehrhahn (1975) and Li (1976) extended Ohta and Kimura's work to the case of nonequilibrium populations. On the other hand, Nei and Chakraborty (1973) studied the expected genetic distance between two populations when these are separated for an arbitrary number of generations, using a similar genetic model. The purpose of the present paper is to study the bottleneck effects on average heterozygosity and genetic distance by using the stepwise mutation model. Some studies on the bottleneck effect (the effect of reduction in population size) on genetic distance have been made by Chakraborty and Nei (1974) and Nei (1976), using the infinite allele model. These authors have shown that the reduction in population size results in an accelerated increase in genetic distance in the early generations. In the present paper we shall employ the method of generating function first used by Nei and Chakraborty (1973) in the study of protein variation and extended considerably by Wehrhahn ( 19 7 5).

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C 0 is a constant depending on the migration law, K0 (y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log m i in two. For is known in one dimension and C 0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

Migration and mutation in stochastic models of gene frequency change. I. The island model.

Migration has in the past been introduced deterministically into stochastic gene frequency models. Migration at rate m then reduces the between-population variability by a factor of (1 - m)2 each generation. We show that with stochastic migration, whether of fixed or variable numbers of individuals, a positive term delta m is added to the variance. As a result of the delta m term, the equilibrium value of the between-population variability is increased compared to the corresponding value for deterministic migration by a factor of approximately (1 - m)-2 for small m. An equivalent result is derived for mutation, using the infinite allele model for a single population. We show in addition that these results may be derived much more simply by use of identity-by-descent probability methods, but only if a modified definition of the probability of identity-by-descent is used, involving the sampling with instead of without replacement of pairs of genes from the population.

Infinite allele model with varying mutation rate.

Available data suggest that the variation in mutation rate among protein loci follows the gamma distribution. Thus, taking into account this variation, formulae are developed for the distribution of allele frequencies, mean and variance of heterozygosity, expected number of alleles, proportion of polymorphic loci, and genetic distance. These formulae should be more appropriate for the analysis of gene frequency data for protein loci than equivalent formulae with constant mutation rate.