MATERIALS AND METHODS

Population structure and selection: The material for this study is Norwegian cattle, which is a commercial breed specialized in both meat and milk production. Elite sires produce large groups of young bulls. Every year, 400 young bulls (sons of elite sires) are tested at station for growth (measured in grams/day and recorded between 3 and 11 months of age) and conformation. Genetic evaluations for growth utilize the bull’s own performance and ∼30 half-sibs. Of the 400 bulls entering in the station, 200 are selected for high growth. Of these, ∼10 young bulls are eliminated for low fertility and another 70 bulls are eliminated for bad conformation. This process resulted in ∼130 young bulls (∼30% selected) with 250 daughters each that were progeny tested for milk. Table 1 shows the six bull sires used and the number of selected and culled sons.

DNA isolation and molecular markers: Semen for genotyping of sires and their selected half-sib sons (after selection for growth and conformation has taken place) was available from the Norwegian breeding organization. Semen from culled sons was not available and DNA was extracted from serum samples routinely utilized in paternity testing of young bulls. Serum samples were lysated at 96° for 10 min and used directly in the PCR. A total of 282 genetic markers covering 2807 cM of the cattle genome (261 microsatellites, one protein polymorphism, and 20 coding genes) were typed in six elite bull sires and 285 sons were selected for high growth and good conformation in the commercial production of Norwegian cattle. A previously constructed genetic map (Vågeet al. 2000; http://www.nlh.no/ihf/Genkartstorfe/) was used for testing for transmission disequilibrium among selected sons.

Distorted segregation of alleles among sperm cells for each of the six sires was tested using single-sperm typing (Liet al. 1988), which utilizes one single-sperm cell and repeated cycles of PCR, allowing amplification of the marker alleles. Single-sperm typing allows testing for a potentially unlimited number of meioses per donor. Approximately 50 sperm cells were isolated for each sire and DNA extraction followed by PCR amplification was carried out for the markers showing significant results in the previous tests. The percentage of success in amplifying single-sperm cells was between 89 and 96%.

Testing of genomic response to selection: Sequential testing for distorted segregation of alleles from widely used commercial bull sires is carried out in three stages: (1) among progeny selected for high growth using a full coverage of the genome, (2) among culled progeny on the basis of low growth, and (3) among sperm cells of sires. Typing in steps 2 and 3 was carried out only for markers yielding significant results in step 1 to reduce the amount of genotyping and reduce the problem of multiple testing at the last stage.

Testing for distorted segregation of alleles among either selected or culled young bulls can be attained by the following method. Consider an elite sire heterozygous for a marker with alleles A and a. DNA for testing dams’ genotypes was not available. Alleles inherited from dams to sons could be either of the sire alleles (A or a) or any other allele segregating in the population (denoted by a^*). If this marker is nearby a QTL affecting a trait under selection, then the frequency among selected sons inheriting either A or a alleles from their sires would be different from the expected 50:50. This approach is similar to a transmission disequilibrium test developed for mapping loci affecting diseases in which the frequency of transmission of alleles from heterozygous parents to their affected offspring deviates from the expected 50% (Spielmanet al. 1993). We developed a maximum-likelihood approach to test for transmission disequilibrium of alleles to half-sib sons. Following Gomez-Raya (2001), the likelihood equation to be maximized is L(v,fA)=K(vfA)nAA(v(1−fA−fa∗)+(1−v)fA)nAa×(vfa∗)nAa∗((1−v)(1−fA−fa∗))naa((1−v)fa∗)naa∗, where K is a constant, v is the transmission disequilibrium parameter, f_A and f_a are frequencies of alleles A and a in the dam population, f_a^* is the frequency of any other allele different from A and a in the dam population, and n_i is the count for genotype i among sons (i = AA, Aa, aa, Aa^*, aa^*). Solutions were obtained using the grid search method after maximizing the above equation for v and f_A. The maximum-likelihood solution for f_a^* was (n_Aa^* + n_aa^*)/(n_Aa^* + n_aa^* + n_aa + n_Aa + n_AA). Therefore, this method uses the expected frequency of each genotype to estimate the transmission parameter and allele frequencies in the dam population. The null hypothesis (marker not linked to QTL for growth) is v = 0.5. Rejection of the null hypothesis (linked QTL) is tested using the likelihood-ratio statistics test (LRT=−2ln(L(0.5,f^A)∕L(v^,f^A))) , where the numerator is the maximum likelihood under the null hypothesis (fixing v = 0.5 and estimating fˆ_A) and the denominator is the maximum likelihood estimating both v̂ and fˆ_A. The distribution of LRT is a χ² with 1 d.f. when the sample size is large. A test was carried out for each marker and sire family. Following Gomez-Raya (2001), an explicit solution for the transmission parameter can be obtained when estimates of allele frequencies in the dam population are available, v=−(A−2B)−(A−2B)2−4B(B+C−A)2(B+C−A), where A=[(nAA+nAa∗)fa−(naa+naa∗)fA+(1−2fA−fa∗)nAa]B=[(nAa+nAa∗)fA]C=−[(naa+naa∗)(1−fA−fa∗)].

The genome-wide transmission disequilibrium testing was performed estimating both the transmission parameter and the dam allele frequencies. A joint test for all families for a particular marker was possible using the properties of the χ² distribution. That is, the sum of independent variables having central χ² distributions follows a central χ² distribution with degrees of freedom equal to the sum of the degrees of freedom of the former central χ² distributions. A test for selection at the genome level was possible under the assumption of independence of χ² across the genome. This is strictly speaking not true because linked markers will have a tendency to cosegregate their alleles. However, the impact of this assumption will be small since each marker will be “unlinked” with most of the other markers in the genome.

A statistical test for distorted segregation of alleles among sperm cells was carried out for those markers and bull sires showing significant results in the testing among selected young bulls. A test for distorted segregation using single-sperm typing was performed by counting the sperm cell carriers of alleles A and a (O_A and O_a, respectively) and by computing χ² = (O_A - E_A)²/E_A + (O_a - E_a)²/E_a, where E_A = E_a = ¹/₂(O_A + O_a). This test is χ² distributed with 1 d.f.

A t-test contrasting growth performance among all sons (selected and culled) inheriting alternative alleles from their sires was carried out by t=x‒−y‒Spooled1∕NA+1∕Na, where Spooled=Σi=1NAxi2−(1∕NA)(Σi=1NAxi)2+Σj=1Nayj2−(1∕Na)(Σj=1Nayj)2NA+Na−2 and x_i and y_j are the growth performances of son i inheriting allele A and son j inheriting allele a, respectively. x and y are the average growths of sons inheriting A and a alleles. The total numbers of sons inheriting alleles A and a were N_A and n_a, respectively.

To account for multiple testing, a permutation test (Churchill and Doerge 1994) was performed by shuffling 500,000 times the observations on growth performance and by randomly assigning those observations to sons. The shuffling of the observations was carried out within the offspring of each sire but keeping the same genotype at all tested markers for each son. The steps for computing experimentwise P values were, (1) finding the maximum t-test among all tested markers for each permuted datum, (2) ordering of the maximum t-test among all tested markers, (3) computing the number of times (ntimes) out of 500,000 permutations that the t-test in the real data was higher than the value of the permuted data, and (4) computing the experimentwise P value, (1 - ntimes)/500,000.

Changes in the transmission disequilibrium parameter by random sampling: The sampling of gametes from a heterozygous sire can lead to changes in the transmission disequilibirium parameter by chance. The approximated accumulated probability of random changes in the transmission parameter can be computed by taking advantage of the normal approximation to the binomial distribution, prob=2∫z∞f(x)dx, where f(x) is the standard normal density and z=(x−0.5n)∕0.25n=(2x−n)∕n . The integral is multiplied by 2 to account for the probability of distorted segregation of either allele.

Statistical power for a transmission disequilibrium test: For simplicity, it is assumed that a transmission disequilibrium test (TDT) is performed in which inheritance of the allele from the sire can always be traced. Let the sire be heterozygous A/a at a genetic marker and the total number of offspring be n. Assume also that a QTL with alleles Q and q affecting growth is linked to the marker with a recombination fraction c. The linkage phase in the sire is AQ/aq. The distribution of phenotypic values is assumed normal with means μ_Q and μ_q for sons inheriting alleles Q and q from their sire. The expected means among sons inheriting alleles A and a are (1 - c)μ_Q + cμ_q and (1 - c)μ_q + cμ_Q, respectively. Therefore, the difference between sons inheriting alleles A and a is (1 - 2c)(μ_Q -μ_q). The sons are evaluated for growth performance using relatives’ information. It is assumed that heritability of growth (h²) was 0.40 and accuracy of estimated breeding values (r) was 0.80. Gene effects are usually given in phenotypic standard deviations rather than in the scale of the estimated breeding values using relatives’ information. The variance of estimated breeding values is r²σ²_A. Thus, if g_P is the gene effect in phenotypic standard deviations then the gene effect in standard deviations of estimated breeding values is g^â = g_Pr/h. If the marker is linked to a QTL affecting growth then the proportion of sons inheriting either A or a will depart from 50% among sons selected for high growth. A mixture of two normal densities corresponding to the sons inheriting alternative alleles from their sire was used for the power calculation. The means of the two distributions used the expected value of gene effects in standard deviations of estimated breeding values. The variance was assumed to be the same in the two groups. Truncation selection makes different the relative contributions of the two groups of offspring to the selected group of sons. The proportion of sons inheriting A and a alleles can be computed by fixing the total proportion selected (p_s) and by using numerical integration on ps=12pA+12pa, with pA=∫t∞f(x∣Gi=A),pa=∫t∞f(x∣Gi=a) , and f(x G_i) being the normal density given the allele inherited from the sire (G_i = A or a). In this equation, ¹/₂’s are used to weigh the contributions to the total selected proportion of the two normal densities and t is the selection truncation point. Under the null hypothesis, the two normal densities would have the same mean and variance. Under the alternative hypothesis, the two normal densities are assumed to have different means but the same variance. In the latter situation, values p_A and p_a can be found for a unique truncation point, t, analytically. The Simpson rule was used for increasing values of the abscissa until a value of t was found to satisfy the above equation for a given p_s. The values used for computing power corresponded to the actual proportion selected in Norwegian cattle (p_s = 0.325). The expected frequencies of A and a alleles among selected sons were pA∗=0.5pA∕psandpa∗=0.5pa∕ps.

The normal approximation to the binomial distribution was used to compute power. It is expected that 50% of the sons inherited either allele A or a from the sire under the null hypothesis. Therefore, the value of the abscissa under the null hypothesis is z0=(0.5+nA−(0.5n))∕0.25n , where n_A is the number of sons inheriting allele A. Numerical integration of the standard normal distribution for increasing values of n_A until the significance level (α) was performed, such as α=12∫z0∞f(x)dx , where f(x) is the standard normal density. For the alternative hypothesis, z_A is computed as if the samples of sons would come from a population with the expected frequencies of alleles A and a among selected sons, pA∗ and pa∗ , respectively. Integration of the normal distribution to compute the probability of type II error was performed by β=∫−∞zAf(x)dx , where zA=(0.5+nA−(pA∗n))∕pA∗pa∗n , with n_A having the same value as obtained for α under the null hypothesis. Finally, power was computed by power_TDT = 1 -β.

Power of the mapping strategy for TDT followed by a daughter design: The mapping strategy consisted, first, of carrying out a TDT using selected offspring. The significant markers and families were then tested using records of performance. The selection at the TDT stage has two main advantages in terms of power: (1) The frequency of heterozygotes at the QTL is increased in the selected families in the t-test and (2) the multiple-testing problem might be reduced since the number of tests in the last stage is much reduced. Power of the experiment attributable to increased heterozygosity at the QTL among sire families can be computed following the approach of Weller et al. (1990) but with an increased heterozygosity.

The frequency of heterozygotes at the QTL among selected families based on TDT results is fH∗=fHpowerTDTfHpowerTDT+α(1−fH), where f_H is the frequency of heterozygous animals at the QTL before the TDT stage, which under Hardy-Weinberg is 2f_Qf_q. The numerator is the probability of detecting a real QTL (power_TDT) multiplied by the probability of being heterozygous. The denominator also includes another term for false positives. Power was computed following the daughter design of Weller et al. (1990) but using the appropriate frequency of heterozygotes (fH∗) and the expected number of families to be tested using the t-test, Nf∗=Nf(fhpowerTDT+α(1−fH)), where N_f is the number of families tested at the TDT stage. The expected power of the TDT followed by the daughter design (DD) was calculated by multiplying the power computed following Weller et al. (1990) for each possible family size (according to a binomial with probability fH∗ and size Nf∗ by its probability and summing over all family sizes. The resulting power, referred to as TDT-DD, was compared to the power for testing with a simple daughter design, ignoring the testing at the TDT stage, with six half-sib families and 100 sons (referred to as DD).

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Figure 1.

—Accumulated probability of obtaining a given transmission disequilibrium parameter under random sampling of 50 sons.