New Mining Regs, New Discontent

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. Background Understanding the genetic architecture of complex traits controlled by many genes and environmental factors is currently one of the grand challenges in genetics. In this quest for the deciphering of the genetic code, Quantitative Trait Loci (QTL) mapping can be a powerful statistical tool. The basic idea of QTL analysis is to trace the inheritance of alleles from founders through a pedigree by using genetic markers. After estimating this gene flow through the pedigree, the allelic effects are estimated by relating Published: 08 January 2007 BMC Genetics 2007, 8:1 doi:10.1186/1471-2156-8-1 Received: 10 October 2006 Accepted: 08 January 2007 This article is available from: http://www.biomedcentral.com/1471-2156/8/1 © 2007 Rönnegård and Carlborg; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 1 of 14 (page number not for citation purposes) BMC Genetics 2007, 8:1 http://www.biomedcentral.com/1471-2156/8/1 the phenotypes with the different alleles. The position in the genome having the greatest statistical evidence for large allelic effects is the most likely position of a QTL. In QTL studies of pedigrees in outbred populations, variance component (VC) models are commonly used to estimate the variance of the allelic effects [1], rather than the effect of each individual allele. The studied phenotype is the explanatory variable and the QTL effect is assumed to be a random part of the phenotype. It is random because the founders of the mapping population are assumed to have QTL alleles with effects drawn from a distribution of allelic effects in the entire population and also because the alleles are transmitted from ancestor to descendent by a random process. The model assumes that the random effect is sampled from a multivariate normal distribution with an infinite number of different alleles, and the model is therefore called the infinite alleles model. Simulations have shown that the model is capable of giving unbiased estimates also when the QTL is biallelic [2-5]. To be able to estimate the variance of the random QTL effect, a between-individual covariance structure has to be specified. For a non-inbred population this is equal to the proportion of genes that individuals share identical-bydescent at a specific position [6], and the matrix describing the covariance structure is therefore called the identity-by-descent (IBD) matrix. Since the IBD matrix is not known a priori, it has to be estimated from marker information. This matrix has applications beyond the VC QTL model [7,8], however, and a lot of effort has been put into developing IBD estimation algorithms. For small pedigrees and few markers, the most likely IBD matrix can be estimated [9], but for large pedigrees this is too computationally demanding and approximate algorithms have therefore also been developed [10-12]. Hence, the IBD matrix estimation algorithms have been described in detail, but explicit definitions of the random QTL effect in terms of independent levels are more difficult to find (see however [13,14]). Several review articles have been published where the assumptions of the model are addressed [2,15-18]. The focus of these papers was on statistical testing of QTL effects and interpretation of genome scans. Several other papers have focused on the biological interpretation of the random QTL effects. Goddard [14] compares the assumptions of the biallelic and the infinite alleles model. He also explains how uncertainty is included in the infinite alleles model by adding a random sampling term to the expected allelic effects and that the variance of the sampling terms are proportional to the QTL variance under the infinite alleles model. Furthermore, Meuwissen and Goddard [19] developed a VC model where they related phenotypes with QTL allele effects by means of an incidence matrix. None of these papers show, however, how the random QTL effect can be given in terms of independent levels. Such a development would more clearly show the definitions of the levels in the random QTL effect and thereby help us to interpret the results from the VC QTL model. The aim of this paper is to develop a simple presentation of the VC QTL model for identification of base generation allele effects in QTL linkage analysis. Although our main objective is to better understand the biological functions of the random QTL effect, we argue that our model formulation may also enhance the development genome scan methods. Results In this section we show how an alternative incidence matrix based VC QTL model is formulated, we give the prerequisites for the model to be equivalent to the IBD matrix based model, and we also present a general algorithm for constructing the incidence matrix. The choice of notation affects the results that can be obtained from the model in terms of best linear unbiased predictions (BLUP) [20]. This is shown with simulations and is also illustrated with an analysis of real data from a wilddomestic chicken cross. The theoretical details are given in the Methods section. An incidence matrix based VC QTL model The alternative incidence matrix notation breaks down the VC model to its most basic form where the levels of the random effect are independent. An advantage of this approach is that it is easy to identify the assumptions directly from the model. The present paper is restricted to VC models where there are additive and dominance effects but the models are easily extended to include polygenic, family specific effects, epistasis and genotype by environmental interactions following the models given in [1,21-23]. A VC model may consist of fixed and random effects (a mixed model) but the main parameter of interest is the variance of the random QTL effect, and the fixed effects are therefore ignored without loss of generality in the presentation below. A restriction on the random QTL effect v can be given either by the covariance structure between individuals, i.e. the IBD matrix Π or alternatively by an incidence matrix Z relating individuals with the QTL alleles in the base generation. In the latter case, the elements in the vector of random effects v* are independent. The trait vector y is multivariate normal and the distribution of the random effects, i.e. the QTL allele effects, is given by Q ~ MVN(0, I) where I is the identity matrix and is the QTL 1 2 2 σ v σ v 2 Page 2 of 14 (page number not for citation purposes) BMC Genetics 2007, 8:1 http://www.biomedcentral.com/1471-2156/8/1 genotypic effect. The genotypic value vi of individual i in the base generation is the sum of the pair of QTL allele effects at a specific position vi = Qk + Qk+1, where the QTL alleles are arbitrarily numbered k = 2i-1 in the base. Hence, by defining the variance of the random QTL genotypic effects as , the variance of the QTL allele effects is . The QTL alleles are all assumed to be independent in the base generation, i.e. Cov(Qi, Qj) = 0 where i and j are different indices for the m base alleles. The VC QTL model assumes that all alleles are different in the base generation so that m equals twice the number of base generation individuals. The incidence presentation of the VC QTL model is: