Prediction of Multilocus Identity-by-Descent

Abstract

Previous studies have enabled exact prediction of probabilities of identity-by-descent (IBD) in randommating populations for a few loci (up to four or so), with extension to more using approximate regression methods. Here we present a precise predictor of multiple-locus IBD using simple formulas based on exact results for two loci. In particular, the probability of non-IBD X ABC at each of ordered loci A, B, and C can be well approximated by X ABC ¼ X AB X BC /X B and generalizes to X 123. . .k ¼ X 12 X 23. . . X kÀ1,k /X kÀ2 , where X is the probability of non-IBD at each locus. Predictions from this chain rule are very precise with population bottlenecks and migration, but are rather poorer in the presence of mutation. From these coefficients, the probabilities of multilocus IBD and non-IBD can also be computed for genomic regions as functions of population size, time, and map distances. An approximate but simple recurrence formula is also developed, which generally is less accurate than the chain rule but is more robust with mutation. Used together with the chain rule it leads to explicit equations for non-IBD in a region. The results can be applied to detection of quantitative trait loci (QTL) by computing the probability of IBD at candidate loci in terms of identity-by-state at neighboring markers. I N a recent article formulas for computing probabilities of identity-by-descent (IBD) at multiple loci in random-mating populations were obtained (Hill and Weir 2007) by extending methods of Cockerham (1969, 1974) for a haploid model. Recurrence equations were presented for multilocus non-IBD, from which IBD can be computed; but the number of terms involved quickly becomes impracticably large to compute. For example, prediction of nonidentity at three loci requires recurrence equations for a total of 16 non-IBD measures defined for loci sampled on two, three, four, five, and six different haplotypes. For four loci the number of measures rises to 139 (Hill and Weir 2007). Hernández-Sánchez et al. (2004) have developed approximations based on multiple regression to compute IBD at multiple loci from that at two loci, but the formulas become increasingly less tractable and accurate as the number of loci increases.Here we develop a straightforward method (the chain rule) for predicting probabilities of multilocus non-IBD, and thus IBD, which uses exact results only on two-locus non-IBD probabilities. Assuming a known population history, this predictor can be very precise for many loci and can enable IBD for a whole chromosome region to be computed. We also develop simple approximate recurrence equations that are generally less precise, except in the presence of mutation.An application of multiple-locus extensions of Wright's inbreeding coefficient is in gene or quantitative trait loci (QTL) mapping on the basis of the association between phenotypic similarity of individuals and shared IBD at a particular genomic region (Meuwissen et al. 2002;Hernández-Sánchez et al. 2006). The magnitude of IBD at a QTL is c...

Mapping quantitative trait loci in a wild population using linkage and linkage disequilibrium analyses

“…For example, Meuwissen & Goddard (2007) developed theory to calculate IBD probabilities under a mutation-drift equilibrium model, thus eliminating the need to estimate, or guess, T. However, QTLs might not be in equilibrium or proving that they are may be difficult. GridQTL could also incorporate theory to accommodate variation in Ne and migration (Hill & Herna´ndez-Sa´nchez, 2007). More complex population histories may be accommodated using simulations, e.g.…”

“…Both gridLA and gridLDLA use a deterministic approach to estimate these matrices given pedigree and markers (Pong-Wong et al, 2001). In addition, grid-LDLA uses population information to infer relationships among pedigree founders (Meuwissen & Goddard, 2000 ;Hill & Herna´ndez-Sa´nchez, 2007). Our approach is much faster than LOKI and almost as accurate (Sørensen et al, 2002).…”

Whole population, genome-wide mapping of hidden relatedness

“…Indeed, renewed interest in IBD (Thompson 2008) has led to the development of novel methods to quantify relatedness, by either genome-wide estimates (Mao and Xu 2005), segment-bysegment analysis (Hill and Hernandez-Sanchez 2007;Hill and Weir 2007), or both (Purcell et al 2007). Applying such methods, the recently published Human Haplotype Map (The International HapMap Consortium 2005) reported the surprising discovery of abundant long haplotype segments shared between individuals purported to be unrelated (Frazer et al 2007) coining them as hidden relatives.…”

A Multilocus Model for Constructing a Linkage Disequilibrium Map in Human Populations

“…Our model allows the relatedness of sampled families from a population through the implementation of identical-by-descent (IBD) probabilities. Hill and Hernandez-Sanchez (2007) recently proposed a model for analyzing the probability of multilocus IBD. A new approach for calculating probabilities of IBD for pairs of haplotypes has been derived by Browning (2008).…”