Genetic Response from Marker Assisted Selection in an Outbred Population for Differing Marker Bracket Sizes and with Two Identified Quantitative Trait Loci

Corresponding author: Richard Spelman, Department of Animal Science, Massey University Private Bag 11222, Palmerston North, New Zealand, r.j.spelman{at}massey.ac.nz (E-mail).

Communicating editor: B. S. WEIR

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS LITERATURE CITED

Effect of flanking quantitative trait loci (QTL)-marker bracket size on genetic response to marker assisted selection in an outbred population was studied by simulation of a nucleus breeding scheme. In addition, genetic response with marker assisted selection (MAS) from two quantitative trait loci on the same and different chromosome(s) was investigated. QTL that explained either 5% or 10% of phenotypic variance were simulated. A polygenic component was simulated in addition to the quantitative trait loci. In total, 35% of the phenotypic variance was due to genetic factors. The trait was measured on females only. Having smaller marker brackets flanking the QTL increased the genetic response from MAS selection. This was due to the greater ability to trace the QTL transmission from one generation to the next with the smaller flanking QTL-marker bracket, which increased the accuracy of estimation of the QTL allelic effects. Greater negative covariance between effects at both QTL was observed when two QTL were located on the same chromosome compared to different chromosomes. Genetic response with MAS was greater when the QTL were on the same chromosome in the early generations and greater when they were on different chromosomes in the later generations of MAS.

QUANTITATIVE trait loci (QTL) are being detected in many species using many different experimental designs. In outbred livestock populations, half-sib experimental designs (WELLER et al. 1990 ) have been successfully used to identify QTL (e.g., GEORGES et al. 1995 ; SPELMAN et al. 1996 ). Similar experimental designs have also been used successfully for QTL detection in forest trees (GRATTAPAGLIA et al. 1996 ). In other livestock species, such as poultry and pigs, crosses between divergent lines have been used in two and three generation experimental designs (ANDERSSON et al. 1994 ; VAN DER BEEK et al. 1995 ). Use of QTL detected in these livestock and forest populations through marker assisted selection (MAS) is still at the theoretical level but will most probably be applied in the near future.

Theoretical evaluation of MAS in breeding schemes has been undertaken starting with the work of NEIMANN-SORENSON and ROBERTSON 1961 . Sporadically over the next 25 years further papers addressed MAS (e.g., SMITH 1967 ; SOLLER 1978 ; SOLLER and BECKMANN 1982 ; SMITH and SIMPSON 1986 ; STAM 1986 ). In the 1990's, there have been many papers evaluating MAS. These studies have investigated MAS for dairy cattle (e.g., KASHI et al. 1990 ; MEUWISSEN and VAN ARENDONK 1992 ; BRASCAMP et al. 1993 ; SPELMAN and GARRICK 1997 ), forestry (e.g., WILLIAMS and NEALE 1992 ; STRAUSS et al. 1992 ), poultry (e.g., VAN DER BEEK and VAN ARENDONK 1995 ) and for other situations (e.g., LANDE and THOMPSON 1990 ; GIMFELARB and LANDE 1994A , GIMFELARB and LANDE 1994B ; RUANE and COLLEAU 1995 , RUANE and COLLEAU 1996 ). The theoretical genetic responses from MAS in these studies have varied among studies as many different QTL sizes, genetic models, and breeding schemes have been modeled. However, the near unanimous conclusion from these studies is that extra genetic responses through use of MAS can be made. Larger increases in genetic response with MAS are seen for low heritability traits (SMITH 1967 ), and for traits where selection is undertaken before the phenotype is observed on selection candidates, or the trait is sex limited or carcass limited (MEUWISSEN and GODDARD 1996 ).

Molecular geneticists are continually developing and applying different methods in trying to get closer to the QTL of interest (GEORGES and ANDERSSON 1996 ). From a scientific point of view, this is important and interesting. However, for the application of MAS in a breeding program the benefits from this extra work have not been quantified. SMITH and SMITH 1993 advocated the need to have close marker QTL linkages in outbred populations (1–2 cM) so that selection could exploit linkage disequilibrium between marker and QTL. However, the benefits of this were not quantified and have been questioned by others (e.g. VAN ARENDONK et al. 1994a).

Genome scans have identified multiple QTL that affect the same trait (e.g., GEORGES et al. 1995 ). Plant breeding programs have not limited themselves to using MAS for only one QTL but have selected for many QTL at the same time (STUBER and EDWARDS 1986 ). However, STUBER and EDWARDS' (1986) MAS selection was solely on marker information and did not account for the genetic variation not explained by the markers. Livestock and forestry breeding program are also likely to implement MAS for multiple QTL that affect the same trait. To date, the use of more than one QTL in MAS has not been extensively investigated.

The objective of this study is to quantify the effect of differing sizes of flanking QTL-marker brackets on genetic response from MAS. In addition, genetic response from two QTL on the same and different chromosome(s) is investigated. Furthermore for the two QTL situation, genetic responses are investigated for two QTL of the same size, and also one large QTL and one small QTL.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS LITERATURE CITED

Simulation model:
A stochastic simulation modeling a closed nucleus breeding scheme with discrete generations (each animal present as parent for only one generation) was developed. The initial generation of animals (termed base population) were unselected, unrelated and non-inbred. Each generation had 1024 animals with equal numbers of males and females. A single trait was simulated with base population heritability of 0.35, where heritability is the additive genetic variance divided by the phenotypic variance. The additive genetic variance was divided between unmarked additive polygenic variation (which will be referred to as polygenic variance) and variation due to the marked chromosomal region(s) (which will be referred to as QTL). Phenotypic records were recorded on females only. The highest ranking 12.5% of males and 50% of females for estimated genetic merit were selected as parents of the next generation. As phenotypes were only available on females, male genetic merit was estimated from pedigree information (e.g., sire, dam, and full- and half-sib information) and female genetic merit from own performance and pedigree information. Selection of males and females was undertaken after the single phenotypic record for females was available. Each sire was mated to four females (avoiding half-sib and closer matings) and each mating resulted in four offspring (two male and two female). Each female was mated to one sire only.

QTL alleles for the unselected base population were drawn from the distribution N(0, ¹/₂V_QTL), where V_QTL is the variance explained by the QTL. Two QTL variances were used in this study: 5% and 10% of phenotypic variance. The additive genetic variance (polygenic variance plus QTL variance) was 35% throughout the study. The number of QTL alleles in the base population was twice the number of parents selected from this generation. The large number of alleles represents the situation where the assumed QTL effect is actually due to a cluster of closely linked QTL.

A polygenic effect for each animal in the base population was sampled from the distribution N(0, V_a), where V_a is the polygenic variance. In subsequent generations, the polygenic component was sampled from the distribution N{¹/₂a_s + ¹/₂a_d , ¹/₂[1 - ¹/₂(F_s + F_d)]V_a}, where s and d denote sire and dam, a is the true polygenic value, and F is the inbreeding coefficient that was calculated using the algorithm presented by TIER 1990 . The inbreeding coefficient is the probability that the two genes at any locus in an individual are identical by descent (FALCONER and MACKAY 1996 , p. 52). Residual components from the distribution N(0, V_e), where V_e is the residual variance, were sampled for females and added to the previously sampled polygenic and QTL effects to complete the phenotypic observations. Phenotypic variance in the base population, that comprised of V_a + V_QTL + V_e , had an expected value of 100, and V_a + V_QTL had an expected value of 35.

Marker alleles were simulated for all animals in the base population. It was assumed that the linkage map had six markers that bracketed the postulated QTL position (Figure 1). For the individuals in the base population, marker genotypes were simulated for each of the marker loci assuming five alleles with equal frequency. HALDANE 1919 mapping function was assumed for the construction of the marker-QTL haplotypes transmitted to the offspring.

View larger version (9K): In this window In a new window Download PPT slide   Figure 1. —Marker haplotype that surrounds the postulated location of the QTL.

The required number of sires (64) and dams (256) were simulated for the base population and mated to produce the first generation. Three generations of selection were undertaken without using marker genotypes in the estimation of an animal's genetic merit. Polygenic variance decreases while selection is undertaken because of induced negative covariance between polygenes (BULMER 1971 ). The level of polygenic variance stabilizes over time and the three generations of conventional breeding (without markers) were undertaken to enable this to occur before using MAS. MAS was introduced after the three generations of conventional breeding and, therefore, the MAS genetic responses represent MAS in an ongoing breeding program. MAS was undertaken for seven generations in total. The generation number for offspring born from the first application of MAS will be termed generation one in this paper. Therefore the base population is generation -4.

Breeding value estimation:
Breeding value estimation (estimation of genetic merit) of polygenic and marker linked effects for MAS was undertaken using the model described by MEUWISSEN and GODDARD 1996

Mixed model equations (HENDERSON 1984 ) are used for best linear unbiased predictions (BLUP) of b, u, and q (for one QTL)

This model is an extension of the methods of FERNANDO and GROSSMAN 1989 that was developed for single markers and GODDARD 1992 method that adapted the previous model for marker haplotypes.

In brief, the computational method for marked-QTL considers that in the base population, the number of QTL alleles is equal to twice the number of base animals. In the next generation, the transmission of the parental QTL alleles is followed by inference on the marker haplotype. When transmission of marker haplotype can be followed, the Q matrix links the progeny's phenotype to the transmitted parental QTL allelic effect. When it is uncertain which QTL allele was transmitted, a new QTL allelic effect is formed in the evaluation procedure. The progeny's phenotype is linked via the Q matrix to the new QTL allelic effect and the new QTL allelic effect is linked to its parents through the G matrix i.e., the expectation of the new QTL allelic effect is equal to the mean of the parental QTL allelic effects.

The evaluation model does not assume that the exact location of the QTL within a marker bracket is known, but postulates that it is within the marker bracket. Probability statements are either that QTL transmission can be followed by inference on the marker haplotype, or it cannot. Thus probability statements, other than 0 or 1, are not made about transmission based on recombination events between flanking markers (double recombination) and postulated position relative to single markers (for further description of model see MEUWISSEN and GODDARD 1996 ). The described MAS breeding value estimation method will be referred to as MA-BLUP for the rest of the paper.

If the origin of the marker allele could not be established at the closest flanking markers around the postulated QTL, based on parental and offspring marker genotypes, then the next informative marker in the haplotype was used. If allele origin could not be determined for at least one side of the marker haplotype, QTL transmission could not be determined according to the rules of MEUWISSEN and GODDARD 1996 . Also, if a recombination was observed between markers, QTL transmission could not be determined.

From generation -3, conventional mixed model equations (marker information not used) (HENDERSON 1984 ) were used to estimate b and u. After three generations of conventional selection, MAS was undertaken, using marker information and phenotypic observations, from generation zero with the aforementioned MA-BLUP model. Markers were available on all animals. As a control, conventional selection was also continued for seven generations from generation zero. The additive genetic variance used in solving the mixed model equations for situations without MAS was the sum of polygenic variation and QTL variation in the base population.

Estimates on polygenic and QTL effects were obtained using iteration of the data (SCHAEFFER and KENNEDY 1986 ). Iterations were continued until solutions were stable, i.e., when convergence criterion, which equals the sum of squares of differences in solutions between iterations divided by the sum of squares of the most recent solutions, was less than 10^-10.

Differing flanking marker-QTL size:
The size of the interval between the two flanking QTL-markers was varied to determine the genetic benefit for MAS of localizing a QTL to a small chromosomal area. The four distances studied were 15 cM, 10 cM, 5 cM, and 2 cM. Distance to markers outside the flanking QTL-markers was kept constant in all simulations at 5 cM (Figure 1). One hundred and sixty replicates were simulated for both MAS and the control for each scenario investigated.

Two QTL:
Two QTL were simulated either on the same or on different chromosomes. The number of alleles simulated for each QTL was twice the number of base parents. The variances due to QTL were either the same size or one accounted for 75% of the QTL variance and the other 25%. The combined variance of the two QTL was either 5% or 10% of the phenotypic variance, i.e., the same levels as used for the one QTL models. When the two QTL were placed on the same chromosome the distance between the two QTL was 30 cM. Thirty centimorgans was chosen as this distance is the approximate level of resolution that one can identify two separate QTL in current livestock QTL experiments (HALEY and KNOTT 1992 ). The flanking QTL-marker distance was 5 cM in all cases. QTL allelic effects were estimated separately for both QTL by extending the MA-BLUP model. Negative covariance generated by selection, between the two QTL, and also between the polygenic and QTL components was evaluated. The negative covariance between the two QTL was calculated as half of the difference between total QTL variance less the sum of the two individual QTL variances. The negative covariance between the QTL component and polygenic component was calculated each generation as half of the difference between total additive genetic variance less the sum of the QTL variance and polygenic variance.

The control for the two QTL scenarios was conventional selection on the genetic model of polygenic variance and variance at two QTL. One hundred and sixty replicates were simulated for both MAS and the control for each scenario investigated.

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS LITERATURE CITED

Genetic gain with base model:
The rate of genetic gain for the breeding scheme modeled for a trait of 35% heritability, which consisted solely of polygenic variance, was close to 0.3 _P per generation. Equilibrium response with this model was reached after three to four generations of conventional BLUP selection, confirming that three generations of conventional breeding was sufficient to mimic the introduction of MAS into an ongoing breeding scheme.

Flanking QTL-marker size:
The smaller the flanking QTL-marker bracket the greater the cumulative superiority of MAS over the control (Table 1 and Table 2). The 5-cM bracket had, on average, 90% and 85% of the genetic superiority of MAS (over the control), which was achieved with the 2-cM bracket for the 5% and 10% QTL, respectively. The 10-cM bracket achieved an average genetic response of some 80% relative to that of the 2-cM bracket for both sized QTL (results not shown). For the 5% QTL and 15-cM bracket, the MAS superiority was quite variable, relative to the 2-cM bracket (Table 1) and lower than that of the 10% QTL (Table 1 and Table 2). The relative superiority of the 5% QTL for the 15-cM bracket is similar to that of the 20-cM bracket for the 10% QTL (results not shown).

View this table: In this window In a new window   Table 1. Effect of differing flanking QTL-marker bracket size on cumulative superiority of MAS over the control for a QTL that explains 5% of phenotypic variance (P)

View this table: In this window In a new window   Table 2. Effect of differing flanking QTL-marker bracket size on cumulative superiority of MAS over the control for a QTL that explains 10% of phenotypic variance (P)

The difference in relative response of the 15-cM bracket to the 2-cM bracket between the 5% and 10% QTL, after generation one, may reflect that the value of phenotypes is a curvilinear function, i.e., the first phenotypes per QTL allelic effect have a larger effect on accuracy than the additional ones. The number of phenotypes needed per QTL allelic effect to get a certain accuracy will be larger for the 5% QTL than the 10% QTL, since the 5% QTL explains less of the phenotypic variance. Thus, for the 5% QTL, the 15-cM flanking QTL-marker bracket may move the accuracy of QTL estimation off the plateau-like level of the curvilinear slope. However, for the 10% QTL, the reduction in number of phenotypes per allelic effect when going from a 10-cM bracket to a 15-cM bracket may only reduce accuracy a little. This was observed with the reduction in QTL accuracy decreasing more for the 5% QTL than the 10% QTL when going from a 10-cM to a 15-cM bracket (not shown).

The source of the extra genetic gain with the smaller marker brackets was from extra gain made at the QTL when moving from a 15-cM bracket to a 5-cM bracket for the 5% and 10% QTL (Table 1 and Table 2). Moving from a 5-cM to a 2-cM bracket, for the 10% QTL, the increase in overall genetic gain was from extra QTL response in the first two generations. In the next three generations the extra gain was from both QTL and polygenic and in the last two generations it came from a reduction in polygenic loss (Table 2). For the 5% QTL the extra genetic gain from going from a 5-cM bracket to a 2-cM bracket came from primarily an increase in QTL response with polygenic loss staying stable (Table 1).

Ability to follow transmission of QTL: The ability to unambiguously follow QTL transmission from parent to offspring based on marker haplotype decreased over generations (Table 3). The size of the flanking QTL-marker bracket affected the ability to follow QTL transmission in the first four to five generations of MAS but after seven generations there were only minor differences (Table 3). Reduction in ability to follow QTL transmission was greater for the 10% QTL compared to the 5% QTL due to greater QTL selection pressure and therefore faster fixation (results not shown).

Correlation of estimated and true QTL effects: The smaller the flanking QTL-marker bracket the higher the correlation between estimated and true QTL effects for the 5% QTL (Table 4). This was also observed for the 10% QTL where the correlation between estimated effects and true effects was higher than that for the 5% QTL (results not presented). The correlation increased in the first three to four generations of MAS as more information (phenotypes) accumulated for the estimation of QTL allelic effects. In the last three to four generations of MAS, the correlation decreased as the ability to follow QTL transmission decreased and, therefore, new QTL allelic effects were formed in the evaluation method. The new allelic effects were allocated the average of the parental effects that resulted in lower accuracy.

For the 10% QTL, the BLUP evaluation method was slightly biased in the later generations and genetic gain at the QTL was overestimated. This is probably due to the decrease in QTL variation through changes in allele frequencies, which violates the assumptions of the model. MAKI-TANILA and KENNEDY 1986 commented that this type of bias can occur when fixation or, equivalently, a selection limit is reached. Accuracy of polygenic estimates increased slightly as the QTL-marker bracket size decreased. This may be because of the greater accuracy of estimated QTL allelic effects. When estimating the polygenic value, the phenotype is adjusted for the fixed effect and the QTL allelic effects. With greater accuracy for QTL effects, the phenotype will be adjusted more correctly, resulting in more accurate estimate of polygenic value.

Two QTL:
For the two QTL that together explained 10% of the phenotypic variance, the genetic response was similar regardless of the relative size of the two QTL (Table 5). In the early generations of MAS, the genetic response with MAS was greater when the two QTL were located on the same chromosome than when they were on different chromosomes. In the later generations, the rate of genetic gain when the two QTL were on the same chromosome was less than when they were on different chromosomes. Comparing the two QTL that had a cumulative variance of 10% to one 10% QTL, the genetic superiority over no MAS was nearly the same for the first five generations. In the last two generations, the two QTL model had greater superiority over the control compared to the one QTL model. This was due to there being more QTL variance for the two QTL genetic model in the later generations compared to the single 10% QTL.

For the 5% QTL, the relative size of the two QTL had an effect on the percentage superiority of MAS over the control (Table 5). Having two QTL that were unequal in size resulted in lower percentage superiority in the later generations than that achieved with QTL of equal size. The lower response for the unequal QTL size for the 5% QTL was due to the size of the smaller QTL explaining only 1.25% of the phenotypic variance. MAS with a single QTL of this size (1.25%) was not superior to that without MAS (results not shown) as the accuracy of the QTL allelic effects was low for the breeding scheme structure simulated.

When the two QTL were positioned on the same chromosome, the level of negative covariance between the two QTL was greater than when the QTL were on different chromosomes (Figure 2). The negative covariance increased in the generations previous to the introduction of MAS. With the introduction of MAS, the level of negative covariance between the QTL increased and the negative covariance remained at a higher level when the two QTL were on the same chromosome. When one QTL comprised 75% of the QTL variance and the other 25%, the level of negative covariance was less than that observed for two QTL of equal size (not shown). The level of negative covariance between the polygenic component and the QTL component was not affected by the relative location of the two QTL nor relative size (not shown). The same trends were observed for two QTL that had a cumulative variance of 5%.

View larger version (15K): In this window In a new window Download PPT slide   Figure 2. —Negative covariance between the two QTL and between the polygenic component and the QTL component. Two equally sized QTL that explain 10% of phenotypic variance cumulatively are located on the same or different chromosomes.

When the two QTL were of unequal size (75% and 25%), greater selection response was made at the larger QTL, as was expected. The level of contribution to the QTL variance from the two QTL changed over the generations. For the 10% QTL, the QTL variance in generation four was comprised of 66% from the larger QTL and 34% from the smaller, and by generation seven it was 50:50. For the 5% QTL, the QTL variance in generation seven was comprised of 60% from the larger QTL and 40% from the smaller QTL. In comparison, the level of variance contributed in the control was some 70:30 after seven generations for both sized QTL.

	DISCUSSION AND CONCLUSIONS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS LITERATURE CITED

Negative covariance between two QTL was maintained at a higher level when the two QTL were on the same chromosome in contrast to being on different chromosomes. This is to be expected as the decay of negative covariance is slowed by linkage (BULMER 1971 ). That is, the unfavorable linkages between QTL alleles can only be broken by recombination when the QTL are on the same chromosome. It is interesting to note that the genetic response was higher in the early generations of MAS for the situation where the two QTL were on the same chromosome despite the higher negative covariance. In the later generations, the genetic response was greater when the QTL were situated on differing chromosomes, which would be expected. The level of negative covariance is affected by population size, selection intensity and mating structure (WEIR and HILL 1980 ). Therefore, the results presented on the effect of negative covariance may alter for different breeding scheme structures.

The accuracy of allelic effect estimates was reasonably high at the start of MAS (Table 4). This was because of marker genotypes being present on all five generations prior to the start of MAS. When MAS started with fewer previous generations of marker genotypes and phenotypes the genetic response to MAS was reduced (MEUWISSEN and GODDARD 1996 ; SPELMAN and VAN ARENDONK 1997 ) as the accuracy of estimated allelic effects was lower. Increasing the accuracy of QTL allelic effects can also be achieved by genotyping and phenotyping more full and half-sibs. This may be important for QTL that only explain a small percentage of the variance, as the breeding structure simulated in this study did not have enough observations to accurately estimate QTL effects and use them successfully via MAS for a QTL that explained 1.25% of phenotypic variance. Therefore, breeding schemes may have different optimal sizes for QTL of differing variances. This will also depend on how many previous generations of phenotypes and genotypes are available. Therefore, for a given breeding scheme, you may decide not to select for a QTL below a certain size.

The greater accuracy in estimation of QTL effects with the smaller flanking brackets resulted in greater gain at the QTL when reducing bracket size from 15 cM to 10 cM and subsequently to 5 cM as would be expected. However, the greater polygenic response, or equivalently, the reduction in polygenic loss when reducing the bracket from 5 cM to 2 cM for the 10% QTL, was not expected. In the last two generations the greater response from the smaller bracket was solely from the polygenic component. The polygenic response may be due to the QTL allele being more accurately estimated in the 2-cM bracket situation and, therefore, the adjustment of phenotype in estimation of polygenic value is more correct. In the last two generations, when one QTL may be predominant, the same QTL allele may be selected for both bracket sizes but it is selected in animals with better polygenic value for the 2-cM bracket situation.

The genetic evaluation system used in this study (MEUWISSEN and GODDARD 1996 ), may be slightly more sensitive to flanking QTL-marker bracket sizes than other MAS evaluation methods proposed. This is due to the model in this study requiring that the marker haplotype is informative on both sides of the QTL location. Other methods (FERNANDO and GROSSMAN 1989 ; VAN ARENDONK et al. 1994b; WANG et al. 1995 ) make probability statements about QTL transmission from single markers. Therefore, when markers on one side of the haplotype could not be followed, probability statements about QTL transmission would be made from a single marker rather than forming a new QTL effect. Making the probability statements from one side of the haplotype requires an estimate of the QTL location within the QTL-flanking marker bracket. However, by simulating relatively informative markers and three marker loci on each side of the QTL, the effect of noninformity has been reduced in this study.

In the MA-BLUP method that was used in this study, a shortcoming was when the two QTL effects for a parent were the same and QTL transmission from the marker haplotype could not be followed. In this situation, a new QTL effect was formed in MA-BLUP for the offspring. An improvement would be to identify via the evaluation method if two QTL effects were presumed to be the same in a parent and offspring of this parent get allocated this QTL effect in the Q matrix regardless of the marker haplotype information. This may have improved the accuracy of estimation of QTL effects in later generations.

EDWARDS and PAGE 1994 showed through simulation that the benefits for MAS when using flanking markers instead of single markers was 11% for markers close to the QTL and 38% for markers loosely linked to the QTL. This study has demonstrated and quantified that getting closer to the QTL and having smaller flanking QTL-marker brackets further increases the genetic response from MAS. The close flanking markers used in this study for MAS are different from the MAS scheme outlined by MEUWISSEN and VAN ARENDONK 1992 . Those authors had only two markers on each chromosome forming the marker haplotype for estimation of QTL effects. As shown by this study, MAS schemes will benefit genetically from getting closer to the QTL or chromosomal segment. The improved genetic responses should be balanced against the costs of achieving it, particularly as the amount of work and cost required to get another centimorgan closer is invariably more than it was for the previous centimorgan.

	ACKNOWLEDGMENTS

The authors acknowledge the helpful comments of JOHAN VAN ARENDONK in the preparation of this manuscript. R. J. SPELMAN thanks Livestock Improvement Corporation for financial support.

Manuscript received July 14, 1997; Accepted for publication November 24, 1997.

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