Components of the genetic variance

Let's assume for the moment that we can actually measure the genotypic values. Later, we'll relax that assumption and see how to use the resemblance among relatives to estimate the genetic components of variance. But it's easiest to see where they come from if we assume that the genotypic value of each genotype is known. If it is then, writing $V_g$ for $\mbox{Var}(G)$

$$V_g = \ p^2[x_{11} - {\bar x}]^2 + 2pq[x_{12} - {\bar x}]^2 + q^2[x_{22} - {\bar x}]^2$$ (6)

$$= \ p^2[x_{11} - 2\alpha_1 + 2\alpha_1 - {\bar x}]^2 + 2pq[x_{12} - (\alpha_1 + \alpha_2) + (\alpha_1 + \alpha_2) - {\bar x}]^2$$

$$\ \ + q^2[x_{22} - 2\alpha_2 + 2\alpha_2 - {\bar x}]^2$$

$$= \ p^2[x_{11} - 2\alpha_1]^2 + 2pq[x_{12} - (\alpha_1+\alpha_2)]^2 + q^2[x_{22} - 2\alpha_2]^2$$

$$\ + p^2[2\alpha_1 - {\bar x}]^2 + 2pq[(\alpha_1 + \alpha_2) - {\bar x}]^2 + q^2[2\alpha_2 - {\bar x}]^2$$

$$\ + p^2[2(x_{11} - 2\alpha_1)(2\alpha_1 - {\bar x})] +2pq[2(x_{12} - \{\alpha_1+\alpha_2\})(\{\alpha_1+\alpha_2\} - {\bar x})]$$

$$\ +q^2[2(x_{22} - 2\alpha_2)(2\alpha_2 - {\bar x})] \quad .$$ (7)

There are two terms in (7) that have a biological (or at least a quantitative genetic) interpretation. The term on the first line is the average squared deviation between the genotypic value and the additive genotypic value. It will be zero only if the effects of the alleles can be decomposed into strictly additive components, i.e., only if the pheontype of the heterozygote is exactly intermediate between the phenotype of the two homozygotes. Thus, it is a measure of how much variation is due to non-additivity (dominance) of allelic effects. In short, the dominance genetic variance, $V_d$, is

$$V_d = p^2[x_{11} - 2\alpha_1]^2 + 2pq[x_{12} - (\alpha_1+\alpha_2)]^2 + q^2[x_{22} - 2\alpha_2]^2 \quad .$$ (8)

Similarly, the term on the second line of (7) is the average squared deviation between the additive genotypic value and the mean genotypic value in the population. Thus, it is a measure of how much variation is due to differences between genotypes in their additive genotype. In short, the additive genetic variance, $V_a$, is

$$V_a = p^2[2\alpha_1 - {\bar x}]^2 + 2pq[(\alpha_1 + \alpha_2) - {\bar x}]^2 + q^2[2\alpha_2 - {\bar x}]^2 \quad .$$ (9)

What about the terms on the third and fourth lines of the last equation in 7? Well, they can be rearranged as follows:

$$\begin{eqnarray*} p^2[2(x_{11} &-& 2\alpha_1)(2\alpha_1 - {\bar x})] + 2pq[2(... ..._{11}-2\alpha_1)p + q(x_{12}-\{\alpha_1+\alpha_2\})] \\ &=& 0 \end{eqnarray*}$$

Where we have used the identities ${\bar x} = 2(p\alpha_1 + q\alpha_2)$ [see equation (3)] and

$$\begin{eqnarray*} p(x_{11} - 2\alpha_1) + q(x_{12} - \alpha_1 - \alpha_2) &=& 0... ... q(x_{22} - 2\alpha_2) + p(x_{12} - \alpha_1 - \alpha_2) &=& 0 \end{eqnarray*}$$

[see equations (4) and (5)]. In short, we have now shown that the total genotypic variance in the population, $V_g$, can be subdivided into two components - the additive genetic variance, $V_a$, and the dominance genetic variance, $V_d$. Specifically,

$$V_g = V_a + V_d \quad ,$$

where $V_g$ is given by the first line of (6), $V_a$ by (9), and $V_d$ by (8).