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.     

Estimating mutation rate and generation time from longitudinal samples of DNA sequences

We present in this paper a simple method for estimating the mutation rate per site per year which also yields an estimate of the length of a generation when mutation rate per site per generation is known. The estimator, which takes advantage of DNA polymorphisms in longitudinal samples, is unbiased under a number of population models, including population structure and variable population size over time. We apply the new method to a longitudinal sample of DNA sequences of the env gene of human immunodeficiency virus type 1 (HIV-1) from a single patient and obtain 1.62 x 10(-2) as the mutation rate per site per year for HIV-1. Using an independent data set to estimate the mutation rate per generation, we obtain 1.8 days as the length of a generation of HIV-1, which agrees well with recent estimates based on viral load data. Our estimate of generation time differs considerably from a recent estimate by Rodrigo et al. when the same mutation rate per site per generation is used. Some factors that may contribute to the difference among different estimators are discussed.     

Linkage disequilibrium, mutational analysis and natural selection in the repetitive region of the clock gene, period, in Drosophila melanogaster

We have used the method of disequilibrium pattern analysis to examine associations between the threonine-glycine (Thr-Gly) encoding repeat region of the clock gene period (per) of Drosophila melanogaster, and polymorphic sites both upstream and downstream of the repeat, in a number of European fly populations. The results are consistent with the view that selection may be operating on various haplotypes which share the Thr-Gly length alleles encoding 17, 20 and 23 dipeptide pairs, and that the repeat itself may be the focus for selection. These conclusions lend support to a number of other population and behavioural investigations which have provided evidence that selection is acting on the Thr-Gly region. The linkage analysis was also used to infer an approximate mutation rate (mu) for the repeat, of 10(-5) < mu < 4 x 10(-5) per gamete per generation. Direct measurements of the mutation rate using the polymerase chain reaction in a pedigree analysis of tens of thousands of individuals do not contradict this value. Consequently, the Thr-Gly repeat does not have a mutation rate that is as high as some of the non-coding minisatellites, but it is several orders of magnitude higher than the nucleotide substitution rate. The implications of this elevated mutation rate for linkage disequilibria and selection are discussed.     

Gene and Allelic Genealogies at a Gametophytic Self-Incompatibility Locus

The properties of gene and allelic genealogies at a gametophytic self-incompatibility locus in plants have been investigated analytically and checked against extensive numerical simulations. It is found that, as with overdominant loci, there are two genealogical processes with markedly different time scales. First, functionally distinct allelic lines diverge on an extremely long time scale which is inversely related to the mutation rate to new alleles. These alleles show a genealogical structure which is similar, after an appropriate rescaling of time, to that described by the coalescent process for genes at a neutral locus. Second, gene copies sampled within the same functional allelic line show genealogical relationships similar to neutral gene genealogies but on a much shorter time scale, which is on the same order of magnitude as the harmonic mean of the number of gene copies within an allelic line. These results are discussed in relation to data showing trans-specific polymorphisms for alleles at the gametophytic self-incompatibility locus in the Solanaceae. It is shown that population sizes on the order of 4 X 10(5) and a mutation rate per locus per generation as high as 10(-6) could account for estimated allelic divergence times in this family.     

The average number of sites separating DNA sequences drawn from a subdivided population

The "infinite sites" model in the absence of recombination is examined in a subdivided population in which there is arbitrary migration among demes. It is shown that, if the migration matrix is symmetric and irreducible, the average number of sites that differ in two alleles chosen from the same deme depends only on an effective size of the whole population and not on either the elements of the migration matrix or the size of each deme separately. If there are n demes all of size N, the average number of sites that differ in two alleles chosen from the same deme is 4nN mu, where mu is the average mutation rate per site. This is the same value as for two alleles drawn from a panmictic population of size nN. The average number of sites that differ in alleles drawn from the same and from different demes can provide some information about the degree of population subdivision, as is illustrated by using the data of Kreitman and Aquadé (1986, Proc. Nat. Acad. Sci. U.S.A., 83, 3562) on Drosophila melanogaster.     

Persistence time of loss-of-function mutations at nonessential loci affecting eye color in Drosophila melanogaster

Persistence time of a mutant allele, the expected number of generations before its elimination from the population, can be estimated as the ratio of the number of segregating mutations per individual over the mutation rate per generation. We screened two natural populations of Drosophila melanogaster for mutations causing clear-cut eye phenotypes and detected 25 mutant alleles, falling into 19 complementation groups, in 1164 haploid genomes, which implies 0.021 eye mutations/genome. The de novo haploid mutation rate for the same set of loci was estimated as 2 x 10(-4) in a 10-generation mutation-accumulation experiment. Thus, the average persistence time of all mutations causing clear-cut eye phenotypes is approximately 100 generations (95% confidence interval: 61-219). This estimate shows that the strength of selection against phenotypically drastic alleles of nonessential loci is close to that against recessive lethals. In both cases, deleterious alleles are apparently eliminated by selection against heterozygous individuals, which show no visible phenotypic differences from wild type.     

Quasi-equilibrium theory for the distribution of rare alleles in a subdivided population: justification and implications

This paper examines a quasi-equilibrium theory of rare alleles for subdivided populations that follow an island-model version of the Wright-Fisher model of evolution. All mutations are assumed to create new alleles. We present four results: (1) conditions for the theory to apply are formally established using properties of the moments of the binomial distribution; (2) approximations currently in the literature can be replaced with exact results that are in better agreement with our simulations; (3) a modified maximum likelihood estimator of migration rate exhibits the same good performance on island-model data or on data simulated from the multinomial mixed with the Dirichlet distribution, and (4) a connection between the rare-allele method and the Ewens Sampling Formula for the infinite-allele mutation model is made. This introduces a new and simpler proof for the expected number of alleles implied by the Ewens Sampling Formula.     

Polymorphism and Balancing Selection at Major Histocompatibility Complex Loci

Amino acid replacements in the peptide-binding region (PBR) of the functional major histocompatibility complex (Mhc) genes appear to be driven by balancing selection. Of the various types of balancing selection, we have examined a model equivalent to overdominance that confers heterozygote advantage. As discussed by A. Robertson, overdominance selection tends to maintain alleles that have more or less the same degree of heterozygote advantage. Because of this symmetry, the model makes various testable predictions about the genealogical relationships among different alleles and provides ways of analyzing DNA sequences of Mhc alleles. In this paper, we analyze DNA sequences of 85 alleles at the HLA-A, -B, -C, -DRB1 and -DQB1 loci with respect to the number of alleles and extent of nucleotide differences at the PBR, as well as at the synonymous (presumably neutral) sites. Theory suggests that the number of alleles that differ at the sites targeted by selection (presumably the nonsynonymous sites in the PBR) should be equal to the mean number of nucleotide substitutions among pairs of alleles. We also demonstrate that the nucleotide substitution rate at the targeted sites relative to that of neutral sites may be much larger than 1. The predictions of the presented model are in surprisingly good agreement with the actual data and thus provide means for inferring certain population parameters. For overdominance selection in a finite population at equilibrium, the product of selection intensity (s) against homozygotes and the effective population size (N) is estimated to be 350-3000, being largest at the B locus and smallest at the C locus. We argue that N is of the order of 10(5) and s is several percent at most, if the mutation rate per site per generation is 10(-8).     

Male-biased mutation rates and the overestimation of extrapair paternity: problem, solution, and illustration using thick-billed murres (Uria lomvia, Alcidae)

The widespread utility of hypervariable loci in genetic studies derives from the high mutation rate, and thus the high polymorphism, of these loci. Recent evidence suggests that mutation rates can be extremely high and may be male biased (occurring in the male germ-line). These two factors combined may result in erroneous overestimates of extrapair paternity, since legitimate offspring with novel alleles will have more mismatches with respect to the biological father than the biological mother. As mutations are male driven, increasing the number of hypervariable loci screened may simply increase the number of mismatches between fathers and their legitimate offspring. Here we describe a simple statistic, the probability of resemblance (PR), to distinguish between mismatches due to parental misassignment versus mutation in either sex or null alleles. We apply this method to parentage data on thick-billed murres (Uria lomvia), and demonstrate that, without considering either mutations or male-biased mutation rates, cases of extrapair paternity (7% in this study) would be grossly overestimated (14.5%-22%). The probability of resemblance can be utilized in parentage studies of any sexually reproducing species when allele or haplotype frequency data are available for putative parents and offspring. We suggest calculating this probability to correctly categorize legitimate offspring when mutations and null alleles may cause mismatches.     

Using molecular markers with high mutation rates to obtain estimates of relative population size and to distinguish the effects of gene flow and mutation: a demonstration using data from endemic Mauritian skinks

We propose a method of analysing genetic data to obtain separate estimates of the size (N(p)) and migration rate (m(p)) for the sampled populations, without precise prior knowledge of mutation rates at each locus ( micro(L)). The effects of migration and mutation can be distinguished because high migration has the effect of reducing genetic differentiation across all loci, whereas a high mutation rate will only affect the locus in question. The method also takes account of any differences between the spectra of immigrant alleles and of new mutant alleles. If the genetic data come from a range of population sizes, and the loci have a range of mutation rates, it is possible to estimate the relative sizes of the different N(p) values, and likewise the m(p) and the micro(L). Microsatellite loci may also be particularly appropriate because loci with a high mutation rate can reach mutation-drift-migration equilibrium more quickly, and because the spectra of mutants arriving in a population can be particularly distinct from the immigrants. We demonstrate this principle using a microsatellite data set from Mauritian skinks. The method identifies low gene flow between a putative new species and populations of its sister species, whereas the differentiation of two other populations is attributed to small population size. These distinct interpretations were not readily apparent from conventional measures of genetic differentiation and gene diversity. When the method is evaluated using simulated data sets, it correctly distinguishes low gene flow from small population size. Loci that are not at mutation-migration-drift equilibrium can distort the parameter estimates slightly. We discuss strategies for detecting and overcoming this effect.     

Rare variant alleles in the light of the neutral theory

Based on the neutral theory of molecular evolution and polymorphism, and particularly assuming "the model of infinite alleles," a method is proposed which enables us to estimate the fraction of selectively neutral alleles (denoted by Pneut) among newly arisen mutations. It makes use of data on the distribution of rare variant alleles in large samples together with information on the average heterozygosity. The formula proposed is Pneut = [He/(1-He)] [loge(2nq)/n alpha (x less than q)], where n alpha(x less than q) is the average number of rare alleles per locus whose frequency, x, is less than q; n is the average sample size used to count rare alleles; He is the average heterozygosity per locus; and q is a small preassigned number such as q = 0.01. The method was applied to observations on enzyme and other protein loci in plaice, humans (European and Amerindian), Japanese monkeys, and fruit flies. Estimates obtained for them range from 0.064 to 0.21 with the mean and standard error Pneut = 0.14 +/- 0.06. It was pointed out that these estimates are consistent with the corresponding estimate Pneut(Hb) = 0.14 obtained independently based on the neutral theory and using data on the evolutionary rate of nucleotide substitutions in globin pseudogenes together with those in the normal globins.      

Genetic profile of cosmopolitan populations: effects of hidden subdivision

Natural populations of many organisms exhibit excess of rare alleles in comparison with the predictions of the neutral mutation hypothesis. It has been shown before that either a population bottleneck or the presence of slightly deleterious mutations can explain this phenomenon. A third explanation is presented in this work, showing that hidden subdivision within a population can also lead to an excess of rare alleles in the total population when the expectations of the neutral model are based on the allele frequency profile of the entire population data. With two examples (mitochondrial DNA-morph distribution and isozyme allele frequency distributions), it is shown that most cosmopolitan human populations exhibit excess of rare as well as total allele counts, when these are compared with the expectations of the neutral mutation hypothesis. The mitochondrial data demonstrate that such excesses can be detected from genetic variation at a single locus as well, and this is not due to stochastic error of allele frequency distributions. Contrast of the present observations with the allele frequency profiles in agglomerated tribal populations from South and Central America shows that even when the neutral expectations hold for individual subpopulations, if all subpopulations are grouped into a single population, the pooled data exhibit an excess of total number of alleles that is mainly due to the excess of rare alleles. Therefore, a primary cause of the excess number of rare alleles could be the hidden subdivision, and the magnitude of the excess indicates the extent of substructuring. The two components of hidden subdivision are: 1) Number of subpopulations, and 2) the average genetic distance among them. The implications of this observation in estimating mutation rate are discussed indicating the difficulties of comparing mutation rates from different population surveys.     

King''s Formula for the Mutation Load with Epistasis

A formula by J. L. King gives the equilibrium mutation load as L = 2 sigma ui(1 - qi)/z - x) in which ui is the mutation rate to deleterious alleles at the ith locus, qi is the frequency of mutant alleles at this locus, x is the mean number of such mutant genes per individual before selection, z is the mean number in individuals eliminated by selection, and the summation is over all relevant loci. We show that this rule is inaccurate for intense selection and that a correct formula is L = 2 sigma ui(1 - qi) w/(z - x) = 2U w/(z - x) = 2U/(z - x + 2U) in which U is the mean number of new mutations per haploid genome in the population and w is the mean relative fitness before selection. If w/(z - x) less than 1/2, the mutation load is less than the Haldane value (U less than or equal to L less than or equal to 2U) and can be considerably less. In a diploid asexual population, however, with independent occurrence of mutations, L = 1 - e-2U regardless of the mode of selection.     

The mutation process of microsatellites during the polymerase chain reaction.

We build a mathematical model for the mutation process of microsatellites during polymerase chain reaction (PCR) using the theory of branching processes. Based on the model, we develop a method to estimate the mutation rate of microsatellites per PCR cycle and the probability of expansion by maximizing a quasi-likelihood of the observed data. We show by simulations that the proposed estimation method can accurately recover the relationship between the mutation rate and number of repeat units. The theoretical basis for the proposed method is also given. We apply the method to experimental data on poly-A and poly-CA repeats.     

Genetic identity between populations when mutation rates vary within and across loci

A formula is derived for the probability that two genes taken at random from the same locus in two populations isolated at time t ago are of the same allelic type. The model assumed is a neutral one where there are possibly different mutation rates between different alleles. Inequalities are derived for this probability. A particular result is that for a fixed overall mutation rate, the probability is least for the infinite alleles model. Inequalities and approximations are found for Nei's genetic identity at one locus when mutation rates vary, and also for the identity across loci when the overall mutation rates per locus vary. Genetic identity at the molecular level is considered and a probability generating function found for the number of segregating sites between two randomly chosen gametes from two divergent populations, under various models.     

Multiple paternity and female-biased mutation at a microsatellite locus in the olive ridley sea turtle (Lepidochelys olivacea)

Multiple paternity in the olive ridley sea turtle (Lepidochelys olivacea) population nesting in Suriname was demonstrated using two microsatellite loci, viz., Ei8 and Cm84. The large number of offspring sampled per clutch (70 on average, ranging from 15 to 103) and the number of alleles found at the two loci (18 and eight alleles, respectively) enabled unambiguous assessment of the occurrence of multiple paternity. In two out of 10 clutches analysed, the offspring had been sired by at least two males, which was confirmed at both loci. In both clutches, unequal paternity occurred: 73% and 92% of the offspring had been sired by the primary male. The probability of detecting multiple paternity was 0.903, and therefore there is a small chance that multiple paternity occurred but remained undetected in some of the eight clutches that appeared to be singly sired. Analysis of 703 offspring revealed a high mutation rate for locus Ei8 (micro = 2.3 x 10(-2)) with all 33 mutations occurring in maternal alleles. In particular, one allele of 274 bp mutated at a high frequency in a clutch to which the mother contributed the allele, but in another clutch where the father contributed the same allele, no such mutations were observed. Inferred allele-specific mutation rates for Ei8 and expected numbers of mutations per clutch confirmed that maternal alleles for Ei8 are more likely to mutate in the olive ridley sea turtle than paternal alleles. Possible explanations are discussed.     

Mutation rate in human microsatellites: influence of the structure and length of the tandem repeat.

In 10,844 parent/child allelic transfers at nine short-tandem-repeat (STR) loci, 23 isolated STR mismatches were observed. The parenthood in each of these cases was highly validated (probability >99.97%). The event was always repeat related, owing to either a single-step mutation (n=22) or a double-step mutation (n=1). The mutation rate was between 0 and 7 x 10(-3) per locus per gamete per generation. No mutations were observed in three of the nine loci. Mutation events in the male germ line were five to six times more frequent than in the female germ line. A positive exponential correlation between the geometric mean of the number of uninterrupted repeats and the mutation rate was observed. Our data demonstrate that mutation rates of different loci can differ by several orders of magnitude and that different alleles at one locus exhibit different mutation rates.     

Estimation of effective population size and migration rate from one- and two-locus identity measures

Standard methods for inferring demographic parameters from genetic data are based mainly on one-locus theory. However, the association of genes at different loci (e.g., two-locus identity disequilibrium) may also contain some information about demographic parameters of populations. In this article, we define one- and two-locus parameters of population structure as functions of one- and two-locus probabilities for the identity in state of genes. Since these parameters are known functions of demographic parameters in an infinite island model, we develop moment-based estimators of effective population size and immigration rate from one- and two-locus parameters. We evaluate this method through simulation. Although variance and bias may be quite large, increasing the number of loci on which the estimates are derived improves the method. We simulate an infinite allele model and a K allele model of mutation. Bias and variance are smaller with increasing numbers of alleles per locus. This is, to our knowledge, the first attempt of a joint estimation of local effective population size and immigration rate.     

An Evaluation of Evolutionary Constraints on Microsatellite Loci Using Null Alleles

A test to evaluate constraints on the evolution of single microsatellite loci is described. The test assumes that microsatellite alleles that share the same flanking sequence constitute a series of alleles with a common descent that is distinct from alleles with a mutation in the flanking sequence. Thus two or more different series of alleles at a given locus represent the outcomes of different evolutionary processes. The higher rate of mutations within the repeat region (10(-3) or 10(-4)) compared with that of insertion/deletion or point mutations in adjacent flanking regions (10(-9)) or with that of recombination between the repeat and the point mutation (10(-6) for sequences 100 bp long) provides the rationale for this assumption. Using a two-phase, stepwise mutation model we simulated the evolution of a number of independent series of alleles and constructed the distributions of two similarity indices between pairs of these allele series. Applying this approach to empirical data from locus AG2H46 of Anopheles gambiae resulted in a significant excess of similarity between the main and the null series, indicating that constraints affect allele distribution in this locus. Practical considerations of the test are discussed.     

Mutation in microsatellite repeats of DNA and embryonal death in humans

In the analysis of tetranucleotide DNA repeats inheritance carried out in 55 families with a history of spontaneous miscarriages and normal karyotypes in respect to 21 loci located on seven autosomes, 8 embryos (14.5%) demonstrating 12 cases of the presence of alleles absent in both parents were described. The study of chromosome segregation using other DNA markers permitted highly probable exclusion of false paternity as well as uniparental disomy as the reasons for parent/child allele mismatches. The high probability of paternity together with the presence of a "new" allele at any offspring locus points to the mutation having occurred during game-togenesis in one of the parents. Examination of mutation in spontaneous abortuses revealed an increased number of tandem repeat units at microsatellite loci in three cases and an decreased number of these repeats in six cases. In two abortuses, a third allele absent in both parents, which resulted from a somatic mutation that occurred during embryonic development, was observed. The prevalence of the male germline mutations, revealed during investigation of the mutation origin, was probably associated with an increased number of DNA replication cycles in sperm compared to the oocytes. In spontaneous abortuses, the mean mutation rate of the tetranucleotide repeat complexes analyzed was 9.8 x 10(-3) per locus per gamete per generation. This was about five times higher than the spontaneous mutation rate of these STR loci. It can be suggested that genome instability detected at the level of repeated DNA sequences can involve not only genetically neutral loci but also active genomic regions crucial for embryonic viability. This results in cell death and termination of embryonic development. Our findings indicate that the death of embryos with normal karyotypes in most cases is associated with an increased frequency of germline and somatic microsatellite mutations. The data of the present study also provide a practical tool for the quantitative evaluation of this phenomenon and for the analysis of the reasons for miscarriages and embryonic death in certain families.