Most traits of agricultural importance, such as growth rate and body composition, milk yield and composition, and egg number and quality show a continuous distribution of quantitative trait measurements in a population. Such traits are termed quantitative traits, in contrast to Mendelian traits, which typically are found in a limited number of qualitatively different forms in a population (e.g., presence or absence of horns, brown or black coat color). Genetic variation in a quantitative trait is generally attributed to allelic variation at a number of genes, in contrast to the one or two genes generally found sufficient to explain genetic variation in a Mendelian trait. Moreover, quantitative trait expression is much affected by environmental variables, again in contrast to Mende-lian traits, which are generally little affected by environment. Consequently, for quantitative traits, the relationship between genotype and phenotype is complex, and the genotype of an individual cannot be inferred from its phenotype or that of its relatives. Instead, the various genes affecting a quantitative trait are individuated by mapping them to specific chromosomal locations (loci). For this reason, the term quantitative trait loci, or QTL, was proposed for the individual mapped genetic factors affecting quantitative trait value.

Mapping the QTL responsible for genetic variation in traits of agricultural importance, and using the map locations to identify the actual genes involved, is a major challenge for animal genetics. Success will provide powerful tools for understanding the physiology of trait variation, and for genetic improvement of animal stocks.

Consequently, showing that a QTL is found in the near vicinity of a specific marker (termed ''linked'' to the marker) is equivalent to mapping the QTL to the location of the marker. Thus, a prior requirement for QTL mapping is the availability of a comprehensive marker map. Such maps, based on DNA-level polymorphic loci, are now available for all of the major farm animals.

At present, QTL can only be mapped by using genetic (as opposed to physical) mapping procedures. Genetic mapping procedures start with an individual that is heterozygous for the marker and for the linked QTL. The genetic distance between a marker and a QTL then stands in direct proportion to the number of recombinant gene combinations (haplotypes) among the progeny of the doubly heterozygous individual, i.e., when a QTL and marker are tightly linked, recombinant haplotypes will be rare. The way in which the proportion of recombinant haplotypes among the progeny of an individual is inferred for a QTL and a linked marker is best explained by example, using the basic half-sib sire-family QTL mapping design.

Let M be a marker locus and Q a nearby linked QTL, with alleles M and m and Q and q, respectively, where allele Q is a positive allele that increases trait value, and allele q is a negative allele that decreases trait value (italics denote genes, and bold type denotes alleles). Consider a sire having haplotypes MQ and mq on a pair of homologous chromosomes carrying these genes. Let r denote the total proportion of daughters that received recombinant haplotypes Mq or mQ from their sire, and (1 — r) the total proportion of daughters that received the parental haplotypes, MQ or mq. Then, the following table shows the proportion of daughters carrying each of the four transmitted sire haplotypes.

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