2 Breeding Experiments: Gametic Crosses

The methods of biometrical genetics are best understood through controlled breeding experiments with inbred strains, in which the results are simple and intuitively obvious. Of course, in the present context we are dealing with continuous variation in humans, where inbred strains do not exist and controlled breeding experiments are impossible. However, the simple results from inbred strains of animals apply directly, albeit in more complex form, to those of free mating organisms such as humans. We feel an appreciation of the simple results from controlled breeding experiments provides insight and lends credibility to the application of the models to human beings. Let us consider a cross between two inbred parental strains, $P_1$ and $P_2$, with genotypes AA and aa, respectively. Since individuals in the $P_1$ strain can produce gametes with only the A allele, and $P_2$ individuals can produce only a gametes, all of the offspring of such a mating will be heterozygotes, Aa, forming what Gregor Mendel referred to as the ``first filial,'' or $F_1$ generation. A cross between two $F_1$ individuals generates what he referred to as the ``second filial'' generation, or $F_2$, and it may be shown that this generation comprises $\frac{1}{4}$ individuals of genotype AA, $\frac{1}{4}$ aa, and $\frac{1}{2}$ Aa. Mendel's first law, the law of segregation, states that parents with genotype Aa will produce the gametes A and a in equal proportions. The pioneer Mendelian geneticist Reginald Punnett developed a device known as the Punnett square, which he found useful in teaching Mendelian genetics to Cambridge undergraduates, that gives the proportions of genotypes that will arise when these gametes unite at random. (Random unions of gametes occur under the condition of random mating among individuals). The result of other matings such as $P_1$ $\times$ $F_1$, the first backcross, $B_1$, and more complex combinations may be elucidated in a similar manner. A simple usage of the Punnett square is shown in Table 3.1 for the mating of two heterozygous parents in a two-allele system. The gamete frequencies in Table 3.1 (shown outside the box) are known as gene or allelic frequencies, and they give rise to the genotypic frequencies by a simple product of independent probabilities. It is this assumption of independence based on random mating that makes the biometrical model straightforward and tractable in more complex situations, such as random mating in populations where the gene frequencies are unequal. It also forms a simple basis for considering the more complex effects of non-random mating, or assortative mating, which are known to be important in human populations.

Table 3.1: Punnett square for mating between two heterozygous parents.

		Male Gametes
		$\frac{1}{2} A$	$\frac{1}{2} a$
Female Gametes	$\frac{1}{2} A$	$\frac{1}{4} AA$	$\frac{1}{4} Aa$
	$\frac{1}{2} a$	$\frac{1}{4} Aa$	$\frac{1}{4} aa$

In the simple case of equal gene frequencies as we have in an $F_2$ population, it is easily shown that random mating over successive generations changes neither the gene nor genotype frequencies of the population. Male and female gametes of the type A and a from an $F_2$ population are produced in equal proportions so that random mating may be represented by the same Punnett square as given in Table 3.1, which simply reproduces a population with identical structure to the $F_2$ from which we started. This remarkable result is known as Hardy-Weinberg equilibrium and is the cornerstone of quantitative and population genetics. From this result, the effects of non-random mating and other forces that change populations, such as natural selection, migration, and mutation, may be deduced. Hardy-Weinberg equilibrium is achieved in one generation and applies whether or not the gene frequencies are equal and whether or not there are more than two alleles. It also holds among polygenic loci, linked or unlinked, although in these cases joint equilibrium depends on a number of generations of random mating. For our purposes the genotypic frequencies from the Punnett square are important because they allow us to calculate the simple first and second moments of the phenotypic distribution that result from genetic effects; namely, the mean and variance of the phenotypic trait. The genotypes, frequencies, and genotypic effects of the biometrical model in Table 3.1 are shown below, and from these we can calculate the mean and variance.

Genotype ($i$)	$AA$	$Aa$	$aa$
Frequency ($f$)	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$
Genotypic effect ($x$)	$d$	$h$	$-d$

The mean effect of the A locus is obtained by summing the products of the frequencies and genotypic effects in the following manner:

$$\mu_A = \sum f_i x_i$$

$$= \frac{1}{4}d + \frac{1}{2}h - \frac{1}{4}d$$

$$= \frac{1}{2}h$$ (7)

The variance of the genetic effects is given by the sum of the products of the genotypic frequencies and their squared deviations from the mean:

$$\sigma^{2}_{A} = \sum f_i(x_i - \mu_A)^2$$

$$= \frac{1}{4}(d-\frac{1}{2}h)^2 + \frac{1}{2}(h-\frac{1}{2}h)^2 + \frac{1}{4}(-d-\frac{1}{2}h)^2$$

$$= \frac{1}{4}d^2 - \frac{1}{4}dh + \frac{1}{16}h^2 + \frac{1}{8}h^2 + \frac{1}{4}d^2 + \frac{1}{4}dh + \frac{1}{16}h^2$$

$$= \frac{1}{2}d^2 + \frac{1}{4}h^2$$ (8)

For this single locus with equal gene frequencies, $\frac{1}{2}d^2$ is known as the additive genetic variance, or $V_A$, and $\frac{1}{4}h^2$ is known as the dominance variance, $V_D$. When more than one locus is involved, perhaps many loci as we envisage in the polygenic model, Mendel's law of independent assortment permits the simple summation of the individual effects of separate loci in both the mean and the variance. Thus, for ($k$) multiple loci,

$$\mu = \frac{1}{2}\sum_{i=1}^{k} h_i \; ,$$ (9)

and

$$\sigma^2 = \frac{1}{2} \sum_{i=1}^{k} d^{2}_{i} + \frac{1}{4} \sum_{i=1}^{k}h^{2}_{i}$$

$$= V_A + V_D\; .$$ (10)

It is the parameters $V_A$ and $V_D$ that we estimate using the structural equations in this book. In order to see how this biometrical model and the equations estimate $V_A$ and $V_D$, we need to consider the joint effect of genes in related individuals. That is, we need to derive expectations for MZ and DZ covariances in terms of the genotypic frequencies and the effects of $d$ and $h$.