2

Breeding Systems ¹

Earl L. Green

Characteristics of mice, like the characteristics of other things, living or nonliving, may be treated as discrete or continuous variables. Litter size, sex, and coat colors are examples of discrete variables. Weight, body size, and lifespan are examples of continuous variables. We may describe groups by the proportions of males, of brown mice, or of brown male mice (discrete variables), or by the means and variances of their weights, lengths, or lifespans (continuous variables).

A collection of samples will usually exhibit some variability in the proportions or means of the characteristics under study. If a sufficient number of characteristics are simultaneously considered, no two mice will be alike. The variability, aside from chance differences between samples, may have either genetic or nongenetic causes. The nongenetic causes of variation in the characteristics under study may be of many kinds, including seasonal fluctuations, differences in kind and amount of food consumed, exposures to noise, light, or humidity, or differences in prior experiences. The genetic causes of variation are simpler, at least in principle, in that they are exclusively the consequences of different genetic contents of zygotes.

Investigators who use mice are concerned with the control of both genetic and nongenetic sources of variation. Some of the nongenetic sources may be controlled by standardizing the management of the mouse colony or, when it is desired, by deliberately imposing two or more different conditions of rearing or treating the mice. The genetic sources of variability may be controlled by the choice of an appropriate breeding system. Various breeding systems useful with mice, their theoretical consequences, and the criteria for choosing one in preference to another are the subjects of this chapter.

More extensive general discussions of breeding systems may be found in the books by Lush ( 1945), Malécot ( 1948), Fisher ( 1949), Mather ( 1949), Lerner ( 1950), Li ( 1955), Kempthorne ( 1957), Falconer ( 1960), and Le Roy ( 1960), and in the papers by Wright ( 1921a, 1963), Bartlett and Haldane ( 1935), Green and Doolittle ( 1963), Kimura and Crow ( 1963), and Robertson ( 1964), as well as others.

The purpose of all breeding systems is to preserve or control the genetic causes of variability in traits of interest. In theoretical populations of infinite size under specified environmental conditions, random mating in the absence of selection or mutation will keep the means and variances of all quantitative traits constant. Inbreeding will subdivide a population and, in the individual subpopulations, will increase or decrease the means and will reduce the genetic variances. Outcrosses (matings between populations) will usually change the mean and increase the genetic variance of the resulting population. Selective breeding of like with like will increase or decrease or stabilize the mean, depending upon the direction of selection, and will also ultimately decrease the genetic variance, but not necessarily eliminate it. If the selected mates are deliberately as unlike as possible, the genetic variance will be kept large.

Combinations of inbreeding and selection systems give geneticists a wide variety of methods for controlling the inherited characteristics of research animals.

Four general types of mice have come into common use: inbred, hybrid, mutant-bearing, and selected. Breeding systems to produce inbred and hybrid mice and to propagate mutant-bearing mice are described in the following sections of this chapter. Systems of selective breeding are described in Chapter 9.

It is necessary to distinguish between two uses of mice in research. Mice for breeding experiments must be individually identified in such a way that the parents and more remote ancestors, the offspring and later descendants, and collateral relatives can, when necessary, also be identified. Methods for keeping breeding records to accomplish this aim are given in Chapter 3. In terminal experiments, the mice are used for investigations of physiology, biochemistry, pathology, behavior, etc., but not for further breeding. They need not necessarily be individually identified, but must be identified by lot. That is, the strain, generation, age, weight, and other characteristics will need to be known, but the exact relationships to other mice in the same or different lots may not always be needed.

The genetic consequences of a system of breeding may be grasped by considering how the system affects the probabilities or theoretical relative frequencies of the alleles or genes, the genotypes, and the mating types at any one locus and hence at all loci.

We shall refer to the a locus as any locus whose heterozygosity is in question under any given system of breeding. For simplicity we shall suppose that there are two alleles, + and a, at the a locus. In certain systems we shall need to refer to the locus of a dominant mutation (D) and to the locus of a recessive mutation (r). These will be denoted as the D locus with alleles + and D and as the r locus with alleles + and r. The a, D, and r loci are distinct and may be linked or not. The D and r loci are called loci of interest in any system of breeding for control of a segregating locus.

From three genotypes, such as +/+, a/+, a/a at a locus with two alleles, there will be nine mating types, which may be grouped into four kinds:

1. Matings of like homozygotes, +/+ x +/+ and a/a, called incrosses
2. Matings of unlike homozygotes +/+ x a/a and a/a x +/+, called crosses
3. Matings of a homozygote and a heterozygote, +/+ x a/a, a/+ x +/+, a/+ x a/a, and a/a x a/+, called backcrosses
4. Matings of heterozygotes, a/+ x a/+ called intercrosses

The theoretical consequences of any breeding system are easily displayed in terms of the probability or frequency of heterozygotes, denoted by h, that is, P(a/+) = h, and of the probability of incrosses, denoted by p, that is, P(+/+ x +/+ or a/a x a/a) = p. The probability of incrosses (p) may increase in a given system of mating only at the expense of the crosses, backcrosses, and intercrosses. The magnitude of p therefore reveals the expected proportion of matings of like homozygotes, and 1-p gives the probabilities of matings of other types.

As h decreases, the probability of homozygotes (1 - h) increases. If these probabilities are interpreted as referring to all loci, we may say that the probability of homozygosity is 1 - h. In the limit when h = 0, the strain may be said to be isogenic. In practice the term isogenic is used to mean homozygous at nearly all loci as well as homozygous at all loci. If two animals or two groups of animals are isogenic (or nearly so) at all loci except one or two at which they are known to be different, either by being opposite homozygotes (+/+ vs. a/a) or by one being homozygous (+/+ or a/a) and the other being heterozygous (a/+), the two animals or groups are called coisogenic ( Chovnik and Fox, 1953). If a new mutation arises in and is propagated within an already inbred strain, the mutant and non-mutant mice are called coisogenic. To avoid the risk of a false claim inherent in the use of coisogenic when in fact h is not zero, Loosli et al. ( 1961) proposed the term congenic. This term is especially useful when referring to stocks produced by repeated crossing of mutant-bearing animals to animals of inbred strains.

Methods of analyzing breeding systems not given here are amply described in several books and papers. The method of path analysis, invented by Sewell Wright, is the most versatile general method of analyzing regular and irregular breeding systems (Wright, 1921a, 1921b, 1934, 1954, 1963). The generation matrix method, first used by Bartlett and Haldane ( 1935), is useful in analyzing regular breeding systems and was adopted by Fisher ( 1949) in an extensive general analysis of inbreeding. The introduction of concepts of probability to the analysis of breeding systems is largely due to Malécot ( 1948). An exposition of the methods and results of the analysis of some regular breeding systems useful with laboratory mice will be found in Green and Doolittle ( 1963).

In the following sections of this chapter are descriptions of random breeding, avoidance of inbreeding, brother-sister inbreeding, other types of inbreeding, strain crossing, breeding systems for putting mutant genes on standard inbred backgrounds and for decomposing complex genotypes, and inbreeding with forced heterozygosis. Special breeding systems using linked markers are described in Chapter 8. The construction and maintenance of linkage testing stocks are described by Carter and Falconer ( 1951).

Random breeding is easy to define, but difficult to achieve. In principle, random breeding requires that the product rule of probability be satisfied. This means that the chance of choosing any one male out of m males must be 1/m and of choosing any one female out of f females be 1/f and that the chance of mating any two specific mice be 1/mf. Tables of random sampling numbers or other equivalent randomizing devices must be used to unsure random choices and random matings. Indiscriminate or nonsystematic ways of making up matings are likely to permit unknown biases to occur in the choices of mates.

The effect of random breeding on the genetic structure of a stock of mice has to be considered separately for theoretical and actual populations.

In theoretical or large populations, random breeding will preserve the gene and genotype frequencies generation after generation. Discrete traits usually determined by one or a few pairs of genes will be retained at stable frequencies in the population, assuming there is no selection and no mutation. The means and variances of continuous or metrical traits will likewise remain constant, under the same assumptions, provided the nongenetic components of variances do not change. These assertions about stability of gene frequencies, genotype frequencies, means, and variances follow from the Hardy-Weinberg principle ( Li, 1955).

In actual populations of small size the expected results of a random breeding system will be slightly different. First, the effective breeding number (N) will determine the rate of random fixation or loss of alleles at each locus, the rate being approximately equal to ½N ( Wright, 1931). Random breeding in populations of finite size will lead to homozygosity (fixation and loss of alleles), slowly if N is large, rapidly if N is small. The proportion of discrete traits will change to higher or lower values, ultimately to all or none, and the means of continuous traits will increase or decrease. The genetic variance will stochastically decrease to zero. These changes arising from the finite size of the population have been variously called the "inbreeding effect," "random drift," "genetic drift," and the "Sewall Wright effect." Second, it is impossible to raise populations of mice in the laboratory in the absence of selection, mutation, and varying nongenetic or environmental causes of variation. Selection may favor one allele or one set of nonalleles, or one or more genotypes, or one or more phenotypes at the expense of all the other alleles, genotypes, and phenotypes. Mutation may be either "forward" or "reverse." Nongenetic sources of variation may not act uniformly on all genotypes. The consequences of all these forces interacting in the population may impair attempts to preserve the gene and genotype frequencies of a population by random breeding. Nonetheless, the experimenter who wishes to maintain the genetic variability of a population has no recourse but to carry out a random breeding system in as large a population as feasible and to try to minimize the effects of selection and of nongenetic causes of variation. There is nothing he can do about natural mutations.

As indicated in the preceding section, random breeding in populations of finite size results in some loss of heterozygosity. Certain systems of breeding minimize the loss of heterozygosity even in finite populations. Wright ( 1921a) showed that regular matings of double first cousins, quadruple second cousins, and octuple third cousins preserve any existing heterozygosity better than random matings in finite populations. Other systems have since been devised that are superior to the regular cousin-mating systems in preserving heterozygosity in later generations, even though the loss of heterozygosity may be greater in early generations. These new systems, called circular, circular pair, and circular subpopulation mating systems, are described and analyzed by Kimura and Crow ( 1963) and Robertson ( 1964).

The system of brother-sister inbreeding is by far the easiest from the standpoint of the investigator. He needs merely to put a male and one of his sisters, usually from the same litter, in a breeding pen. The record-keeping is minimal ( Chapter 3). Repeating this act, generation after generation, will produce an inbred strain of mice. Meanwhile, mice with certain characteristics of interest may be selected in order to fix the character or characters, if possible, within the line, along with the inevitable selection for viability, fertility, and fecundity. The only major difficulty is that mice, like other naturally outbreeding species, exhibit inbreeding depression, that is, loss of reproductive fitness as inbreeding progresses. Astute selection for the more vigorous mice in each generation may enable the line to escape reproductive failure, the most severe consequence of inbreeding.

From the large number of strains of mice, successfully propagated for more than 20 generations and some for more than 100 generations of brother-sister inbreeding, one may gain the impression that mice accept inbreeding easily. Two observations deny this. First, many attempts to inbreed mice have failed. Although the number of failures has never been recorded, it probably exceeds the number of successes about fivefold. Second, crosses between unrelated inbred strains of mice characteristically produce vigorous, healthy progeny which in turn give birth to and rear large families. Hybrid vigor is the opposite of inbreeding depression and can be manifested only if some degree of depression has already occurred.

A strain of mice is called an "inbred strain" when there have been 20 or more consecutive generations of brother-sister matings or 20 or more consecutive generations of parent-offspring matings, provided the offspring was mated to the younger parent ( Staats, 1964). For convenience, strains with fewer generations of inbreeding are said to be "partially inbred" or "on the way to being inbred." It is necessary to understand that this definition of an inbred strain of mice is in no way inconsistent with the more general concept of inbreeding as matings between individuals related by descent more frequently than expected by the product principle of probability.

A word of caution is necessary. When inbred mice are used in terminal experiments, it is often necessary to make up the experimental groups by pooling mice from several different litters, usually of about the same age and therefore born of different parents. The parents of such litters should themselves be closely related through a recent common pair of ancestors. If the common ancestors were in, say, the 30th generation and the mice to be used in the experiment are in the 33rd to 38th generation, they may confidently be expected to be genetically similar. They may not be genetically identical for there is no way of arresting the evolutionary forces that create genetic diversity. On the other hand, if the common ancestors were in the second generation and the mice to be used in the experiment are in the 33rd generation, one should not be surprised to find phenotypic diversity due to genetic diversity in the group as a whole, even though each individual litter satisfies the operational definition of an inbred strain.

The genetic consequences of brother-sister inbreeding are easily seen from the probability of heterozygosity (h) and the probability of incrosses (p) at specified generations with respect to any neutral locus, the a locus, whose heterozygosity is in question. In the case of starting to inbreed following a cross (a/a x +/+), h approaches zero so that at 20 generations only about 1 per cent of neutral loci like a are expected to be heterozygous. Meanwhile p approaches unity so that 98 per cent of the matings are expected to be incrosses ( Table 2-1).

At the same time that all neutral or unselected loci are being driven to homozygosity by brother-sister inbreeding, one or more loci may deliberately be forced to remain heterozygous by matings such as D/+ x +/+ or D/+ x D/+, and r/+ x r/r or r/+ x r/+. Brother-sister inbreeding with forced heterozygosis is described in a later section of this chapter.

Systems of inbreeding, other than brother-sister, may sometimes be necessary. A sequence of mating offspring to the younger parent is genetically equivalent to brother-sister mating.

Any system of breeding within a closed colony will inevitably produce increases in the probability of homozygosity and of incrosses, the rates of increase being dependent upon the size of the colony and the specific system of breeding.

Pen breeding is a system of mating a number of males, usually one, with a number of females, usually two or more, of rearing all the progeny together in a single pen, and of making no distinction among the progeny when mates are chosen to produce the next generation. The mated mice may thus be related as brother and sisters, as brother and half-sisters, or, if there were two or more males, as cousins of some degree. (It is usually not practical to put more than one male mouse in a pen with females because of fighting between males leading to emasculation or death.) If pen breeding starts with mice from an inbred strain, the genetic uniformity already achieved will not only be retained but improved upon. If it starts with noninbred stock, genetic uniformity will be approached less rapidly than under brother-sister inbreeding. At worst, when a large number of females are mated with a single male in each generation, the rate of approach to homozygosity is still more than half the rate of approach under brother-sister inbreeding.

If it is not practical to use a regular system of inbreeding such as brother-sister, parent-offspring, or a fixed number of mates for pen breeding, it is still possible to estimate the probability of homozygosity at the a locus by computing the inbreeding coefficient, F ( Wright, 1931; Malécot, 1948; Emik and Terrill, 1949; Cruden, 1949). F may be interpreted as a probability of homozygosity at any locus.

One further type of mice may be produced by matings of mice within an inbred strain, although the males and females are not necessarily related as brother and sister. The progeny of such matings are called "inbred-derived" to distinguish them from inbred mice. Obviously the mated mice should have recent common ancestors within an inbred line if the genetic variance is to be kept low. In terminal experiments requiring large numbers of mice from a single strain, inbred-derived mice may be superior to inbred mice. Any genetic heterogeneity within an inbred strain will be more uniformly spread over the inbred-derived progeny and so their use will help to avoid accidental differences between experimental groups traceable to genetic heterogeneity.

Crosses between inbred strains of mice may serve any one of six purposes:

1. To produce uniform F₁ hybrids for use in terminal experiments
2. To produce F₁, F₂, B₁, B₂, and other generations to analyze the genetic basis of a variable trait, either discrete or continuous
3. To produce 2-way, 3-way, 4-way, etc., crossbred populations to be used as the starting point for genetic selection experiments
4. To produce all possible F₁ generations (diallel crosses) between several inbred strains to reveal the nature of a multiple factor genetic system determining variations in a continuous trait or to estimate general and specific combining abilities
5. To transfer a mutant gene from one genetic background to another
6. To decompose a complex genotype into its single-locus constituents and thus lead to the discovery and identification of new genes

The first four of these purposes are briefly described in the following subsections. The last two, because of their special usefulness in experiments with mice, are described in the next two sections. See Chapter 9 for more discussion of the third and fourth purposes.

F₁ Hybrids

F₁ hybrids produced by crosses of two inbred strains of mice are usually decidedly more hardy than mice of the inbred strains. They grow faster, survive to maturity in greater proportions, live longer, and in turn reproduce earlier and more abundantly. Hybrid litters are usually, but not always, larger than inbred litters ( Forsthoefel, 1954; Butler, 1958; Franks et al., 1962, McCarthy, 1965). Hybrid mice will usually accept grafts of tumors, ovaries, skin and other tissues from either parental strain. In some traits F₁ hybrids may be less variable than mice of inbred strains ( Yoon, 1955).

F₁ hybrids will be heterogeneous at all loci for which the inbred strains are homozygous for different alleles. But if the inbred strains are isogenic or nearly so, the F₁ hybrids will be uniformly heterozygous or nearly so and thus be as genetically uniform as the mice of the inbred strains. Therefore F₁ hybrids are especially valuable in terminal experiments when genetic and phenotypic uniformity are requisites, but are useless for propagating their kind.

Crosses between two strains

One may cross two inbred strains of mice to analyze the genetic basis of any trait for which the strains are different. The basic steps are the same irrespective of whether the trait is discrete or continuous, but the methods of analysis of the data are decidedly different.

The basic steps are to cross the strains, denoted as P₁ and P₂, reciprocally if possible, to produce one or two first hybrid or F₁ generations. Then the F₁ mice are mated among themselves (intercrossed) to produce a second hybrid or F₂ generation and are also backcrossed to the parental strains to produce two first-generation backcross generations, B₁ and B₂. The matings may stop at this point, but it is usually desirable to produce some additional generations. The choice of which ones to produce depends upon whether the trait is discrete or continuous and upon the results obtained in the hybrid and backcross generations.

If the trait is discrete and appears in sharply alternative states, such as "all" vs. "none," "black" vs. "brown," or "high" vs. "low" in the parental strains, and if, further, a small number of discretely distinguishable types appear in Mendelian proportions in the segregating generations (F₂, B₁, B₂, etc.), one may infer that one or two or three pairs of genes distinguish the two parental strains with respect to the strain in question. It is rarely possible to discern the discrete effects of more than three pairs of alleles. There may be any degree of interaction between alleles (dominance) and any degree of interaction between nonalleles (epistasis). Even if the results suggest only one pair of alleles with dominance, it is desirable and sometimes of crucial importance to perform either of two breeding tests.

One test is to ascertain that the F₂ mice with the dominant phenotype are in fact of two types, homozygous and heterozygous, and that the two types have occurred in the expected frequencies of 1/3 and 2/3. This may be done by mating the F₂ mice with dominant phenotypes to any recessive mice. The other test requires mating B₂ mice with the dominant phenotype to the supposed recessive mice to produce a second backcross B₂₂ in which the two types should again occur in equal frequencies. Yet another backcross of B₂₂ mice with the dominant phenotype to the supposed recessive mice should be performed in doubtful cases to ascertain that the dominant phenotypes in the B₂ and B₂₂ are in fact all heterozygotes.

Examples of breeding experiments of the sort described abound in the literature. Experiments with clear-cut results were reported by Shreffler and Owen ( 1963), Russell and Coleman ( 1963) and Ruddle and Roderick ( 1965).

If the trait is continuous, there should be at least a significant separation of the means of the parental strains to make a breeding experiment worthwhile. The mice of the P₁, P₂, F₁, F₂, B₁, B₂, and other such generations as B₁₁, B₁₂, B₂₁, and B₂₂ are individually measured for the trait. The data may then yield estimates of the amount of additive genetic variance, dominance variance, epistatic variance, and the number of segregating loci. The methods of analysis are set forth in Mather's ( 1949) book and elsewhere. Honeyman ( 1957) used these methods for analyzing a nutritional difference between two strains of mice. Bruell ( 1962) has given an exposition of the method of analysis and has analyzed the genetics of spontaneous activity and of exploratory behavior of two strains of mice.

Crossbred populations

The motive for crossing inbred strains of mice may be to produce a high degree of genetic heterogeneity within a population to serve as the starting point for a selection experiment. If strains are designated as A, B, C, ..., one may start with the progeny of a 2-way cross of A x B, or a 3-way cross of (A x B) x C where females of the F₁ of A x B are chosen to mate with males of C in order to take advantage of the extra vigor, if any, of the F₁ hybrids. Or 4-way, 6-way, or 8-way crosses may be constructed.

Diallel crosses

A diallel cross is the set of all possible matings between several genotypes, defined as individuals, lines, or inbred strains. If there are n genotypes there are n² mating combinations, counting all inbred, crossbred, and reciprocal matings. The results of a diallel cross may be set forth in a diallel table of n² measurements corresponding to the n² mating combinations ( Hayman, 1954). A diallel table may, therefore, be constructed from all first generation data (inbreds and F₁'s) or from all second generation data (inbred and F₂'s or inbreds and backcrosses) for one sex or for both sexes.

The type of analysis of a diallel table will depend in part on the experimenter's objectives. He may be interested in an analysis that yields estimates of the additive genetic variance, the dominance variance (allelic interactions), the epistatic variance (nonallelic interactions), and the environmental variance. Or he may be interested in judging the specific and general combining ability of each strain. Or his interest may lie in maternal effects or differences between reciprocal hybrids. The data from the F₁ generations and the parental strains may yield all the requisite information. Thus the diallel crossing system has an advantage of simplicity of execution since segregating generations will not be required except for the most exhaustive analyses.

The literature on diallel crosses has grown enormously. For discussion of the method and of the interpretation of results, se Allard ( 1956), Griffing ( 1956 and earlier), Jinks ( 1956 and earlier), Dickinson and Jinks ( 1956), Kempthorne ( 1956), Hayman ( 1960 and earlier), Broadhurst ( 1960), Kempthorne and Curnow ( 1961), Falconer and Bloom ( 1962), Wearden ( 1964), and Bloom and Falconer ( 1964).

The purpose of transferring a mutation to the genetic background of an inbred strain of mice is to permit comparisons of the effects of the mutant and nonmutant alleles with the greatest possible precision, that is, as free as possible from the effects of variability due to other unidentified loci. This may be regarded as comparing the results of different doses (0, 1,2) of the mutant gene in the nonmutant homozygote, the heterozygote, and the mutant homozygote. Further, the relative quantitative effects of different mutations may be ascertained only by comparing them against otherwise uniform genetic backgrounds. For this purpose one of the standard inbred strains, such as C57BL/6J, provides the requisite background. Three breeding systems have passed into general use for this purpose ( Green and Doolittle, 1963). The backcross system is most useful if the mutant is a dominant, but may be used if the mutant is a recessive, particularly if the recessive homozygote is inviable or infertile. The cross-intercross and the cross-backcross-intercross systems are useful if the mutation is a recessive. Other breeding and manipulative techniques for maintaining special genetic stocks are described in Chapter 8 and by Lyon ( 1963).

Backcross system

Mice bearing a dominant mutation of interest, either as heterozygotes (D/+) or, if viable, as homozygotes (D/D), are mated with mice of an inbred strain which are +/+ at the D locus. The D/+ progeny are backcrossed to +/+ mice from the inbred strain, following which the D/+ progeny in the next generation are also backcrossed to +/+ mice from the inbred strain. This sequence may be continued as long as desired.

The probability of incrosses with respect to the a locus in the nth generation following a mating in the zeroth generation of D/+ x +/+ is p_n = 1 - (1 - c)^{n - 1}, where c is the probability of crossing over between the a and D loci. The probability of heterozygosity is h_n = (1 - c)^{n - 1}. Values of p_n and h_n in generation n for selected values of c are shown in Table 2-2. In eight generations p exceeds 99 per cent if c is near 50 per cent, but 45 generations will be required for p to exceed 99 per cent if c is near 10 per cent.

After seven or eight generations of backcrossing, nearly all the alleles not closely linked with D will have been replaced by alleles from the inbred strain. At this point it may be desirable to cease backcrossing to the inbred strain and to continue with brother-sister matings of the types D/+ x D/+ or D/+ x +/+. Sib matings of these sorts will allow variable loci linked to the D locus to become homozygous faster than will continued backcrossing to the inbred strain (see below). Further, if D/D is viable, a few generations of intercrossing following seven or eight generations of backcrossing will allow all three genotypes to be examined against a relatively uniform background. As a practical matter, therefore, seven or eight generations of backcrossing to an inbred strain followed by 10 or 12 generations of backcrossing or intercrossing sibs will usually be adequate for producing genetically similar mice.

The backcross system may also be used if the mutation of interest is recessive. Heterozygotes (r/+) are mated with homozygous mice (+/+) of an inbred strain and the heterozygous progeny are again mated to +/+ mice of the inbred strain. Backcrossing in this way may continue as long as desired. If the heterozygotes (r/+) and the homozygotes (+/+) produced in each backcross generation are indistinguishable, it will be necessary to identify the heterozygotes by a breeding test. This requires more time, but not more generations, to reach a given probability of incrosses than backcrossing with a dominant. The use of heterozygotes will be obligatory when the recessive homozygotes (r/r) are inviable or infertile.

Cross-intercross system

Mice homozygous for a recessive mutation (r/r) are crossed with mice of an inbred strain which are +/+ at the r locus. Their progeny (r/+) are intercrossed to recover the recessive homozygotes (r/r) which are used to start another cycle of crossing and intercrossing. The number of such cycles, with two generations of matings per cycle, may be made as large as desired.

The probabilities of intercrosses (p_m) and of heterozygotes (H_m) at the a locus after m cycles of the cross-intercross system are shown in Table 2-3 for selected values of c. Eight cycles or 16 generations will raise the probability of incrosses above 99 per cent if there is loose linkage, but 46 cycles will be required to raise it to 99 per cent if there is close linkage (c = 10 per cent).

Cross-backcross-intercross system

This system starts the same as the cross-intercross system by matings of recessive homozygotes with mice of an inbred strain which is +/+ at the r locus. All the progeny (r/+) are backcrossed to mice of the inbred strain, thus producing two indistinguishable kinds of mice (r/+ and +/+) expected in equal proportions. Among these backcross progeny 12 or more matings should be made up to insure recovery of the homozygotes (r/r). The homozygotes are then crossed with mice of the inbred strain and a new cycle of three generations — cross, backcross, and intercross — is thereby started. The sequence may go on for as many cycles or generations as desired.

The question of how many intercross matings to make up may be treated as follows. The Mendelian probability of a heterozygote in the backcross is ½ and the probability of a mating between two heterozygotes, a "right" mating, is ¼. Thus the probability of the "wrong" type of mating is ¾. If k matings are made up, the probability that all of them are "wrong" is (¾)^k. When k exceeds 11 this probability is less than 5 per cent and when it exceeds 17, the probability is less than 1 per cent. One should plan therefore to set out about 12 intercrosses, with the understanding that the "right" intercross may be found among the first few attempts.

The chief advantage of this system appears when the recessive homozygotes (r/r) are not easily detectable, but rather require expensive time-consuming tests. The tests are necessary only once in each three generations. This has to be weighed against the extra expense and time of maintaining more breeding pens of mice.

The probabilities of incrosses (p) and of heterozygotes (h) after m cycles of three generation each are shown in Table 2-4. Four cycles or 12 generations raise the probability of incrosses to about 99 per cent if there is loose linkage or independence, but 23 cycles will be required to achieve the same probability if there is close linkage (c = 10 per cent).

In the preceding section, three systems for transferring a mutant allele to an inbred background are described under the assumption that a locus with a discrete effect has already been identified by the usual breeding tests. The same three systems of breeding may be used to decompose a complex character into identifiable subunits and thus to isolate single pairs of genes affecting or controlling the character of interest. Even though the breeding systems are identical, the terminology must be changed to avoid confusion. Thus instead of referring to mutant and nonmutant mice, we refer to those with and those without the trait, or to those with high and those with low levels of the trait. The point of the change in terminology is that one does not know, when attempting to decompose a trait into its genetic units, how many loci are responsible for the alternate categories or levels of the trait in any given generation.

The breeding techniques for the isolation of single pairs of genes have been used most effectively by Snell ( 1958 and earlier; Chapter 24) to isolate histocompatibility loci. The histocompatibility reactions fulfill all of the desiderata for the most successful use of the cross-intercross and cross-backcross-intercross breeding systems n uncovering and isolating the histocompatibility loci: discrete all-or-none differences between inbred strains, reasonably small numbers of differential loci distinguishing any pair of strains, and each mouse readily classifiable as having or not having the trait. It is not likely that other genetic systems of such analytic beauty will readily be found in mice.

In the foregoing discussion I have assumed that the trait of interest exhibits discrete alternative. If the trait is continuous, the same breeding procedure may be used, but the outcome is likely to be less clear cut because the environmental sources of variability may blur the differences between alleles at a single locus. Nonetheless, Chai ( 1961) has reported some success in isolating genes affecting body size in mice.

One final word of caution is necessary. These breeding techniques do not actually isolate single loci, but do isolate successively smaller and smaller bits of heterozygous chromosomes which carry at least one pair of alleles affecting the trait of interest. See Haldane's appendix in Snell's ( 1948) paper.

Brother-sister inbreeding may be pursued while heterozygosity is forced upon a specific locus, either by backcrossing or by intercrossing. All matings may be backcrosses, such as r/+ x r/r, if r is a viable recessive, or D/+ x +/+, if d is a dominant or semidominant lethal; or they may be intercrosses, such as r/+ x r/+, if r is a recessive lethal or D/+ x D/+, if D is a dominant or semidominant lethal.

Inbreeding with forced heterosis may be desirable if an inbred strain is not available, making it impossible to try to transfer a mutant allele to the background of a standard inbred strain. Inbreeding with forced heterozygosis may be obligatory if the mutant is incompatible with the genetic backgrounds of available standard inbred strains.

Loci not linked with r (or D) will become homozygous with the same probabilities as under brother-sister inbreeding. Linked loci will become homozygous less rapidly, the rate depending upon the recombination probability c. Further, the probabilities of heterozygosity at linked loci (a) differ in the homozygotes and heterozygotes of the locus of interest (r or D) ( Bartlett and Haldane, 1935; Green and Doolittle, 1963). The probability of incrosses (p) for selected values of c are shown in Table 2-5.

With the exception of intercrossing with a recessive, the systems of brother-sister inbreeding with forced heterozygosis have nearly the simplicity of regular brother-sister inbreeding. Matings of the desired types (r/r x r/+, D/+ x +/+, or D/+ x D/+) may usually be made up from littermates. The task of keeping a standard inbred strain on hand for the sole purpose of putting a mutant gene on an inbred background is thus unnecessary.

Intercrossing with a recessive is somewhat bothersome. The desired matings are r/+ x r/+ and these must be selected from a number of matings which are ?/+ x ?/+. This is so because the probability for r/+ among ?/+ progeny is 2/3 and the probability of r/+ x r/+ is 4/9. Hence to reduce the probability of not getting the right mating to less than 1 per cent requires that k be selected so that (5/9)^k < 0.01.

From Tables 2-2, 2-3, and 2-4 in comparison with 2-5, one may see that the backcross, cross-intercross, and cross-backcross-intercross systems require fewer generations than the systems with forced heterozygosis to achieve a specified high probability of incrosses at the a locus when the a locus is independent of or loosely linked with the r or D locus. When the a locus is closely linked with the r or D locus, the relationship is reversed. This suggests that the two kinds o systems may be used effectively in sequence. To produce a congenic line with segregation at the D locus, for instance, one may use the backcross system for seven or eight generations to bring about a high percentage of incrosses at loci independent of or closely linked with the D locus and follow this with 10 or 12 generations of brother-sister inbreeding with forced heterozygosis to increase the percentage of homozygosity at loci closely linked with the d locus, while not only preserving the percentage of incrosses already achieved but improving at the same time.

Some dominant mutations in mice depress viability and fertility so severely that they can neither be put on standard inbred backgrounds nor inbred with forced heterozygosis. It is therefore necessary, just to preserve the mutation, to propagate it in heterogeneous mice. This may be done either by deliberately avoiding inbreeding within a stock, by routinely backcrossing mutant-bearing mice to a vigorous randombred or noninbred stock, or by routinely backcrossing mutant-bearing mice to vigorous F₁ hybrids between two standard inbred strains.

An experimenter may manipulate the genetic variability in mice by means of the inbreeding systems described in this chapter. Inbreeding will reduce it, outcrossing will increase it, and random breeding will preserve it.

Brother-sister inbreeding has been useful in producing genetically uniform mice and has yielded a large number of inbred strains. Various systems of mating for preserving heterozygosity at a specified locus have also come into general use. These produce mice which are as genetically similar as possible except for the different genotypes at a controlled or segregating locus.

Strains may be crossed to analyze the genetic basis of discrete or of continuous traits. A classic Mendelian experiment with a discrete trait requires the examination of parental, first- and second- generation hybrids, backcrosses, and sometimes other generations. Biometrical techniques will be needed to analyze the genetic constitution of groups of strains differing in quantitative characters. The breeding techniques include selection and diallel crosses.

Mutant genes may be transferred to standard inbred backgrounds by the backcross, the cross-intercross, and the cross-backcross-intercross systems of mating. The same systems may be used to decompose a complex trait into its genetic components and thus to isolate new alleles. Inbreeding with forced heterozygosis may be advantageous when a mutant cannot, for some reason, be transferred to a standard inbred background.

¹The writing of this chapter was supported in part by the Aaron E. Norman Fund and by Contract AT(30-1)-1979 with the U.S. Atomic Energy Commission.

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