Animals from three populations were analysed: (A) 427 individuals from a Yorkshire and Large White synthetic breed, (B) 547 Large White individuals, and (C) 531 Large White pigs individuals (Table 1). The populations were structured as half-sib families and the numbers of sires, dams and progeny across the populations ranged from 10 to 11, 91 to 146, and 326 to 391, respectively. Groups A, B and C had been maintained as closed populations for at least 15, 10 and 16 generations, respectively prior to sampling. Two fat traits (back fat P1 and P2) were analysed in population A and three fat traits (back fat P1 and P3 and loin fat L) in population B. Three growth traits, average daily gain pre test (DGP), average daily gain on test (DGT), and average daily gain through the whole life from birth to the end of test (DGW), were used for population C (Table 1). Birth weights were assumed to be zero in order to estimate DGP and DGW. Phenotypic data were measured in the progeny generation only in populations A and C. In population B, phenotypic values from the parental generation were also used. More details are described in the previous papers 614.

The genotyped markers (relative distance from the first marker: cM) were S0001 (0.0), SW35 (11.9), SW839 (15.6), S0107 (17.1), SW841 (23.9) and S0073 (28.4) on chromosome 4 for population A. SW1354 (0.0), SWR1078 (8.9), TNFB (27.5), SW2019 (29.3) and S0102 (39.3 cM) on chromosome 7 were genotyped for populations A and B. S0064 (6.4) and SW1344 (17.0) were additionally genotyped on chromosome 7 for population C. The distances between markers were estimated using the mapping software Crimap 26.

To estimate the presence of a QTL against the null hypothesis (no QTL) at a test position, the likelihood ratio (LR) test statistic LRT = - 2ln(L₀/L₁) was calculated, where L₀ and L₁ represent the respective likelihood values under the hypothesis of either the absence (H₀) or presence (H₁) of a QTL for the one-QTL model. For the two-QTL model, both the presence of two QTLs against the null hypothesis and the difference between one-QTL and two-QTL models were considered 30. The hypotheses for comparing the presence of one QTL in the first position and two QTLs in both the first and second positions are as follows:

H₁: QTL in the first position but no QTL in the second position

H₂: QTLs in both first and second positions.

When comparing the hypothesis of one QTL in the second position with that of QTLs in both the first and second positions, then H₁: QTL presence in the second position but no QTL in the first position, was applied and H₂ was as before.

For the hypothesis tests 3132, a mixture of χ² distributions with different degrees of freedom was applied. For the two-QTL model, the hypotheses (H₀: no QTL, H₂: two QTLs present) were examined with 1/4 χ₀² +1/2χ₁² +1/4χ₂² distribution, and the hypotheses (H₁: one QTL, H₂: two QTLs) were tested with 1/2 χ₁² + 1/2χ₂² distribution. For the one-QTL model, the hypotheses (H₀: no QTL, H₁: one QTL present) were examined with 1/2 χ₀² + 1/2χ₁² distribution.

When a QTL is found in a similar genomic region for each of two correlated traits, we can consider that either (1) QTLs for the two traits are linked (linkage) or (2) two traits are controlled by the same gene or genes at that location (pleiotropy). There are some tests to find linked loci that affect a single trait 2425. However even if there is no proof of linkage, it does not mean that the trait is controlled by a single pleiotropic locus, because we can still argue that the accuracy and power of the test may be insufficient to discriminate between the two situations. Further information provided from denser markers and/or a larger number of generations may reveal the presence of linked loci. Therefore, we do not have a general statistical test to distinguish a pleiotropic locus from linked loci in our analysis scheme. However, we can indicate a theoretical limit of allelic correlation between two traits in a gamete when it is caused by linkage disequilibrium. If allele frequencies do not change across generations, an allelic correlation is determined by a relative size of linkage disequilibrium D 33. In a large population, linkage disequilibrium of the base population, D₀, changes to D_t at generation t with a recombination rate c 19.

Thus, even if there is a perfect correlation (= |1|) between two alleles at linked loci on a gamete at the foundation of a population, it will be reduced by random mating in future generations unless linkage is complete (i.e. c = 0.0). For example, population A has been closed for at least 15 generations. In this population, if c is 0.01, (meaning 1 cM between loci), the size of D relative to the base generation is reduced to 0.86 with t (=15) generations and allelic correlation also changes from 1 to 0.86. If the allelic correlation is more than 0.86 in the current generation, the distance between loci must be less than 1 cM or we can assume a single locus (pleiotropy). In this case we choose the value of allelic correlation greater than 0.86 as a cut off with values above this as being indicative of pleiotropy. We recognize that this choice is somewhat arbitrary and that linked loci may be linked at distances less than 1 cM apart. Note, however, that this is a relatively small window compared with the normal mapping accuracy of QTL studies, for example Gardner et al. 33 have summarized that the average confidence interval around QTLs is 15.6 cM based on more than 200 mapped QTLs. Populations B and C have been closed for at least 10 and 16 generations, respectively. Thus, we chose as criteria values of the allelic correlation of 0.90 (1 cM linkage distance) and 0.82 (2 cM) in population B and 0.85 (1 cM) and 0.72 (2 cM) in population C. The relatively few parents genotyped only represent a small proportion of the whole breeding populations. Therefore, we assumed that effective population sizes are large enough to apply these criteria.

When the two-QTL model was applied within chromosome 4 and 7, we observed very slight differences in test statistic values between the one-QTL and two-QTL models. Significant differences between the two models were evident when two single loci in the two chromosomes were used. When one-QTL model was applied for P1 on chromosomes 4 and 7, both chromosomes showed QTLs at the 5% significance level (Table 2). The test statistic peaks were at 17 cM on chromosome 4 and 37 cM on chromosome 7 with the one-QTL model (Figures 1 and 2). In the two-QTL model, the peak on chromosome 4 was consistent at 17 cM, but shifted slightly to 34 cM on chromosome 7. The two-QTL model displayed a higher test statistic i.e. 10.32, which was significant against both the one-QTL model using chromosomes 4 and 7 (<0.05), and the null hypothesis, no QTL (<0.01) (Table 2). Here, we call the QTLs on chromosomes 4 and 7 'QTL1 and QTL2', respectively. The percentages of residual (one-QTL model: 38% and 41%, two-QTL model: 43%) and QTL genetic variances (QTL1: 14% to 17% and QTL2: 18% to 21%) were slightly altered from the one-QTL to the two-QTL model. However, polygenic variances were reduced from 48% (chromosome 4) and 41% (chromosome 7) to 19% (Table 2).

The one-QTL model for P2 on chromosome 7 was significant (P < 0.05) (Table 3). Test statistic peaks for P2 were lower than those for P1 (Figures 1 and 2). However, the peak for P2 (17 cM) on chromosome 4 was similar to that for P1. The P2 peak on chromosome 7 was at 35 cM, which was very close to the peak at 37 cM for P1. Polygenic heritabilities in the two-QTL model were much lower than those in the one-QTL model (Tables 2 and 3).

Therefore, the estimated total genetic variance, a sum of polygenic and QTL genotypic variances, was similar between the one-QTL and the two-QTL models. For instance, the genetic variance for P2 was 68% (= 62+6) and 68% (= 54+14) in the one-QTL model and 67% (= 44+9+14) in the two-QTL model (Table 3).

LR test statistic (Figures 1, 2, 3 and 4) curves showed that figures for P1 and P2 were quite similar on chromosomes 4 and 7. QTLs for both P1 and P2 displayed the same peak at 17 cM on chromosome 4. However, the peaks positions on chromosome 7 were slightly different between models and traits, i.e. 37 cM in the one-QTL model and 34 cM in the two-QTL model for P1, and 35 cM in the one-QTL model and 32 cM in the two-QTL model for P2. The differences in test statistic values around these peaks were very small.

The correlation of the allelic effects between P1 and P2 within the gamete from the same parent was highly positive. For instance, the correlation between P1 and P2 was 0.90 within the paternal and 0.94 within the maternal gamete on chromosome 4 (Table 4). These allelic correlations are higher than the criterion of 0.86 calculated earlier for an assumed 1 cM linkage distance. We also examined the allelic correlations between the paternal and maternal gametes for each trait. For instance, the correlation between the paternal allelic effect of P1 and maternal allelic effect of P1 was 0.17 on chromosome 4. These allelic correlations between paternal and maternal gametes were 0.12 for P2 (chromosome 4), 0.13 for P1 and 0.13 for P2 (chromosome 7). They suggest weak assortative mating within a line, which may contribute to the allelic correlation in the same gamete. Therefore, we calculated the partial correlations 34 using the matrix of correlations between all related paternal and maternal allelic effects. The partial correlations between allelic effects on the same gamete allow us to exclude the effect of assortative mating. However, all allelic correlations in the same gamete showed very little variation (maximum change <0.3%).

Three fat traits, P1, P3 and L, from population B, displayed similar peak positions (1 and 2 cM), and two fat traits, P1 and P3, showed significant test statistics (<0.05) on chromosome 7 (Figure 5). The heritabilities for QTLs were 12%, 19% and 10% for P1, P3 and L, respectively (Table 5).

Allelic effects of the three fat traits within the gamete from the same parent were highly correlated. Two of the allelic correlations between P1 and P3 0.92 and between P1 and L 0.91 within the paternal gamete were higher than a criterion of 0.90 (1 cM linkage distance). Allelic correlations 0.84 within the maternal and 0.86 within the paternal gamete were higher than a criterion of 0.82 (2 cM linkage distance) (Table 6). Phenotypic correlations between three fat traits were also very high (0.81 to 0.89). However, the allelic correlations between paternal and maternal alleles between traits were quite low, e.g. 0.09 and 0.16 between P1 and P3 fat. In view of these data (Figure 5) and high allelic and phenotypic correlations, we can conclude that it is likely that the three fat traits are controlled by a single pleiotropic locus.

Three measures of growth rate, DGP, DGT and DGW, from population C showed the same peak position, 6 cM, of LR test statistic on chromosome 7 (Table 7 and Figure 6). Heritabilities for QTLs varied from 6% to 12%, and only DGW displayed significant test statistics (<0.01).

Allelic correlations within a gamete between DGW and DGP (0.94 and 0.87) are higher than a criterion of 0.85 (1 cM linkage distance). Allelic correlations within a gamete between DGW and DGT (0.82 and 0.71) were also high (Table 8). However, the allelic correlations between DGP and DGT were relatively low (0.64 and 0.41) and they did not reach the criterion 0.72 (2 cM linkage distance). The phenotypic correlation between DGP and DGT (0.41) is also lower than that between DGP and DGW (0.84) and that between DGT and DGW (0.78).

The correlations between paternal and maternal alleles suggest the possibility of weak assortative mating in both populations B and C (Tables 6 and 8). These correlations are very low, e.g. 0.14 for P1, 0.09 for P3, 0.16 for L (Table 6). After adjusting for the effect of assortative mating, allelic correlations in the same gamete showed very little variation (maximum change <0.4%) in populations B and C.