Stable linkage disequilibrium without epistasis in subdivided populations

In a large random mating population stable linkage disequilibrium occurs only when there is epistasis. However if a population is divided into a number of subpopulations among which migration occurs, stable linkage disequilibrium in each subpopulation may be produced without epistasis. In the case of two subpopulations a necessary condition for linkage equilibrium in the absence of epistasis is that at least at one of the two loci under consideration the gene frequency must be the same for the two populations. This condition is rather severe and any violation of this will lead to stable linkage disequilibrium. A similar conclusion can be made with more than two populations. In general the presence of linkage disequilibrium does not necessarily imply the existence of epistasis even in equilibrium populations.     

A simple derivation of the partitioning of genetic differentiation within subdivided populations

The genetic differentiation within a subdivided population can be partitioned into two proportions, one due to differentiation within sub-populations and the other due to differentiation among subpopulations. A simple mathematical derivation of this procedure, known as Nei's G _ST -statistics, is presented. The significance of considering the differing relative sizes of subpopulations is stressed. Possible fields of applications for breeders and conservationists who are concerned with the establishment of genetically diverse populations are mentioned.     

Joint linkage and linkage disequilibrium mapping in natural populations

A new strategy for studying the genome structure and organization of natural populations is proposed on the basis of a combined analysis of linkage and linkage disequilibrium using known polymorphic markers. This strategy exploits a random sample drawn from a panmictic natural population and the open-pollinated progeny of the sample. It is established on the principle of gene transmission from the parental to progeny generation during which the linkage between different markers is broken down due to meiotic recombination. The strategy has power to simultaneously capture the information about the linkage of the markers (as measured by recombination fraction) and the degree of their linkage disequilibrium created at a historic time. Simulation studies indicate that the statistical method implemented by the Fisher-scoring algorithm can provide accurate and precise estimates for the allele frequencies, recombination fractions, and linkage disequilibria between different markers. The strategy has great implications for constructing a dense linkage disequilibrium map that can facilitate the identification and positional cloning of the genes underlying both simple and complex traits.     

Characterizing linkage disequilibrium in pig populations

Knowledge of the extent and range of linkage disequilibrium (LD), defined as non-random association of alleles at two or more loci, in animal populations is extremely valuable in localizing genes affecting quantitative traits, identifying chromosomal regions under selection, studying population history, and characterizing/managing genetic resources and diversity. Two commonly used LD measures, r(2) and D', and their permutation based adjustments, were evaluated using genotypes of more than 6,000 pigs from six commercial lines (two terminal sire lines and four maternal lines) at ~4,500 autosomal SNPs (single nucleotide polymorphisms). The results indicated that permutation only partially removed the dependency of D' on allele frequency and that r(2) is a considerably more robust LD measure. The maximum r(2) was derived as a function of allele frequency. Using the same genotype dataset, the extent of LD in these pig populations was estimated for all possible syntenic SNP pairs using r(2) and the ratio of r(2) over its theoretical maximum. As expected, the extent of LD highest for SNP pairs was found in tightest linkage and decreased as their map distance increased. The level of LD found in these pig populations appears to be lower than previously implied in several other studies using microsatellite genotype data. For all pairs of SNPs approximately 3 centiMorgan (cM) apart, the average r(2) was equal to 0.1. Based on the average population-wise LD found in these six commercial pig lines, we recommend a spacing of 0.1 to 1 cM for a whole genome association study in pig populations.     

Mitonuclear linkage disequilibrium in human populations

There is extensive evidence from model systems that disrupting associations between co-adapted mitochondrial and nuclear genotypes can lead to deleterious and even lethal consequences. While it is tempting to extrapolate from these observations and make inferences about the human-health effects of altering mitonuclear associations, the importance of such associations may vary greatly among species, depending on population genetics, demographic history and other factors. Remarkably, despite the extensive study of human population genetics, the statistical associations between nuclear and mitochondrial alleles remain largely uninvestigated. We analysed published population genomic data to test for signatures of historical selection to maintain mitonuclear associations, particularly those involving nuclear genes that encode mitochondrial-localized proteins (N-mt genes). We found that significant mitonuclear linkage disequilibrium (LD) exists throughout the human genome, but these associations were generally weak, which is consistent with the paucity of population genetic structure in humans. Although mitonuclear LD varied among genomic regions (with especially high levels on the X chromosome), N-mt genes were statistically indistinguishable from background levels, suggesting that selection on mitonuclear epistasis has not preferentially maintained associations involving this set of loci at a species-wide level. We discuss these findings in the context of the ongoing debate over mitochondrial replacement therapy.     

The rule of "one migrant per generation" and genetic differentiation in subdivided populations

The genetic differentiation in population with migration according island and two-dimensional stepping stone models was studied with simulation methods. It is shown that migration of one or few individuals per generation is insufficient for leveling differences between subpopulations in allele frequencies. Even if migration is estimated as m = 0.5 (exchange of 50 and 500 individuals per generation) statistically significant differences remain at least in the half of populations with insular structure. Spatial heterogeneity disappears completely only if m = 0.7-0.8. In case of two-dimensional step model the level of genetic differentiation is higher and statistically significant heterogeneity remains at all levels of genetic exchange including that which was estimated as m = 1.     

Detecting linkage disequilibrium in bacterial populations.

The distribution of the number of pairwise differences calculated from comparisons between n haploid genomes has frequently been used as a starting point for testing the hypothesis of linkage equilibrium. For this purpose the variance of the pairwise differences, VD, is used as a test statistic to evaluate the null hypothesis that all loci are in linkage equilibrium. The problem is to determine the critical value of the distribution of VD. This critical value can be estimated either by Monte Carlo simulation or by assuming that VD is distributed normally and calculating a one-tailed 95% critical value for VD, L, L = EVD + 1.645 sqrt(VarVD), where E(VD) is the expectation of VD, and Var(VD) is the variance of VD. If VD (observed) > L, the null hypothesis of linkage equilibrium is rejected. Using Monte Carlo simulation we show that the formula currently available for Var(VD) is incorrect, especially for genetically highly diverse data. This has implications for hypothesis testing in bacterial populations, which are often genetically highly diverse. For this reason we derive a new, exact formula for Var(VD). The distribution of VD is examined and shown to approach normality as the sample size increases. This makes the new formula a useful tool in the investigation of large data sets, where testing for linkage using Monte Carlo simulation can be very time consuming. Application of the new formula, in conjunction with Monte Carlo simulation, to populations of Bradyrhizobium japonicum, Rhizobium leguminosarum, and Bacillus subtilis reveals linkage disequilibrium where linkage equilibrium has previously been reported.     

Three components of genetic drift in subdivided populations

Wright's metaphor of sampling is extended to consider three components of genetic drift: those occurring before, during, and after migration. To the extent that drift at each stage behaves like an independent random sample, the order of events does not matter. When sampling is not random, the order does matter, and the effect of population size is confounded with that of mobility. The widely cited result that genetic differentiation of local groups depends only on the product of group size and migration rate holds only when nonrandom sampling does not occur prior to migration in the life cycle.     

Characterization and conservation of genetic diversity in subdivided populations

We review the available tools for analysing genetic diversity in conservation programmes of subdivided populations. Ways for establishing conservation priorities have been developed in the context of livestock populations, both from the classical population genetic analysis and from the more recent Weitzman's approach. We discuss different reasons to emphasize either within or between-population variation in conservation decisions and the methodology to establish some compromise. The comparison between neutral and quantitative variation is reviewed from both theoretical and empirical points of view, and the different procedures for the dynamic management of conserved subdivided populations are discussed.     

Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populations

Linkage analysis and allelic association (also referred to as linkage disequilibrium) studies are two major approaches for mapping genes that control simple or complex traits in plants, animals, and humans. But these two approaches have limited utility when used alone, because they use only part of the information that is available for a mapping population. More recently, a new mapping strategy has been designed to integrate the advantages of linkage analysis and linkage disequilibrium analysis for genome mapping in outcrossing populations. The new strategy makes use of a random sample from a panmictic population and the open-pollinated progeny of the sample. In this article, we extend the new strategy to map quantitative trait loci (QTL), using molecular markers within the EM-implemented maximum-likelihood framework. The most significant advantage of this extension is that both linkage and linkage disequilibrium between a marker and QTL can be estimated simultaneously, thus increasing the efficiency and effectiveness of genome mapping for recalcitrant outcrossing species. Simulation studies are performed to test the statistical properties of the MLEs of genetic and genomic parameters including QTL allele frequency, QTL effects, QTL position, and the linkage disequilibrium of the QTL and a marker. The potential utility of our mapping strategy is discussed.     

Management of genetic diversity of subdivided populations in conservation programmes

Population subdivision must be explicitly considered in the management of conservation programmes, as most populations of wild species at risk of extinction and those kept in captivity are spatially structured. The partition of gene and allelic diversity in within- and between-subpopulation components allows for the integral management of populations. We summarise the main aspects of this partition and some of its applications in terms of priorisation of populations for conservation and establishment of synthetic populations. The procedures for the maintenance of diversity in subdivided populations making use of molecular markers and its implementation by the software METAPOP are illustrated with empirical data.     

Temporal dynamics and linkage disequilibrium in natural Caenorhabditis elegans populations

Caenorhabditis elegans is a major laboratory model system yet a newcomer to the field of population genetics, and relatively little is known of its biology in the wild. Recent studies of natural populations at a single time point revealed strong spatial population structure and suggested that these populations may be very dynamic. We have therefore studied several natural C. elegans populations over time and genotyped them at polymorphic microsatellite loci. While some populations appear to be genetically stable over the course of observation, others seem to go extinct, with full replacement of multilocus genotypes upon regrowth. The frequency of heterozygotes indicates that outcrossing occurs at a mean frequency of 1.7% and is variable between populations. However, in genetically stable populations, linkage disequilibrium between different chromosomes can be maintained over several years at a level much higher than expected from the heterozygote frequency. C. elegans seems to follow metapopulation dynamics, and the maintenance of linkage disequilibrium despite a low yet significant level of outcrossing suggests that selection may act against the progeny of outcrossings.     

Extensive linkage disequilibrium in small human populations in Eurasia

The extent of linkage disequilibrium (LD) was studied in two small food-gathering populations—Evenki and Saami—and two larger food-producing populations—Finns and Swedes—in northern Eurasia. In total, 50 single-nucleotide polymorphisms (SNPs) from five genes were genotyped using real-time pyrophosphate DNA sequencing, whereas 14 microsatellites were genotyped in two X-chromosomal regions. In addition, hypervariable region I of the mtDNA was sequenced to shed light on the demographic history of the populations. The SNP data, as well as the microsatellite data, reveal extensive levels of LD in Evenki and Saami when compared to Finns and Swedes. mtDNA-sequence variation is compatible with constant population size over time in Evenki and Saami but indicates population expansion in Finns and Swedes. Furthermore, the similarity between Finns and Swedes in SNP allele- and haplotype-frequency distributions indicate that these two populations may share a recent common origin. These findings suggest that populations such as the Evenki and the Saami, rather than the Finns, may be particularly suited for the initial coarse mapping of common complex diseases.     

Assessing the effects of rare alleles and linkage disequilibrium on estimates of genetic diversity in the chicken populations

Phenotypic diversity in poultry has been mainly driven by artificial selection and genetic drift. These led to the adaptation to the environment and the development of specific phenotypic traits of chickens in response to their economic use. This study evaluated genetic diversity within and between Russian breeds and populations using Illumina Chicken 60 K SNP iSelect BeadChip by analysing genetic differences between populations with Hudson's fixation index (F_ST statistic) and heterozygosity. We estimated the effect of rare alleles and linkage disequilibrium (LD) on these measurements. To assess the effect of LD on the genetic diversity population, we carried out the LD-based pruning (LD < 0.5 and LD < 0.1) for seven chicken populations combined (I) or separately (II). LD pruning was specific for different dataset groups. Because of the noticeably large sample size in the Russian White RG population, pruning was substantial for Dataset I, and F_ST values were only positive when LD < 0.1 pruning was applied. For Dataset II, the LD pruning results were confirmed by examining heterozygosity and alleles' frequency distribution. LD between single nucleotide polymorphisms was consistent across the seven chicken populations, except the Russian White RG population with the smallest r² values and the largest effective population size. Our findings suggest to study variability in each population LD pruning has to be carried separately not after merging to avoid bias in estimates.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Highly variable patterns of linkage disequilibrium in multiple soybean populations

Prospects for utilizing whole-genome association analysis in autogamous plant populations appear promising due to the reported high levels of linkage disequilibrium (LD). To determine the optimal strategies for implementing association analysis in soybean (Glycine max L. Merr.), we analyzed the structure of LD in three regions of the genome varying in length from 336 to 574 kb. This analysis was conducted in four distinct groups of soybean germplasm: 26 accessions of the wild ancestor of soybean (Glycine soja Seib. et Zucc.); 52 Asian G. max Landraces, the immediate results of domestication from G. soja; 17 Asian Landrace introductions that became the ancestors of North American (N. Am.) cultivars, and 25 Elite Cultivars from N. Am. In G. soja, LD did not extend past 100 kb; however, in the three cultivated G. max groups, LD extended from 90 to 574 kb, likely due to the impacts of domestication and increased self-fertilization. The three genomic regions were highly variable relative to the extent of LD within the three cultivated soybean populations. G. soja appears to be ideal for fine mapping of genes, but due to the highly variable levels of LD in the Landraces and the Elite Cultivars, whole-genome association analysis in soybean may be more difficult than first anticipated.     

Measuring the extent of linkage disequilibrium in commercial pig populations

To evaluate the extent of linkage disequilibrium in domestic pigs, we genotyped 33 and 44 unrelated individuals from two commercial populations for 29 and five microsatellite markers located on chromosomes 15 and 2 respectively. A high proportion of marker pairs up to 40 cM apart exhibited significant linkage disequilibrium in both populations. Pair-wise r(2) values averaged between 0.15 and 0.50 (depending on chromosome and population) for markers <1 cM apart and declined to values of 0.05 for more distant syntenic markers. Our results suggest that both populations underwent a bottleneck approximately 20 generations ago, which reduced the effective population size from thousands to <200 animals.     

Contribution of genetic effects to genetic variance components with epistasis and linkage disequilibrium

Background

Cockerham genetic models are commonly used in quantitative trait loci (QTL) analysis with a special feature of partitioning genotypic variances into various genetic variance components, while the F_∞ genetic models are widely used in genetic association studies. Over years, there have been some confusion about the relationship between these two type of models. A link between the additive, dominance and epistatic effects in an F_∞ model and the additive, dominance and epistatic variance components in a Cockerham model has not been well established, especially when there are multiple QTL in presence of epistasis and linkage disequilibrium (LD).

Results

In this paper, we further explore the differences and links between the F_∞ and Cockerham models. First, we show that the Cockerham type models are allelic based models with a special modification to correct a confounding problem. Several important moment functions, which are useful for partition of variance components in Cockerham models, are also derived. Next, we discuss properties of the F_∞ models in partition of genotypic variances. Its difference from that of the Cockerham models is addressed. Finally, for a two-locus biallelic QTL model with epistasis and LD between the loci, we present detailed formulas for calculation of the genetic variance components in terms of the additive, dominant and epistatic effects in an F_∞ model. A new way of linking the Cockerham and F_∞ model parameters through their coding variables of genotypes is also proposed, which is especially useful when reduced F_∞ models are applied.

Conclusion

The Cockerham type models are allele-based models with a focus on partition of genotypic variances into various genetic variance components, which are contributed by allelic effects and their interactions. By contrast, the F_∞ regression models are genotype-based models focusing on modeling and testing of within-locus genotypic effects and locus-by-locus genotypic interactions. When there is no need to distinguish the paternal and maternal allelic effects, these two types of models are transferable. Transformation between an F_∞ model's parameters and its corresponding Cockerham model's parameters can be established through a relationship between their coding variables of genotypes. Genetic variance components in terms of the additive, dominance and epistatic genetic effects in an F_∞ model can then be calculated by translating formulas derived for the Cockerham models.

     

The genetic mating structure of subdivided populations I. Open-mating model

The genetic mating structure of a subdivided population can describe how parental genotypes gave rise to zygotes. When parents of the same genotype are considered together as one class (“open-mating”), three independent parameters of inbreeding and mating structure are needed to describe this structure at a diallelic locus. One is Wright's fixation index F. The other two are mating structure parameters, derived herein and termed the “effective selfing” rate E and the “inbreeding assortative selfing” rate D. E is the genetically equivalent proportion of self-fertilization at a single locus, and is given by standardized second and third central moments of gene frequencies of mates. E is a summary measure of inbreeding that includes effects due to self-fertilization and mating to relatives, as well as correlations between mates induced by Wahlund effects and/or selective diversification among neighborhoods. The second parameter D measures the tendency of inbred or more homozygous individuals to effectively self more (or less) than outbred or more heterozygous individuals. D is related to the maintenance of variation of inbreeding among individuals and/or to the prevalence of spatial variation of selection. D is independent of E, but together with E controls the generational change of inbreeding, ΔF. Extensions of the model to unequal allele frequencies in male vs female mates, and to multi-allelic loci, are also examined.