Estimation of the Allele Number in a Natural Population by the Method of Isofemale Lines

The number of alleles present in a natural population of unknown structure is estimated using a sequential sampling procedure applied to isofemale lines. Two questions are raised: how many individuals per isofemale line must be assayed and how many isofemale lines must be sampled to get an adequate sample to estimate the number of alleles, at a given risk, of the natural population? On the one hand, we show that when wild females are inseminated once, only two individuals per line are required. On the other hand, the number of isofemale lines that must be sampled depends on the risk chosen of losing an allele, on the number of alleles present in the population and on their drawing probabilities. When the population structure is known, an accurate answer can be provided. For an unknown population structure, one general sequential sampling previously described by J. Rouault and P. Capy is proposed to estimate the number of alleles in the population from data on isofemale lines.     

Population Bottlenecks and Nonequilibrium Models in Population Genetics. II. Number of Alleles in a Small Population That Was Formed by a Recent Bottleneck

A model is presented in which a large population in mutation/drift equilibrium undergoes a severe restriction in size and subsequently remains at the small size. The rate of loss of genetic variability has been studied. Allelic loss occurs more rapidly than loss of genic heterozygosity. Rare alleles are lost especially rapidly. The result is a transient deficiency in the total number of alleles observed in samples taken from the reduced population when compared with the number expected in a sample from a steady-state population having the same observed heterozygosity. Alternatively, the population can be considered to possess excess gene diversity if the number of alleles is used as the statistical estimator of mutation rate. The deficit in allele number arises principally from a lack of those alleles that are expected to appear only once or twice in the sample. The magnitude of the allelic deficiency is less, however, than the excess that an earlier study predicted to follow a rapid population expansion. This suggests that populations that have undergone a single bottleneck event, followed by rapid population growth, should have an apparent excess number of alleles, given the observed level of genic heterozygosity and provided that the bottleneck has not occurred very recently. Conversely, such populations will be deficient for observed heterozygosity if allele number is used as the sufficient statistic for the estimation of 4Nev. Populations that have undergone very recent restrictions in size should show the opposite tendencies.     

Overdominant alleles in a population of variable size.

An approximate method is developed to predict the number of strongly overdominant alleles in a population of which the size varies with time. The approximation relies on the strong-selection weak-mutation (SSWM) method introduced by J. H. Gillespie and leads to a Markov chain model that describes the number of common alleles in the population. The parameters of the transition matrix of the Markov chain depend in a simple way on the population size. For a population of constant size, the Markov chain leads to results that are nearly the same as those of N. Takahata. The Markov chain allows the prediction of the numbers of common alleles during and after a population bottleneck and the numbers of alleles surviving from before a bottleneck. This method is also adapted to modeling the case in which there are two classes of alleles, with one class causing a reduction in fitness relative to the other class. Very slight selection against one class can strongly affect the relative frequencies of the two classes and the relative ages of alleles in each class.     

Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Estimating the total number of alleles using a sample coverage method.

Previously reported methods for estimating the number of different alleles at a single locus in a population have not described a useful general result. Using the number of alleles observed in a sample gives an underestimate for the true number of alleles. The similar problem of estimating the number of species in a population was first investigated in 1943. In this article we use the sample coverage method proposed by Chao and Lee in 1992 to estimate the number of alleles in a population when there are unequal allele frequencies. Simulation studies under the recurrent mutation model show that, for reasonable sample sizes, a significantly better estimate of the true number can be obtained than that using only the observed alleles. Results under the stepwise mutation model and infinite-allele model are presented. Possible applications include improving the characterization of the prior distribution for the allele frequencies, adjusting the estimates of genetic diversity, and estimating the range of microsatellite alleles.     

Selection and segregation distortion in a sex-differentiated population

We extend the classical model for selection at an autosomal locus in a sex-differentiated population to include segregation distortion. The equations remain the same, but the fitness parameters are interpreted differently and refer to alleles instead of genotypes. We derive conditions for internal and external stability of the equilibria, i.e., stability with respect to perturbations of alleles that are already present at equilibrium and stability with respect to invasion attempts by newly arising alleles. We show that, in a sex-differentiated population, external stability of an equilibrium can be judged on the basis of Shaw--Mohler criteria. Throughout, we compare the situation in populations with and without sex differentiation. Interestingly, internal stability is more difficult to achieve in a population without sex differentiation than in a population in which selection and segregation distortion are restricted to one sex. In a companion paper we show how the general results of the present paper can lead to new insights into specific systems such as the t complex of the house mouse.     

Gene flow's effect on the genetic architecture of a local adaptation and its consequences for QTL analyses

This paper uses computer simulations to determine how gene flow between populations affects (1) the genetic architecture of a local adaptation and (2) properties of alleles segregating in quantitative trait locus (QTL) mapping populations. Results suggest that the average magnitude of an allele that causes a phenotypic difference between populations declines as the migration rate increases, but with an increase in migration, alleles of larger magnitude cause proportionally more of the phenotypic difference between populations. Gene flow between populations that are used in a QTL study tends to cause the average magnitude and percent variance explained (PVE) of an allele in a mapping population to increase. Thus, although the average magnitude of an allele causing a difference declines with migration the average magnitude or PVE of an allele in a QTL mapping population may increase. The reason is that the probability an allele is sampled for a QTL mapping population is in direct proportion to its frequency and alleles of larger magnitude tend to segregate at relatively higher frequencies than alleles of smaller effect with an increased migration. As the rate of gene flow between populations increases, the proportion of the phenotypic difference explained by alleles that are segregating in a QTL mapping population (and therefore potentially detected) decreases. Lastly, results suggest QTL alleles of large effect (>20% PVE) should be commonly found, provided the divergence time between populations is not too long or optima of populations are not too far apart.     

DAT1 VNTR polymorphisms in a European and an African population: identification of a new allele

Polymorphism frequencies of the dopamine transporter gene (DAT1) hypervariable region have been analyzed in a sample of Italian and Ivory Coast individuals. The 3' untranslated region (UTR) of DAT1 includes a variable number of tandem repeats (VNTR) of a 40-bp monomer, ranging from 3 to 13 repeats in Caucasian and African populations. In our sample we found alleles with 3 to 16 repeats, and the most common alleles were the 10-repeat (DAT1*10) and the 9-repeat (DAT1*9) alleles. We also found two rare alleles in the Italian population and four in the Ivory Coast population. For the first time the new allele DAT1*16 is described in the Ivorians. The Ivory Coast population was not in Hardy-Weinberg equilibrium for the DAT1 locus because of a deficit of heterozygote genotypes. The observed heterozygosity of the Ivorian population was half that of the Italians. The lower observed heterozygosity and deviation from Hardy-Weinberg equilibrium could be the result of microevolutionary trends, such as genetic drift and/or inbreeding, acting on the relatively small and isolated population sampled for this study, although some sort of selective pressures acting against the shorter alleles cannot be excluded. This evidence, in association with the reduced polymorphism shown by the DAT1 VNTR compared to other VNTRs, seems to indicate that the DAT1 locus may be under some selective pressure.     

‘True’ null allele detection in microsatellite loci: a comparison of methods,assessment of difficulties and survey of possible improvements

Null alleles are alleles that for various reasons fail to amplify in a PCR assay. The presence of null alleles in microsatellite data is known to bias the genetic parameter estimates. Thus, efficient detection of null alleles is crucial, but the methods available for indirect null allele detection return inconsistent results. Here, our aim was to compare different methods for null allele detection, to explain their respective performance and to provide improvements. We applied several approaches to identify the ‘true’ null alleles based on the predictions made by five different methods, used either individually or in combination. First, we introduced simulated ‘true’ null alleles into 240 population data sets and applied the methods to measure their success in detecting the simulated null alleles. The single best‐performing method was ML‐NullFreq_frequency. Furthermore, we applied different noise reduction approaches to improve the results. For instance, by combining the results of several methods, we obtained more reliable results than using a single one. Rule‐based classification was applied to identify population properties linked to the false discovery rate. Rules obtained from the classifier described which population genetic estimates and loci characteristics were linked to the success of each method. We have shown that by simulating ‘true’ null alleles into a population data set, we may define a null allele frequency threshold, related to a desired true or false discovery rate. Moreover, using such simulated data sets, the expected null allele homozygote frequency may be estimated independently of the equilibrium state of the population.     

Estimation of the Number of Sex Alleles and Queen Matings from Diploid Male Frequencies in a Population of APIS MELLIFERA

The distribution of diploid males in a population of Apis mellifera was obtained by direct examination of the sexual phenotypes of the larvae. Using these data, estimates are derived for the number of sex alleles and the number of matings undergone by the queen. The number of sex alleles is estimated to be 18.9. The estimate is larger than previous ones, which have ranged between 10 and 12. However, the increase in the number of sex alleles can be explained by the large effective population number for our data. The best estimator of the number of matings by a queen is a maximum likelihood type that assumes a prior distribution on the number of matings. For the data presented here, this estimate is 17.3. This estimate is compared to others in the literature obtained by different approaches.     

Variance and covariance of homozygosity in a structured population

The variance of homozygosity for a K-allele model with n partially isolated subpopulations is derived numerically using identity coefficients. The variance of homozygosity within a subpopulation is shown to depend strongly upon the migration rates between subpopulations but is not strongly influenced by the number of alleles possible at a locus. The variance of homozygosity within a subpopulation, given the value of expected homozygosity, is approximately equal to the value of the variance of homozygosity given by Stewart's formula for a single population. If the population is presumed to be panmictic, but is actually subdivided, and the gametes are sampled at random from the total population, the apparent variance of homozygosity depends on the number of alleles possible. With small migration rates and K large, the apparent variance of homozygosity is much smaller than in a single population with the same expected homozygosity. However, when K is small, the variance of homozygosity is approximately given by Stewart's formula. The transient behavior of the variance of homozygosity shows that a large number of generations may be required to approach equilibrium values.     

The spatial distribution of transient alleles in a subdivided population: a simulation study

The spatial distributions of newly introducted alleles in a subdivided population are generated using a computer program to model the processes of selection, gene flow and genetic drift. Advantageous, neutral and deleterious alleles are considered, and certain aspects of the patterns generated by new alleles that are ultimately fixed and ultimately lost are examined. To characterize the spatial pattern of rare alleles, the distribution, P(i), the probability that the new allele is found in exactly i local populations before it is lost, is defined and estimated from the simulations. The shape of the P(i) distribution is surprisingly similar for selected and neutral alleles. For advantageous alleles going to fixation, the "wave of advance" is set up quickly, but stochastic effects reduce the wave speed from Fisher's (1937) value. Gene flow is much more effective in dispersing alleles in a two-dimensional array than in one dimension. Long distance gene flow has a much smaller effect in two dimensions than in one dimension.     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

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.     

Geographical distribution of a neutral allele considered as a branching process

The limiting spacial correlations are derived for a population of neutral alleles migrating among K colonies. The allelic population is modeled as a subcritical branching process and the limiting correlations are obtained conditional on nonextinction of the population.     

Allele frequencies after a bottleneck

The effect that a recent change in population size (a “bottleneck”) has on the genetic composition of a random sample of genes is studied. The population is assumed to evolve as in the Wright-Fisher model with infinitely many neutral alleles. Simple analytic formulas are found for such quantities as the probability distribution and moments of the total number of alleles, the allelic “frequency spectrum,” and the homozygosity, in the sample. Numerical examples are given which compare these results with those obtained previously by a variety of other methods.     

The Effects of Overdominance on Linkage in a Multilocus System

Computer simulations were performed with overdominant multiple alleles among tightly linked multiple loci under a multiplicative fitness model. The quantity X²/N(n — 1) was introduced as a new measure of linkage disequilibrium which, unlike previously available measures, can be applied to multiple allele models, where N is the sample size, and n is the number of alleles at the locus possessing fewest alleles. Simulations showed that (1) With multiple (three or four) alleles, the approach to stable disequilibrium is slower and the amount of disequilibrium established is weaker than in a two allele system. (2) The number of complementary chromosomes is a function of number of alleles and of population size. (3) As population size increases, the rate of the approach to stable disequilibrium is slower. (4) There is an optimum selection coefficient which minimizes the transient fixation probability of alleles when linkage is present. (5) The absence of linkage disequilibrium is in most cases not a practical method of testing the hypothesis of balancing selection of genetic polymorphisms because it depends strongly on population size in determining linkage disequilibria.     

Average number of nucleotide differences in a sample from a single subpopulation: a test for population subdivision

Unbiased estimates of θ = 4Nµ in a random mating population can be based on either the number of alleles or the average number of nucleotide differences in a sample. However, if there is population structure and the sample is drawn from a single subpopulation, these two estimates of θ behave differently. The expected number of alleles in a sample is an increasing function of the migration rates, whereas the expected average number of nucleotide differences is shown to be independent of the migration rates and equal to 4N_Tµ for a general model of population structure which includes both the island model and the circular stepping-stone model. This contrast in the behavior of these two estimates of θ is used as the basis of a test for population subdivision. Using a Monte-Carlo simulation developed so that independent samples from a single subpopulation could be obtained quickly, this test is shown to be a useful method to determine if there is population subdivision.     

Estimation of the frequency of hexosaminidase a variant alleles in the American Jewish population.

There appear to be several alleles of the hexosaminidase A (HEX A) gene that lead to different clinical syndromes. In addition to the infantile-onset Tay-Sachs disease (TSD), there is a juvenile-onset and an adult-onset form, which are also characterized by low HEX A levels. There are also apparently healthy adults with low HEX A activity. Based primarily on data from population screening for TSD carrier status, we estimate the allele frequency of the combined variant alleles for which data are available to be about 4.5 x 10(-4) and the frequency of adults showing zero HEX A levels (when tested using artificial substrate) to be about 1:67,000. The implications for population screening and prenatal diagnosis are discussed.     

Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is 1/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from 1/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.