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Institute of Plant Breeding, Seed Science, and Population Genetics, University of Hohenheim, 70593 Stuttgart, Germany
* Corresponding author (melchinger{at}uni-hohenheim.de)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: BCt, t-th backcross generation • cM, centimorgan • MDP, marker data points • RPG recurrent parent genome
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Monitoring the parental origin of alleles at markers throughout the genome in backcrossing was originally proposed by Tanksley et al. (1989) and later termed background selection (Hospital and Charcosset, 1997). Background selection for introgression of a single gene with known map position was investigated by various authors (Hospital et al., 1992; Openshaw et al., 1995; Frisch et al., 1999a,b; Frisch and Melchinger, 2001a,b). Hospital and Charcosset (1997) provided theoretical and simulation results on background selection for the introgression of favorable alleles at quantitative trait loci. To our knowledge, no conceptual frame-work exists for background selection for introgression of two genes with known map positions.
In this study, we extend results on the efficiency of background selection for the introgression of one target gene (Frisch et al., 1999b) to the simultaneous introgression of two genes. We consider phenotypic selection for presence of two dominant target genes combined with background selection for a high RPG at markers covering the whole genome. Our research objectives were to (i) examine alternative breeding plans, (ii) compare different selection strategies, and (iii) examine the effects of varying the population size from early to late generations with respect to the level of RPG attained and the number of MDP required.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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We investigated two scenarios: (i) the two target loci are unlinked, one being located on Chromosome 5, 30 cM from the most distal marker, the second being located on Chromosome 7, 40 cM from the most distal marker; and (ii) the two target loci are linked and located on Chromosome 5 with map distances 30 and 70 cM from the most distal marker.
Breeding Plans
We compared a breeding scheme (Breeding Plan 1) for a backcross program with selection only for presence of the target genes with five alternative breeding schemes (Breeding Plans 2–6), which employ selection for presence of the target genes as well as background selection (Fig. 1)
. In Breeding Plan 1, the target genes A and B are introgressed in separate branches of the breeding program from the donor genotypes DA and DB into the genetic background of the recipient genotype R. After an initial cross, six to eight generations of backcrossing with selection for presence of the respective target genes are carried out. The BCA6 to BCA8 individuals are crossed with the respective BCB6 to BCB8 individuals and a carrier of both target genes (denoted by BC6–BC8) is selfed to generate homozygous carriers of the target genes (denoted by BC6-S1–BC8-S1).
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Population Size
For conducting a successful breeding program, it is necessary that both target genes are recovered in each cross and backcross generation, and that a homozygous carrier of both target genes is present in the progeny of the selfing generation. These conditions can be used to determine the minimum population size for each generation in Breeding Plans 1 to 6.
Let p denote the probability that an individual has the desired genotype in a crossing, backcrossing, or selfing generation. We first consider generations before merging the target genes, in which the desired genotype is heterozygous for one target gene: p = 1 if the parent carrying the target gene is homozygous and p = 1/2 if it is heterozygous. In the generation of merging the target genes, the desired genotype is heterozygous for both target genes, and each parent carries one target gene. If the parents are homozygous, p = 1; if both are heterozygous, p = 1/4. In generations after merging the target genes, the desired genotype carries both target genes, which are originating from one of its parents. The probability p depends on (i) the linkage between the two target genes and (ii) whether both target genes originate from the same ancestor of the crossing parent (coupling phase) or each target gene originates from a different ancestor of the crossing parent (repulsion phase). Repulsion phase occurs in the first generation after merging the two target genes into one genotype; coupling phase occurs in all subsequent generations. For these cases the probabilities p are calculated with the equations given in Table 1 in terms of the recombination frequency r, which is obtained from the map distance d under the assumption of no interference (Stam, 1979) with the inverse of Haldane's (1919) mapping function
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In generations after merging the target genes
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The (a priori) probability that at least one selfing progeny is homozygous for both target genes is obtained by multiplying the values for the s generations of a breeding program:
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When q1 = 1 and qt = q
for all t 
{2, ..., s}, Eq. [2] can be solved to find the minimum population size for generations before merging the target genes in one individual as
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For generations after merging the target genes, solving Eq. [4] yields
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Applying Eq. [6] and [7] to Breeding Plans 2 to 6 yields, for two unlinked target loci and q = 0.99, a minimum populations size of 22 individuals (p = 1/2) in crossing and backcrossing generations and 97 individuals in selfing generations (p = 1/4). The minimum population sizes required for linked target loci (d = 0.4, r = 0.275) are shown in Table 2.
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For Breeding Plans 2 to 6, we employed altogether 3 x 2 x 60 = 360 individuals for the three backcross generations. They were allocated to the generations in three variants: (i) constant population size—in each backcross generation 120 individuals were generated; (ii) increasing population size—in generations BC1 to BC3 a total of 60, 120, and 180 individuals, respectively, were generated; (iii) decreasing population size—in generations BC1 to BC3 a total of 180, 120, and 60 individuals, respectively, were generated. In generations before merging the target genes, the total number of individuals per generation was split equally among the two branches of the breeding program. None of these population sizes is smaller than the respective minimum population sizes determined above.
We employed 120 individuals in the crossing generation to merge the target genes and in the selfing generation. This resulted in a constant total number of individuals applied in the breeding program. However, when two linked target genes are merged in generation BC3, the minimum population size according to Table 2 is not reached and, hence, in an applied breeding program, a larger population size should be used.
Selection Strategy
We applied the two-, three-, and four-stage selection strategies, originally developed for studying the introgression of one target gene (Frisch et al., 1999b) to the above breeding plans. We considered c chromosomes of length li (i = 1 ... c), those carrying at least one target locus are referred to as "carrier chromosomes", the remaining are referred to as "non-carrier chromosomes". Positions on the chromosomes are represented by a scale in Morgan units ranging from 0 to li. We considered 2 target loci at positions xi,j (j = 1, 2) on the respective carrier chromosomes and two flanking markers at positions yi,j,n1 and yi,j,n2, and m additional markers on the carrier chromosomes are located at positions zi,k (k = 1 ... m). In total e markers at positions ui,f (f = 1 ... e) are located on the non-carrier chromosomes. Let Xi,j, Yi,j,n1, Yi,j,n2, Zi,k, and Ui,f be indicator variables, which take the value 1, if the corresponding locus is homozygous for the recurrent parent allele and 0 otherwise. From these random variables we obtain the count variables X = 
i,j Xi,j, Y = 
, and U = Y + i,k Zi,k + i,f Ui,f. Furthermore, the indicator variable Z is 1, if all m additional markers on the carrier chromosomes are homozygous for the recurrent parent allele and 0 otherwise.
By using the random variables X, Y, Z, and U as selection indices, three sequential selection strategies were applied. The first step always involved selection of individuals carrying the target allele (X = 1). Subsequently followed one, two, or three steps with background selection. In each selection step, only those individuals selected in the previous step are subjected to marker analyses. In the individual selected for producing the next backcross generation, all markers not fixed in the previous generation(s) are analyzed to determine which of them are homozygous and, hence, need not be examined in the subsequent generation(s).
In two-stage selection, selection in the second step is based on the index U, which takes into account all marker loci irrespective of their position in the genome. In three-stage selection, the second selection step rests on the flanking markers (index Y), while the final step is again based on all markers (index U) irrespective of their genomic location. Four-stage selection is similar to three-stage selection, but adds after the second step one additional selection step exclusively based on the markers located on the carrier chromosome (index Z).
We further investigated a modification of two-, three-, and four-stage selection, which we call reduced two-, three-, and four-stage selection. In generations before merging the target genes, in each branch of the breeding program, selection is performed for presence of one target gene. In such generations, the markers located on the carrier chromosome of the second target gene were not analyzed.
Simulations
Each backcross program was simulated 10000 times with PLABSIM (Frisch et al., 2000). In the simulations, the entire map was additionally covered with equally spaced (1 cM) background loci to monitor the parental origin of the entire genome. For each simulation run, the percentage of the RPG was assessed in the selected individuals of each generation by dividing the number of loci (marker and background loci) homozygous for the recurrent parent allele by the total number of loci monitored. The values obtained from the 10000 repetitions can be regarded as realizations of random variables that describe the proportion of RPG attained in a backcross program with the parameter settings considered. The 10% percentile of the empirical distribution of the RPG in the selected individual (Q10) was used as an estimator for the proportion of RPG reached after selection in the final generation with probability 0.90. For example, a Q10 value of 98% means that "with probability 0.90 an RPG proportion greater than 98% is attained" under the parameter combination being considered.
The number of MDP required in each generation was counted and summed over the entire backcross program. The mean value over the 10000 repetitions was used to characterize the required number of MDP for any one breeding plan.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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The selection strategy had a considerable effect on the RPG content only when merging the two target genes in generations P or F1 (Table 5). In this case, the Q10 values with two-stage selection were up to one percent greater than with four-stage selection. No differences in the Q10 values were observed when the target genes were merged in later generations. However, two-stage selection required two times (merging target genes in generation BC3, increasing population size) to five times (merging target genes in generation P, decreasing population size) more MDP than did three-stage selection. Four-stage selection further reduced the number of required MDP by about 10 to 20% compared with three-stage selection.
The reduced selection strategies reached practically the same Q10 values as the ordinary selection strategies (Table 5). While the ordinary and reduced four-stage selection required the same number of MDP when the target genes were merged in generation BC1, applying the reduced two-stage selection saved about 10% of the MDP required for ordinary two-stage selection when targets were merged in generation BC3.
Merging the target genes in advanced generations resulted in a greater RPG content than in early generations (Table 5). The difference was about 2% in four-stage selection and 1% in two-stage selection. The greater Q10 values resulting from merging the target genes in advanced compared with early generations required a greater number of MDP. For example, merging the target genes in generation BC3 required two to three times more MDP than merging them in generation P.
For two linked target genes (d = 0.4 M), the Q10 values for the RPG of selected BC3-S1 individuals (Table 6) were greater than those of BC6-S1 individuals with out background selection (Table 3) only when (i) applying three- or four-stage selection and merging the target genes in generation F1 (Breeding Plan 3), or (ii) merging the target genes in generations BC1 to BC3 (Breeding Plans 4–6). In contrast, when (i) merging the target genes in generation P (Breeding Plan 2), or (ii) merging the target genes in generation F1 (Breeding Plan 3) and applying two-stage selection, the Q10 values for the RPG of selected BC3-S1 individuals were smaller than those reached without background selection in generation BC6-S1.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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We used Haldane's (1919) mapping function due to its mathematical simplicity and the stochastic independence of crossover formations in adjacent chromosome regions, which allowed us to derive closed analytical formulas for the problems addressed in this study. Applying gamma inter-event distributions would in most instances yield unwieldy formulas, which could only be numerically approximated. Moreover, as pointed out by Stam and Zeven (1981), dropping the assumption of no interference would reduce the generality of the presented approach because it would be necessary to know the type and degree of interference for the chromosome region of each target gene.
With positive crossover interference, multiple crossover events in a given chromosome region occur less frequently than with no interference (Stam, 1979). Assuming positive interference, (i) the reduction in the number of MDP required (unlinked and linked target genes) and (ii) the increase of RPG values (linked target genes), which are observed in three- and four-stage selection compared with two-stage selection, are expected to be lower in magnitude. The reverse holds true under the assumption of negative interference. In conclusion, the reader should be aware that the model presented (as with most mathematical models of biological systems) is not capable of capturing every detail of the underlying biological process, and our results should be interpreted accordingly.
Saving Backcross Generations
Introgression of one gene is usually accomplished by six generations of backcrossing (Allard, 1960). Hence, a straightforward method for introgression of two genes is to transfer each of them into the recipient genotype and then cross these converted lines, leading to the creation of one homozygous individual. We used this concept as motivation for Breeding Plan 1, which was used as a reference to determine the RPG proportion reached in a classical gene introgression program without background selection (Table 3).
With background selection, the number of backcross generations required for the introgression of one target gene was reduced from six to three when using 60 individuals per generation (Frisch et al., 1999b). Consequently, when applying background selection, using 60 individuals per generation in each branch of the breeding program and merging the target genes in generation BC3 (Breeding Plan 6), RPG values comparable with those reached after six generations of backcrossing without background selection (Breeding Plan 1) is attainable. This expectation proved true for both unlinked and linked target genes (Tables 5 and 6), indicating that saving of three backcross generations due to background selection is a realistic goal for simultaneous introgression of two genes.
Another trend observed for the introgression of one target gene (Frisch et al., 1999b) was confirmed for two genes (Tables 3 and 4): Even with limited resources (population size nt = 80 and 500 to 1000 MDP), background selection can save one or more generations in a backcross program. However, a direct comparison of the RPG values across both studies was not possible, because in the present study background selection was employed not only in backcross generations but also in crossing and selfing generations. This contrasts with the introgression of one target gene (Frisch et al., 1999b), where background selection was employed only in backcross generations.
Increasing, Constant, or Decreasing Population Size
Variation in the population size nt for constant total number of individuals nt only marginally affected the RPG proportion in the last generation of the breeding program (Tables 5 and 6). This result can be attributed to two effects that compensated for each other (Frisch et al., 1999b): First, selection from large populations in early backcross generations takes advantage of the large variance in RPG. Second, by contrast, the selection response due to large populations in late backcross generation has a greater carry-over rate with respect to the RPG content in the final breeding product. The latter effect was explained by Hospital et al. (1992) with a mathematical derivation for the special case of a backcross program with one generation of background selection. These effects suggest that in breeding programs with at least three backcross generations, increasing population sizes can be used to minimize the number of MDP required. In contrast, in breeding programs with the goal of maximizing the RPG after two backcross generations, a constant population size in generations BC1 and BC2 
is preferable (Table 4).
Selection Strategy
For unlinked target genes, two-stage selection resulted in greater RPG values in early generations than three- and four-stage selection. However, in advanced backcross and selfing generations these differences were marginal (Table 4). A high selection pressure was placed on carrier chromosomes in early generations with three- and four-stage selection, and a low selection pressure was placed on noncarrier chromosomes (Frisch et al., 1999b). This results in low overall RPG values for these selection strategies in early generations, because the noncarrier chromosomes form the major part of the genome. Consequently, when the goal of a breeding program is to generate high RPG values after two backcross generations, two-stage selection is preferable. However, with three backcross generations, three- or four-stage selection provides an option to minimize the number of MDP required.
For two linked target genes, three- and four-stage selection yielded greater RPG values in the final generation than two-stage selection (Table 6), indicating that the chromosome segment directly attached to linked target genes is hardly reduced when treating all markers as equal. Individuals showing recombination between the target genes and the flanking markers are preferably selected in three- and four-stage selection. Hence, the probability that the finally selected individual carries a chromosome segment from the recipient between the linked target genes is considerably greater than for two-stage selection. This results in greater Q10 values for the RPG reached for three- and four-stage selection compared with two-stage selection. This result suggests that for linked target genes, three- and four-stage selection increases the proportion of the RPG that can be recovered and decreases the number of MDP required.
The reduced selection strategies for unlinked target genes resulted in the same RPG values observed for the corresponding regular strategies (Table 5). In the generation of merging the target genes, the carrier chromosomes of the selected individuals are inherited from the parent, which is originating from the branch of the breeding program in which background selection on the carrier chromosome was applied. Therefore, omitting background selection on the respective chromosome in the other branch of the breeding program has practically no effect on the RPG values after the merging of the target genes. Consequently, for unlinked target genes with merging of target genes in generation BC1 to BC3, the reduced selection strategies are preferable. They allow a reduction in the total number of required MDP, without any appreciable effect on the final recovery of the RPG.
Generation of Merging the Target Genes
Breeding Plans 2 to 6 differ with regard to the number of generations in which background selection is applied, and the expected number of carriers of the target genes per generation. The latter determines the number of individuals that are subjected to background selection.
When merging the target genes in generation P (Breeding Plan 2), only four generations of background selection can be carried out, whereas in other breeding plans five generations are possible (Fig. 1). This explains (i) the lower RPG values reached with Breeding Plan 2 compared with Breeding Plans 3 to 6, and also (ii) the greater RPG values reached with two-stage selection compared with three- and four-stage selection, because two-stage selection yields greater RPG values in early generations.
In a backcross generation before merging the target genes, about one-half of the backcross individuals are expected to carry one target gene and, hence, to be marker assayed. In contrast, in a backcross generation after merging two unlinked target genes, only 1/4 of the individuals are expected to carry both target genes. For linked target genes the expected portion of carriers of both target genes depends on the degree of linkage and the gametic phase of the target genes. For loose linkage (r 
1/2), it converges to 1/4, whereas for tight linkage (r 0) it converges to 1/2, when the target genes are in coupling, or to 0, when they are in repulsion phase. The greater portion of individuals expected to be marker-assayed in generations before merging the target genes than in generations after merging the target genes results in a higher selection intensity. This explains the greater RPG values attained when the target genes are merged in advanced generations.
Consequently, when the primary objective is reaching a maximum RPG value with a given number of individuals, merging the target genes in generation BC2 or BC3 is advantageous. However, when the target genes are merged in generation BC3, they are in repulsion phase in the individual to be selfed. Depending on the recombination frequency r between the target genes, the population size of the selfing progeny must be fairly large in order to recover with sufficient probability of success q at least one carrier of both target genes (Eq. [7]).
In contrast, when the primary objective is to save three backcross generations with minimum expenditures, merging the target genes in generation F1 combined with three- or four-stage selection is advantageous. This scheme combines five generations of background selection, resulting in a saving of three backcross generations, with a low consumption of MDP for both unlinked and linked target genes.
Number of Marker Data Points Required
By applying two-stage selection and a constant population size of nt = 60 individuals in generations BC1 to BC3 introgression of one target gene required a total of 3340 MDP (Frisch et al., 1999b, Table 3). Introgression of two unlinked target genes with Breeding Plan 2 and three-stage selection required 3500 MDP, when using 120 individuals per generation (Table 5).
In introgression of one target gene, the expected portion of carriers of the target gene in a backcross population, which are undergoing marker analyses, is 1/2, whereas for introgression of two unlinked genes the expected portion is 1/4 according to Breeding Plan 2. Therefore, in a backcross program for two genes, in which twice the number of individuals are employed than in a backcross program for one gene, the number of required MDP is expected to be the same as in the backcross program for one gene. The additional marker analyses required in the selfing generation account for a slight increase (3500 vs. 3340) in the total number of MDP required with two genes. Thus, with Breeding Plan 2 introgression of two unlinked target genes can be accomplished with only a few more MDP than required for the transfer of one target gene, but at the expense of doubling the population size.
The smaller number of MDP required with three- and four-stage selection in comparison with two-stage selection (Tables 3 and 5) is due to a reduction in the number of individuals, which are analyzed for the entire set of markers in the first generation of background selection (Frisch et al., 1999b). The reduction of the number of individuals for which the complete set of markers is analyzed in generation BC1 explains also that the number of required MDP is reduced when (i) reducing the population size in early generations and increasing it in advanced generations in two-stage selection, or (ii) merging target genes in generation BC1.
Omitting the marker analyses for one chromosome in the reduced strategies results in fewer MDP required by these selection strategies than by the respective ordinary selection strategies.
Transferability to Other Situations in Breeding
Our simulation results depend on the population sizes employed and the chosen genetic map. In addition to the presented results, we investigated two alternative methods to determine the population size in backcross and selfing generations: First, the population size in the crossing generation was varied in accordance with the varying population size for the backcross generations (e.g., for Breeding Plan 3 for the scenario with increasing population size we used 2 x 1, 48, 96, 144, 192, 120 individuals in generations C1, C2, BC1, BC2, BC3, BC3-S1, respectively). Second, the population size in the crossing generation was chosen according to minimum population size calculated with Eq. [7] (e.g., 2 x 1, 22, 60, 120, 180, 120 individuals in generations C1, C2, BC1, BC2, BC3, BC3-S1, respectively). While the absolute Q10 values for the RPG under these scenarios differed slightly from the results presented here, the effects of selection strategy, breeding plan, and generation of merging the target genes were essentially the same (data not shown). For both of the above alternatives, the number of MDP required for Breeding Plan 3 were about 10% below the values of the presented results, whereas the differences were only marginal for the other breeding plans.
Important characteristics of the genetic map used in this study are that (i) it consists of 10 chromosomes with a total map length of 16 M, (ii) map positions of markers and target loci are known, and (iii) the average marker density is 20 cM. The ratio between carrier and noncarrier chromosomes determines the differences between the selection strategies with respect to RPG values attained and number of MDP required. For instance, in crops with less than 10 chromosomes, the differences are expected to be smaller, whereas with more than 10 chromosomes, the proportion of genome on the noncarrier chromosomes increases, and, consequently, the differences between the selection strategies are expected to be greater.
When the map position of markers or target genes is unknown, two-stage selection is the only option. In this situation applying increasing population sizes from early to advanced generations and merging the target genes in early generations can reduce the number of MDP required.
Previous simulations for one target gene (Frisch et al., 1999b) have shown that when considering marker densities higher than 20 cM, increases the RPG values are difficult to attain and require substantially more MDP. This can be explained by the fact that given a high marker density, the low frequency of recombination events is the limiting factor for background selection but not the marker coverage to detect the recombination events. However, using evenly spaced markers resulted in greater RPG values without requiring more MDP (Frisch et al., 1999b).
For linked target genes we assumed a recombination frequency r = 0.275 (map distance d = 0.4 M, no interference). If the target genes are more tightly linked, then it is expected that the observed effects of the breeding plan and selection strategy on the proportion of RPG reached and the number of required MDP converge to the values observed for one target gene. In contrast, with increasing map distance between the target genes it is expected that these trends converge to those observed for unlinked target genes. In particular, the increase in RPG values reached with three- and four-stage selection compared with two-stage selection is expected to be smaller with extremely tight but also loose linkage between the target genes.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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A reduction in the number of backcross generations from six to three can be attained with 1000 to 1500 MDP in maize.
Applying three- or four-stage selection for linked target genes and reduced three- or four-stage selection for unlinked target genes requires fewer MDP than two-stage selection. For unlinked target genes, the saving in MDP is realized without reduction of the RPG content, unless the target genes are merged in generation P. For linked target genes, the saving in MDP is accompanied with an increase in the RPG content.
Small population sizes in early generations and large population sizes in advanced generations require less MDP than constant or decreasing population sizes while attaining the same RPG content.
When merging the target genes in generation P, only four generations of marker-assisted background selection are possible compared with five generations when merging the target genes in generations F1 to BC3. Merging the two target genes in an advanced generation results in greater RPG values than merging them in an early generation but requires more MDP.
Received for publication September 1, 2000.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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selection only for presence of the target gene(s), 
selection for presence of the target gene(s) and marker-assisted background selection, x crossing, and 
selfing, 
, 
, and 
indicate generations with crossing, backcrossing, and selfing, respectively.