Crosses between inbred strains are the bread and butter of experimental genetics. Inbred strains are specially developed to be homozygous at all loci throughout the genome, in contrast to outbred populations which are often polymorphic. Standard approaches for dissecting an heritable trait in experimental genetics often involve crossing inbred strains. The genetic contribution to a trait cannot be demonstrated by looking at individuals from a single inbred strain alone, since in principle all members of the same strain are genetically identical and the observed phenotypical variability among them has to be attributed to environmental factors. Therefore, in order to map genetic factors one must select at least two different strains, which show different levels of expression of the phenotype under the same environmental conditions. In many cases one can screen the commercially available inbred strains for an appropriate pair of strains, which are phenotypically distinct. Careful crosses between the selected strains produce the population that is used for the genetic mapping.

The two most popular protocols for crossing inbred strains are the back-cross and the intercross, which we describe below. However, before going into the details of the two different breeding schemes, let us introduce the standard terminology. Practically all breeding experiments start with an outcross, a mating between two animals or strains considered unrelated to each other. Specifically, we consider here an outcross between two distinct inbred strains that show a phenotypic difference. The resulting offspring are called the first filial generation, or F\. Consider any genetic locus where the two strains differ. Recall that inbred strains are homozygous at all loci. Say that the genotype of one of the inbred strains at a given locus is AA, and the genotype of the other inbred strain is aa. It follows that the genotype of the F\ generation at the polymorphic site must be Aa, since each parent passes on one of its alleles to the offspring. Consequently, F\ mice have a fixed genetic composition - all are heterozygous at all polymorphic loci.

A backcross is obtained by mating the Fi offspring with mice from either one of the original inbred strains. Note that there are two possible types of backcross, depending on the choice of the inbred strain for the cross, denoted here by the AA x Aa backcross and the aa x Aa backcross. (As a matter of fact, one can further divide the backcross breeding scheme based on the sex of the Fi mice in the cross. This may be important if a sex-linked trait or imprinting is considered. However, we ignore these possibilities in the sequel.)

Consider the aa x Aa backcross design in the context of the simple QTL model described above in (2.2). Assume the mother is inbred and the father is the F1. Then xM is always equal zero, but xF may be zero or one. The backcross offspring are either aa homozygous, which corresponds to xF = 0, or Aa heterozygous, which corresponds to xF = 1. The probability of each of the values is 1/2, with corresponding phenotypic mean levels of j and j+a+5. The resulting regression model is given by with a similar equation for the AA x Aa backcross. The offspring are either AA homozygous or Aa heterozygous. In this case, the variable 1 — xF has a Binomial(1, 0.5) distribution, and the regression model takes the form:

Both models (2.6) and (2.7) have a similar form. However, their statistical properties may be quite different depending on the relations between a and 5. Since a Bernoulli random variable with p = 1/2 has variance p(1 — p) = 1/4, the variance component associated with the genetic factor is (a + 5)2/4 for the first model and (a — 5)2/4 for the second. Consequently, the locus specific heritability is larger for the first model if a and 5 have the same sign, and vice versa if they have opposite signs. For the additive model (5 = 0) the ratios are equal.

The intercross is a result of the mating of an F1 male and an F1 female. In terms of the notation, we will refer to the intercross as the Aa x Aa cross. The y = n + (a + S)xF + e ,

term F2 may also be used. (Subsequent generations of mating are denoted Fn, where n is the number of generations since the initial outcross.) The offspring of the AaxAa intercross can have any one of three genotypes. The distribution of the genotypes follows the ratios 1:2:1, thus x has a binomial distribution, B(2,1/2). It was shown above (Equation (2.5) and the following sentences) that the variance component associated with the given locus is a2/2 + S2/4.

We turn now, with the aid of a small simulation study, to a demonstration of the segregation of the trait in a cross. The simulation will generate segregation of the alleles from parents to offspring and the resulting expression of the phenotype in the offspring. Recall that a parent carries two alleles at a given locus, one inherited from the grandfather and the other inherited from the grandmother. Only one of these two alleles will be passed on to the parent's offspring. According to Mendel's first law of segregation each of the alleles has an equal chance to be passed on. For example, if the parent is Aa at the given locus, then it will pass with equal probabilities either the allele A or the allele a. Of course, if the parent's genotype is AA (aa) it will pass on the allele A (respectively a) with certainty.

Assume the parent is an Fi mouse:

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