Linkage disequilibrium in two‐stage marker‐assisted selection
Authors: Schulman By N. F. , De Vries M. J. , Dentine M. R. 
Published: March 2003

Introduction

DNA markers on genetic linkage maps can be used to search for quantitative trait loci (QTL) which affect economically important traits in breeding populations. Several such QTL have already been found in cattle ( Cowan et al. 1990
;
Ron et al. 1994
;
Georges et al. 1995
;
Blattman et al. 1996 ). One way of utilizing this information is marker‐assisted selection (MAS) ( Soller 1978; Soller & Beckmann 1983; Lande & Thompson 1990) to increase the accuracy of selection and improve genetic merit of livestock.

Dairy cattle provide a particular opportunity for markers to be used to distinguish between full sibs prior to entry into progeny testing. Pre‐sorting based on markers within animals of identical pedigree should provide an additional selection round and improve the merits of animals entering progeny test ( Kashi et al. 1990 ; Ruane & ; Colleau 1995) .

Bulmer (1971) has shown that directional selection induces negative gametic disequilibrium. The use of MAS may lower the total genetic response compared with traditional selection schemes because of this disequilibrium ( Gibson et al. 1990 ; Gomez‐Raya & Gibson 1993; Spelman & Garrick 1997). In this paper preselection based on marker genotypes within full sibs selected on pedigree merit as a strategy to improve the genetic merit of progeny‐tested sires is investigated. In particular, the disequilibrium induced by this scheme is compared with that expected from the traditional pedigree selection of candidates.

Methods

Genetic model

A single sex‐limited trait (phenotype P) with a polygenic component (PG), a non‐genetic component (NG), and two QTL physically unlinked to the polygenes or to each other (QTL1, QTL2) was modelled. The phenotypic data and estimated breeding values (EBV) were a linear combination of these terms. Polygenic effects in the base population were generated from a normal distribution. Polygenic variance was derived assuming a polygenic heritability () of 0.3, defined as the ratio of the polygenic variance to the sum of the polygenic and non‐genetic variance. The non‐genetic component was generated from an independent normal distribution. The variance of the non‐genetic component was derived without inclusion of QTL effects and was thus not dependent on changes of genetic variance due to QTL allele frequencies.

The two QTL were modelled as additive, diallelic and physically unlinked. QTL alleles were assumed to be completely identified by the use of close flanking markers and to have effects on the quantitative trait that were known without error. These assumptions were made to maximize the influence of selection using markers on linkage disequilibrium (as in Spelman & Garrick 1997). Recombination events between less closely linked markers or with estimation errors should lessen the impact of marker‐assisted selection on response and linkage equilibrium.

Parameters

The base population was in linkage equilibrium for the different parts of the genome. The sizes of the QTL were either 0.55 σPG or 1 σPG where σPG is the standard deviation of the polygenic component of the trait. Alternatively the size can be expressed as 0.3 σPG+NG and 0.55 σPG+NG, where σPG+NG is the standard deviation of the polygenic plus non‐genetic component in the original population. These sizes were considered to be moderate and large QTL effects and were modelled for equal size QTL or for one moderate and one large QTL. The initial frequency of favourable alleles in the unselected base was varied (0.01, 0.02, 0.05, 0.1, 0.3, 0.5, 0.9). For most runs the frequencies were the same for both QTL but some different frequencies (0.05 and 0.3) were used.

Breeding values for the different QTL genotypes were computed as additive effects following Falconer (1989) and their mean was set to zero in the base generation. Variation associated with the two QTL depended on allele frequencies and size of effects. Thus the effective total genetic heritability available for selection (ratio of the sum of polygenic variance and QTL variance to the total phenotypic variance) was higher than 0.3. Maximum total heritabilities with QTL of these sizes occur at favourable allele frequencies of 0.5 and would reach 0.46 for both QTL of moderate size and 0.65 for two larger QTL.

General simulation scheme

Stochastic simulation, programmed in FORTRAN, was used to generate the estimated breeding values for a large unselected base generation of 200 000 unrelated animals, half females and half males. Selection was applied on this population on the basis of family index selection for a single trait with the use of progeny testing. Animals with the highest estimated breeding value based on information that might be typically available for national improvement schemes were selected. From the total tested population, 1 000 dams and 100 sires were selected as parents of young bulls and one son per dam entered the progeny test. Final selection resulted in 100 progeny‐tested bulls based on estimated breeding values using pedigree information and records of daughters in a selection index. Bull dams were modelled to have correlations of estimated and true breeding values of 0.45 based on pedigree information and a small number of individual records. Correlations of estimated and true breeding values were 0.80 for progeny‐tested bulls based on both pedigree and progeny information. High selection intensities for bull dams and sires were used to be sure that linkage disequilibrium from intense selection was already present in the parents of the young bulls presented for progeny testing. Although QTL were physically unlinked to polygenes, the breeding value for QTL and polygenes were statistically (negatively) correlated in the young bull candidates.

Offspring

Selected animals (parents of young bulls eligible for progeny test) were mated at random, 10 females per male, assuming no inbreeding. Multiple ovulation and embryo transfer were used to produce six full sib offspring per family. The sex of the offspring was assigned at random with a probability of either sex being 0.5. Hence, some families may have offspring of only one sex. The polygenic value of an offspring (PGo) was modelled as half the polygenic value of each parent ((PGdam+PGsire)/2) plus a random and uncorrelated polygenic Mendelian sampling term (MS).

The Mendelian term was generated from an independent normal distribution with mean zero and variance 0. . For each QTL, one of parents’ alleles were randomly assigned to the offspring with equal probability.

The relative breeding values for QTL genotypes were computed using new allele frequencies in the offspring population. A mean value (M) was added to all breeding values for each QTL, where

image

where a is half the deviation between homozygotes, p is the allele frequency of the favourable allele, q of the unfavourable allele. Thus the population mean reflected gains from increases in frequency of favourable QTL alleles and allowed adjustments to EBV after genotyping to include QTL effects as differences in genotype value within the present population.

Selection schemes

Two stages of selection among male offspring were used ( Fig. 1). In the first stage, marker genotypes for the two unlinked QTL were used to adjust EBV in order to select one male among the full sibs (if available) for entry into progeny testing (MAS). If no male was available, the family was not utilized; families with one available male were utilized for both schemes although no MAS was possible. Adjustments to EBV were made given known allele frequencies and sizes of QTL effects. Alternatively, a male candidate was selected within the family (within identical pedigree EBV) at random (Traditional selection). When using QTL information, selection was on the sum of the two QTL segregation terms using knowledge of offspring inheritance of sire and dam alleles. This selection includes two independent pieces of information on the young bull: the pedigree merit (combined information from polygenes and QTL based on family phenotypes) and the Mendelian segregation deviations of the QTL.

In the second stage of selection both strategies used the results of daughter records in the same way. Selected young bulls were progeny tested and selected on the results of family index without considering genotypes at the QTL markers. Although the use of marker‐information on QTL would have resulted in higher correlations of EBV and genetic merit, this information is not likely to be used in the present industry and would have minimal impact given the large numbers of daughter records. This situation would also be the case if genotypes were privately used for MAS and were not publically available for EBV calculations after progeny test.

The disequilibrium (ρ) between the QTL was quantified using the correlation parameter suggested by Hill & Robertson (1968).

image

where, D = f (A1B1) f (A2B2)−f (A1B2) f (A2B1);Ai = alternative alleles of locus QTL1; Bi = alternative alleles of locus QTL2; f ( ) = frequency; p1 = allele frequency of favourable allele (A1) at QTL1; p2 = allele frequency of favourable allele (B1) at QTL2; q1 = 1–p1; q2 = 1–p2.

Computations

Parents were selected on the basis of EBV and one generation of young bulls were selected using one of the two schemes and progeny tested. Results were summarised using parameters for the final selected progeny‐test bulls for polygenic trait value, QTL trait value, disequilibrium between the polygenic and QTL components and disequilibrium between the two QTL, and the results were averaged across 250 replicates and reported as mean±standard error of mean. (For disequilibrium measures between the two QTL no standard errors were computed.)

Results

Total genetic response

Total genetic response, measured as the true breeding values of bulls after the final selection, based on progeny test including the response in QTL effects and polygenes, was always higher with MAS ( Fig. 2). When comparing different starting values for the frequencies of the favourable alleles in the base generation, the largest total response with moderate‐sized QTL was obtained at moderate frequencies of favourable alleles (p1 = p2 = 0.3) with mean genetic gains of 4.50±0.00 σPG for MAS and 4.29±0.01 σPG with traditional pedigree selection followed by progeny testing. The largest difference between the selection schemes was seen at lower frequencies (p1 = p2 = 0.1), where difference in MAS versus Traditional = 0.22 σPG. The larger QTL showed similar trends; but the response was higher for the larger QTL (5.66±0.01 σPG for MAS and 5.39±0.01 σPG with Traditional selection, starting at p1 = p2 = 0.3). At initial frequencies of p1 = p2 = 0.1, the advantage of MAS in breeding value of the selected progeny‐tested bulls was 5.3 % with two moderate QTL and 7.5 % with the larger QTL.

Polygenic response

The polygenic response was very similar for both selection schemes, but was larger for the smaller QTL sizes where MAS was not as useful ( Table 1). The lowest polygenic response was seen when the QTL were at low to intermediate frequencies. At low initial frequencies of the favourable allele, the Traditional scheme showed slightly less depression of polygenic response than MAS while the opposite was true for higher starting frequencies. Polygenic response was higher with the MAS strategy for frequencies >0.3 for moderate QTL and frequencies >0.05 for larger QTL. Larger QTL effects resulted in more depression of polygenic response in MAS than moderate QTL effects. This depression in polygenic response was overcome by QTL gains so that the total response was always higher with MAS.

QTL response

MAS always resulted in higher QTL response than Traditional selection as expected ( Table 2). The response was obtained after the marker selection within full sibs and the advantage was still present after the final selection following progeny test. Higher responses were seen at moderate frequencies of the favourable alleles in the initial populations. Larger size effects of QTL gave a higher response. The largest difference between the schemes in QTL BV after final selection with moderate QTL effects was 32.9 % with p1 = p2 = 0.1.

Disequilibrium

Polygenes and two QTL

The disequilibrium between polygenes and QTL was always negative as expected under selection. The most negative value (−0.693±0.004) was seen for the larger size of QTL with initial frequencies of favourable alleles at 0.05. MAS resulted in slightly larger negative disequilibrium at low starting frequencies and less negative disequilibrium at higher starting frequencies of favourable alleles compared with the Traditional scheme ( Fig. 3). This reversal of the sign of the differences in the two schemes paralleled that of the polygenic response with the MAS scheme resulting in less disequilibrium at higher frequencies.

QTL1 and QTL2

Disequilibrium measures between the two QTL were also negative, but not large in absolute size ( Table 3). Larger QTL resulted in more negative disequilibrium. In most cases MAS resulted in more negative disequilibrium than Traditional selection but the differences were not very large. The disequilibrium was most negative when the initial frequencies of favourable alleles were close to 0.5 in the parents.

Effects of different allele frequencies or size of QTL

When QTL had different starting frequencies for the favourable allele (p1 = 0.05 and p2 = 0.3), the responses were intermediate between those seen with p1 = p2 = 0.05 and p1 = p2 = 0.3. The same is true for the disequilibrium values. The QTL with frequency of favourable allele closer to 0.5 explained a greater part of the total response ( Table 4). When the two QTL were of different size, the results, except disequilibrium between the QTL, resemble more those seen with two QTL of large effect. The portion of the response explained by the larger QTL was greater. With higher initial frequencies for the favourable alleles, the gains due to each QTL were more similar than with lower starting frequencies ( Table 5).

Allele frequencies

Allele frequencies of the favourable alleles increased rapidly with selection. Initial selection of parents increased the frequency compared with the base population. For example with an initial frequency of 0.10 for each QTL, favourable allele frequencies were increased to 0.21 in the selected dams and 0.33 in the sires when both QTL had equal moderate‐sized effects. When marker information was used, there was an additional increase of the favourable alleles in the bulls entering progeny test (with starting frequencies of 0.1 to 0.38). After the final selection based on progeny test results, both schemes led to a large increase in the frequencies but the scheme using markers showed still higher frequencies due to the advantage from selection among full sibs (p1 = p2 = 0.51 in MAS and =0.41 in Traditional, with starting frequencies of 0.1).

Discussion

Traditional selection schemes have utilized pedigree selection for young bulls and discovered differences in Mendelian segregation within candidates of nearly identical pedigree merit by utilizing progeny testing. Using our selection schemes, MAS within animals of high pedigree merit had a clear advantage over Traditional selection especially at low frequencies of favourable alleles in the initial population. Both strategies increased the frequency of the favourable alleles but MAS accelerated the increase. When the frequency of the favourable allele was high at the beginning of selection, both strategies moved frequencies close to fixation especially when the size of the QTL was large. Similar superior response in QTL effects using MAS has been reported in other studies ( Gibson et al. 1990 ; Ruane & Colleau 1995; Meuwissen & Goddard 1996; Spelman & Garrick 1997).

Other studies using various strategies have reported much lower polygenic response with MAS compared with traditional schemes ( Gibson et al. 1990
;
Gibson 1994;
Ruane & Colleau 1996;
Spelman & Garrick 1997). In most of these studies MAS was started in a population at linkage equilibrium between polygenes and QTL (
Gibson et al. 1990
;
Gibson 1994;
Ruane & Colleau 1995,
1996;
Spelman & Garrick 1997). This differed from the scheme described herein where one round of intense selection created disequilibrium in the population before introduction of MAS. There should be disequilibrium if the QTL affects a trait that has been under selection such as a milk trait. Also
Meuwissen & ; Goddard (1996) started MAS in a population at disequilibrium. In the present study a two‐stage selection strategy was used where MAS was applied between full brothers and the bulls selected with MAS were progeny tested and the best selected.
Ruane & Colleau (1996) also applied within‐family selection. Other studies have used one‐step selection with QTL information included in an index or BLUP methods (
Gibson et al. 1990 ; Gibson 1994; Ruane & Colleau 1995; Meuwissen & Goddard 1996; Spelman & Garrick 1997). The authors’ scheme allowed markers to improve selection for QTL at a stage where estimated merit was not highly variable among candidates and where information on Mendelian segregation of markers was independent of pedigree EBV within family. Thus, the disadvantage in polygenic response was only seen at very low starting frequencies of favourable alleles. MAS increases the frequencies of the favourable alleles in the offspring. As there were more young bulls with favourable alleles before progeny testing this resulted in less polygenetic response than traditional selection at low starting frequencies of the favourable alleles. Both strategies showed lower polygenic response when QTL frequencies were close to intermediate values where selection operates most intensely on the QTL. Once the favourable QTL alleles are at a higher frequency, polygenic response was responsible for proportionally more of the gains in both strategies with the MAS scheme having higher polygenic responses. The strategy of using MAS for selection within full sibs allows an additional round of selection on QTL and allows the Mendelian sampling component of polygenic breeding value to become more of a determinant of progeny‐test superiority than selection only on pedigree merit. Because of the similarity of polygenic response in the two schemes, the advantage of MAS in QTL response was reflected in the total genetic response.

The disadvantage of MAS in total genetic response reported by others has usually been seen only after several generations of selection. In the present study only first generation responses were investigated however, comparing different starting frequencies would give some idea of the situation in later generations. At higher initial frequencies of favourable QTL alleles, MAS still had an advantage in the first generation. More generations of selection are however needed to get clear answers to this question and additional simulation is being carried out using these schemes for several generations. Primary results from this second study support the findings reported herein.

Gibson et al. (1990) suggested that negative correlations between polygenes and QTL would lead to lower polygenic response with MAS. The present study showed that under typical selection intensities, both strategies induce negative disequilibrium and the differences between them are quite small. The disequilibrium values depend on the allele frequencies in the population and also on the size of QTL. The present results do not support the idea of MAS creating much larger negative disequilibrium values. Ruane & Colleau (1995) also concluded that the disequilibrium did not cause the lower response seen with MAS. They suggested that the reduction was due to reduced covariances between polygenic effects and estimated breeding values which probably depends on the evaluation model that they used.

We also looked at the disequilibrium between QTL which showed small negative values with MAS creating more negative values at low frequencies of favourable alleles. Comparing the values at different starting frequencies should be done with caution because the disequilibrium measure used is not completely independent of allele frequencies in the population ( Hedrick 1987).

For most simulations a family size of six full sibs was used, but a contrast was made with eight full sibs to test for results when increased selection intensity on markers was possible. Only small differences were seen between these schemes. Larger family size allowed for slightly higher selection intensity in the MAS phase, resulting in a higher total response (1 % higher with moderate QTL and p1 = p2 = 0.05), more response in QTL value (7 % higher with moderate QTL and p1 = p2 = 0.05), slightly higher polygenic response and sometimes more linkage disequilibrium. Responses in the Traditional scheme were unchanged since random selection of one of eight full sibs was not expected to be different from random selection of one of six.

In the simulation model described herein the QTL were assumed to be the same as the markers. This would result in a best‐case situation for success of MAS, but also the most extreme disequilibrium. Relaxing this assumption would probably lead to more similar results for both strategies for response and disequilibrium. The polygenic heritability used in this study was 0.3. The advantage of MAS is higher at low heritabilities ( Lande & Thompson 1990; Ruane & Colleau 1995), but would not involve such a severe disequilibrium.

The strategy proposed herein is already being implemented by artificial insemination organizations ( Cowan et al. 1997 ). Although additional gains could be made by the use of parental genotypes for QTL when selecting dams and sires for young bulls, selection within full sibs provides a convenient way to distinguish between candidates that would otherwise have identical EBV. Those breeders concerned with risk aversion due to uncertain estimates of QTL position or the size of effects may utilize within‐family selection more readily. In addition, Ruane & Colleau (1996) proposed within‐family selection to reduce the accumulation of inbreeding in nucleus populations. The results from this study suggest that using MAS in a within‐family selection scheme would be a useful way of increasing the frequencies of the favourable alleles and improving the total genetic merit of the selected animals.

Acknowledgements

This research was supported by the Finnish Cultural Foundation and the University of Wisconsin‐Madison USDA Hatch Grant #WIS‐3648 (NC‐209). We wish to thank S andra R
odriguez‐Z
as for the Spanish translation and M
artin L
idauer for the German translation of the summary.

The advantage of using marker‐assisted selection (MAS) to pre‐select between animals of identical pedigree prior to progeny testing was investigated for a single generation. In particular, the disequilibrium induced by MAS among full sibs was compared with that of traditional pedigree selection without knowledge of Mendelian segregation. Stochastic simulation was used to model a situation similar to dairy cattle improvement schemes to generate a large base population from which dams and sires were selected on a single sex‐limited index trait. The trait was modelled with a polygenic component, a non‐genetic component and two additive, diallelic and physically unlinked loci of major effect. A two stage selection was applied on the offspring. In the first stage, QTL (Quantitative Trait Loci) information was used to select within each full‐sib family from highly selected parents. Alternatively, selection was at random within elite families. In the second stage, selected males were progeny tested and results of progeny performance used for selection. Total genetic response and QTL response were always higher with MAS. With QTL of moderate size and starting frequencies of 0.1 for the favourable alleles, the advantage of MAS over traditional schemes was 5.3 % for total genetic response and 32.9 % for QTL response. The polygenic response was very similar for both selection schemes. At low initial frequencies of favourable alleles, the traditional scheme showed slightly less depression of polygenic response than MAS while the opposite was true for higher starting frequencies. Disequilibrium values between polygenes and QTL and between the two QTL were always negative in both schemes. Marker‐assisted selection often resulted in less negative disequilibrium between polygenes and QTL than traditional selection.