The structure of linkage disequilibrium around a selective sweep

The fixation of advantageous mutations by natural selection has a profound impact on patterns of linked neutral variation. While it has long been appreciated that such selective sweeps influence the frequency spectrum of nearby polymorphism, it has only recently become clear that they also have dramatic effects on local linkage disequilibrium. By extending previous results on the relationship between genealogical structure and linkage disequilibrium, I obtain simple expressions for the influence of a selective sweep on patterns of allelic association. I show that sweeps can increase, decrease, or even eliminate linkage disequilibrium (LD) entirely depending on the relative position of the selected and neutral loci. I also show the importance of the age of the neutral mutations in predicting their degree of association and describe the consequences of such results for the interpretation of empirical data. In particular, I demonstrate that while selective sweeps can eliminate LD, they generate patterns of genetic variation very different from those expected from recombination hotspots.     

QUANTITATIVE GENETIC VARIANCE MAINTAINED BY FLUCTUATING SELECTION WITH OVERLAPPING GENERATIONS: VARIANCE COMPONENTS AND COVARIANCES

The quantitative genetic variance-covariance that can be maintained in a random environment is studied, assuming overlapping generations and Gaussian stabilizing selection with a fluctuating optimum. The phenotype of an individual is assumed to be determined by additive contributions from each locus on paternal and maternal gametes (i.e., no epistasis and no dominance). Recurrent mutation is ignored, but linkage between loci is arbitrary. The genotype distribution in the evolutionarily stable population is generically discrete: only a finite number of polymorphic alleles with distinctly different effects are maintained, even though we allow a continuum of alleles with arbitrary phenotypic contributions to invade. Fluctuating selection maintains nonzero genetic variance in the evolutionarily stable population if the environmental heterogeneity is larger than a certain threshold. Explicit asymptotic expressions for the standing variance-covariance components are derived for the population near the threshold, or for large generational overlap, as a function of environmental variability and genetic parameters (i.e., number of loci, recombination rate, etc.), using the fact that the genotype distribution is discrete. Above the threshold, the population maintains considerable genetic variance in the form of positive linkage disequilibrium and positive gamete covariance (Hardy-Weinberg disequilibrium) as well as allelic variance. The relative proportion of these disequilibrium variances in the total genetic variance increases with the environmental variability.     

The deviation from linkage equilibrium with multiple loci varying in a stepping-stone cline

The balance between the creation of associations between alleles at different loci by immigration and the convergence to linkage equilibrium due to the recombination process is studied in a theoretical model. The geographical structure of the model is a stepping-stone chain of populations linking two genetically constant source populations. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The gene frequencies at each locus will then show a linear cline through the stepping-stone chain of populations. The deviation from linkage equilibrium through the stepping-stone cline is characterized by an equation for linear measures that provide the linkage disequilibrium measures for a given set of loci in terms of the gene frequencies and the linkage disequilibria in the source populations and in terms of the linkage disequilibrium measures through the cline for lower numbers of loci. Numerical examples of this iterative solution are given, and it is shown that the build-up of the higher order Bennett-disequilibria through the cline is considerably more pronounced than the build-up of two-locus disequilibria.     

Analytical results concerning linkage disequilibrium in models with genetic transformation and conjugation

Expressions are obtained for the expected levels of linkage disequilibrium under three different equilibrium neutral models that make different assumptions about how recombination takes place. A transformation model is considered in which exchange events involve only one locus at a time. Two conjugation models are considered one with a linear genome and one with a circular genome. In the conjugation models large blocks of genes can be transferred with each conjugation. Consistent with published simulation results, it is found that if the transformation rate per locus is more than twenty times the mutation rate per locus, then the levels of linkage disequilibrium are quite low. If the number of loci being sampled is greater than 10, conjugation with a circular genome can be considerably more effective than transformation in reducing linkage disequilibrium. When recombination rates are high, expected linkage disequilibrium is shown to be proportional to the inverse of the transformation rate (or conjugation rate.)     

The predisposition to type 1 diabetes linked to the human leukocyte antigen complex includes at least one non-class II gene

The human leukocyte antigen (HLA) complex, encompassing 3.5 Mb of DNA from the centromeric HLA-DPB2 locus to the telomeric HLA-F locus on chromosome 6p21, encodes a major part of the genetic predisposition to develop type 1 diabetes, designated "IDDM1." A primary role for allelic variation of the class II HLA-DRB1, HLA-DQA1, and HLA-DQB1 loci has been established. However, studies of animals and humans have indicated that other, unmapped, major histocompatibility complex (MHC)-linked genes are participating in IDDM1. The strong linkage disequilibrium between genes in this complex makes mapping a difficult task. In the present paper, we report on the approach we have devised to circumvent the confounding effects of disequilibrium between class II alleles and alleles at other MHC loci. We have scanned 12 Mb of the MHC and flanking chromosome regions with microsatellite polymorphisms and analyzed the transmission of these marker alleles to diabetic probands from parents who were homozygous for the alleles of the HLA-DRB1, HLA-DQA1, and HLA-DQB1 genes. Our analysis, using three independent family sets, suggests the presence of an additional type I diabetes gene (or genes). This approach is useful for the analysis of other loci linked to common diseases, to verify if a candidate polymorphism can explain all of the association of a region or if the association is due to two or more loci in linkage disequilibrium with each other.     

Linkage disequilibrium in natural populations of Trypanosoma cruzi (flagellate), the agent of Chagas' disease

We have studied linkage disequilibrium in natural populations of Trypanosoma cruzi, the agent of Chagas' disease, by analyzing (i) a set of 524 stocks from the whole geographical range of the parasite, characterized at four gene loci coding for enzymes; (ii) a subsample of 121 stocks characterized at 12 enzyme loci; and (iii) a subset of 386 stocks from six locations in Bolivia, characterized by four enzyme loci. Our results show that the linkage disequilibrium reaches the maximum possible value, given the observed allelic frequencies, for almost all the locus pairs. This result is most consistent with the hypothesis that genetic recombination is absent or very rare in T. cruzi natural populations. Partition of the linkage disequilibrium variance for the six Bolivian populations shows that both inter- and intrapopulation components are substantial and that the relationships among the components are D2IS less than D2ST, and D'2IS less than D'2ST. These inequalities are interpreted as the result of an interplay between genetic drift, rare or absent mating, and clonal selection in generating linkage disequilibrium in T. cruzi populations.     

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.     

Characterization of 12 polymorphic microsatellite loci for the whitespotted bamboo shark (Chiloscyllium plagiosum Bennett)

Twelve polymorphic microsatellite loci without linkage disequilibrium were isolated and characterized from a microsatellite DNA-enriched DNA library for the whitespotted bamboo shark (Chiloscyllium plagiosum Bennett, 1830). These loci show polymorphism information content ranging from 0.384 to 0.885, allele number ranging from 4 to 12, effective allele number ranging from 1.686 to 9.438, and observed and expected heterozygosities from 0.229 to 1.000 and from 0.407 to 0.894 respectively. Four loci show strong Hardy-Weinberg deviations. We expect that these markers would be useful for population genetic and breeding studies of the whitespotted bamboo shark.     

Disequilibrium among several linked neutral genes in finite population. II. Variances and covariances of disequilibria

A method is derived for computing the variances and covariances of linkage disequilibria between neutral genes in finite populations, which is based on a linear transformation of results given previously for the mean values of disequilibria. The formulae obtained are limited to moments of sixth order or less, such as the variance of the three-locus disequilibrium. It is shown that there is no covariance between any pair of disequilibria in populations starting equilibrium. The pattern of change with time in variance of the three-locus disequilibrium from populations initially in equilibrium is similar to that for two loci, except that the highest values are achieved rather earlier and are smaller.     

Disequilibrium Pattern Analysis. I. Theory

We have developed a method, disequilibrium pattern analysis, for examining the disequilibrium distribution of the entire array of two locus multiallelic haplotypes in a population. It is shown that a selected haplotype will produce a distinct pattern of linkage disequilibrium values for all generations while the selection is acting. This pattern will also presumably be maintained for many generations after the selection event, until the disequilibrium pattern is eventually broken down by genetic drift and recombination. Related haplotypes, sharing an allele with a selected haplotype, assume a value of linkage disequilibrium proportional to the frequency of the unshared allele and have a single negative value of the normalized linkage disequilibrium. The analysis assumes zero linkage disequilibrium for all allelic combinations initially. The same basic results continue to apply if the selection involves a new mutant, the occurrence of which creates linkage disequilibrium for some haplotypes. The disequilibrium pattern predicted under selection is robust with respect to the influence of migration and random genetic drift. This method is applicable to population data having linked polymorphic loci including that determined from protein or DNA sequencing.     

Nonuniform linkage disequilibrium within a 1,500-kb region of the human immunoglobulin heavy-chain complex.

We have characterized 10 VH polymorphic loci of the VH2, VH3, VH4, and VH5 families. Eight of 10 VH polymorphisms were found to be insertion/deletion polymorphisms, probably the result of nonhomologous recombination over the course of evolution of the current human VH repertoire. The 10 VH polymorphic loci were analyzed in 10 three-generation and 10 two-generation Canadian caucasoid families. Linkage disequilibrium (allelic association) was measured between pairs of VH polymorphic loci, and 12 significant associations were found. The degree of linkage disequilibrium measured between IGH polymorphic loci was then compared with the physical distance separating the loci. The physical distance between IGH polymorphic loci does not entirely determine the degree of linkage disequilibrium between polymorphic loci. Two regions, one in the VH region (between VH3f-2 and VH5-2 and one in the CH region (between C delta and C gamma 3), were found to have linkage disequilibrium values approximately 1/3,000 of that observed in other portions of the IGH region. The previous identification of recombinants in the C delta-to C gamma 3 region indicates that these areas of low linkage disequilibrium are consistent with the presence of recombination hot spots. The observed high amount of recombination in the subtelomeric portion of chromosome 14 therefore appears to be the result of specific hot spots for recombination, rather than a general increase in recombination in this region.     

Estimation of multilocus linkage disequilibria in diploid populations with dominant markers

Analysis of population structure and organization with DNA-based markers can provide important information regarding the history and evolution of a species. Linkage disequilibrium (LD) analysis based on allelic associations between different loci is emerging as a viable tool to unravel the genetic basis of population differentiation. In this article, we derive the EM algorithm to obtain the maximum-likelihood estimates of the linkage disequilibria between dominant markers, to study the patterns of genetic diversity for a diploid species. The algorithm was expanded to estimate and test linkage disequilibria of different orders among three dominant markers and can be technically extended to manipulate an arbitrary number of dominant markers. The feasibility of the proposed algorithm is validated by an example of population genetic studies of hickory trees, native to southeastern China, using dominant random amplified polymorphic DNA markers. Extensive simulation studies were performed to investigate the statistical properties of this algorithm. The precision of the estimates of linkage disequilibrium between dominant markers was compared with that between codominant markers. Results from simulation studies suggest that three-locus LD analysis displays increased power of LD detection relative to two-locus LD analysis. This algorithm is useful for studying the pattern and amount of genetic variation within and among populations.     

Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). 1 demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.Supported by PHS Grant ROI GM 32130     

Restriction fragment length polymorphism of bovine lysozyme genes

Genetic polymorphism of bovine lysozyme (LYZ) genes was investigated by analysing restriction fragment length polymorphism (RFLP). The analysis revealed three RFLP loci designated LYZ1, LYZ2 and LYZ2. Each system included two or three allelic variants. Evidence for close genetic linkage of the three loci was found. There was also a significant linkage disequilibrium among the three loci in a sample of about 200 breeding bulls from one breed. No statistically significant association was found between LYZ RFLPs and breeding values of bulls for disease or milk production traits.     

Decomposing multilocus linkage disequilibrium

We present a mathematically precise formulation of total linkage disequilibrium between multiple loci as the deviation from probabilistic independence and provide explicit formulas for all higher-order terms of linkage disequilibrium, thereby combining J. Dausset et al.'s 1978 definition of linkage disequilibrium with H. Geiringer's 1944 approach. We recursively decompose higher-order linkage disequilibrium terms into lower-order ones. Our greatest simplification comes from defining linkage disequilibrium at a single locus as allele frequency at that locus. At each level, decomposition of linkage disequilibrium is mathematically equivalent to number theoretic compositions of positive integers; i.e., we have converted a genetic decomposition into a mathematical decomposition.     

Linkage disequilibrium for DNA haplotypes near the cystic fibrosis locus in two South European populations

Summary Three polymorphic DNA marker loci (INT1L1, D7S23 and D7S399) map to a chromosomal region that is very close to the cystic fibrosis (CF) locus in terms of genetic distance. These marker loci have been used to analyse the linkage disequilibrium in 137CF families from two South European countries (Italy and Spain). The markers can be analysed for differences in linkage disequilibrium more easily in these populations than in North Europeans, in whom the disequilibrium between the allelic systems defined by the probes and CF is much greater and on a plateau through the genetic region. The different levels of disequilibrium found in the studied populations suggest that D7S399 and D7S23 are both closer to CF than INT1L1, and provide additional information on the origins and homogeneity of the CF defect.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

Allelic recombination and linkage disequilibrium within Msp-1 of Plasmodium falciparum, the malignant human malaria parasite

The C-terminal, cysteine-rich 19kDa domain of merozoite surface protein-1 (MSP-1) of Plasmodium falciparum is a target of the host's humoral immunity and thus a malaria vaccine candidate. Although variation in the 19kDa domain is limited among parasite isolates, tertiary structure-dependent intramolecular associations between the 19kDa domain and other parts of MSP-1 are suggested to be involved in immune evasion by allowing competitive binding of protective and non-protective antibodies directed to their epitopes, which are conformationally in close proximity but separated at the primary structure. Since allelic recombination can account for the major variability of the Msp-1 gene, we examined whether linkage disequilibrium occurs between polymorphic loci in the 5'- and the 3'-region, the latter encoding the 19kDa domain. From 184 Thai field isolates, we selected 69 isolates with a single allelic type in six variable blocks of Msp-1 as determined by PCR-based allelic typing. All the isolates showed no evidence of recombination in blocks 6 to 16, whereas recombination was apparent in blocks 2 to 6. Sequencing of the 3'-region revealed two potential recombination sites in block 17. Strong linkage disequilibrium was seen between polymorphic loci in the 5'- and 3'-regions. The strength of this disequilibrium did not correlate with distance between loci. We discuss the possible role of epistatic selection on particular association types (haplotypes) of Msp-1.     

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.     

Allozymic Variation and Linkage Disequilibrium in Some Laboratory Populations of DROSOPHILA MELANOGASTER

Nine laboratory populations of D. melanogaster were surveyed by starch gel electrophoresis for variation at 17 enzyme loci. A single-fly extract could be assayed for all 17 enzymes, so that the data consist of 17-locus genotypes.--Pairwise linkage disequilibria were estimated from the multilocus genotypic frequencies, using both Burrows' and Hill's methods. Large amounts of linkage disequilibrium were found, in contrast to the results reported for natural populations.-Knowledge of the approximate sizes of these populations was used to compare the observed heterozygosities and linkage disequilibria with predictions of the neutral allele hypothesis. The relatively large amount of linkage disequilibrium is consistent with the small sizes of the populations. However, the levels of heterozygosity in at least some populations suggest that some mechanism has been operating to retard the rate of decay by random drift. Several examples of significant deviation from Hardy-Weinberg frequencies and the large amount of linkage disequilibrum present in these populations indicate that a likely mechanism is selective effects associated with neutral alleles because of linkage disequilibrium with selected loci (e.g., "associative overdominance"). The results are therefore consistent with both neutralist, and selectionist hypotheses, but suggest the importance of considering linkage disequilibrium between neutral and selected loci when attempting to explain the dynamics of enzyme polymorphisms.