Use of Multiple Genetic Markers in Prediction of Breeding Values

Genotypes at a marker locus give information on transmission of genes from parents to offspring and that information can be used in predicting the individuals' additive genetic value at a linked quantitative trait locus (MQTL). In this paper a recursive method is presented to build the gametic relationship matrix for an autosomal MQTL which requires knowledge on recombination rate between the marker locus and the MQTL linked to it. A method is also presented to obtain the inverse of the gametic relationship matrix. This information can be used in a mixed linear model for simultaneous evaluation of fixed effects, gametic effects at the MQTL and additive genetic effects due to quantitative trait loci unlinked to the marker locus (polygenes). An equivalent model can be written at the animal level using the numerator relationship matrix for the MQTL and a method for obtaining the inverse of this matrix is presented. Information on several unlinked marker loci, each of them linked to a different locus affecting the trait of interest, can be used by including an effect for each MQTL. The number of equations per animal in this case is 2m + 1 where m is the number of MQTL. A method is presented to reduce the number of equations per animal to one by combining information on all MQTL and polygenes into one numerator relationship matrix. It is illustrated how the method can accommodate individuals with partial or no marker information. Numerical examples are given to illustrate the methods presented. Opportunities to use the presented model in constructing genetic maps are discussed.     

A model for marker-assisted selection among single crosses with multiple genetic markers

 Trait means of marker genotypes are often inconsistent across experiments, thereby hindering the use of regression techniques in marker-assisted selection. Best linear unbiased prediction based on trait and marker data (TM-BLUP) does not require prior information on the mean effects associated with specific marker genotypes and, consequently, may be useful in applied breeding programs. The objective of this paper is to present a flanking-marker, TM-BLUP model that is applicable to interpopulation single crosses that characterize maize (Zea mays L.) breeding programs. The performance of a single cross is modeled as the sum of testcross additive and dominance effects at unmarked quantitative trait loci (QTL) and at marked QTL (MQTL). The TM-BLUP model requires information on the recombination frequencies between flanking markers and the MQTL and on MQTL variances. A tabular method is presented for calculating the conditional probability that MQTL alleles in two inbreds are identical by descent given the observed marker genotypes (G ^k _obs) at the kth MQTL. Information on identity by descent of MQTL alleles can then be used to calculate the conditional covariance of MQTL effects between single crosses given G ^k _obs. The inverse of the covariance matrix for dominance effects at unmarked QTL and MQTL can be written directly from the inverse of the covariance matrices of the corresponding testcross additive effects. In practice, the computations required in TM-BLUP may be prohibitive. The computational requirements may be reduced with simplified TM-BLUP models wherein dominance effects at MQTL are excluded, only the single crosses that have been tested are included, or information is pooled across several MQTL. Received: 22 June 1997 / Accepted: 25 February 1998     

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.     

Genetic evaluation by best linear unbiased prediction using marker and trait information in a multibreed population.

Genetic evaluation by best linear unbiased prediction (BLUP) requires modeling genetic means, variances, and covariances. This paper presents theory to model means, variances, and covariances in a multibreed population, given marker and breed information, in the presence of gametic disequilibrium between the marker locus (ML) and linked quantitative trait locus (MQTL). Theory and algorithms are presented to construct the matrix of conditional covariances between relatives (Gv) for the MQTL effects in a multibreed population and to obtain the inverse of Gv efficiently. Theory presented here accounts for heterogeneity of variances among pure breeds and for segregation variances between pure breeds. A numerical example was used to illustrate how the theory and algorithms can be used for genetic evaluation by BLUP using marker and trait information in a multibreed population.     

Genetic response from marker assisted selection in an outbred population for differing marker bracket sizes and with two identified quantitative trait loci.

Effect of flanking quantitative trait loci (QTL)-marker bracket size on genetic response to marker assisted selection in an outbred population was studied by simulation of a nucleus breeding scheme. In addition, genetic response with marker assisted selection (MAS) from two quantitative trait loci on the same and different chromosome(s) was investigated. QTL that explained either 5% or 10% of phenotypic variance were simulated. A polygenic component was simulated in addition to the quantitative trait loci. In total, 35% of the phenotypic variance was due to genetic factors. The trait was measured on females only. Having smaller marker brackets flanking the QTL increased the genetic response from MAS selection. This was due to the greater ability to trace the QTL transmission from one generation to the next with the smaller flanking QTL-marker bracket, which increased the accuracy of estimation of the QTL allelic effects. Greater negative covariance between effects at both QTL was observed when two QTL were located on the same chromosome compared to different chromosomes. Genetic response with MAS was greater when the QTL were on the same chromosome in the early generations and greater when they were on different chromosomes in the later generations of MAS.     

Combining the meiosis Gibbs sampler with the random walk approach for linkage and association studies with a general complex pedigree and multimarker loci

A linkage analysis for finding inheritance states and haplotype configurations is an essential process for linkage and association mapping. The linkage analysis is routinely based upon observed pedigree information and marker genotypes for individuals in the pedigree. It is not feasible for exact methods to use all such information for a large complex pedigree especially when there are many missing genotypic data. Proposed Markov chain Monte Carlo approaches such as a single-site Gibbs sampler or the meiosis Gibbs sampler are able to handle a complex pedigree with sparse genotypic data; however, they often have reducibility problems, causing biased estimates. We present a combined method, applying the random walk approach to the reducible sites in the meiosis sampler. Therefore, one can efficiently obtain reliable estimates such as identity-by-descent coefficients between individuals based on inheritance states or haplotype configurations, and a wider range of data can be used for mapping of quantitative trait loci within a reasonable time.     

The covariance between relatives conditional on genetic markers

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.     

Simulation of marker-assisted selection utilizing linkage disequilibrium: the effects of several additional factors

Use was made of our published model and methods to investigate the effects of several additional factors on marker-assisted selection (MAS) utilizing linkage disequilibrium. The additional factors were: size of the sample used to estimate the marker quantitative trait locus (MQTL) association effects, the method used to estimate the MQTL effects, use of the average of the top MQTL estimates in selection rather than individual estimates, size of the selection population, and the crossing of duplicate selection lines to generate further linkage disequilibrium and further selection response. The average map distance between the quantitative trait loci (QTLs) and their nearest marker was 0.15 Morgans. Use of estimates of MQTL effects derived by least squares yielded smaller selection responses than estimates derived by mixed-model methods. Selection responses were also reduced by using a smaller sample for estimating the associations because MQTL effects were less well estimated. This applied to selection on the MQTL effects themselves and to selection combining the MQTL with phenotypic information. Thus, poorly estimated MQTL effects added noise to the system and reduced selection response in combined selection. Using the average of the top MQTL estimates, rather than individual estimates, also reduced selection response. New linkage disequilibrium, generated by crossing two lines selected from the same population, did not lead to additional selection response in the cross line. These results show limitations to MAS using linkage disequilibrium until close linkages of markers and QTLs are available.     

X chromosome effects and their interactions with mitochondrial effects

We report a simple and rapid method for detecting additive genetic variance due to X-linked loci in the absence of marker data for this chromosome. We examined the interaction of this method with an established method for detecting mitochondrial linkage (another source of sex-asymmetric genetic covariance). When applied to data from the Collaborative Study on the Genetics of Alcoholism, this method found evidence of X-chromosomal linkage for one continuous trait (ntth1) and one discrete trait (SPENT). Evidence of mitochondrial contribution was found for one discrete trait (CRAVING) and three continuous traits (ln(CIGPKYR), ecb21, and tth1). Results for ntth1 suggest that methods that do not also allow for male-female heterogeneity in environmental variance may be overly conservative in detection of X-chromosomal effects.     

The genetic variance for multiple linked quantitative trait loci conditional on marker information in a crossed population

In the prediction of genetic values and quantitative trait loci (QTLs) mapping via the mixed model method incorporating marker information in animal populations, it is important to model the genetic variance for individuals with an arbitrary pedigree structure. In this study, for a crossed population originated from different genetic groups such as breeds or outbred strains, the variance of additive genetic values for multiple linked QTLs that are contained in a chromosome segment, especially the segregation variance, is investigated assuming the use of marker data. The variance for a finite number of QTLs in one chromosomal segment is first examined for the crossed population with the general pedigree. Then, applying the concept of the expectation of identity-by-descent proportion, an approximation to the mean of the conditional probabilities for the linked QTLs over all loci is obtained, and using it an expression for the variance in the case of an infinite number of linked QTLs marked by flanking markers is derived. It appears that the approach presented can be useful in the segment mapping using, and in the genetic evaluation of, crosses with general pedigrees in the population of concern. The calculation of the segregation variance through the current approach is illustrated numerically, using a small data-set.     

Quantitative genetic variance associated with chromosomal markers in segregating populations

Summary Use of chromosomal markers can accelerate genetic progress for quantitative traits in pedigree selection programs by providing early information on Mendelian segregation effects for individual progeny. Potential effectiveness of selection using markers is determined by the amount of additive genetic variance traced from parents to progeny by the markers. Theoretical equations for the amount of additive genetic variance associated with a marker were derived at the individual level and for a segregating population in joint linkage equilibrium. Factors considered were the number of quantitative trait loci linked to the marker, their individual effects, and recombination rates with the marker. Subsequently, the expected amount of genetic variance associated with a marker in a segregating population was derived. In pedigree selection programs in segregating populations, a considerable fraction of the genetic variance on a chromosome is expected to be associated with a marker located on that chromosome. For an average chromosome in the bovine, this fraction is approximately 40% of the Mendelian segregation variance contributed by the chromosome. The effects of interference and position of the marker on this expectation are relative small. Length of the chromosome has a large effect on the expected variance. Effectiveness of MAS is, however, greatly reduced by lack of polymorphism at the marker and inaccuracy of estimation of chromosome substitution effects. The size of the expected amount of genetic variance associated with a chromosomal marker indicates that, even when the marker is not the active locus, large chromosome substitution effects are not uncommon in segregating populations.     

Association between a genetic trait and a marker: Discrimination between epistasis and gametic disequilibrium

Summary We reconsider the method of Ott and Falk (1982) for the analysis of genetic linkage and of epistasis in the presence of phenotypic association. Their approach is extended to allow for gametic disequilibrium between marker and trait loci. We show that epistasis and tight linkage with gametic disequilibrium may be indistinguishable as explanations of association even in a very large pedigree.     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

Information gain for genetic parameter estimation with incorporation of marker data

Genetic marker data has been increasingly incorporated into segregation analysis, as combined segregation and linkage analysis has been performed more frequently. In this article, we study the extent of information gains with incorporation of marker data in segregation analysis, a topic that has not been investigated rigorously. Specifically, the current study is to investigate the influence of marker data on genetic model parameter estimation. A variance matrix criterion (as the inverse of the Fisher information matrix) and a relative entropy criterion (a measure of flatness of expected log-likelihood surface) are used to quantify the information gains. Our results indicate that substantial information gain can be achieved with the incorporation of marker data. The amount of variance reduction increases as the heterozygosity of the linked marker increases and as the trait gets closer to the linked marker(s). Incorporation of marker data in larger pedigrees also yields greater information gains based on both criteria. The effect of pedigree structure is also studied.     

A Monte Carlo algorithm for computing the IBD matrices using incomplete marker information

Identity-By-Descent (IBD) is a general measurement of the relationship between two groups of genes. If the two groups consist of two homologous genes, one from each individual, the IBD is called the coancestry between the two individuals. Coancestry is an important concept in both population and quantitative genetics. It is the probability that both genes are copies of the same gene in the genealogy. The average coancestry value at a random locus in a population reflects the level of population diversity, effective population size, the level of inbreeding and other attributes. Coancestry is also the building block for the covariance structure used to estimate the additive genetic variance component for a quantitative trait. There are many other types of IBD matrices, depending on the natures of the genes included in each group, and these IBD matrices vary from locus to locus. Molecular markers distributed along the genome provide information that can be used to infer these locus-specific IBD matrices. As a result, we can estimate and test the variance components of a quantitative trait contributed by these loci using the inferred IBD matrices. In this study, we develop the concept of locus-specific epistatic IBD matrices and a Monte Carlo method to infer these IBD matrices. The method is suitable for large pedigrees with arbitrary complexity and various levels of missing marker information. With these locus-specific IBD matrices, we are ready to search for quantitative trait loci along the genome in complicated pedigrees.     

Diagnosis of susceptibility to malignant hyperthermia with flanking DNA markers.

OBJECTIVE--To define the region on human chromosome 19 carrying the gene for malignant hyperthermia susceptibility and to evaluate the use of flanking DNA markers in diagnosing susceptibility. DESIGN--Prospective molecular genetic linkage studies in a large malignant hyperthermia pedigree. SETTING--Irish malignant hyperthermia testing centre. SUBJECTS--A large Irish malignant hyperthermia pedigree. MAIN OUTCOME MEASURES--Routine diagnosis of susceptibility to malignant hyperthermia with in vitro contracture test on muscle biopsy specimens and genetic linkage between susceptibility and polymorphic DNA markers in a malignant hyperthermia family. RESULTS--Genetic typing of polymorphic DNA markers in a large Irish malignant hyperthermia pedigree generated a lod score of greater than 3 for the marker D19S9 and showed that the gene for susceptibility is flanked by the markers D19S9 and D19S16. These tightly linked flanking markers allowed non-invasive presymptomatic diagnosis of susceptibility in five untested subjects in the large pedigree with an accuracy of greater than 99.7%. CONCLUSIONS--DNA markers flanking the gene for susceptibility to malignant hyperthermia can be used with high accuracy to diagnose susceptibility in subjects in large known malignant hyperthermia pedigrees and may replace the previous in vitro contracture test for diagnosing this inherited disorder in large families with malignant hyperthermia.     

Mapping quantitative trait loci using naturally occurring genetic variance among commercial inbred lines of maize (Zea mays L.)

Many commercial inbred lines are available in crops. A large amount of genetic variation is preserved among these lines. The genealogical history of the inbred lines is usually well documented. However, quantitative trait loci (QTL) responsible for the genetic variances among the lines are largely unexplored due to lack of statistical methods. In this study, we show that the pedigree information of the lines along with the trait values and marker information can be used to map QTL without the need of further crossing experiments. We develop a Monte Carlo method to estimate locus-specific identity-by-descent (IBD) matrices. These IBD matrices are further incorporated into a mixed-model equation for variance component analysis. QTL variance is estimated and tested at every putative position of the genome. The actual QTL are detected by scanning the entire genome. Applying this new method to a well-documented pedigree of maize (Zea mays L.) that consists of 404 inbred lines, we mapped eight QTL for the maize male flowering trait, growing degree day heat units to pollen shedding (GDUSHD). These detected QTL contributed >80% of the variance observed among the inbred lines. The QTL were then used to evaluate all the inbred lines using the best linear unbiased prediction (BLUP) technique. Superior lines were selected according to the estimated QTL allelic values, a technique called marker-assisted selection (MAS). The MAS procedure implemented via BLUP may be routinely used by breeders to select superior lines and line combinations for development of new cultivars.     

Calculation of multipoint likelihoods using flanking marker data: a simulation study

The calculation of multipoint likelihoods is computationally challenging, with the exact calculation of multipoint probabilities only possible on small pedigrees with many markers or large pedigrees with few markers. This paper explores the utility of calculating multipoint likelihoods using data on markers flanking a hypothesized position of the trait locus. The calculation of such likelihoods is often feasible, even on large pedigrees with missing data and complex structures. Performance characteristics of the flanking marker procedure are assessed through the calculation of multipoint heterogeneity LOD scores on data simulated for Genetic Analysis Workshop 14 (GAW14). Analysis is restricted to data on the Aipotu population on chromosomes 1, 3, and 4, where chromosomes 1 and 3 are known to contain disease loci. The flanking marker procedure performs well, even when missing data and genotyping errors are introduced.     

Pedigree-free animal models: the relatedness matrix reloaded

Animal models typically require a known genetic pedigree to estimate quantitative genetic parameters. Here we test whether animal models can alternatively be based on estimates of relatedness derived entirely from molecular marker data. Our case study is the morphology of a wild bird population, for which we report estimates of the genetic variance-covariance matrices (G) of six morphological traits using three methods: the traditional animal model; a molecular marker-based approach to estimate heritability based on Ritland's pairwise regression method; and a new approach using a molecular genealogy arranged in a relatedness matrix (R) to replace the pedigree in an animal model. Using the traditional animal model, we found significant genetic variance for all six traits and positive genetic covariance among traits. The pairwise regression method did not return reliable estimates of quantitative genetic parameters in this population, with estimates of genetic variance and covariance typically being very small or negative. In contrast, we found mixed evidence for the use of the pedigree-free animal model. Similar to the pairwise regression method, the pedigree-free approach performed poorly when the full-rank R matrix based on the molecular genealogy was employed. However, performance improved substantially when we reduced the dimensionality of the R matrix in order to maximize the signal to noise ratio. Using reduced-rank R matrices generated estimates of genetic variance that were much closer to those from the traditional model. Nevertheless, this method was less reliable at estimating covariances, which were often estimated to be negative. Taken together, these results suggest that pedigree-free animal models can recover quantitative genetic information, although the signal remains relatively weak. It remains to be determined whether this problem can be overcome by the use of a more powerful battery of molecular markers and improved methods for reconstructing genealogies.     

Derivation of single-locus relationship coefficients conditional on marker information

The coefficient of relationship is defined as the correlation between the additive genetic values of two individuals. This coefficient can be defined specifically for a single quantitative trait locus (QTL) and may deviate considerably from the overall expectation if it is taken conditional on information from linked marker loci. Conditional halfsib correlations are derived under a simple genetic model with a biallelic QTL linked to a biallelic marker locus. The conditional relationship coefficients are shown to depend on the recombination rate between the marker and the QTL and the population frequency of the marker alleles, but not on parameters of the QTL, i.e. number and frequency of QTL alleles, degree of dominance etc., nor on the (usually unknown) QTL genotype of the sire. Extensions to less simplified cases (multiple alleles at the marker locus and the QTL, two marker loci flanking the QTL) are given. For arbitrary pedigrees, conditional relationship coefficients can also be derived from the conditional gametic covariance matrix suggested by Fernando and Grossman (1989). The connection of these two approaches is discussed. The conditional relationship coefficient can be used for marker-assisted genetic evaluation as well as for the detection of QTL and the estimation of their effects.