Two-locus,fourth-order gene frequency moments: Implications for the variance of squared linkage disequilibrium and the variance of homozygosity

Identity coefficients are used to construct a sufficient set of equations to determine the fourth-order moments of gene frequencies for two linked loci. This allows the variance of the expected squared linkage disequilibrium to be found. It is shown that the coefficient of variation is generally greater than one and if the mutation rate is small, the standard deviation is more than four times the size of the mean. This demonstrates that squared linkage disequilibrium is a highly variable quantity. The variance of homozygosity for a gene which consists of two sites can also be obtained. Recombination between these sites increases the variance of homozygosity, suggesting that intragenic recombination significantly changes all the expected moments of gene frequencies if 4Nμ > 1.0 and r > μ (where N is the population size, μ is the mutation rate of the gene to neutral alleles, and r is the recombination rate between two sites within the gene).     

Allele Frequencies at Microsatellite Loci: The Stepwise Mutation Model Revisited

We summarize available data on the frequencies of alleles at microsatellite loci in human populations and compare observed distributions of allele frequencies to those generated by a simulation of the stepwise mutation model. We show that observed frequency distributions at 108 loci are consistent with the results of the model under the assumption that mutations cause an increase or decrease in repeat number by one and under the condition that the product Nu, where N is the effective population size and u is the mutation rate, is larger than one. We show that the variance of the distribution of allele sizes is a useful estimator of Nu and performs much better than previously suggested estimators for the stepwise mutation model. In the data, there is no correlation between the mean and variance in allele size at a locus or between the number of alleles and mean allele size, which suggests that the mutation rate at these loci is independent of allele size.     

Microsatellite Polymorphism in Natural Populations of the Wild Plant Arabidopsis Thaliana

Variation in repeat number at 20 microsatellite loci of Arabidopsis thaliana was studied in a worldwide sample of 42 ecotypes to investigate the pattern and level of polymorphism in repetitive sequences in natural plant populations. There is a substantial amount of variation at microsatellite loci despite the selfing nature of this plant species. The average gene diversity was 0.794 and the average number of alleles per locus was 10.6. The distribution of alleles was centered around the mean of repeat number at most loci, but could not be regarded as normal. There was a significantly positive correlation between the number of repeats and the amount of variation. For most loci, the observed number of alleles was between the expected values of the infinite allele and stepwise mutation models. The two models were rejected by the sign test. Linkage disequilibrium was detected in 12.1% of the pairwise comparisons between loci. In phylogenetic tree, there was no association between ecotype and geographic origin. This result is consistent with the recent expansion of A. thaliana throughout the world.     

Equilibrium Values of Measures of Population Subdivision for Stepwise Mutation Processes

Expected values of WRIGHT's F-statistics are functions of probabilities of identity in state. These values may be quite different under an infinite allele model and under stepwise mutation processes such as those occurring at microsatellite loci. However, a relationship between the probability of identity in state in stepwise mutation models and the distribution of coalescence times can be deduced from the relationship between probabilities of identity by descent and the distribution of coalescence times. The values of F(IS) and F(ST) can be computed using this property. Examination of the conditional probability of identity in state given some coalescence time and of the distribution of coalescence times are also useful for explaining the properties of F(IS) and F(ST) at high mutation rate loci, as shown here in an island model of population structure.     

A genealogical interpretation of linkage disequilibrium

The degree of association between alleles at different loci, or linkage disequilibrium, is widely used to infer details of evolutionary processes. Here I explore how associations between alleles relate to properties of the underlying genealogy of sequences. Under the neutral, infinite-sites assumption I show that there is a direct correspondence between the covariance in coalescence times at different parts of the genome and the degree of linkage disequilibrium. These covariances can be calculated exactly under the standard neutral model and by Monte Carlo simulation under different demographic models. I show that the effects of population growth, population bottlenecks, and population structure on linkage disequilibrium can be described through their effects on the covariance in coalescence times.     

Linkage disequilibrium under skewed offspring distribution among individuals in a population

Correlations in coalescence times between two loci are derived under selectively neutral population models in which the offspring of an individual can number on the order of the population size. The correlations depend on the rates of recombination and random drift and are shown to be functions of the parameters controlling the size and frequency of these large reproduction events. Since a prediction of linkage disequilibrium can be written in terms of correlations in coalescence times, it follows that the prediction of linkage disequilibrium is a function not only of the rate of recombination but also of the reproduction parameters. Low linkage disequilibrium is predicted if the offspring of a single individual frequently replace almost the entire population. However, high linkage disequilibrium can be predicted if the offspring of a single individual replace an intermediate fraction of the population. In some cases the model reproduces the standard Wright-Fisher predictions. Contrary to common intuition, high linkage disequilibrium can be predicted despite frequent recombination, and low linkage disequilibrium under infrequent recombination. Simulations support the analytical results but show that the variance of linkage disequilibrium is very large.     

Simultaneous estimation of all the parameters of a stepwise mutation model.

Minisatellite and microsatellite are short tandemly repetitive sequences dispersed in eukaryotic genomes, many of which are highly polymorphic due to copy number variation of the repeats. Because mutation changes copy numbers of the repeat sequences in a generalized stepwise fashion, stepwise mutation models are widely used for studying the dynamics of these loci. We propose a minimum chi-square (MCS) method for simultaneous estimation of all the parameters in a stepwise mutation model and the ancestral allelic type of a sample. The MCS estimator requires knowing the mean number of alleles of a certain size in a sample, which can be estimated using Monte Carlo samples generated by a coalescent algorithm. The method is applied to samples of seven (CA)n repeat loci from eight human populations and one chimpanzee population. The estimated values of parameters suggest that there is a general tendency for microsatellite alleles to expand in size, because (1) each mutation has a slight tendency to cause size increase and (2) the mean size increase is larger than the mean size decrease for a mutation. Our estimates also suggest that most of these CA-repeat loci evolve according to multistep mutation models rather than single-step mutation models. We also introduced several quantities for measuring the quality of the estimation of ancestral allelic type, and it appears that the majority of the estimated ancestral allelic types are reasonably accurate. Implications of our analysis and potential extensions of the method are discussed.SINCE the discovery that a large number of loci with tandemly repeated sequences in human and many eukaryote species are highly polymorphic because of copy number variation of the repeats in different individuals (Jeffreys 1985; Litt and Luty 1989; Weber and May 1989), allele size data from such loci are rapidly becoming the dominant source of genetic markers for genome mapping, forensic testing, and population studies. Loci with repeat sequences longer than 5 bp are generally referred to as minisatellite or variable number tandem repeat loci, and those with repeat sequences between 2 to 5 bp are referred to as microsatellite or short tandem repeat loci (Tautz 1993). Because mutations change the copy number of such loci in a stepwise fashion, rapid accumulation of population samples from minisatellite and microsatellite loci has resurrected the interest of the stepwise mutation model (SMM), which was popular in the 1970s.     

The Sampling Distribution of Linkage Disequilibrium under an Infinite Allele Model without Selection

The sampling distributions of several statistics that measure the association of alleles on gametes (linkage disequilibrium) are estimated under a two-locus neutral infinite allele model using an efficient Monte Carlo method. An often used approximation for the mean squared linkage disequilibrium is shown to be inaccurate unless the proper statistical conditioning is used. The joint distribution of linkage disequilibrium and the allele frequencies in the sample is studied. This estimated joint distribution is sufficient for obtaining an approximate maximum likelihood estimate of C = 4Nc, where N is the population size and c is the recombination rate. It has been suggested that observations of high linkage disequilibrium might be a good basis for rejecting a neutral model in favor of a model in which natural selection maintains genetic variation. It is found that a single sample of chromosomes, examined at two loci cannot provide sufficient information for such a test if C less than 10, because with C this small, very high levels of linkage disequilibrium are not unexpected under the neutral model. In samples of size 50, it is found that, even when C is as large as 50, the distribution of linkage disequilibrium conditional on the allele frequencies is substantially different from the distribution when there is no linkage between the loci. When conditioned on the number of alleles at each locus in the sample, all of the sample statistics examined are nearly independent of theta = 4N mu, where mu is the neutral mutation rate.     

Microsatellite loci for the Chinese bamboo rat Rhizomus sinensis

We describe 19 polymorphic microsatellite loci for the bamboo rat Rhizomus sinensis. When tested with 20 samples from a single population, these loci exhibited a mean of 4.5 alleles per locus and a mean expected heterozygosity of 0.612. All loci were in Hardy-Weinberg equilibrium and no evidence for linkage disequilibrium was detected between any loci. These loci will be useful for studying population genetic diversity and differentiation in the Chinese bamboo rat.     

Polymorphic microsatellite loci for the cardinal fish (<Emphasis Type="Italic">Apogon imberbis</Emphasis>)

Eight polymorphic microsatellite loci were isolated and characterized for the Cardinal fish (Apogon imberbis), a coastal-reef fish endemic to the Mediterranean Sea. Characterization of 30 Cardinal fish individuals form the western Mediterranean showed moderate to high allelic diversity ranging from 6 to 19 alleles per locus. Two loci showed significant departures from Hardy-Weinberg equilibrium presumably due to null alleles. No evidence of linkage disequilibrium was found for any locus pairwise comparasions. This microsatellite set could be useful for any basic population genetic studies of this species.     

An Evaluation of Genetic Distances for Use with Microsatellite Loci

Mutations of alleles at microsatellite loci tend to result in alleles with repeat scores similar to those of the alleles from which they were derived. Therefore the difference in repeat score between alleles carries information about the amount of time that has passed since they shared a common ancestral allele. This information is ignored by genetic distances based on the infinite alleles model. Here we develop a genetic distance based on the stepwise mutation model that includes allelic repeat score. We adapt earlier treatments of the stepwise mutation model to show analytically that the expectation of this distance is a linear function of time. We then use computer simulations to evaluate the overall reliability of this distance and to compare it with allele sharing and Nei's distance. We find that no distance is uniformly superior for all purposes, but that for phylogenetic reconstruction of taxa that are sufficiently diverged, our new distance is preferable.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

Evidence for complex mutations at microsatellite loci in Drosophila.

Fifteen lines each of Drosophila melanogaster, D. simulans, and D. sechellia were scored for 19 microsatellite loci. One to four alleles of each locus in each species were sequenced, and microsatellite variability was compared with sequence structure. Only 7 loci had their size variation among species consistent with the occurrence of strictly stepwise mutations in the repeat array, the others showing extensive variability in the flanking region compared to that within the microsatellite itself. Polymorphisms apparently resulting from complex nonstepwise mutations involving the microsatellite were also observed, both within and between species. Maximum number of perfect repeats and variance of repeat count were found to be strongly correlated in microsatellites showing an apparently stepwise mutation pattern. These data indicate that many microsatellite mutation events are more complex than represented even by generalized stepwise mutation models. Care should therefore be taken in inferring population or phylogenetic relationships from microsatellite size data alone. The analysis also indicates, however, that evaluation of sequence structure may allow selection of microsatellites that more closely match the assumptions of stepwise models.     

Dynamics of Repeat Polymorphisms under a Forward-Backward Mutation Model: within- and between-Population Variability at Microsatellite Loci

Suggested molecular mechanisms for the generation of new tandem repeats of simple sequences indicate that the microsatellite loci evolve via some form of forward-backward mutation. We provide a mathematical basis for suggesting a measure of genetic distance between populations based on microsatellite variation. Our results indicate that such a genetic distance measure can remain proportional to the divergence time of populations even when the forward-backward mutations produce variable and/or directionally biased alleles size changes. If the population size and the rate of mutation remain constant, then the measure will be proportional to the time of divergence of populations. This genetic distance is expressed in terms of a ratio of components of variance of allele sizes, based on expressions developed for studying population dynamics of quantitative traits. Application of this measure to data on 18 microsatellite loci in nine human populations leads to evolutionary trees consistent with the known ethnohistory of the populations.     

Estimating Effective Population Size from Linkage Disequilibrium between Unlinked Loci: Theory and Application to Fruit Fly Outbreak Populations

There is a substantial literature on the use of linkage disequilibrium (LD) to estimate effective population size using unlinked loci. The estimates are extremely sensitive to the sampling process, and there is currently no theory to cope with the possible biases. We derive formulae for the analysis of idealised populations mating at random with multi-allelic (microsatellite) loci. The ‘Burrows composite index’ is introduced in a novel way with a ‘composite haplotype table’. We show that in a sample of diploid size , the mean value of or from the composite haplotype table is biased by a factor of , rather than the usual factor for a conventional haplotype table. But analysis of population data using these formulae leads to estimates that are unrealistically low. We provide theory and simulation to show that this bias towards low estimates is due to null alleles, and introduce a randomised permutation correction to compensate for the bias. We also consider the effect of introducing a within-locus disequilibrium factor to , and find that this factor leads to a bias in the estimate. However this bias can be overcome using the same randomised permutation correction, to yield an altered with lower variance than the original , and one that is also insensitive to null alleles. The resulting formulae are used to provide estimates on 40 samples of the Queensland fruit fly, Bactrocera tryoni, from populations with widely divergent expectations. Linkage relationships are known for most of the microsatellite loci in this species. We find that there is little difference in the estimated values from using known unlinked loci as compared to using all loci, which is important for conservation studies where linkage relationships are unknown.     

Influence of mutational and sampling factors on the estimation of demographic parameters in a "continuous" population under isolation by distance

In numerous species, individual dispersal is restricted in space so that "continuous" populations evolve under isolation by distance. A method based on individual genotypes assuming a lattice population model was recently developed to estimate the product Dsigma2, where D is the population density and sigma2 is the average squared parent-offspring distance. We evaluated the influence on this method of both mutation rate and mutation model, with a particular reference to microsatellite markers, as well as that of the spatial scale of sampling. Moreover, we developed and tested a nonparametric bootstrap procedure allowing the construction of confidence intervals for the estimation of Dsigma2. These two objectives prompted us to develop a computer simulation algorithm based on the coalescent theory giving individual genotypes for a continuous population under isolation by distance. Our results show that the characteristics of mutational processes at microsatellite loci, namely the allele size homoplasy generated by stepwise mutations, constraints on allele size, and change of slippage rate with repeat number, have little influence on the estimation of Dsigma2. In contrast, a high genetic diversity (approximately 0.7-0.8), as is commonly observed for microsatellite markers, substantially increases the precision of the estimation. However, very high levels of genetic diversity (>0.85) were found to bias the estimation. We also show that statistics taking into account allele size differences give unreliable estimations (i.e., high variance of Dsigma2 estimation) even under a strict stepwise mutation model. Finally, although we show that this method is reasonably robust with respect to the sampling scale, sampling individuals at a local geographical scale gives more precise estimations of Dsigma2.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

Linkage Disequilibrium in Growing and Stable Populations

Nonrandom associations between alleles at different loci can be tested for using Fisher's exact test. Extensive simulations show that there is a substantial probability of obtaining significant nonrandom associations between closely or completely linked polymorphic neutral loci in a population of constant size at equilibrium under mutation and genetic drift. In a rapidly growing population, however, there will be little chance of finding significant nonrandom associations even between completely linked loci if the growth has been sufficiently rapid. This result is illustrated by the analysis of mitochondrial DNA sequence data from humans. In comparing all pairs of informative sites, fewer than 5% of the pairs show significant disequilibrium in Sardinians, which have apparently undergone rapid population growth, while 20% to 30% in !Kung and Pygmies, which apparently have not undergone rapid growth, show significance. The extent of linkage disequilibrium in a population is closely related to the gene genealogies of the loci examined, with ``starlike' genealogies making significant linkage disequilibrium unlikely.     

Genetic analysis of microsatellite polymorphism in the Elliot's Pheasant (Syrmaticus ellioti) in China

In this study, we reported the population genetic analyses in the Elliot's Pheasant(Syrnaticus ellioti) using seven polymorphism microsatellite loci based on 105 individuals from 4 geographical populations. Departures from Hardy-Weinberg equilibrium were found in four geographical populations. The average number of alleles was 8.86, with a total of 62 alleles across 7 loci; observed heterozygosity (HO) was generally low and the average number was 0.504. For the seven microsatellite loci, the polymorphism information content ranged from 0.549 to 0.860, with an average number 0.712. Population bottlenecks of the four geographical populations were tested by infinite allele mutation model, step-wise mutation model and two-phase mutation model, which found that each population had experienced bottleneck effect during the recent period. Fst analysis across all geographical populations indicated that the genetic differentiaton between the Guizhou geographical population and the Hunan geographical population was highly significant (P<0.001), a finding supported by the far genetic relationship showed by the neighbor-joining tree of four geographical populations based on Nei's unbiased genetic distances. Using hierarchical analysis of molecular variance (Guizhou geographical population relative to all others pooled), we found a low level of the genetic variation among geographical populations and that between groups. However, differences among populations relative to the total sample explained most of the genetic variance (92.84%), which was significant.     

Microsatellite variation and evolution in the Mimulus guttatus species complex with contrasting mating systems

Mutational variability at microsatellite loci is shaped by both population history and the mating system. In turn, alternate mating systems in flowering plants can resolve aspects of microsatellite loci evolution. Five species of yellow monkeyflowers (Mimulus sect. Simiolis) differing for historical rates of inbreeding were surveyed for variation at six microsatellite loci. High levels of diversity at these loci were found in both outcrossing and selfing taxa. In line with allozyme studies, inbreeders showed more partitioning of diversity among populations, and diversity in selfing taxa was lower than expected from reductions in effective population size due to selfing alone, suggesting the presence of either population bottlenecks or background selection in selfers. Evaluation of the stepwise mutation model (a model of DNA replication slippage) suggests that these loci evolve in a stepwise fashion. Inferred coalescent times of microsatellite alleles indicate that past bottlenecks of population size or colonization events were important in reducing diversity in the inbreeding taxon.