# Quantitative Genetics - Basic Principles

Basic Principles

The phenotypic value (P) of an individual is the combined effect of the genotypic value (G) and the environmental deviation (E):

P = G + E

The genotypic value is the combined effect of all the genetic effects, including nuclear genes, mitochondrial genes and interactions between the genes. It is worthwhile to note that the mathematics is related to the genetics: for which the brief following revision may be useful. In diploid organisms, a nucleus gene is represented twice in the genotype, one provided by each parent during sexual reproduction. Each gene is located at a particular place (a locus) on homologous chromosomes, one from each parent. Functional forms are called alleles. If both alleles at a gene have the same phenotypic effect, the gene is homozygous: if each allele at a gene is different, the gene is heterozygous. The average phenotypic outcome may be affected by dominance and by how genes interact with genes at other loci ("epistasis"). The founder of Quantitative Genetics - Sir Ronald Fisher - perceived all of this when he proposed the first mathematics of this branch of genetics. He sought to define a single statistical summary of all the variance arising from phenotypic change during the course of genetic assortment and segregation, which he called "genetic" variance. His residual genotypic variance (called simply "residual" by Fisher) represented assortment which did not lead to phenotypic change. Subsequently, these partitions became known respectively as the "additive"(σ²A) and "dominance" (σ²D) variances, which titles do not appear to convey Fisher's partitions. However, an exposition in Falconer and Mackay, pp. 112–116, shows how the assortment/substitution partition of Fisher can be viewed as the average effects of genes after inheritance from parents (remember they pass through meiosis and fertilisation to be inherited - the very mechanisms effecting assortment/substitution). So, "additive" actually means "average inherited effect", and is equivalent to Fisher's assortative disequilibrium. Recently it has been shown to contain the homozygote variance, a portion of the heterozygote variance, and a covariance between homozygote and heterozygote effects. Falconer and Mackay next show that Fisher's "residual" (depicted in their Fig.7.2, p. 117) is due to heterozygosis, ie "dominance"; but not all of it, as some is embedded in the "additive" component, as we noted earlier.

This is not the only approach to defining and partitioning genotypic variances. An alternative was advanced by Mather and Jinks. Their notation was entirely different to that of Fisher, and his method predominates. However, when their approach is translated into Fisher's notation, the relationships between the two approaches are clear. The Mather and Jinks approach is more "genetical" than Fisher's, being based on variances arising straightforwardly from homozygotes and heterozygotes. The inter-conversion between the Fisher and Mather & Jinks methods are given in Gordon (2003).

The mating-system assumed in deriving these genotypic variances is panmixia : which implies random fertilisation with uniform distribution of gametes in a very large population (theoretically, infinity). This rarely occurs in nature, as gamete distribution may be limited, for example by dispersal restrictions, or preferential matings, or chance sampling in small populations of gametes (gamodemes). Each gamete restriction leads to a descendant small-population (line). Individuals within a line will not all be the same, but they will be more similar than individuals from panmictic populations. In any source breeding group, many separate gamete restrictions will occur during a mating cycle, each one leading to a line. These lines also will vary with respect to their mean phenotypess, and the process is called dispersion. The inbreeding coefficient quantifies the increase in homozygosity which results. The values of this coefficient for a wide variety of situations (e.g. islands, "onion-skin" aggregates, linear strips, matings of related parents) are available. As well as a general rise in homozygosity, the dispersed lines vary in their allele frequencies because of gamete sampling. However, the mean of the frequencies across all lines from the one source will be the same as the original frequencies in the source population. The phenotypic mean of all of these lines is less than that of the original source, this being inbreeding depression. The genetic variances also change relative to those of panmixia. Variance-within-lines decreases, but the variance-amongst-lines and the total-variance-in-the-system both increase (Mackay et al.; Gordon 2003). The first of these facts is common knowledge, but the latter two are not. Many of these lines will be inferior in phenotype: but, some lines will be superior, and some will be about average (Chapter 13 in Falconer et al.). Selection assisted by dispersion leads to maximum genetic advance (see previous references). Plant and animal breeders utilise these properties routinely, and have devised breeding methods especially to do so (e.g. line breeding, pure-line breeding, backcrossing). The role of dispersion in natural selection has not received much attention.

The Environmental variance is much more straightforward. This can be subdivided into a pure environmental component (E) and an interaction component (I) describing the gene-environment interaction. The overall "single gene" model can be written as:

P = a + d + E + I.

Expansion of the model to multiple genes (loci) is still not resolved satisfactorily, and until that is solved it is not possible to account for epistasis. The problem is being tackled currently. The contribution of those components cannot be determined in a single individual, but they can be estimated for whole populations by estimating the variances for those components, denoted as:

VP = Va + Vd + VE + VI

The heritability of a trait is the proportion of the total (i.e. phenotypic) variance (VP) that is explained by the total genotypic variance (VG). This is known also as the "broad sense" heritability (H2). If only Additive genetic variance (VA) is used in the numerator, the heritability is "narrow sense" (h2). The broadsense heritability indicates the genotypic determination of the phenotype: while the latter estimates the degree of assortative disequlibrium in the trait. Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on disequilibrium: and it has been used also for predicting the results of artificial selection.

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