## The Mendelian basis of quantitative trait variation

10.1 The connection between particulate inheritance and quantitative trait variation

• Establishing a scale for genotypic values.

• Phenotypic values as population averages.

• Why we can neglect environmental variation

This chapter will develop the concepts needed to understand the detailed connections between quantitative trait variation and particulate inheritance. Although the components of quantitative trait variation were described in Chapter 9 as population-level phenomena, the variance is ultimately caused by different alleles and genotypes possessed by individuals. The goal of this chapter is to show how additive and dominance components of variation in quantitative traits (VA and VD) are caused by allele and genotype frequencies in a population as well as by the nature of gene action when alleles are combined into genotypes. To accomplish this goal, we will work with a hypothetical quantitative trait that is the product of a single locus with two alleles throughout the chapter. While use of a single diallelic locus as an example does not approximate the multilocus basis of quantitative traits and the multiallelic state of many loci, it greatly simplifies the resulting mathematical expressions while still illustrating key biological concepts. Bear in mind that the epistatic component of genetic variance (Vj) arises due to interaction between two or more loci and therefore cannot be represented in a single-locus model. Therefore, the use of a single locus is an implicit assumption that VT is zero. This chapter will start by constructing expressions that predict the phenotypic mean value of some type of population. The population of interest will initially be all individuals and later be only those individuals with genotypes that contain a certain allele. The population mean value will also be divided into components due to the additive action of alleles and the dominance effect of genotypes. Ultimately, these mean values will be used to build expressions for the variance in phenotypic values expected in a population, specifically the additive and dominance components of genetic variation (VA and VD). The chapter will conclude with a section on the expected phenotypic resemblance among populations of relatives based on the probabilities that related individuals share alleles or genotypes in common.

### Scale of genotypic values

The hypothetical single locus used throughout this section will have two alleles, A1 and A2. By convention, the A1 allele contributes to larger phenotypic values and the A2 allele to smaller phenotypic values. A conceptual scale of measurement is used to represent the genotypic values of each of the three genotypes (Fig. 10.1a). On this scale, the genotypic value of the A1A1 genotype is +a, whereas the genotypic value of the A2A2 genotype is -a. The genotypic

Genotype

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