Mapping quantitative trait loci (QTL) is often based on a functional or statistical model of gene action involving two alleles per locus. Such model is adequate for mapping populations derived from a cross between two inbred lines as in many plants and some laboratory animals and for di-allelic molecular markers (e.g., single nucleotide polymorphisms, SNPs ). However, many mapping populations (e.g., those derived from crosses between more than two inbred lines) and multi-allelic molecular markers (e.g., microsatellites) require a model to describe functional genetic effects for multiple alleles. This paper has two objectives. The primary objective is to develop a set of new models of functional genetic effects with multiple alleles. The secondary objective is to establish the relationship between these functional genetic effects with well-known statistical genetic effects. Our multi-allelic models reveal three new features that do not exist in the di-allelic models. First, with r (>2) alleles, there are r(r-1)/2 functional additive effects and r(r-1)/2 functional dominance effects, but only (r-1) functional additive effects need to be specified and the remaining (r-1)(r-2)/2 additive effects can be derived. Second, the presence of functional dominance effect for one pair of alleles is sufficient to cause the presence of statistical dominance deviations for all the genotypes. Third, the equality of gene frequencies is no longer a sufficient condition for any direct relationship between physiological and statistical genetic effects in the multi-allelic case. Thus, our new multi-allelic models will have a wider range of applications to QTL mapping and quantitative genetic studies.
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