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Recovery of Superior Homozygous Progeny from Biparental Crosses and Backcrosses.

Breeders of self-pollinated species or of species that re customarily self pollinated to generate inbred lines for use in hybrid production are faced with a need to identify homozygous progeny superior to either parent of a cross population. In the case of crosses between elite parents (so-called "adapted-by-adapted" or "good-by-good" crosses), breeders of many species have relied on pedigree selection or early generation testing to shift the frequency of desirable alleles above one half (Fehr, 1987). Others use modified pedigree selection (Goulden, 1941; Brim, 1966) to increase the frequency of homozygotes prior to selection. Bailey and Comstock (1975) addressed the effect of selection on fixation of desirable alleles in selfed populations under strong intraline selection and concluded that the probability of fixation of the desirable allele would be less than 0.6 in most cases. This does not represent a great departure from the probability of 0.5 expected without selection.

Many breeders utilize exotic parents that exhibit some specific desirable trait. The classic backcrossing procedure for transferring simply inherited traits is straightforward, well known, and will not be discussed further in this work. Other "wide" crosses are made with the objective of incorporating alleles of polygenes from an exotic parent that is often unadapted to the breeder's targeted region of crop production. The epiphytotic of Bipolaris maydis (Nisikado & Miyake) Shoemaker (= Helminthosporium maydis Nisikado & Miyake) in 1970 heightened awareness of genetic vulnerability of crop cultivars in the USA (NAS, 1972). Many breeders began to incorporate exotic germplasm into breeding populations simply to increase genetic diversity (Fehr and de Cianzio, 1981). It is common to find elite-by-unadapted crosses and not unprecedented to find unadapted-by-unadapted crosses in breeding programs. Such populations would be expected to have large numbers of segregating loci. In elite-by-unadapted crosses, the proportion of loci deriving the desirable allele from the elite parent would generally be large unless adaptation is conferred by only a few genes such as genes affecting photoperiodism. In such crosses, most breeders make one or more backcrosses to the elite parent to increase the frequency of alleles contributed by that parent before imposing selection.

A key question in developing biparental populations is the probability of recovering progeny homozygous for more desirable alleles than either parent. This is a particularly important question when crossing adapted with unadapted parents. The probability is dependent on several factors: the number of allelic differences between the parents used to generate the population, the relative contributions of desirable alleles from the parents, the probability of fixation of the desirable allele at an individual genetic locus, and the number of genetic differences necessary to allow the breeder to distinguish between the better parent and a superior plant selected from the population. This work is presented to assist plant breeders in choosing procedures to maximize the chance of recovering desirable plants from biparental populations.

THEORETICAL DEVELOPMENT

The stochastic model is based on the following assumptions: (i) parents of the biparental populations are completely homozygous so that each contributes only one allele to each segregating locus in the population, (ii) all genes are of equal effect so that genotypic value of homozygotes is a linear function of the number of loci bearing desirable alleles in the homozygous state, (iii) the genes assort independently, and (iv) there is no selection, i.e., no differential reproduction, in the segregating generations. A particular pair of homozygous parents is polymorphic at N = [n.sub.b] + [n.sub.w] loci where [n.sub.b] and [n.sub.w] are the numbers of loci in the [F.sub.1] progeny at which the desirable allele is derived from the better and worse parents, respectively. Note that with equal effects of all genes, [n.sub.b] [is greater than] [n.sub.w]. Let [p.sub.b] be the probability of fixation of a desirable allele derived from the better parent (a function of the number of backcrosses and level of inbreeding), [p.sub.w] be the probability of fixation of a desirable allele from the worse parent, and [n.sub.d] be the number of loci bearing homozygous desirable alleles necessary to differentiate between plants. In other words, two plants must differ by at least [n.sub.d] loci to be detectably different at the phenotypic level. For a given plant, let X and Y be the numbers of loci homozygous for a desirable allele derived from the better and worse parents, respectively. The probability of occurrence of a plant with X = i is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the probability of occurrence of a plant with Y = j is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because the loci are independent, the probability of occurrence of a plant with X = i and Y = j is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To detect a desirable plant, it must carry [n.sub.b] + [n.sub.d] desirable homozygous alleles. Combinations of X and Y that satisfy this requirement occur when X ranges from [n.sub.b] + [n.sub.d] - [n.sub.w] to [n.sub.b] and Y ranges from [n.sub.b] + [n.sub.d] - X to [n.sub.w]. The probability of occurrence of a detectable desirable plant is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The breeder must grow a population of size [N.sub.p] [is greater than or equal to] ln([Alpha])/ ln(1 - P) to achieve a probability of 1 - [Alpha] of recovering at least one desirable plant in the population. Any number of scenarios may be examined by varying the number of segregating loci (N), the relative contribution of desirable alleles by the better parent (k = [n.sub.b]/N), the probabilities of fixation of desirable alleles ([p.sub.b] and [p.sub.w]) which are determined by the nature of the population under scrutiny, and the number of homozygous genetic differences required to detect a superior plant ([n.sub.d]).

DISCUSSION

Rarely does a plant breeder know the exact number of genetic differences between two parents. The number of allelic differences between parents is a function of their degree of relatedness as well as of the frequencies of alleles in the entire breeding population. In some species, molecular markers can be used to obtain an estimate of the number of polymorphic loci in a particular cross, but the exact number will remain unknown. Most breeders make crosses among elite lines and cultivars to create populations with high probabilities of producing superior new lines. One would expect the number of segregating loci in a particular cross between elite parents to be low relative to a cross between an elite parent and an unrelated, unadapted parent. Clearly, as N increases, so does the probability that segregating loci affecting the trait will be linked. One may consider any block of tightly linked genes to be inherited as a unit and interpret the theory set forth herein as relating to "effective factors" rather than to individual loci (Mather and Jinks, 1977). Within a set of elite-by-elite crosses, the deviation from unity of the coefficient of coancestry between parents might be used as an indicator of the number of segregating loci. In elite-by-elite crosses, the average proportion of desirable alleles contributed by an individual parent is probably near one half, but in specific crosses the proportions might vary considerably. Populations derived from elite-by-unadapted crosses or unadapted-by-unadapted crosses would be expected to have large numbers of segregating loci. In the case of elite-by-unadapted crosses, the proportion of loci deriving the desirable allele from the elite parent would generally be large unless adaptation is conferred by only a few genes such as genes affecting photoperiodism.

The probability of fixation of a desirable allele is dependent on gene frequencies and level of inbreeding in the population (Table 1). For these probabilities to apply to backcross populations, one would ideally maximize the number of [BC.sub.1][F.sub.1] plants produced and produce only one [BC.sub.i][S.sub.j] for each (Isleib, 1997). It is widely recognized that inbreeding a population to an essentially homozygous state enhances the effect of selection by increasing the frequency of homozygous plants and reducing the masking effect of dominance. Most breeders understand intuitively that it is beneficial to backcross to the better parent when working with elite-by-unadapted populations, but there is some debate as to how many backcrosses would be required to maximize the chance of recovery of a plant superior to the elite parent. Few would advocate backcrossing to either parent in an elite-by-elite cross.

Table 1. Probabilities of fixation of desirable alleles from the better parent ([p.sub.b]) and the worse parent ([p.sub.w]) for selfed and backcrossed populations.
Population                           [p.sub.b]         [p.sub.w]

[F.sub.2]                           1/4 (0.25)        1/4 (0.25)
[F.sub.3]                           3/8 (0.38)        3/8 (0.38)
[F.sub.4]                          7/16 (0.44)       7/16 (0.44)
[F.sub.5]                         15/32 (0.47)      15/32 (0.47)
[F.sub.6]                         31/64 (0.48)      31/64 (0.48)
[F.sub.[infinity]]                  1/2 (0.50)        1/2 (0.50)
[BC.sub.1][S.sub.1]                 5/8 (0.62)        1/8 (0.12)
[BC.sub.1][S.sub.2]               11/16 (0.69)       3/16 (0.19)
[BC.sub.1][S.sub.[infinity]]        3/4 (0.75)        1/4 (0.25)
[BC.sub.2][S.sub.1]               13/16 (0.81)       1/16 (0.06)
[BC.sub.2][S.sub.2]               27/32 (0.84)       3/32 (0.09)
[BC.sub.2][S.sub.[infinity]]        7/8 (0.88)        1/8 (0.12)
[BC.sub.3][S.sub.1]               29/32 (0.91)       1/32 (0.03)
[BC.sub.3][S.sub.2]               59/64 (0.92)       3/64 (0.05)
[BC.sub.3][S.sub.[infinity]]      15/16 (0.94)       1/16 (0.06)
[BC.sub.4][S.sub.2]               61/64 (0.95)       1/64 (0.02)
[BC.sub.4][S.sub.2]             123/128 (0.96)      3/128 (0.02)
[BC.sub.4][S.sub.[infinity]]      31/32 (0.97)       1/32 (0.03)
[BC.sub.5][S.sub.1]             125/128 (0.98)      1/128 (0.01)
[BC.sub.5][S.sub.2]             251/256 (0.98)      3/256 (0.01)
[BC.sub.5][S.sub.[infinity]]      63/64 (0.98)       1/64 (0.02)
[BC.sub.6][S.sub.1]             253/256 (0.99)      1/256 (0)
[BC.sub.6][S.sub.2]             507/512 (0.99)      3/512 (0)
[BC.sub.6][S.sub.[infinity]]    127/128 (0.99)      1/128 (0.01)


The number of homozygous genetic differences required to differentiate between plants phenotypically is dependent on the magnitude of the effects of individual genes, the broad-sense heritability of the trait under consideration, and the precision with which the breeder can compare plants. In generations with appreciable levels of residual heterozygosity, each plant is genetically unique and replication is not possible in species lacking some mode of asexual reproduction. In later generations, the breeder may replicate inbred families and take advantage of the concomitant increase in precision. Nevertheless, it is probably unreasonable to expect a breeder to be able to identify a plant carrying only one more desirable homozygous gene than the better parent of the population. On the other hand, such a plant should not be significantly inferior to the better parent.

Consider, for example, an [F.sub.2] population ([p.sub.b] = [p.sub.w] = 0.25) segregating at 10 loci (N = 10), seven of which derive the desirable allele from the better parent ([n.sub.b] = 7, [n.sub.w] = 3). Assume that the breeder's acumen is highly developed so that he can detect a plant superior to the better parent by a single homozygous locus. The classes of desirable plants are as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The sum of the probabilities is 436/([4.sup.10]) = 4.158 x [10.sup.-4]. The breeder would have to grow an [F.sub.2] population ofln (.1)/ln [1 - (4.158 x [10.sup.-4])] = 5537 plants to achieve a 90% probability of recovery of such a plant. If the population were selfed to the [F.sub.4] before selection ([p.sub.b] = [p.sub.w] = 0.4375), the probability of occurrence of a desirable plant would increase to 2.267 x [10.sup.-2] and the population size necessary to achieve a 90% probability of recovery would decrease to 101. A similar improvement in the chance of recovery could be achieved by backcrossing the [F.sub.1] to the better parent and selecting in the [BC.sub.1][S.sub.1] ([p.sub.b] = 0.625, [p.sub.w] = 0.125) where the probability of occurrence of a desirable plant would be 1.957 x [10.sup.-2] and the necessary population size would be 117. The usefulness of a backcross was unforeseen in a cross between parents differing in numbers of "good" alleles by only a few loci. The breeder may choose to select in the population type most convenient from the standpoint of cost of development and the urgency of the improvement program. Compared with selfing to the [F.sub.4], use of the [BC.sub.1][S.sub.1] generation saves the time required for one plant life cycle but necessitates a second round of crossing. For most self-pollinated species, selfing requires fewer resources than backcrossing while in cross-pollinated species there may be little difference. The breeder may be able to grow two or more selfed generations of plants per year by using a greenhouse, growth chamber, or winter nursery. Clearly the choice of procedure is dependent on the reproductive nature of the plant species as well as on the resources available to the breeder.

For the sake of further illustration, consider an array of selfed and backcrossed populations with N = 2, 3, ..., 100; k = 0.5, 0.6, 0.7, 0.8, and 0.9; and [n.sub.d] = 1, 5, or 10 (Fig. 1-3). It is clear that when the two parents are similar in terms of the numbers of benefical alleles carried (k [approximately equals] 0.5), then selfing rather than backcrossing to either parent is the preferred path toward homozygosity. However, as k approaches 1, especially for larger numbers of segregating loci, then the increased probability of recovery of the good alleles from the better parent associated with backcrossing to the better parent outweighs the reduced probability of recovering good alleles from the worse parent. This is particularly so when selecting in highly inbred generations (Fig. 1bdfhj, 2bdfhj, 3bdfhj). As N increases, the number of backcrosses required to maximize the probability of recovery of the desired plants also increases.

[Figures 1-3 ILLUSTRATION OMITTED]

The advantage of backcrossing is apparent at larger values of N when [n.sub.d] is larger than 1, but more cycles of backcrossing are required to achieve the advantage. The advantage may be moot in wide crosses (k [is greater than] 0.7) at large values of [n.sub.d] because the probability of recovery of the desired plant is extremely low even with repeated backcrossing (Fig. 2ij, 3ij). These conclusions are similar to those derived by Dudley (1982) for random-mating populations of cross-pollinated species.

In cases where the probability of recovering a superior plant is extremely low, direct selection for the trait of interest may result in no genetic gain. In such cases, application of marker-assisted selection would be advantageous. In species exhibiting much genetic polymorphism among molecular markers, the markers may be used to identify useful chromosomal segments even when the segregating population contains no individuals superior to the better parent. Because of the prohibitive cost of assaying all plants in a population for their molecular profiles generation after generation, it may be possible to utilize markers to identify combinations of parents with large numbers of genetic differences, then apply the theory developed in this work to choose an optimal breeding strategy for producing desirable progeny.

REFERENCES

Bailey, T.B., Jr., and R.E. Comstock. 1975. Linkage and the synthesis of better genotypes in self-fertilizing species. Crop Sci. 15:363-370.

Brim, C.A. 1966. A modified pedigree method of selection in soybeans. Crop Sci. 6:220.

Dudley, J.W. 1982. Theory for the transfer of alleles. Crop Sci. 22: 631-636.

Fehr, W.R., and S.R. de Cianzio. 1981. Registration of soybean germplasm populations AP10 to AP14. Crop Sci. 21:477-478.

Fehr, W.R. 1987. Principles of cultivar improvement. Vol. 1. Theory and technique. Macmillan, New York.

Goulden, C.H. 1941. Problems in plant selection, p. 132-133. In R.C. Punnett (ed.) Proc. 7th Int. Genet. Cong., Edinburgh. Cambridge University Press, Cambridge, England.

Isleib, T.G. 1997. Cost-effective transfer of recessive traits via the backcross procedure. Crop Sci. 37:139-144.

Mather, K., and J.L. Jinks. 1977. Introduction to biometrical genetics. Cornell Univ. Press, Ithaca, NY.

National Academy of Sciences. 1972. Genetic vulnerability of major crops. Nat. Acad. Sci., Washington, DC.

Dep. of Crop Science, North Carolina State Univ., Raleigh, NC 27695-7629. Received 20 Jan. 1998.

T. G. Isleib, Corresponding author (tisleib@cropservl.cropsci.ncsu.edu).

Published in Crop Sci. 39:558-563 (1999).
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Date:Mar 1, 1999
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