The Age of a Neutral Mutant Persisting in a Finite Population

Formulae for the mean and the mean square age of a neutral allele which is segregating with frequency x in a population of effective size N(e) have been obtained using the diffusion equation method, for the case of 4N(e)v<1 where v is the mutation rate. It has been shown that the average ages of neutral alleles, even if their frequencies are relatively low, are quite old. For example, a neutral mutant whose current frequency is 10% has the expected age roughly equal to the effective population size N(e) and the standard deviation 1.4N(e) (in generations), assuming that this mutant has increased by random drift from a very low frequency. Also, formulae for the mean "first arrival time" of a neutral mutant to a certain frequency x have been presented. In addition, a new, approximate method has been developed which enables us to obtain the condition under which frequencies of "rare" polymorphic alleles among local populations are expected to be uniform if the alleles are selectively neutral.-It was concluded that exchange of only a few individuals on the average between adjacent colonies per generation is enough to bring about such a uniformity of frequencies.     

The first arrival time and mean age of a deleterious mutant gene in a finite population.

The mean and standard deviation of the first arrival time for a single mutant to reach a certain frequency and the mean age of a mutant persisting in a population have been studied using diffusion methods. These quantities are shown to be highly dependent on both the heterozygous effect and the population size. For partially recessive deleterious mutations, both the mean first arrival time and the mean age decrease with increasing selection coefficient against heterozygotes. For overdominant mutations, the mean age always increases very rapidly with increasing heterozygous advantage, while the mean first arrival time first increases rapidly with increasing heterozygous advantage to a maximum and then decreases rapidly with increasing heterozygous advantage. The standard deviation of the first arrival time is small while that of the age is large. The results of this study have been applied to study the case of the sickle cell anemia mutant in Africa. It is argued that the present prevalence may be explained without the necessity of quite so great a heterozygous advantage as .25 or higher as proposed by some workers. A reasonable range for the heterozygous advantage seems to be from .05 to .18.     

A simple proof that certain quantities are independent of the geographical structure of population

This paper gives a proof that certain quantities are independent of the geographical structure of a population. The quantities are: (1) the fixation probability of a mutant; (2) the sum of the quantity x(1 ? x), where x is the mutant frequency, while the mutant is segregating; and (3) the quantity x(1 ? x) summed over the generations during which the gene frequency in the whole population assumes a specified value. The independence of geographical structure for the latter two quantities is not exact if there is selection, but is a close approximation.The model is a geographically structured version of Moran's haploid overlapping generation model. The population consists of colonies connected genetically by migration. Each individual has the same negative exponential lifetime distribution. When an individual dies, it is immediately replaced by an individual born in the same colony with a probability proportional to the frequency and fitness of the type giving birth. In a diploid population the quantity x(1 ? x) is proportional to the heterozygosity.     

Reversibility and the age of an allele. I. Moran's infinitely many neutral alleles model

A population of constant size is subjected to mutation, such that each mutant is of a new allelic type. For the particular population model studied in this paper, the age of an allele, whose present frequency is known, is a random variable with distribution independent of the frequencies of other alleles. As a consequence of reversibility of the population process, the age of an allele, from the past to the present, has the same distribution as its time to extinction, from the present into the future. This verifies, and re-interprets, certain diffusion approximations found by Kimura and Ohta [Genetics 75, 199–212 (1973)] and Maruyama [Genet. Res. Cambridge 23, 137–143 (1974)].     

Population stratification may bias analysis of PGC-1α as a modifier of age at Huntington disease motor onset

Huntington’s disease (HD) is an inherited neurodegenerative disorder characterized by motor, cognitive and behavioral disturbances, caused by the expansion of a CAG trinucleotide repeat in the HD gene. The CAG allele size is the major determinant of age at onset (AO) of motor symptoms, although the remaining variance in AO is highly heritable. The rs7665116 SNP in PPARGC1A, encoding the mitochondrial regulator PGC-1α, has been reported to be a significant modifier of AO in three European HD cohorts, perhaps due to affected cases from Italy. We attempted to replicate these findings in a large collection of (1,727) HD patient DNA samples of European origin. In the entire cohort, rs7665116 showed a significant effect in the dominant model (p value?=?0.008) and the additive model (p value?=?0.009). However, when examined by origin, cases of Southern European origin had an increased rs7665116 minor allele frequency (MAF), consistent with this being an ancestry-tagging SNP. The Southern European cases, despite similar mean CAG allele size, had a significantly older mean AO (p?<?0.001), suggesting population-dependent phenotype stratification. When the generalized estimating equations models were adjusted for ancestry, the effect of the rs7665116 genotype on AO decreased dramatically. Our results do not support rs7665116 as a modifier of AO of motor symptoms, as we found evidence for a dramatic effect of phenotypic (AO) and genotypic (MAF) stratification among European cohorts that was not considered in previously reported association studies. A significantly older AO in Southern Europe may reflect population differences in genetic or environmental factors that warrant further investigation.     

Empirical Distributions of F ST from Large-Scale Human Polymorphism Data

Studies of the apportionment of human genetic variation have long established that most human variation is within population groups and that the additional variation between population groups is small but greatest when comparing different continental populations. These studies often used Wright’s F _ST that apportions the standardized variance in allele frequencies within and between population groups. Because local adaptations increase population differentiation, high-F _ST may be found at closely linked loci under selection and used to identify genes undergoing directional or heterotic selection. We re-examined these processes using HapMap data. We analyzed 3 million SNPs on 602 samples from eight worldwide populations and a consensus subset of 1 million SNPs found in all populations. We identified four major features of the data: First, a hierarchically F _ST analysis showed that only a paucity (12%) of the total genetic variation is distributed between continental populations and even a lesser genetic variation (1%) is found between intra-continental populations. Second, the global F _ST distribution closely follows an exponential distribution. Third, although the overall F _ST distribution is similarly shaped (inverse J), F _ST distributions varies markedly by allele frequency when divided into non-overlapping groups by allele frequency range. Because the mean allele frequency is a crude indicator of allele age, these distributions mark the time-dependent change in genetic differentiation. Finally, the change in mean-F _ST of these groups is linear in allele frequency. These results suggest that investigating the extremes of the F _ST distribution for each allele frequency group is more efficient for detecting selection. Consequently, we demonstrate that such extreme SNPs are more clustered along the chromosomes than expected from linkage disequilibrium for each allele frequency group. These genomic regions are therefore likely candidates for natural selection.     

Frequency analysis of the delta32ccr5 HIV resistance allele in a medieval plague mass grave

The 32 basepair deletion in the gene for the human chemokine receptor CCR5 (delta32ccr5) conferring resistance against HIV-1 infection is present in Caucasian populations. The mutant allele is believed to have originated by a single mutational event in historic times and to have reached its present population frequency of an average 10 % in Europe through selective pressure by a pathogenic agent. Because of their great impact on European populations, the medieval Plague epidemics have been considered as a possible candidate. To test this hypothesis, we studied the delta32ccr5-frequency in 35 individuals from a mass grave containing victims of the 14th century Plague pandemic in Lübeck, Northern Germany, and compared them to the frequency in a control group from the same burial site, dating from the time before the first Plague pandemic. If the delta32ccr5 allele conferred an at least partial resistance against the medieval Plague, its frequency would be expected to be lower in those that died in the pandemic, than it was in the local population before the arrival of the Plague. The CCR5 locus could be typed successfully for 14 Plague victims and for 20 individuals from the medieval control group. We found a delta32ccr5 allelic frequency of 14.2% and 12.5%, respectively. The difference between these figures is not statistically significant. Furthermore, they are comparable to the delta32ccr5 frequency for nowadays Northern Europe. We therefore conclude that the medieval Plague pandemic has not contributed to an increase in the allelic frequency of the mutant delta32ccr5 allele and that, if there has been a positive selection of this allele, it is likely to have occurred before the 14th century and thus before the arrival of the Plague in Europe.     

Extinction time and age of an allele in a large finite population

For a single locus with two alleles we study the expected extinction and fixation times of the alleles under the influence of selection and genetic drift. Using a diffusion model we derive asymptotic approximations for these expected exit times for large populations. We consider the case where the fitness of the heterozygote is in between the fitnesses of the homozygotes (incomplete dominance). The asymptotic analysis not only yields new quantitative results but also reveals interesting features that remain hidden in the exact solution. Some of the outcomes are extensions of results known in the literature. The asymptotic approximations also apply to the expected first arrival time of an allele at a specified frequency and to the expected age of an allele.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.     

An Age-of-Allele Test of Neutrality for Transposable Element Insertions

How natural selection acts to limit the proliferation of transposable elements (TEs) in genomes has been of interest to evolutionary biologists for many years. To describe TE dynamics in populations, previous studies have used models of transposition–selection equilibrium that assume a constant rate of transposition. However, since TE invasions are known to happen in bursts through time, this assumption may not be reasonable. Here we propose a test of neutrality for TE insertions that does not rely on the assumption of a constant transposition rate. We consider the case of TE insertions that have been ascertained from a single haploid reference genome sequence. By conditioning on the age of an individual TE insertion allele (inferred by the number of unique substitutions that have occurred within the particular TE sequence since insertion), we determine the probability distribution of the insertion allele frequency in a population sample under neutrality. Taking models of varying population size into account, we then evaluate predictions of our model against allele frequency data from 190 retrotransposon insertions sampled from North American and African populations of Drosophila melanogaster. Using this nonequilibrium neutral model, we are able to explain ∼80% of the variance in TE insertion allele frequencies based on age alone. Controlling for both nonequilibrium dynamics of transposition and host demography, we provide evidence for negative selection acting against most TEs as well as for positive selection acting on a small subset of TEs. Our work establishes a new framework for the analysis of the evolutionary forces governing large insertion mutations like TEs, gene duplications, or other copy number variants.     

Frequency distributions for chloroplast genes in Chlamydomonas zygote clones: Evidence for random drift

Mitochondria and chloroplasts of eucaryotic cells contain populations of DNA molecules. In certain cases, e.g., the chloroplasts of Chlamydomonas reinhardtii and the mitochondria of Saccharomyces cerevisiae, organelles contributed by the two parents are known to fuse in the zygote, creating a single population of DNA molecules. In a cross, this population will include molecules of both parental genotypes. There is reason to suspect that organelle DNA molecules in this population are selected randomly for replication and recombination. This would result in random changes in the frequency of a particular allele or genotype within the organelle gene pool of a single zygote and also within its clone of progeny cells. A given gene frequency would increase in some zygote clones and decrease in others, analogous to random drift of gene frequencies in small Mendelian populations. To test this, we have examined the distribution of chloroplast gene frequencies among the zygote clones produced in each of a number of crosses of Chlamydomonas. These distributions are typically U or L shaped as predicted by the random drift hypothesis. They include uniparental zygote clones, in which a chloroplast allele from one parent has been fixed (frequency 100%) and the alternative allele from the other parent has been lost (frequency 0%). Among the remaining (biparental) zygote clones, there is a linear distribution of allele frequencies, showing a great increase in variance over the input frequencies. In these experiments both biparental and uniparental zygotes show a bias favoring chloroplast alleles from the mt⁺ (maternal) parent, and there is no statistically significant mode at the allele frequency of 0.5 corresponding to the equal input of alleles from the maternal and paternal (mt^?) parents. The observed distributions support the hypothesis that both uniparental inheritance and the high variance of allele frequencies among zygote clones are due to random drift of allele frequencies, coupled with a directional force which favors fixation of the maternal allele. In addition, statistical analysis of the data shows a strong but incomplete tendency for linked chloroplast markers to be fixed or lost together in uniparental zygotes. Possible cellular and molecular mechanisms for these observations are discussed.     

TLR4 896A/G gene polymorphism, rather than the TLR4 1196C/T and TLR2 2258G/A gene polymorphisms, determines the severe and aggravated course of atopic dermatitis in children

The frequencies of the TLR2 gene 2258G/A and the TLR4 gene 1196C/T and 896A/G polymorphisms in children with atopic dermatitis (AD) were studied against controls. It was established after analyzing the distribution of the TLR2 and TLR4 genotypes and alleles that the TLR4 gene’s mutant 896G allele is reliably more frequently detected in children with AD who are susceptible to acute respiratory viral infections (9.3%), compared to the control group (χ² = 4.33; p = 0.038). An analysis of the clinical manifestations of the disease and their associations has shown a higher frequency of its mild course (p = 0.0001) in children with AD who have normal body resistance and of a moderately severe course (p = 0.0033), as well as of concomitant allergic rhinitis (AR) and/or bronchial asthma (BA) (p = 0.0355) and concomitant AR (p = 0.0673), in AD patients with higher susceptibility to acute respiratory viral infections. A severe course of the disease (p = 0.0485), associated with adenoid vegetation in combination with AR and/or BA (p = n0.0248) and concomitant adenoid vegetation in combination with AR (p = 0.0053), was more frequent in AD patients with the TLR4 gene’s mutant 896G allele, compared to patients with a “wild”-type allele.     

Determinants of Genetic Structure in a Nonequilibrium Metapopulation of the Plant Silene latifolia

Population genetic differentiation will be influenced by the demographic history of populations, opportunities for migration among neighboring demes and founder effects associated with repeated extinction and recolonization. In natural populations, these factors are expected to interact with each other and their magnitudes will vary depending on the spatial distribution and age structure of local demes. Although each of these effects has been individually identified as important in structuring genetic variance, their relative magnitude is seldom estimated in nature. We conducted a population genetic analysis in a metapopulation of the angiosperm, Silene latifolia, from which we had more than 20 years of data on the spatial distribution, demographic history, and extinction and colonization of demes. We used hierarchical Bayesian methods to disentangle which features of the populations contributed to among population variation in allele frequencies, including the magnitude and direction of their effects. We show that population age, long-term size and degree of connectivity all combine to affect the distribution of genetic variance; small, recently-founded, isolated populations contributed most to increase F _ST in the metapopulation. However, the effects of population size and population age are best understood as being modulated through the effects of connectivity to other extant populations, i.e. F _ST diminishes as populations age, but at a rate that depends how isolated the population is. These spatial and temporal correlates of population structure give insight into how migration, founder effect and within-deme genetic drift have combined to enhance and restrict genetic divergence in a natural metapopulation.     

On the evolution of behavioral reproductive isolation: The Wallace effect

A model of speciation is constructed in which two equally fit but cross-sterile interbreeding species W and C meet in a zone of overlap Z. Within the type W population, which is held at a constant proportion γ of Z by migration from the outside, a mutation arises which causes a partial aversion to individuals of type C. Formulas for the expected frequency of the various possible mating pairs in Z are derived, and an approximation for small aversion is found. The increase in frequency of the mutant gene in the type W population in Z is the found to be approximately pq(1 ? γ)[r₁(1 ? 2p) + r₂p], where p is the relative proportion of the mutant gene in the type W population, q = 1 ? p, and r₁ and r₂ are the aversion probabilities for mutant heterozygotes and homozygotes, respectively. The resulting selective pressure towards behavioral reproductive isolation is discussed.     

Conditional ancestral times and related properties of an extant mutant at a locus A given the number of segregating sites observed at a linked locus B

 It is shown how the mean ancestral times at one locus are affected in a two- locus model with recombination when information is given regarding the number of segregating sites at another locus. For samples of n genes, recursive equations are derived that describe precisely the evolution of the time-depth of such a linked genealogy. Exact numerical solutions and Markov chain Monte Carlo simulations are discussed and compared. The dependence of some properties of a singleton mutation on waiting times between events in the two-locus genealogy is quantified and illustrates the effect of recombination on these properties. The following cases are presented: (1) the distribution of the number of mutant genes in a sample arising from a singleton mutation; (2) the probability that an allele observed in a genes of a sample of size n is the ancestral type (the oldest); (3) the expectation and variance of the age of a mutant having b copies in a sample of n genes. Received: 1 September 2000 / Revised version: 1 October 2001 / Published online: 8 May 2002     

Rate of Gene Silencing at Duplicate Loci: A Theoretical Study and Interpretation of Data from Tetraploid Fishes

A large-scale simulation has been conducted on the rate of gene loss at duplicate loci under irreversible mutation. It is found that tight linkage does not provide a strong sheltering effect, as thought by previous authors; indeed, the mean loss time for the case of tight linkage is of the same order of magnitude as that for no linkage, as long as Nu is not much larger than 1, where N is the effective population size and u the mutation rate. When Nu is 0.01 or less, the two loci behave almost as neutral loci, regardless of linkage, and the mean loss time is about only half the mean extinction time for a neutral allele under irreversible mutation. However, the former becomes two or more times larger than the latter when Nu ≥ 1.——In the simulation, the sojourn times in the frequency intervals (0, 0.01) and (0.99, 1) and the time for the frequency of the null allele to reach 0.99 at one of the two loci have also been recorded. The results show that the population is monomorphic for the normal allele most of the time if Nu ≤ 0.01, but polymorphic for the null and the normal alleles most of the time if Nu ≥ 0.1.——The distribution of the frequency of the null allele in an equilibrium tetraploid population has been studied analytically. The present results have been applied to interpret data from some fish groups that are of tetraploid origin, and a model for explaining the slow rate of gene loss in these fishes is proposed.     

The average frequency of private alleles in a partially isolated population

An analytic model is developed to explore the relationship between gene flow, selection, and genetic drift. We assume that a single copy of a mutant allele appears in a finite, partially isolated population and allow for the effects of immigration, genic selection, and mutation on the frequency of the mutant. Our concern is with the distribution of the mutant's frequency before it either is lost from the population or emigrates. Before either of these events, the allele will be a “private allele” and would be found in only one of several populations in a larger collection. Slatkin [(1985) Evolution 39, 53–65] found several simple properties of private alleles in his simulations. We use the method developed by Karlin and Tavaré [(1980) Genet. Res. 37, 33–46; (1981a), Theor. Pop. Biol. 19, 187–214; (1981b) Theor. Pop. Biol. 19, 215–229] for a model similar to ours to obtain a diffusion equation with a “killing term” and obtain the mean and variance of the mutant's frequency and its expected frequency in samples of a specified size. There is only fair agreement between the analytic results from this model and those from Slatkin's (loc. cit.) simulations. The rescaling method used to obtain the results indicates that if emigration is relatively frequent, the distribution of rare alleles is governed largely by the balance between genetic drift and emigration, with selection, mutation, and immigration playing a lesser role.     

The Distribution of Mutant Alleles in a Subdivided Population

The results are presented from a simulation study of the spatial distribution of mutant alleles in a subdivided population. Statistical measures of the spatial pattern are defined in such a way that the same quantities could be measured in a geographic survey of allele frequencies in natural populations. Two types of quantities are discussed in this paper: (1) the occupancy distribution provides information on the presence or absence of the mutant in different numbers of demes; and (2) the conditional frequency distribution provides information about the extent of local differentiation when the mutant is present in different numbers of demes. Properties of these distributions are found for different types of natural selection acting on the mutant. Some results are presented for the same statistical measures based on samples of individuals from a fraction of the total number of demes. The simulation results for intermediate levels of the migration rates are compared with analytic results obtained on the limits of high and low migration rates. The main conclusion is that these measures of the spatial distribution of mutants in a subdivided population have simple properties that could provide a new perspective on data from natural populations.