A hidden Markov model approach to multilocus linkage analysis in a full-sib family

Statistical packages for constructing genetic linkage maps in inbred lines are well developed and applied extensively, while linkage analysis in outcrossing species faces some statistical challenges because of their complicated genetic structures. In this article, we present a multilocus linkage analysis via hidden Markov models for a linkage group of markers in a full-sib family. The advantage of this method is the simultaneous estimation of the recombination fractions between adjacent markers that possibly segregate in different ratios, and the calculation of likelihood for a certain order of the markers. When the number of markers decreases to two or three, the multilocus linkage analysis becomes traditional two-point or three-point linkage analysis, respectively. Monte Carlo simulations are performed to show that the recombination fraction estimates of multilocus linkage analysis are more accurate than those just using two-point linkage analysis and that the likelihood as an objective function for ordering maker loci is the most powerful method compared with other methods. By incorporating this multilocus linkage analysis, we have developed a Windows software, FsLinkageMap, for constructing genetic maps in a full-sib family. A real example is presented for illustrating linkage maps constructed by using mixed segregation markers. Our multilocus linkage analysis provides a powerful method for constructing high-density genetic linkage maps in some outcrossing plant species, especially in forest trees.     

Efficient computations in multilocus linkage analysis.

This paper describes efficient methods for likelihood calculations and maximum-likelihood estimation in multilocus linkage analysis of reference families and general disease pedigrees, and it documents their performance as implemented in the LINKAGE programs. This information should be of considerable value in determining computing needs for linkage investigations, and in evaluating the merits of alternative algorithms.     

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.     

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.     

A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis

For tightly linked loci, cosegregation may lead to nonrandom associations between alleles in a population. Because of its evolutionary relationship with linkage, this phenomenon is called linkage disequilibrium. Today, linkage disequilibrium-based mapping has become a major focus of recent genome research into mapping complex traits. In this article, we present a new statistical method for mapping quantitative trait loci (QTL) of additive, dominant, and epistatic effects in equilibrium natural populations. Our method is based on haplotype analysis of multilocus linkage disequilibrium and exhibits two significant advantages over current disequilibrium mapping methods. First, we have derived closed-form solutions for estimating the marker-QTL haplotype frequencies within the maximum-likelihood framework implemented by the EM algorithm. The allele frequencies of putative QTL and their linkage disequilibria with the markers are estimated by solving a system of regular equations. This procedure has significantly improved the computational efficiency and the precision of parameter estimation. Second, our method can detect marker-QTL disequilibria of different orders and QTL epistatic interactions of various kinds on the basis of a multilocus analysis. This can not only enhance the precision of parameter estimation, but also make it possible to perform whole-genome association studies. We carried out extensive simulation studies to examine the robustness and statistical performance of our method. The application of the new method was validated using a case study from humans, in which we successfully detected significant QTL affecting human body heights. Finally, we discuss the implications of our method for genome projects and its extension to a broader circumstance. The computer program for the method proposed in this article is available at the webpage http://www.ifasstat.ufl.edu/genome/~LD.     

Improvement of mapping accuracy by unifying linkage and association analysis

It is well known that pedigree/family data record information on the coexistence in founder haplotypes of alleles at nearby loci and the cotransmission from parent to offspring that reveal different, but complementary, profiles of the genetic architecture. Either conventional linkage analysis that assumes linkage equilibrium or family-based association tests (FBATs) capture only partial information, leading to inefficiency. For example, FBATs will fail to detect even very tight linkage in the case where no allelic association exists, while a violation of the assumption of linkage equilibrium will result in biased estimation and reduced efficiency in linkage mapping. In this article, by using a data augmentation technique and the EM algorithm, we propose a likelihood-based approach that embeds both linkage and association analyses into a unified framework for general pedigree data. Relative to either linkage or association analysis, the proposed approach is expected to have greater estimation accuracy and power. Monte Carlo simulations support our theoretical expectations and demonstrate that our new methodology: (1) is more powerful than either FBATs or classic linkage analysis; (2) can unbiasedly estimate genetic parameters regardless of whether association exists, thus remedying the bias and less precision of traditional linkage analysis in the presence of association; and (3) is capable of identifying tight linkage alone. The new approach also holds the theoretical advantage that it can extract statistical information to the maximum extent and thereby improve mapping accuracy and power because it integrates multilocus population-based association study and pedigree-based linkage analysis into a coherent framework. Furthermore, our method is numerically stable and computationally efficient, as compared to existing parametric methods that use the simplex algorithm or Newton-type methods to maximize high-order multidimensional likelihood functions, and also offers the computation of Fisher's information matrix. Finally, we apply our methodology to a genetic study on bone mineral density (BMD) for the vitamin D receptor (VDR) gene and find that VDR is significantly linked to BMD at the one-third region of the wrist.     

Joint estimation of recombination fractions and interference coefficients in multilocus linkage analysis.

Estimation of recombination fractions and interference coefficients is of importance in multilocus linkage analysis. With the development of molecular genetic technologies such as RFLP, multilocus data are readily available to researchers. Several methods have been developed to analyze such data, and each performs well under restrictive conditions. The present paper proposes a method based on a multiplicative model and maximum-likelihood estimation of recombination fractions and interference coefficients. The estimators are consistent regardless of the model assumptions and are efficient if the model is a good approximation. The estimators are tractable even when there are incomplete observations. Furthermore, the interference between nonadjacent chromosomal regions or those among three chromosomal regions can be modeled and tested by a simple Z-test. The proposed method was applied to linkage analysis of four-locus data obtained from Drosophila and that of seven-locus data obtained again from Drosophila. Reanalysis of the first example revealed that there is interference between chromosomal regions 2 and 3. Analysis of the second example suggested that there is triple interference as well as pairwise interference between nonadjacent chromosomal regions; the genetic interpretation of these findings remains to be developed.     

Joint tests of linkage and association for quantitative traits

In this paper we describe various study designs and analytic techniques for testing the joint hypothesis that a genetic marker is both linked to and associated with a quantitative phenotype. Issues of power and sampling are addressed. The distinction between methods that explicitly examine association and those that infer association by examining the distribution of allelic transmissions from a heterozygous parent is examined. Extensions to multivariate, multiallelic, and multilocus situations are addressed. Recent approaches that combine variance-components-based linkage analyses with joint tests of linkage in the presence of association for disentanglement of the linkage and association and the application of such methods to fine mapping are discussed. Finally, new classes of joint tests of linkage and association that do not require samples of related individuals are described.     

Efficient multipoint linkage analysis through reduction of inheritance space

Computational constraints currently limit exact multipoint linkage analysis to pedigrees of moderate size. We introduce new algorithms that allow analysis of larger pedigrees by reducing the time and memory requirements of the computation. We use the observed pedigree genotypes to reduce the number of inheritance patterns that need to be considered. The algorithms are implemented in a new version (version 2.1) of the software package GENEHUNTER. Performance gains depend on marker heterozygosity and on the number of pedigree members available for genotyping, but typically are 10-1,000-fold, compared with the performance of the previous release (version 2.0). As a result, families with up to 30 bits of inheritance information have been analyzed, and further increases in family size are feasible. In addition to computation of linkage statistics and haplotype determination, GENEHUNTER can also perform single-locus and multilocus transmission/disequilibrium tests. We describe and implement a set of permutation tests that allow determination of empirical significance levels in the presence of linkage disequilibrium among marker loci.     

A molecular genetic linkage map of mouse chromosome 2

Interspecific backcross mice were used to create a molecular genetic linkage map of chromosome 2. Genomic DNAs from N2 progeny were subjected to Southern blot analysis using molecular probes that identified the Abl, Acra, Ass, C5, Cas-1, Fshb, Gcg, Hox-5.1, Jgf-1, Kras-3, Ltk, Pax-1, Prn-p, and Spna-2 loci; these loci were added to the 11 loci previously mapped to the distal region of chromosome 2 in the same interspecific backcross to generate a composite multilocus linkage map. Several loci mapped near, and may be the same as, known mutations. Comparisons between the mouse and the human genomes indicate that mouse chromosome 2 contains regions homologous to at least six human chromosomes. Mouse models for human diseases are discussed.     

Multilocus linkage analysis of affected sib pairs

OBJECTIVE: The conventional affected sib pair methods evaluate the linkage information at a locus by considering only marginal information. We describe a multilocus linkage method that uses both the marginal information and information derived from the possible interactions among several disease loci, thereby increasing the significance of loci with modest effects. METHODS: Our method is based on a statistic that quantifies the linkage information contained in a set of markers. By a marker selection-reduction process, we screen a set of polymorphisms and select a few that seem linked to disease. RESULTS: We test our approach on genome scan data for inflammatory bowel disease (InfBD) and on simulated data. On real data we detect 6 of the 8 known InfBD loci; on simulated data we obtain improvements in power of up to 40% compared to a conventional single-locus method. CONCLUSION: Our extensive simulations and the results on real data show that our method is in general more powerful than single-locus methods in detecting disease loci responsible for complex traits. A further advantage of our approach is that it can be extended to make use of both the linkage and the linkage disequilibrium between disease loci and nearby markers.     

Modeling the effect of inbreeding among founders in linkage analysis

In this paper, we present a unified mathematical model for linkage analysis that allows for inbreeding among founders in all families. The identical by descent (IBD) configuration of each pedigree is modeled as a Markov process containing two parameters; the inverse inbreeding and kinship coefficient and a rate parameter proportional to the inverse expected length of chromosome segments shared IBD by two different founder haplotypes. We use hidden Markov models and define a forward-backward algorithm for computing the conditional IBD-distribution given marker data, thereby extending the multipoint method of Lander and Green [1987. Construction of multilocus genetic maps in humans, Proc. Natl. Acad. Sci. USA 84, 2363-2367] to situations where founders are inbred. Our methodology is valid for arbitrary pedigree structures. Simulation and theoretical approximations for nonparametric linkage (NPL) analysis based on affected sib pairs reveal that NPL scores are inflated and type 1 errors increased when the inbreeding coefficient or rate parameter is underestimated. When the parents are genotyped, we present a general way of modifying the score function to drastically reduce this effect.     

Genetic linkage in the estimation of pairwise relationship

 New types of markers, such as RAPDs, microsatellite markers, AFLPs, and SNPs provide the opportunity to obtain information on individuals at multiple genetic loci across the genome. This increase in the number of marker loci has provided enhanced opportunities for statistical analysis of the genetic consequences of genealogical relationship among individuals. In place of the classical models, we can now investigate empirical multilocus segregation patterns. Linkage among loci decreases the precision of relationship estimation but permits additional dimensions of genome sharing to be explored. In this paper we consider the effect of linkage on the pattern of genome sharing among relatives who share (on average) 25% of their dipolid genomes using the empirical meioses giving rise to 58 gametophytes from a single maternal plant of the species Pinus taeda (loblolly pine). The genome sharing among relatives is quantified in terms of the linkage map of the markers. Received: 26 November 1997 / Accepted: 3 March 1998     

Waardenburg syndrome (WS): the analysis of a single family with a WS1 mutation showing linkage to RFLP markers on human chromosome 2q.

Waardenburg syndrome type I (WS1; MIM 19350) is caused by a pleiotropic, autosomal dominant mutation with variable penetrance and expressivity. Of individuals with this mutation, 20%-25% are hearing impaired. A multilocus linkage analysis of RFLP data from a single WS1 family with 11 affected individuals indicates that the WS1 mutation in this family is linked to the following four marker loci located on the long arm of chromosome 2: ALPP (alkaline phosphatase, placental), FN1 (fibronectin 1), D2S3 (a unique-copy DNA segment), and COL6A3 (collagen VI, alpha 3). For the RFLP marker loci, a multilocus linkage analysis using MLINK produced a peak lod (Z) of 3.23 for the following linkage relationships and recombination fractions (theta i): (ALPP----.000----FN1)----.122----D2S3----.267----CO L6A3. A similar analysis produced a Z of 6.67 for the following linkage relationships and theta i values among the markers and WS1: (FN1----.000----WS1----.000----ALPP)----.123----D2S 3----.246----COL6A3. The data confirm the conclusion of Foy et al. that at least some WS1 mutations map to chromosome 2q.     

Influence of aberrant observations on high-resolution linkage analysis outcomes

Because of the availability of efficient, user-friendly computer analysis programs, the construction of multilocus human genetic maps has become commonplace. At the level of resolution at which most of these maps have been developed, the methods have proved to be robust. This may not be true in the construction of high-resolution linkage maps (3-cM interlocus resolution or less). High-resolution meiotic maps, by definition, have a low probability of recombination occurring in an interval. As such, even low frequencies of errors in typing (1.5% or less) may influence mapping outcomes. To investigate the influence of aberrant observations on high-resolution maps, a Monte Carlo simulation analysis of multipoint linkage data was performed. Introduction of error was observed to reduce power to discriminate orders, dramatically inflate map length, and provide significant support for incorrect over correct orders. These results appear to be due to the misclassification of nonrecombinant gametes as multiple recombinants. Chi 2-Like goodness-of-fit analysis appears to be quite sensitive to the appearance of misclassified gametes, providing a simple test for aberrant data sets. Multiple pairwise likelihood analysis appears to be less sensitive than does multipoint analysis and may serve as a check for map validity.     

On the probability of loss of new mutations in the presence of linkage disequilibrium

A new selectively neutral mutation occurs in a multilocus genetic background that has achieved a stable equilibrium at which there is a linkage disequilibrium. Perturbation techniques are applied to an extension of the branching process formulation of Fisher in order to address the question of extinction probabilities. We show that under appropriate conditions the probability of extinction of the new mutant is increased by the existence of linkage disequilibrium in the genetic background.Research supported in part by NIH grant GM 28016     

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.     

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.     

On the use of DNA fingerprints for linkage studies in cattle

To find a marker for the bovine "muscular hypertrophy" gene and for the "roan" locus, we have typed six cattle pedigrees totaling 540 animals for nine blood group systems, for 12 biochemical markers, for RFLPs at four loci, and with five probes revealing multilocus DNA fingerprints. Segregation analysis of the fingerprint bands showed that, in cattle, a fingerprint probe will reveal a mean of 7.6 clearly resolvable bands, behaving as simple, highly informative Mendelian entities characterized by a mean mutation rate of +/- 1/4500 gametes. For one of the bands, we observed a "mutation burst" generating germline mosaicism. Because some of the fingerprint bands were allelic or corresponded to clustered minisatellites, a mean of only 5.7 independent loci is explored per probe. Fingerprint bands revealed by different probes also show a clear propensity for close linkage, pointing toward nonrandom distribution of minisatellite sequences or the existence of minisatellite clusters. Although this reduces the power of fingerprints for linkage analysis substantially, we were able to demonstrate genetic linkage between fingerprint bands and at least three of the classical markers, to exclude the roan locus from 4.5 Morgans of the bovine genome with the DNA fingerprints and for an additional 2.5 Morgans with the classical markers, and to identify a solid candidate marker for the bovine muscular hypertrophy gene, yielding a lod score greater than or equal to 2.84 without any obliged recombinant.     

Multilocus linkage analysis with the human argininosuccinate synthetase gene

We have identified three restriction fragment length polymorphisms (RFLPs) from within the argininosuccinate synthetase (ASS) gene which maps to human chromosome 9q34-qter. Although RFLPs at pseudogene loci are detected by the cDNA, these are the first polymorphisms reported at the ASS locus. The three RFLPs are in linkage equilibrium with each other, and haplotypes for the ASS locus are highly informative. Two-locus recombination estimates between ASS and seven other 9q markers indicated that ASS is closest to the ABO blood group with a recombination fraction of 0.04 (0.005-0.11). A multilocus lod score analysis with these seven 9q markers indicated that ASS maps between ABL and MCT136 close to ABO, but it is uncertain if ASS is centromeric or telomeric to ABO.