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.     

Testing for homogeneity of gametic disequilibrium across strata

Background

Variance component (VC) models are commonly used for Quantitative Trait Loci (QTL) mapping in outbred populations. Here, the QTL effect is given as a random effect and a critical part of the model is the relationship between the phenotypic values and the random effect. In the traditional VC model, each individual has a unique QTL effect and the relationship between these random effects is given as a covariance structure (known as the identity-by-descent (IBD) matrix).

Results

We present an alternative notation of the variance component model, where the elements of the random effect are independent base generation allele effects and sampling term effects. The relationship between the phenotypic vales and the random effect is given by an incidence matrix, which results in a novel, but statistically equivalent, version of the traditional VC model. A general algorithm to estimate this incidence matrix is presented. Since the model is given in terms of base generation allele effects and sampling term effects, these effects can be estimated separately using best linear unbiased prediction (BLUP). From simulated data, we showed that biallelic QTL effects could be accurately clustered using the BLUP obtained from our model notation when markers are fully informative, and that the accuracy increased with the size of the QTL effect. We also developed a measure indicating whether a base generation marker homozygote is a QTL heterozygote or not, by comparing the variances of the sampling term BLUP and the base generation allele BLUP. A ratio greater than one gives strong support for a QTL heterozygote.

Conclusion

We developed a simple presentation of the VC QTL model for identification of base generation allele effects in QTL linkage analysis. The base generation allele effects and sampling term effects were separated in our model notation. This clarifies the assumptions of the model and should also enhance the development of genome scan methods.     

LIAN 3.0: detecting linkage disequilibrium in multilocus data. Linkage Analysis

SUMMARY: LIAN is a program to test the null hypothesis of linkage equilibrium for multilocus data. LIAN incorporates both a Monte Carlo method as well as a novel algebraic method to carry out the hypothesis test. The program further returns the genetic diversity of the sample and the pairwise distances between its members.     

Seasonal cycles of allozyme-by-chromosomal-inversion gametic disequilibrium in Drosophila subobscura

Allozyme loci are frequently found non randomly associated to the chromosomal inversions in which they are included in Drosophila. Two opposite views compete to explain strong allozyme-by-inversion gametic disequilibria: they result from natural selection or, conversely, merely represent remnants of associations accidentally established at the origin of inversions. Empirical efforts aimed at deciding between adaptive and historical scenarios have focused on the spatial distribution of disequilibria. Yet, the evolutionary significance of these associations remains uncertain. I report here the results of a time-series analysis of the seasonal variation of alleles at six allozyme loci (Acph, Lap, Pept-1, Ao, Mpi, and Xdh) in connection with the O chromosomal polymorphisms of D. subobscura. The findings were: (1) in the segment I of the O chromosome, Lap and Pept-1 allozymes changed seasonally in a cyclical fashion within the ST gene arrangement, but they changed erratically within the 3 + 4 gene configuration; (2) the frequencies of Lap1.11 and Pept-1(0.40) within ST dropped to their lowest values in early and late summer, respectively, when the seasonal level of the ST arrangement is lowest. Furthermore, Lap1.11 and Pept-1(0.40) covary with ST only within these seasons, yet in a fashion inconsistent with these alleles having a major influence on the dynamics of the inversion; (3) seasonal cycling of alleles within inversions were not detected at Acph, Ao, Mpi, and Xdh, yet these loci are nearly monomorphic at the study population, and/or their sampled series were shorter than those for Lap and Pept-1; and (4) simply monitoring allozyme frequencies separately for each inversion proved to be superior, for evidencing the seasonal cycles of the disequilibria, to the use of the D' coefficient of association. Observed seasonal cycles of allozymes within inversions likely reflect natural selection.     

Effect of gametic disequilibrium on selection in an autotetraploid population

Summary The effects of a gametic disequilibrium (DSE) in an autotetraploid population on response to selection as measured by the covariance of selection were investigated. The theoretical responses were calculated for mass selection [Mass (1)] and half-sib progeny test selection (HSPT) in a two-allele (B and b), single locus, autotetraploid population. The complexity of calculations precluded analytical expressions for the covariances so numerical analysis was used assuming the following genetic models: monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and additive gene action.The results indicated the DSE could greatly affect the covariance of selection. For a constant allele frequency the DSE might double the covariance expected with selection in a population at random mating equilibrium (RME) of gametes, but in other instances approach zero. For all genetic models and the two breeding methods investigated the covariance of selection was always increased when the frequency of BB gamete exceeded p² (where p is frequency of allele B) and decreased when the frequency of BB gamete was less than p². The possible incorporation of this information into a long term breeding program and some other ramifications were briefly discussed.With the DSE the covariances of selection with HSPT and Mass (1) had a proportionality of 1:2, respectively, with the additive genetic model, but this relationship rarely occurred for other genetic models. The deviations from this ratio were not large in comparison to differences between selection in populations in DSE and RME.Cooperative investigations of the Alfalfa Production Research Unit, United State Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada. Paper No. 512. Scientific Journal Series, Nevada Agricultural Experiment Station     

The D' measure of overall gametic disequilibrium between pairs of multiallelic loci

Abstract The D ' coefficient is one of the most commonly used measures of the extent of gametic disequilibrium between multiallelic loci. It has been suggested that the range of the D ' measure of overall disequilibrium between pairs of multiallelic loci depends on allele frequencies, except under some very restricted conditions. Nevertheless, the problem of dependence of the range of D ' has not been characterized under a wide set of possible polymorphisms. Evaluation of the utility of D ' as a measure of the strength of overall disequilibrium between all possible pairs of alleles at two multiallelic loci requires better knowledge of its range than is currently available. In this work, the conditions of polymorphism under which the range of D ' is frequency independent are given. It is found that the range of D ' is more often independent of allelic frequencies than is commonly thought. Furthermore, the range of D ' undergoes only small fluctuations as a function of the polymorphisms at the loci. Numerical cases and microsatellite data from humans are used for illustration. These observations indicate that the D ' coefficient is a useful tool for the estimation and comparison of the extent of overall disequilibrium across pairs of multiallelic loci.     

Combined linkage disequilibrium and linkage mapping: Bayesian multilocus approach

Quantitative trait loci (QTL) affecting the phenotype of interest can be detected using linkage analysis (LA), linkage disequilibrium (LD) mapping or a combination of both (LDLA). The LA approach uses information from recombination events within the observed pedigree and LD mapping from the historical recombinations within the unobserved pedigree. We propose the Bayesian variable selection approach for combined LDLA analysis for single-nucleotide polymorphism (SNP) data. The novel approach uses both sources of information simultaneously as is commonly done in plant and animal genetics, but it makes fewer assumptions about population demography than previous LDLA methods. This differs from approaches in human genetics, where LDLA methods use LA information conditional on LD information or the other way round. We argue that the multilocus LDLA model is more powerful for the detection of phenotype–genotype associations than single-locus LDLA analysis. To illustrate the performance of the Bayesian multilocus LDLA method, we analyzed simulation replicates based on real SNP genotype data from small three-generational CEPH families and compared the results with commonly used quantitative transmission disequilibrium test (QTDT). This paper is intended to be conceptual in the sense that it is not meant to be a practical method for analyzing high-density SNP data, which is more common. Our aim was to test whether this approach can function in principle.     

Using multilocus sequence data to assess population structure, natural selection, and linkage disequilibrium in wild tomatoes

We employed a multilocus approach to examine the effects of population subdivision and natural selection on DNA polymorphism in 2 closely related wild tomato species (Solanum peruvianum and Solanum chilense), using sequence data for 8 nuclear loci from populations across much of the species' range. Both species exhibit substantial levels of nucleotide variation. The species-wide level of silent nucleotide diversity is 18% higher in S. peruvianum (pi(sil) approximately 2.50%) than in S. chilense (pi(sil) approximately 2.12%). One of the loci deviates from neutral expectations, showing a clinal pattern of nucleotide diversity and haplotype structure in S. chilense. This geographic pattern of variation is suggestive of an incomplete (ongoing) selective sweep, but neutral explanations cannot be entirely dismissed. Both wild tomato species exhibit moderate levels of population differentiation (average F(ST) approximately 0.20). Interestingly, the pooled samples (across different demes) exhibit more negative Tajima's D and Fu and Li's D values; this marked excess of low-frequency polymorphism can only be explained by population (or range) expansion and is unlikely to be due to population structure per se. We thus propose that population structure and population/range expansion are among the most important evolutionary forces shaping patterns of nucleotide diversity within and among demes in these wild tomatoes. Patterns of population differentiation may also be impacted by soil seed banks and historical associations mediated by climatic cycles. Intragenic linkage disequilibrium (LD) decays very rapidly with physical distance, suggesting high recombination rates and effective population sizes in both species. The rapid decline of LD seems very promising for future association studies with the purpose of mapping functional variation in wild tomatoes.     

Visualization of pairwise and multilocus linkage disequilibrium structure using latent forests

Linkage disequilibrium study represents a major issue in statistical genetics as it plays a fundamental role in gene mapping and helps us to learn more about human history. The linkage disequilibrium complex structure makes its exploratory data analysis essential yet challenging. Visualization methods, such as the triangular heat map implemented in Haploview, provide simple and useful tools to help understand complex genetic patterns, but remain insufficient to fully describe them. Probabilistic graphical models have been widely recognized as a powerful formalism allowing a concise and accurate modeling of dependences between variables. In this paper, we propose a method for short-range, long-range and chromosome-wide linkage disequilibrium visualization using forests of hierarchical latent class models. Thanks to its hierarchical nature, our method is shown to provide a compact view of both pairwise and multilocus linkage disequilibrium spatial structures for the geneticist. Besides, a multilocus linkage disequilibrium measure has been designed to evaluate linkage disequilibrium in hierarchy clusters. To learn the proposed model, a new scalable algorithm is presented. It constrains the dependence scope, relying on physical positions, and is able to deal with more than one hundred thousand single nucleotide polymorphisms. The proposed algorithm is fast and does not require phase genotypic data.     

Unbiased application of the transmission/disequilibrium test to multilocus haplotypes

When the transmission/disequilibrium test (TDT) is applied to multilocus haplotypes, a bias may be introduced in some families for which both parents have the same heterozygous genotype at some locus. The bias occurs because haplotypes can only be deduced from certain offspring, with the result that the transmissions of the two parental haplotypes are not independent. We obtain an unbiased TDT for individual haplotypes by calculating the correct variance for the transmission count within a family, using information from multiple siblings if they are available. An existing correction for dependence between siblings in the presence of linkage is retained. To obtain an unbiased multihaplotype TDT, we must either count transmissions from one randomly chosen parent or count all transmissions and estimate the significance level empirically. Alternatively, we may use missing-data techniques to estimate uncertain haplotypes, but these methods are not robust to population stratification. An illustration using data from the insulin-gene region in type 1 diabetes shows that the validity and power of the TDT may vary by an order of magnitude, depending on the method of analysis.     

Entropy as a measure for linkage disequilibrium over multilocus haplotype blocks

OBJECTIVE: The presence of linkage disequilibrium (LD) forms the basis for a range of uses, including the fine-mapping of diseases and studies on human genealogy. Recent findings indicate that single nucleotide polymorphisms (SNP) can occur in blocks of limited haplotypic diversity with high degrees of LD. Commonly used measures for LD, such as r(2) and D', consider only two loci and might miss information to appropriately describe LD in larger haplotypic structures. METHODS: We introduce the Normalized Entropy Difference, epsilon, as a new multilocus measure for LD. A related quantity, deltaS, provides an approximate chi(2) test for the significance of LD. The ability of the measure to detect haplotype blocks is investigated using simulated data sets as well as a real data set previously analyzed by Daly et al. (2001). RESULTS: epsilon allows for arbitrary numbers of loci, describes LD with regard to the loci sequence, and can be interpreted as a multilocus extension of r(2). The application of epsilon to the data sets demonstrated the measure's ability to appropriately describe simultaneous multilocus LD and to detect haplotype blocks. CONCLUSIONS: epsilon is a reasonable multilocus LD measure and might be of potential use in the construction of the human haplotype map.     

RETRACTED ARTICLE: An analysis of population structure and linkage disequilibrium using multilocus data in 187 maize inbred lines

This study analyzes population structure and linkage disequilibrium (LD) among 187 commonly used Chinese maize inbred lines, representing the genetic diversity among public, commercial and historically important lines for corn breeding. Seventy SSR loci, evenly distributed over 10 chromosomes, were assayed for polymorphism. The identified 290 alleles served to estimate population structure and analyze the genome-wide LD. The population of lines was highly structured, showing 6 subpopulations: BSSS (American BSSS including Reid), PA (group A germplasm derived from modern U.S. hybrids in China), PB (group B germplasm derived from modern U.S. hybrid in China), Lan (Lancaster Surecrop), LRC (derivative lines from Lvda Reb Cob, a Chinese landrace) and SPT (derivative lines from Si-ping-tou, a Chinese landrace). Forty lines, which formerly had an unknown and/or miscellaneous origin and pedigree record, were assigned to the appropriate group. Relationship estimates based on SSR marker data were quantified in a Q matrix, and this information will inform breeder’s decisions regarding crosses. Extensive inter- and intra-chromosomal LD was detected between 70 microsatellite loci for the investigated maize lines (2109 loci pairs in LD with D′ > 0.1 and 93 out of them at P < 0.01).This suggests that rapidly evolving microsatellites may track recent population structure. Interlocus LD decay among the diverse maize germplasm indicated that association studies in QTLs and/or candidate genes might avoid nonfunctional and spurious associations since most of the LD blocks were broken between diverse germplasm. The defined population structure and the LD analysis present the basis for future association mapping. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Effect of gametic disequilibrium on means and on genetic variances of autotetraploid synthetic varieties

Summary The occurrence and effects of a gametic disequilibrium (DSE) in the first generation of a theoretical two-population synthetic variety were investigated. Theoretical development was limited to the genetics at a single locus with two alleles in an autotetraploid species with random chromosome inheritance. Algebraic expressions were developed for the differences between the mean genotypic values of the two-population synthetic variety at generation one and in random mating equilibrium (RME). For the situation where both parents of the synthetic were in RME, a numerical analysis was performed for all possible allele frequencies assuming the following types of genic action: monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and additive. The result indicated that with non-additive genic action the DSE could, in some cases, greatly depress or inflate the mean genotypic value of the first generation (Syn-1_(RME)). Thus, any change of means over advancing generations with loss of DSE could be positive or negative. When additive genic action was assumed, there was no effect associated with DSE and when both parents had the same allele frequencies there was no DSE. The DSE, with only a minor exception, decreased the genetic variance and in numerous cases forced it near zero. Expressions were developed for mean genotypic values of a first generation synthetic with DSE in one parent (Syn-1_(DSE/RME)) or both parents (Syn-1_(DSE)). The deviation of these means from those of Syn-1_(RME) was a function of digenic and quadragenic population effects. An inspection of the response equations for Syn-1_(RME) indicated that in a series of crosses with one common parent the rankings of first generation means would be the same as the ranking of populations at equilibrium though the individual means would be biased. More importantly with DSE of one or both parents there are situations when a ranking of first generation mean genotypic values would not reflect relative frequency of desirable alleles in the populations. These results indicate that statistical analyses and selections based on means of the Syn-1 generation can have an error which is not avoidable by improvement in precision of evaluation.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada, USAPaper No. 590 Scientific Journal Series, Nevada Agricultural Experiment Station     

Computationally efficient sibship and parentage assignment from multilocus marker data

Quite a few methods have been proposed to infer sibship and parentage among individuals from their multilocus marker genotypes. They are all based on Mendelian laws either qualitatively (exclusion methods) or quantitatively (likelihood methods), have different optimization criteria, and use different algorithms in searching for the optimal solution. The full-likelihood method assigns sibship and parentage relationships among all sampled individuals jointly. It is by far the most accurate method, but is computationally prohibitive for large data sets with many individuals and many loci. In this article I propose a new likelihood-based method that is computationally efficient enough to handle large data sets. The method uses the sum of the log likelihoods of pairwise relationships in a configuration as the score to measure its plausibility, where log likelihoods of pairwise relationships are calculated only once and stored for repeated use. By analyzing several empirical and many simulated data sets, I show that the new method is more accurate than pairwise likelihood and exclusion-based methods, but is slightly less accurate than the full-likelihood method. However, the new method is computationally much more efficient than the full-likelihood method, and for the cases of both sexes polygamous and markers with genotyping errors, it can be several orders faster. The new method can handle a large sample with thousands of individuals and the number of markers limited only by the computer memory.     

Estimating selfing rates from reconstructed pedigrees using multilocus genotype data

Several methods have been developed to estimate the selfing rate of a population from a sample of individuals genotyped for several marker loci. These methods can be based on homozygosity excess (or inbreeding), identity disequilibrium, progeny array (PA) segregation or population assignment incorporating partial selfing. Progeny array-based method is generally the best because it is not subject to some assumptions made by other methods (such as lack of misgenotyping, absence of biparental inbreeding and presence of inbreeding equilibrium), and it can reveal other facets of a mixed-mating system such as patterns of shared paternity. However, in practice, it is often difficult to obtain PAs, especially for animal species. In this study, we propose a method to reconstruct the pedigree of a sample of individuals taken from a monoecious diploid population practicing mixed mating, using multilocus genotypic data. Selfing and outcrossing events are then detected when an individual derives from identical parents and from two distinct parents, respectively. Selfing rate is estimated by the proportion of selfed offspring in the reconstructed pedigree of a sample of individuals. The method enjoys many advantages of the PA method, but without the need of a priori family structure, although such information, if available, can be utilized to improve the inference. Furthermore, the new method accommodates genotyping errors, estimates allele frequencies jointly and is robust to the presence of biparental inbreeding and inbreeding disequilibrium. Both simulated and empirical data were analysed by the new and previous methods to compare their statistical properties and accuracies.