snp.plotter: an R-based SNP/haplotype association and linkage disequilibrium plotting package

snp.plotter is a newly developed R package which produces high-quality plots of results from genetic association studies. The main features of the package include options to display a linkage disequilibrium (LD) plot below the P-value plot using either the r2 or D' LD metric, to set the X-axis to equal spacing or to use the physical map of markers, and to specify plot labels, colors, symbols and LD heatmap color scheme. snp.plotter can plot single SNP and/or haplotype data and simultaneously plot multiple sets of results. R is a free software environment for statistical computing and graphics available for most platforms. The proposed package provides a simple way to convey both association and LD information in a single appealing graphic for genetic association studies. AVAILABILITY: Downloadable R package and example datasets are available at http://cbdb.nimh.nih.gov/~kristin/snp.plotter.html and http://www.r-project.org.     

A genealogical interpretation of linkage disequilibrium

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

Exploring linkage disequilibrium

Linkage disequilibrium (LD, association of allelic states across loci) is poorly understood by many evolutionary biologists, but as technology for multilocus sampling improves, we ignore LD at our peril. If we sample variation at 10 loci in an organism with 20 chromosomes, we can reasonably treat them as 10 ‘independent witnesses’ of the evolutionary process. If instead, we sample variation at 1000 loci, many are bound to be close together on a chromosome. With only one or two crossovers per meiosis, associations between close neighbours decay so slowly that even LD created far in the past will not have dissipated, so we cannot treat the 1000 loci as independent witnesses (Barton 2011 ). This means that as marker density on genomes increases classic analyses assuming independent loci become mired in the problem of overconfidence: if 1000 independent witnesses are assumed, and that number should be much lower, any conclusion will be overconfident. This is of special concern because our literature suffers from a strong publication bias towards confident answers, even when they turn out to be wrong (Knowles 2008 ). In contrast, analyses that take into account associations across loci both control for overconfidence and can inform us about LD generating events far in the past, for example human/Neanderthal admixture (Fu et al. 2014 ). With increased marker density, biologists must increase their awareness of LD and, in this issue of Molecular Ecology Resources, Kemppainen et al. ( 2015 ) make software available that can only help in this process: LDna allows patterns of LD in a data set to be explored using tools borrowed from network analysis. This has great potential, but realizing that potential requires understanding LD.     

Haplotype evolution and linkage disequilibrium: A simulation study

We simulated the evolution of a three-site haplotype system, two restriction fragment length polymorphisms flanking one short tandem repeat polymorphism, under five different demographic scenarios, three with constant population size and two with population growth. The simulation was designed to observe the effects of population history, recombination fraction, and mutation rate on allele and haplotype frequencies, haplotype diversity, frequency of ancestral alleles, and linkage disequilibrium. The known ancestral haplotypes were often found at low frequencies and even became extinct after 5, 000 generations, especially with small effective population sizes. The original linkage disequilibrium was eroded and even reversed.     

Homozygosity and linkage disequilibrium

We illustrate how homozygosity of haplotypes can be used to measure the level of disequilibrium between two or more markers. An excess of either homozygosity or heterozygosity signals a departure from the gametic phase equilibrium: We describe the specific form of dependence that is associated with high (low) homozygosity and derive various linkage disequilibrium measures. They feature a clear biological interpretation, can be used to construct tests, and are standardized to allow comparison across loci and populations. They are particularly advantageous to measure linkage disequilibrium between highly polymorphic markers.     

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.     

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.     

A multilocus linkage disequilibrium measure based on mutual information theory and its applications

Evaluating the patterns of linkage disequilibrium (LD) is important for association mapping study as well as for studying the genomic architecture of human genome (e.g., haplotype block structures). Commonly used bi-allelic pairwise measures for assessing LD between two loci, such as r ² and D′, may not make full and efficient use of modern multilocus data. Though extended to multilocus scenarios, their performance is still questionable. Meanwhile, most existing measures for an entire multilocus region, such as normalized entropy difference, do not consider existence of LD heterogeneity across the region under investigation. Additionally, these existing multilocus measures cannot handle distant regions where long-range LD patterns may exist. In this study, we proposed a novel multilocus LD measure developed based on mutual information theory. Our proposed measure described LD pattern between two chromosome regions each of which may consist of multiple loci (including multi-allele loci). As such, the proposed measure can better characterize LD patterns between two arbitrary regions. As potential applications, we developed algorithms on the proposed measure for partitioning haplotype blocks and for selecting haplotype tagging SNPs (htSNPs), which were helpful for follow-up association tests. The results on both simulated and empirical data showed that our LD measure had distinct advantages over pairwise and other multilocus measures. First, our measure was more robust, and can capture comprehensively the LD information between neighboring as well as disjointed regions. Second, haplotype blocks were better described via our proposed measure. Furthermore, association tests with htSNPs from the proposed algorithm had improved power over tests on single markers and on haplotypes.     

Risks of fetal cystic fibrosis based on linkage disequilibrium data

Summary First-trimester prenatal diagnosis of cystic fibrosis depends on tissues being available from a previous affected child for determination of the phase relationship of DNA markers. If no such tissues are available, it is possible to estimate the risks of a couple producing an affected child from the distribution of haplotypes showing linkage disequilibrium with the cystic fibrosis gene. We have calculated all the fetal risk subsets from the various parental haplotype combinations for the restriction fragment length polymorphisms identified by the KM. 19/PstI and XV-2c/TaqI systems. We conclude that only in a limited number of parental combinations are the fetal risks sufficiently high or sufficiently low to be used in prenatal diagnosis.     

Positional cloning by linkage disequilibrium

Recently, metric linkage disequilibrium (LD) maps that assign an LD unit (LDU) location for each marker have been developed (Maniatis et al. 2002). Here we present a multiple pairwise method for positional cloning by LD within a composite likelihood framework and investigate the operating characteristics of maps in physical units (kb) and LDU for two bodies of data (Daly et al. 2001; Jeffreys et al. 2001) on which current ideas of blocks are based. False-negative indications of a disease locus (type II error) were examined by selecting one single-nucleotide polymorphism (SNP) at a time as causal and taking its allelic count (0, 1, or 2, for the three genotypes) as a pseudophenotype, Y. By use of regression and correlation, association between every pseudophenotype and the allelic count of each SNP locus (X) was based on an adaptation of the Malecot model, which includes a parameter for location of the putative gene. By expressing locations in kb or LDU, greater power for localization was observed when the LDU map was fitted. The efficiency of the kb map, relative to the LDU map, to describe LD varied from a maximum of 0.87 to a minimum of 0.36, with a mean of 0.62. False-positive indications of a disease locus (type I error) were examined by simulating an unlinked causal SNP and the allele count was used as a pseudophenotype. The type I error was in good agreement with Wald's likelihood theorem for both metrics and all models that were tested. Unlike tests that select only the most significant marker, haplotype, or haploset, these methods are robust to large numbers of markers in a candidate region. Contrary to predictions from tagging SNPs that retain haplotype diversity, the sample with smaller size but greater SNP density gave less error. The locations of causal SNPs were estimated with the same precision in blocks and steps, suggesting that block definition may be less useful than anticipated for mapping a causal SNP. These results provide a guide to efficient positional cloning by SNPs and a benchmark against which the power of positional cloning by haplotype-based alternatives may be measured.     

Stabilizing selection and linkage disequilibrium

An approach to the investigation of the evolution of quantitative traits on the basis of analysis of two-locus marginal systems dynamics has been developed. It has been shown that under stabilizing selection the "quasi-stationary" state is quickly reached and maintained continuously. The "quasi-stationary" state is characterized by small changes in allele frequencies and by linkage disequilibrium that significantly decreases genotypic variance. Equations defining the role of linkage disequilibrium in the stationary state of mutation-selection balance are derived.     

GOLD--graphical overview of linkage disequilibrium

SUMMARY: We describe a software package that provides a graphical summary of linkage disequilibrium in human genetic data. It allows for the analysis of family data and is well suited to the analysis of dense genetic maps. AVAILABILITY: http://www.well.ox.ac.uk/asthma/GOLD CONTACT: goncalo@well.ox.ac.uk     

Inferences about linkage disequilibrium.

Existing theory for inferences about linkage disequilibrium is restricted to a measure defined on gametic frequencies. Unless gametic frequencies are directly observable, they are inferred from genotypic frequencies under the assumption of random union of gametes. Primary emphasis in this paper is given to genotypic data, and disequilibrium coefficients are defined for all subsets of two or more of the four genes, two at each of two loci, carried by an individual. Linkage disequilibrium coefficients are defined for genes within and between gametes, and methods of estimating and testing these coefficients are given for gametic data. For genotypic data, when coupling and repulsion double heterozygotes cannot be distinguished. Burrows' composite measure of linkage disequilibrium is discussed. In particular, the estimate for this measure and hypothesis tests based on it are compared to the usual maximum likelihood estimate of gametic linkage disequilibrium, and corresponding likelihood ratio or contingency chi-square tests. General use of the composite measure, whether or not random union of gametes is an appropriate assumption, is recommended. Attention is given to small samples, where the non-normality of gene frequencies will have greatest effect on methods of inference based on normal theory. Even tools such as Fisher's z-transformation for the correlation of gene frequencies are found to perform quite satisfactorily.     

Volume measures for linkage disequilibrium

Background

A major QTL for fatness and growth, denoted FAT1, has previously been detected on pig chromosome 4q (SSC4q) using a Large White – wild boar intercross. Progeny that carried the wild boar allele at this locus had higher fat deposition, shorter length of carcass, and reduced growth. The position and the estimated effects of the FAT1 QTL for growth and fatness have been confirmed in a previous study. In order to narrow down the QTL interval we have traced the inheritance of the wild boar allele associated with high fat deposition through six additional backcross generations.

Results

Progeny-testing was used to determine the QTL genotype for 10 backcross sires being heterozygous for different parts of the broad FAT1 region. The statistical analysis revealed that five of the sires were segregating at the QTL, two were negative while the data for three sires were inconclusive. We could confirm the QTL effects on fatness/meat content traits but not for the growth traits implying that growth and fatness are controlled by distinct QTLs on chromosome 4. Two of the segregating sires showed highly significant QTL effects that were as large as previously observed in the F₂ generation. The estimates for the remaining three sires, which were all heterozygous for smaller fragments of the actual region, were markedly smaller. With the sample sizes used in the present study we cannot with great confidence determine whether these smaller effects in some sires are due to chance deviations, epistatic interactions or whether FAT1 is composed of two or more QTLs, each one with a smaller phenotypic effect. Under the assumption of a single locus, the critical region for FAT1 has been reduced to a 3.3 cM interval between the RXRG and SDHC loci.

Conclusion

We have further characterized the FAT1 QTL on pig chromosome 4 and refined its map position considerably, from a QTL interval of 70 cM to a maximum region of 20 cM and a probable region as small as 3.3 cM. The flanking markers for the small region are RXRG and SDHC and the orthologous region of FAT1 in the human genome is located on HSA1q23.3 and harbors approximately 20 genes. Our strategy to further refine the map position of this major QTL will be i) to type new markers in our pigs that are recombinant in the QTL interval and ii) to perform Identity-By-Descent (IBD) mapping across breeds that have been strongly selected for lean growth.     

LDA--a java-based linkage disequilibrium analyzer

SUMMARY: We describe an integrated java-based program that provides elaborate graphic and plain-text output of pairwise linkage disequilibrium (LD) analysis of single nucleotide polymorphisms genotypic data. It is most suitable for molecular geneticists, who are focusing on LD measures estimation, statistical significance test and extent prediction. AVAILABILITY: The software is available at: http://www.chgb.org.cn/lda/lda.htm. SUPPLEMENTARY INFORMATION: Detailed tutorials, LDA help system and examples are distributed within LDA software. For Macintosh OS X user, the Jre version 1.4 can be downloaded from http://connect.apple.com.     

Testing linkage disequilibrium in sibships

We describe the use of multivariate regression for testing allelic association in the presence of linkage, using marker genotype data from sibships. The test is valid, provided that the correct mean structure is modeled but does not require the correlation structure within families to be specified. The test can be implemented using standard statistical software such as the SAS programming language. In a simulation study, we evaluated this new test in comparison with one from a standard, matched-case-control analysis. First, we noted that the genetic effect needed to be quite extreme before residual familial correlation due to linkage led to false inference using the standard, matched-pair analysis. Second, we showed that under examples of extreme residual familial correlation, the new test had the correct test size. Third, we found that the test was more powerful than the sibship disequilibrium test of Horvath and Laird. Finally, we concluded that although the standard analysis may lead to correct inference for practical purposes, the new test is valid, even under extreme residual familial correlation and with no cost in power at the causal locus.     

Human prehistory: the message from linkage disequilibrium

The vast amount of information being gathered about human DNA sequence variation raises the question of what these data can tell us about events in our past. A new way has been found by which patterns of linkage disequilibrium can be used to detect the effects of natural selection in human prehistory.     

Population genomics: linkage disequilibrium holds the key

Current efforts to find disease-causing genes depend on patterns of linkage disequilibrium in human populations. Recent work has shown that linkage disequilibrium can extend over much larger genomic regions than expected, and that the patterns of linkage disequilibrium can differ markedly among populations.     

High-resolution joint linkage disequilibrium and linkage mapping of quantitative trait loci based on sibship data

This paper proposes variance component models for high resolution joint linkage disequilibrium (LD) and linkage mapping of quantitative trait loci (QTL) based on sibship data; this can include population data if independent individuals are treated as single sibships. One application of these models is late onset complex disease gene mapping, when parental data are not available. The models simultaneously incorporate both LD and linkage information. The LD information is contained in mean coefficients of sibship data. The linkage information is contained in the variance-covariance matrices of trait values for sibships with at least two siblings. We derive formulas for calculating the probability of sharing two trait alleles identical by descent (IBD) for sibpairs in interval mapping of QTL; this is the coefficient of dominant variance of the trait covariance of sibpairs on major QTL. To investigate the performance of the formulas, we calculate the numerical values via the formulas and get satisfactory approximations. We compare the power and sample sizes for both LD and linkage mapping. By simulation and theoretical analysis, we compare the results with those of Fulker and Abecasis "AbAw" approach. It is well known that the resolution of linkage analysis can be low for complex disease gene mapping. LD mapping, on the other hand, can increase mapping precision and is useful in high resolution mapping. Linkage analysis is less sensitive to population subdivisions and admixtures. The level of LD is sensitive to population stratification which may easily lead to spurious association. Performing a joint analysis of LD and linkage mapping can help to overcome the limits of both approaches. Moreover, the advantages of the two complementary strategies can be utilized maximally. In practice, linkage analysis may be performed using pedigree data to identify suggestive linkage between markers and trait loci based on a sparse marker map. In the presence of linkage, joint LD and linkage mapping can be carried out to do fine gene mapping based on a dense genetic map using both pedigree and population data. Population and pedigree data of any type can be combined to perform a joint analysis of high resolution LD and linkage mapping of QTL by generalizing the method.