Linkage Disequilibrium in Subdivided Populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.     

Amino Acid Covariation in a Functionally Important Human Immunodeficiency Virus Type 1 Protein Region Is Associated With Population Subdivision

The frequently reported amino acid covariation of the highly polymorphic human immunodeficiency virus type 1 (HIV-1) exterior envelope glycoprotein V3 region has been assumed to reflect fitness epistasis between residues. However, nonrandom association of amino acids, or linkage disequilibrium, has many possible causes, including population subdivision. If the amino acids at a set of sequence sites differ in frequencies between subpopulations, then analysis of the whole population may reveal linkage disequilibrium even if it does not exist in any subpopulation. HIV-1 has a complex population structure, and the effects of this structure on linkage disequilibrium were investigated by estimating within- and among-subpopulation components of variance in linkage disequilibrium. The amino acid covariation previously reported is explained by differences in amino acid frequencies among virus subpopulations in different patients and by nonsystematic disequilibrium among patients. Disequilibrium within patients appears to be entirely due to differences in amino acid frequencies among sampling time points and among chemokine coreceptor usage phenotypes of virus particles, but not source tissues. Positive selection explains differences in allele frequencies among time points and phenotypes, indicating that these differences are adaptive rather than due to genetic drift. However, the absence of a correlation between linkage disequilibrium and phenotype suggests that fitness epistasis is an unlikely cause of disequilibrium. Indeed, when population structure is removed by analyzing sequences from a single time point and phenotype, no disequilibrium is detectable within patients. These results caution against interpreting amino acid covariation and coevolution as evidence for fitness epistasis.     

Linkage disequilibrium in a population undergoing periodic fragmentation and admixture

Glacial and interglacial cycles are considered to have caused the fragmentation and admixture of populations in many organisms. A simple model incorporating such periodic changes of the population structure is analysed in order to investigate the behaviour of neutral genetic variation at one and two loci. The equilibrium is reached very quickly in terms of cycles if the length of a cycle is long, as would be expected of the glaciation cycles. Heterozygosity and linkage disequilibrium are shown to depend on the length of time of the fragmented and admixed phases, population sizes, and number (n) of subpopulations in the fragmented phase. If the population size is small in the fragmented phase and its duration is long, the squared correlation coefficient of two loci (a measure of linkage disequilibrium) just after the admixture is approximated by 1/(n-1) for n > 1. After admixture, the correlation decays at a rate of approximately twice the recombination rate. Therefore, if post-glaciation admixture created linkage disequilibrium, we expect to observe linkage disequilibrium even between moderately linked loci, and its decay pattern along the chromosome is very different from that in a random mating population at equilibrium. This is especially true in organisms with long generation times such as trees.     

Evidence for Linkage Disequilibrium Maintained by Selection in Two Natural Populations of DROSOPHILA SUBOBSCURA

One island and one mainland population of Drosophila subobscura were found polymorphic at the XDH (xanthine dehydrogenase) and the AO (aldehyde oxidase) loci. It was observed that one allele at the XDH locus, which has a low frequency in both populations, is nonrandomly associated with the alleles at the AO locus. Two lines of evidence support the thesis that this linkage disequilibrium is due to epistasis rather than random drift: (1) D or r, measures of the disequilibrium, have the same sign and magnitude in both populations. (2) The linkage disequilibrium is not due to inversions. Inversions segregating on the chromosome carrying XDH and AO have been separated into two classes, between which exchange of alleles at the two loci is suppressed. Linkage disequilibrium for XDH and AO was observed within each class. In the absence of any exchange of alleles, these disequilibria must have arisen and been maintained independently. The suggestion is made that the epistatic disequilibrium results from the close structural and physiological relationship which exists between the two enzymes.     

Inter-locus interactions: A review of experimental evidence

In quantitative genetics, experiments designed to elucidate the nature of gene action and hence the importance of epistasis, have included analysis of genetic differences among individuals in random mating populations (partitioning of genetic variation, analysis of selection responses), of differences among inbred lines or selected populations (variance components in crosses among lines, chromosome analysis using genetic markers and crossover suppression), of the effects of inbreeding, and of population structure. Evidence in population genetic studies has come from studies of linkage disequilibrium and co-adaptation in natural populations, and of multilocus fitness estimation and linkage disequilibrium and associative overdominance in experimental populations. While it is clear that epistasis does contribute to the genetic variation in some quantitative characters, and in particular reproductive fitness, much of the evidence is equivocal and unsatisfying.     

Limits to the Relationship among Recombination, Disequilibrium and Epistasis in Two-Locus Models

I determine limits to the equilibrium relationship among epistasis, recombination and disequilibrium in two-locus, two-allele models using linear programming techniques. I show that when allele frequencies are one-half at each locus, the symmetric model is the fitness pattern that generates the most disequilibrium for the smallest level of epistasis. When allele frequencies deviate from one-half much larger levels of epistasis are required to generate similar levels of disequilibrium. I determine the level of epistasis required to generate observed significant levels of disequilibrium in natural populations. The overall implication is that disequilibrium will be large at equilibrium only between strongly interacting, closely linked loci.     

Simultaneous detection of linkage disequilibrium and genetic differentiation of subdivided populations

We propose a new method for simultaneously detecting linkage disequilibrium and genetic structure in subdivided populations. Taking subpopulation structure into account with a hierarchical model, we estimate the magnitude of genetic differentiation and linkage disequilibrium in a metapopulation on the basis of geographical samples, rather than decompose a population into a finite number of random-mating subpopulations. We assume that Hardy-Weinberg equilibrium is satisfied in each locality, but do not assume independence between marker loci. Linkage states remain unknown. Genetic differentiation and linkage disequilibrium are expressed as hyperparameters describing the prior distribution of genotypes or haplotypes. We estimate related parameters by maximizing marginal-likelihood functions and detect linkage equilibrium or disequilibrium by the Akaike information criterion. Our empirical Bayesian model analyzes genotype and haplotype frequencies regardless of haploid or diploid data, so it can be applied to most commonly used genetic markers. The performance of our procedure is examined via numerical simulations in comparison with classical procedures. Finally, we analyze isozyme data of ayu, a severely exploited fish species, and single-nucleotide polymorphisms in human ALDH2.     

Normative genetic profiles of RAAS pathway gene polymorphisms in North Indian and South Indian populations

Population-based genetic association studies, popularly known as case-control studies, have continued to be the most preferred method for deciphering the genetic basis of various complex diseases, even in the post-human genome sequencing era. However, interpopulation differences in allele, genotype, and haplotype frequencies and linkage disequilibrium patterns lead to inconsistent results in candidate gene association studies. Therefore, for any meaningful disease association study, knowledge of the normative genetic background of the baseline population is a prerequisite. In addition, such genetic variation data also provide a ready-made menu of allele frequencies and linkage disequilibrium patterns of various polymorphisms in specific candidate genes in a particular population, which is a useful reference for further genetic association studies. Such genetic variation data are lacking for the Indian population, which represents about one-sixth of the world's population. In the present study we have reported the allele, genotype, and haplotype frequencies, Hardy-Weinberg equilibrium status, and linkage disequilibrium patterns of 12 polymorphisms in six candidate genes from the renin-angiotensin-aldosterone system among Indians. Because of their different history of origin, the Indian population is broadly divided into two subpopulations: North Indians (Caucasian Europeans) and South Indians (Dravidians). Considering this well-documented difference in gene pools, we have presented a comparative account of the normative genetic data of North Indian and South Indian populations with at least four individuals of urban and suburban origin from each of the representative states of northern and southern India.     

An inbreeding model of associative overdominance during a population bottleneck

Associative overdominance, the fitness difference between heterozygotes and homozygotes at a neutral locus, is classically described using two categories of models: linkage disequilibrium in small populations or identity disequilibrium in infinite, partially selfing populations. In both cases, only equilibrium situations have been considered. In the present study, associative overdominance is related to the distribution of individual inbreeding levels (i.e., genomic autozygosity). Our model integrates the effects of physical linkage and variation in inbreeding history among individual pedigrees. Hence, linkage and identity disequilibrium, traditionally presented as alternatives, are summarized within a single framework. This allows studying nonequilibrium situations in which both occur simultaneously. The model is applied to the case of an infinite population undergoing a sustained population bottleneck. The effects of bottleneck size, mating system, marker gene diversity, deleterious genomic mutation parameters, and physical linkage are evaluated. Bottlenecks transiently generate much larger associative overdominance than observed in equilibrium finite populations and represent a plausible explanation of empirical results obtained, for instance, in marine species. Moreover, the main origin of associative overdominance is random variation in individual inbreeding whereas physical linkage has little effect.     

Analysis of genetic structure in soil populations of Rhizobium leguminosarum recovered from the USA and the UK

Although many studies have shown that animal-associated bacterial species exhibit linkage disequilibrium at chromosomal loci, recent studies indicate that both animal-associated and soil-borne bacterial species can display a nonclonal genetic structure in which alleles at chromosomal loci are in linkage equilibrium. To examine the situation in soil-borne species further, we compared genetic structure in two soil populations of Rhizobium leguminosarum bv. trifolii and two populations of R. leguminosarum bv. viciae from two sites in Oregon, with genetic structure in R. leguminosarum bv. viciae populations recovered from peas grown at a site in Washington, USA, and at a site in Norfolk, UK. A total of 234 chromosomal types (ET) were identified among 682 strains analysed for allelic variation at 13 enzyme-encoding chromosomal loci by multilocus enzyme electrophoresis (MLEE). Chi-square tests for heterogeneity of allele frequencies showed that the populations were not genetically uniform. A comparison of the genetic diversity within combined and individual populations confirmed that the Washington population was the primary cause of genetic differentiation between the populations. Each individual population exhibited linkage disequilibrium, with the magnitude of the disequilibrium being greatest in the Washington population and least in the UK population of R. leguminosarum bv. viciae. Linkage disequilibrium in the UK population was created between two clusters of 9 and 23 ETs, which, individually, were in linkage equilibrium. Strong linkage disequilibrium between the two major clusters of 8 and 12 ETs in the Washington population was caused by the low genetic diversity of the ETs within each cluster relative to the inter-cluster genetic distance. Because neither the magnitude of genetic diversity nor of linkage disequilibrium increased as hierarchical combinations of the six local populations were analysed, we conclude that the populations have not been isolated from each other for sufficient time, nor have they been exposed to enough selective pressure to develop unique multilocus genetic structure.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.     

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.     

On the formula for admixture linkage disequilibrium

Admixture linkage disequilibrium (ALD), a phenomenon created by gene flow between genetically distinct populations, has for some time been used as a tool in gene mapping. It is therefore important to analyze the pattern of ALD over generations. In this study we explore two models of admixture: the gradual admixture (GA) model, in which admixture occurs at a variable rate in every generation; and the immediate admixture (IA) model, a special case of the GA model, in which admixture occurs in a single generation. In the case of ALD, the well-known formula of linkage disequilibrium (Delta(t)=(1-r)t Delta(0)) is not applicable under these two models. We note the effect of a random mating population (RMP) to the gametic frequencies from the parental population to the offspring population, and provide the correct formula for ALD.     

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Population genetics of modifiers of meiotic drive III. Equilibrium analysis of a general model for the genetic control of segregation distortion

Prout, Bungaard and Bryant (1973, Theor. Popul. Biol. 4, 446–465) presented the first formal treatment of a model of meiotic drive involving a modifier locus which controls the intensity of drive. They studied the equilibrium behavior in the simplest model where it is assumed that drive is maximal when not suppressed. In that case there is one polymorphic equilibrium at which there is linkage disequilibrium. The equilibrium solutions in the general model of meiotic drive proposed by Prout, et al. are given in this paper together with a stability analysis. It is shown that up to three polymorphic equilibria may exist, two of which are in linkage disequilibrium and one in linkage equilibrium. These equilibria exhibit behavior qualitatively opposite to what is widely accepted as the usual for two locus systems and which is not seem in the simple case originally treated. The polymorphic equilibria with linkage disequilibrium may be stable for loose linkage and not for tight while that with linkage equilibrium is stable in an interval of relatively tight linkage values.     

Epistatic selection opposed by immigration in multiple locus genetic systems

A model of “complete” epistatis is considered in which all “plus” alleles must be present in an individual before the adaptive phenotype is expressed. The conditions under which the plus alleles and hence the adaptive phenotype can increase and reach a stable equilibrium in the presence of immigration of gametes carrying minus alleles are found. In haploids and diploids in which the plus alleles are recessive, frequencies of the plus alleles are the same at all loci, regardless of the linkage relationships. Tight linkage favors the existence of a locally stable polymorphic equilibrium, but the equilibrium with only minus alleles is locally stable unless there is very tight linkage or very strong selection. Thus, this kind of epistasis, which provides a simple model for a character that requires several components to be present at the same time, is very sensitive to even a small amount of immigration. Hence, the evolution of such characters is likely only in completely rather than partially isolated populations.     

Recombination disequilibrium in ideal and natural populations

Following Hardy-Weinberg disequilibrium (HWD) occurring at a single locus and linkage disequilibrium (LD) between two loci in generations, we here proposed the third genetic disequilibrium in a population: recombination disequilibrium (RD). RD is a measurement of crossover interference among multiple loci in a random mating population. In natural populations besides recombination interference, RD may also be due to selection, mutation, gene conversion, drift and/or migration. Therefore, similarly to LD, RD will also reflect the history of natural selection and mutation. In breeding populations, RD purely results from recombination interference and hence can be used to build or evaluate and correct a linkage map. Practical examples from F₂, testcross and human populations indeed demonstrate that RD is useful for measuring recombination interference between two short intervals and evaluating linkage maps. As with LD, RD will be important for studying genetic mapping, association of haplotypes with disease, plant breading and population history.     

Selection for Linkage Modification II. a Recombination Balance for Neutral Modifiers

A stable polymorphic equilibrium may be established at a selectively-neutral gene locus which controls the extent of recombination between two other selected loci. The condition for the existence of the stable polymorphism is analogous to heterozygous advantage. The heterozygote at the modifying locus should produce a recombination fraction allowing the greatest linkage disequilibrium. In the models treated this has the effect of producing the highest mean fitness. The relationship of these findings to general problems of coadaptation is discussed.