Exploring Alternative Models for Sex-Linked Quantitative Trait Loci in Outbred Populations: Application to an Iberian x Landrace Pig Intercross

Corresponding author: Miguel Pérez-Enciso, Station d'Amélioration Génétique des Animaux, BP27, 31326 Castanet-Tolosan, France., mperez{at}toulouse.inra.fr (E-mail)

Communicating editor: C. HALEY

	ABSTRACT

TOP ABSTRACT THEORY APPLICATION TO AN F2... RESULTS AND DISCUSSION LITERATURE CITED

We present a very flexible method that allows us to analyze X-linked quantitative trait loci (QTL) in crosses between outbred lines. The dosage compensation phenomenon is modeled explicitly in an identity-by-descent approach. A variety of models can be fitted, ranging from considering alternative fixed alleles within the founder breeds to a model where the only genetic variation is within breeds, as well as mixed models. Different genetic variances within each founder breed can be estimated. We illustrate the method with data from an F₂ cross between Iberian x Landrace pigs for intramuscular fat content and meat color component a*. The Iberian allele exhibited a strong overdominant effect for intramuscular fat in females. There was also limited evidence of one or more regions affecting color component a*. The analysis suggested that the QTL alleles were fixed in the Iberian founders, whereas there was some evidence of segregation in Landrace for the QTL affecting a* color component.

THE mammalian sex chromosomes, X and Y, are relatively poor in gene content. About 690 genes are known in the human X chromosome and only 77 in the Y chromosome as of April 2002 (HUBBARD et al. 2002 ). Nevertheless, >200 diseases are currently listed in the OMIM database as being caused by deficiencies in genes located in the Human X chromosome (HAMOSH et al. 2002 ), and 40 diseases in domestic animals (NICHOLAS et al. 2000 ) are listed. There also exists increasing evidence of quantitative trait loci (QTL) linked to chromosome X in mice as well as in livestock species. RANCE et al. 1997A , RANCE et al. 1997B reported a QTL affecting growth in the mouse X chromosome. In pigs, a sex-linked QTL affecting backfat has been reported in at least five independent experiments (KNOTT et al. 1998 ; HARLIZIUS et al. 2000 ; ROHRER 2000 ; BIDANEL et al. 2001A ; MALEK et al. 2001B ). Further, the porcine X chromosome seems also to be involved in meat quality-related traits (MALEK et al. 2001A ) as well as in follicle-stimulating hormone levels and testes size (BIDANEL et al. 2001B ; ROHRER et al. 2001 ).

The aforementioned porcine QTL results were obtained in crosses between divergent breeds, and the methodology employed was a regression-based approach (KNOTT et al. 1998 ). This analysis is done within sex and alternative fixed alleles within each founder breed are assumed. Nevertheless, there exists genetic variation within as well as between lines in crosses between divergent outbred populations. Furthermore, the QTL analysis of sex chromosomes poses special statistical challenges because of the dosage compensation phenomenon and because of the limited homology between X and Y chromosomes. The dosage compensation phenomenon consists of the random inactivation of one of the two female X chromosomes in different cell lineages. This results in female mosaicism for loci located in the differential chromosome region. A review of the molecular mechanisms involved in the X inactivation process is in AVNER and HEARD 2001 . From a QTL mapping point of view, a consequence is that the female genotype can be modeled as the sum of one-half its individual allelic effects (LYNCH and WALSH 1998 , p. 716). This model results in a female genetic variance that is one-half that of males although no differences in mean between sexes are predicted. This model also implicitly assumes that the allelic effect in females is strictly proportional to the percentage of cells on which that allele is active. Departures from this simple model can be allowed for by adding a term for dominance interaction in females.

All in all, it is important to allow for genetic variation within breeds in analyzing outbred lines, but it is also desirable to model dosage compensation explicitly. The objectives of this work are (1) to present a coherent and general theory that allows us to analyze X-linked QTL in general pedigrees, including crosses between outbred lines and (2) to report the QTL analysis of sex chromosomes in an F₂ cross between the Iberian x Landrace pigs, the so-called IBMAP cross. Results of an autosome scan using a simple regression approach have been presented elsewhere (CLOP et al. 2000 ; OVILO et al. 2000A ; PEREZ-ENCISO et al. 2000B ).

	THEORY

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Our approach is based on the theory developed by PEREZ-ENCISO and VARONA 2000 to analyze crosses between divergent outbred breeds. First we recall the main results from this work and later we generalize them to account for the sex chromosome particularities. Suppose the breeding values (g) of two breeds, A and B, are distributed as g_A N(a/2, ²_A) and g_B N(-a/2, ²_B). The following mixed model can then be used to analyze the records of crossed individuals, say, from the F₂,

(1)

where y is a vector containing the phenotypic records, X and Z are incidence matrices relating y to fixed effects (ß) and genetic values (g_F2), respectively, and e contains the residuals. The distribution of the random variables is

(2)

where c is a vector containing the terms (2p_i - 1), p being the probability, for each ith individual, of an allele being of breed A origin, and is the covariance genetic matrix between individuals i and i'. Further suppose that the total genome is partitioned into n_seg genome segments, a segment being delimited between any two genome positions defined arbitrarily. Then, approximately,

(3)

(PEREZ-ENCISO and VARONA 2000 ), where p_si is the average probability for the ith individual of a gene within segment s being of breed A origin; and _As(i,i') (_Bs(i,i')) is the probability of individuals i and i' having received identical-by-descent (IBD) alleles of breed A (B). Similarly, .

Equation 1Equation 2Equation 3 are also valid to analyze sex chromosome data; the key issue lies in modeling mammalian dosage compensation and in computing the p_s, _sA, and _sB coefficients given marker information. Genes in the pseudoautosomal region do not pose any special modeling problem. Suppose now for a gene located in the differential X region, the male and female phenotypes, y_M and y_F, respectively, can be expressed, simplifying (1), as

(4)

and

(5)

where µ is the sex mean; the superscript in g, the genetic effect, indicates the haplotype origin, 1 for male and 2 for females; ^h is the dosage compensation effect for the hth haplotype allele effect; and d is the dominance interaction. Note that the allele contributing to a male's phenotype is always of the dam's origin (g²). The parameters ¹ and ² should add up to 1; it is biologically possible to think of under- or overexpression of female alleles with respect to male alleles but this effect would be captured by the overall sex effect (µ_M and µ_F). The usual model to account for mammal dosage compensation as explained before (LYNCH and WALSH 1998 ) is retrieved setting . Nevertheless, (5) is general and allows for values other than .

From (4) and (5), together with a generalization of (3), it can be seen that the genetic covariances between any two crossed individuals are

(6a)

if i and i' are males,

(6b)

when i is a male and i' is a female, and

(6c)

for both i and i' being females, where Pr(g^h_i g^h'_i' A) is the probability of alleles g^h_i and g^h'_i' being IBD and being of breed origin A, similarly for Pr(g^h_i g^h'_i' B), and ²_Ag (²_Bg) is the variance of the gene effects in breed A (B). We define, similarly, when i is a male and otherwise. Equation 6aEquation 6b HREF="#FD6c">Equation 6c can then be included into (3) to obtain the genetic covariance matrix G of a sex-linked gene in crosses; likewise, we compute c, the means' vector. As stated above, some interesting consequences follow from (6). Provided that , female genetic variance is halved with respect to that in males (LYNCH and WALSH 1998 ), except if the female is completely inbred. Similarly, the expected genetic covariance between female full-sibs is 3²_g/4 with limits [²_g/2, ²_g], because they always share the male allele, or ²_g/4 between female and male full-sibs, with limits [0, ²_g/2].

The coefficients Pr(g^h_i g^h'_i' A), Pr(g^h_i g^h'_i' B), Pr(g²_i A), and Pr(g²_i B) were obtained via a modification of the algorithm fully described previously (PEREZ-ENCISO et al. 2000B ). This is a Monte Carlo Markov chain approach that allows us to analyze any pedigree structure resulting from an outbred or a crossed population and provide the probabilities p, _A, and _B at the desired positions or the average probabilities over a user-defined segment. The algorithm was modified to account for the dosage compensation parameters () using (6a–6c) plus the fact that males recombine only in the pseudoautosomal region.

	APPLICATION TO AN F2 PIG CROSS

TOP ABSTRACT THEORY APPLICATION TO AN F2... RESULTS AND DISCUSSION LITERATURE CITED

Experimental design and genotyping:
Full details of the experiment are given elsewhere (OVILO et al. 2000B ; PEREZ-ENCISO et al. 2000A ). Briefly, the pedigree consisted of three pure Iberian boars, 31 pure Landrace sows, 79 F₁ individuals (6 sires and 73 sows), and 577 F₂ animals, the so-called IBMAP pedigree. A number of meat quality-related traits were measured. Here we illustrate the proposed method with the analyses corresponding to two traits highly relevant to meat quality: intramuscular fat percentage in longissimus muscle (IMF) and Minolta meat color component, a*. The two founder breeds were highly divergent for these traits (SERRA et al. 1998 ). Founder, F₁, and 338 F₂ individuals were typed for five microsatellites. The female map was SW949–50–SW2126–27–SW2456–7–SW2476–25–SW1608 (distances in cM). These distances were obtained with the CRIMAP software (GREEN et al. 1990 ). Only the first marker (SW949) was in the pseudoautosomal region.

Statistical analysis strategy:
A variety of analysis strategies can be envisaged using the mixed-model methodology described here and in previous works, depending on the aim of the study (PEREZ-ENCISO and VARONA 2000 ; PONZ et al. 2001 ). Here we used a two-step analysis, the purpose being to illustrate a possible way to spare computing cost but minimizing the risk of discarding genome regions containing significant QTL. Thus, first we fitted the two most extreme models that can be envisaged in (1), a model where alternative alleles are assumed to be fixed in each founder breed ,

and a model that presupposes no genetic differences between breeds ,

Above, u₀ is the polygenic genetic value, distributed as N(0, G₀ ²₀), where G₀ is the usual numerator relationship matrix, ²₀ is the infinitesimal genetic variance, and u_s is a random genetic effect for segment s. Model c1 did not allow for dominance in the female genotype; i.e., model (6) was simplified as . In Model v1, it is assumed that g_A and g_B are distributed identically , with G_g computed as shown in (6) setting . The effects included in ß were sex, batch (five levels), and carcass weight as covariate. These two models were fitted at successive segments of 5 cM, i.e., IBD probabilities were obtained for segments 0–5, 5–10, ... , 105–110 cM. This preliminary analysis can be carried out to preselect traits showing promissory QTL.

In a second step, a more thorough analysis was done. Model c1 as above plus three additional models were fitted in successive 2-cM segments in the chromosome region where the previous analysis was suggested as more likely to contain the QTL. The remaining models fitted were

and

Model c2 is a model where dominance in female genotypes is allowed for as shown in (5), c^'_s is a vector containing the probabilities of having received an allele from each of the two founder breeds, and it contains a zero for all male coefficients. Model c2 assumes fixed alleles within breed. The latter two models, m1 and m2, are mixed models where additive genetic differences between and within breeds are allowed for. Model m1 still presupposes that additive variances are equal in both founder breeds , although in this case it is assumed that mean allelic effects between breeds can be different. Finally, Model m2 also allows for the possibility that ²_A and ²_B are different and they are estimated separately. In this latter case , where G_sA is a matrix consisting of elements Pr(g^h_i g^h'_i' A) obtained from (6), and similarly for G_sB. A brief description of the models is outlined in Table 1.

The likelihood ratio (LR) between hierarchical models provides insight about the genetic nature of the QTL. Model c2 compared to c1 permits us to evaluate whether dominance occurs. It should be noted, nonetheless, that only two genotypes are possible in F₂ females in the particular design of our experiment, "AB" and "BB," where B stands for the Landrace allele, and thus the dominance parameter should be interpreted broadly to include sex x allele interaction. Similarly, the LR of model m1 over c1, or m2 over c1, is aimed at assessing whether there is evidence of genetic segregation within breeds. The comparison of m1 with v1 can be used to identify differences between mean allelic effects for each breed. Finally, unequal genetic variances within breed are tested comparing m2 with m1. Note that these two latter models are hierarchized, given that ²_As and ²_Bs can be reparameterized as ²_gs + ² and ²_gs - ², respectively. Finally, we found evidence that there might be two chromosome regions affecting color component a* (see below) and we thus fitted a two-segment model to refine the statistical evidence. Details are presented as necessary in the next section.

It should be stressed that the method uses all available marker and pedigree information, which is particularly relevant for estimating the genetic variances ²_As, ²_Bs, and ²₀. All individuals in our IBMAP pedigree were directly or indirectly related and the exact relationship coefficients were computed, except that founders were taken as unrelated. The parameters of interest, ²_As, ²_Bs, ²₀, and the residual variance were obtained by maximizing the log-likelihood , using a simplex algorithm.

	RESULTS AND DISCUSSION

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Application to pig data:
The plots of the likelihood ratios for every chromosome position are shown in Fig 1, using the two extreme models, a completely fixed model (c1) and a random model (v1). It should be recalled that Models c1 and v1 are not hierarchized and thus the LR values for each model are not comparable. The trend was similar with either v1 or c1 models for IMF but there was also a striking difference in the maxima positions identified for each model in color component a*. The results of the detailed analysis are in Fig 2 and in Table 2. Neither Model m1 nor m2 improved upon Model c1 for IMF, thus strongly suggesting that alternative alleles are fixed in each founder breed. The most relevant result concerning this trait is that the model including a dominance effect in (5), Model c2, was far more likely than the simpler strictly additive Model c1. The value 12.7 corresponds, approximately, to a critical value of 0.1% for a chi square with 2 d.f. Interestingly, the Iberian allele was distinctly overdominant: the females, being heterozygous for Iberian/Landrace alleles, had a significantly larger amount of IMF than the average males and females homozygous for the Iberian allele. Of course the terms "dominant" and "overdominant" should not be interpreted literally given that dominance is not possible in males and that only two genotypes are present per sex. It is stressed, nevertheless, that all a, d, and the mean sex differences are estimable functions. A QTL effect on IMF had been already reported in chromosome X (HARLIZIUS et al. 2000 ) although the primary effect was on backfat thickness, in agreement with other studies, which have found an effect on backfat but not on intramuscular fat (KNOTT et al. 1998 ; ROHRER 2000 ; BIDANEL et al. 2001B ; MALEK et al. 2001B ). We did not find any clear-cut significant result on backfat in this data set (results not presented).

View larger version (15K): In this window In a new window Download PPT slide   Figure 1. Preliminary 5-cM segment scans for the traits analyzed. Results with a fixed model (c1) are represented by a solid line and random model results (v1) by a dashed line. Twice the likelihood ratio of the corresponding model over a model that contained only the fixed effects plus the infinitesimal genetic value is plotted. (a) Intramuscular fat percentage (IMF); (b) a* meat color components. The markers were located in positions 0, 51, 78, 85, and 110; the differential chromosome region begins in position 51. See Table 1 for model description.

View larger version (21K): In this window In a new window Download PPT slide   Figure 2. Detailed 2-cM scan. Twice the likelihood ratio of the corresponding model over a model that contained only the fixed effects plus the infinitesimal genetic value is plotted. (a) Intramuscular fat (IMF); (b) color component a*. Model c1 includes only the fixed additive effect and assumes that alleles are fixed within breed; Model c2 includes in addition a dominance effect and assumes alleles fixed within breed; m1 is a mixed additive mixed model where genetic variances are equal in both founder breeds (), and Model m2 is an additive model where genetic variances within each breed, 2A and 2B, are estimated separately. Note that not the whole chromosome is represented. The two horizontal lines represent the values 3.8 and 6.0, which are the chi-square 5% significance thresholds with 1 and 2 d.f., respectively. The markers were located in positions 0, 51, 78, 85, and 110; the differential chromosome region begins in position 51. See Table 1 for model description.

As far as the color component a* is concerned, the likelihood maxima are quite distinct depending on the model of choice. The bordering areas between the pseudoautosomal and the differential regions are more significant when the data are analyzed via the fixed Model c2, whereas the mixed Model m2 suggests instead the distal part of the differential region, between the SW2476 and SW1608 markers. Model m2 also resulted in an estimate of (Table 2). We can hypothesize that there might be two QTL affecting the trait, each with a different genetic action. To evaluate this hypothesis, we fitted the following two-segment model,

i.e., a first QTL with dominant action and fixed alleles plus a second additive QTL segregating only in the Landrace breed. The first segment was fitted between positions 44 and 64 cM and the second between 94 and 110 cM, again in 2-cM steps. A bivariate plot of the results is in Fig 3. The positions corresponding to the maximum likelihood were 54 and 104 cM for segments 1 and 2, respectively. The LR of the two-segment model over the model containing only the infinitesimal genetic effect was 10.7 and, thus, the LR over either Model c2 or m2 (setting ²_As to 0) was 3. Thus, we cannot conclude beyond reasonable doubt that a two-QTL model is much more likely than a single-QTL model. If only a single QTL exists, it seems that it would be located in the border of the pseudoautosomal region with a confidence interval that would certainly include that region. There is not, to our knowledge, any QTL detected in that chromosomal region in the pig or in any other domestic species. In humans, the nonheterochromatic pseudoautosomal part contains only a few genes and, a priori, it seems to be more likely that the genes affecting a*, or any other trait, are located in the differential rather than in the pseudoautosomal region.

View larger version (74K): In this window In a new window Download PPT slide   Figure 3. Two-QTL scan for color component a*. A first QTL with dominant action and fixed alleles was fitted between 44 and 64 cM, plus a second additive QTL segregating in the Landrace breed was fitted between 94 and 110 cM at 2-cM steps. Twice the likelihood ratio of the two-QTL model over the model that contained only the fixed effects plus the infinitesimal genetic value is plotted. () 10.5–11, () 10–10.5, () 9.5–10, () 9–9.5.

Application to general pedigrees:
This two-step method (a step to compute relationship coeffcients and a step to obtain the maximum-likelihood estimates) can be applied to pedigrees of any complexity. We and others (GEORGE et al. 2000 ; PEREZ-ENCISO and VARONA 2000 ; PRATT et al. 2000 ) have shown it to be quite a robust and powerful approach in a variety of situations. The only limitation, either for autosome or sex-linked QTL, lies in computing the IBD coefficients in (6). The Markov chain Monte Carlo (MCMC) method that we have applied is valid for any pedigree complexity as long as markers are available for all individuals. Nevertheless, the method is able to deal with a limited number of missing markers if there are at least parents or offspring genotyped (i.e., the method may get stuck if several generations of individuals have not been typed). We are currently developing a more general MCMC algorithm that allows for a far larger flexibility in terms of missing information and that also block updates a series of marker phases jointly, which is important if markers are tightly linked. We plan also to generalize a Bayesian fine-mapping strategy (M. PÉREZ-ENCISO and D. MILAN, unpublished data) to sex-linked QTL.

Estimating dosage compensation parameter ():
The most straightforward assumption in mammals is to set (LYNCH and WALSH 1998 ). This was the value that we used here but the reader should be cautious in interpreting the results. This model implies that the fact that one allele is inactivated in 50% of the cells is equivalent to halving its effect on the phenotypic scale, which is by no means evident, even allowing for dominance. Another consequence of this model is that the additive genetic variance in the females is one-half that of the males, as discussed above. In principle, one could estimate the dosage compensation computing G from (6) at different values of and maximizing the likelihood. The justification for not taking for granted is that there are some genes, at least a dozen in humans, that are known to escape X-inactivation (HEARD et al. 1997 ), but also allow a more flexible relationship between the phenotypic scale and the degree of inactivation. Unfortunately, our experimental design prevents us from estimating when treating the QTL effect as fixed because no reciprocal crosses of Iberian females by Landrace males were carried out and thus the dominance effect and are confounded. This is not the case in the random IBD approach (e.g., Model v1) because then matrix V depends nonlinearly on but is independent of d. We computed G for two extreme values, and , and the trait IMF in a limited number of mixed models and genome positions but we observed that the likelihood was barely affected. Probably as expected, cannot be estimated accurately from these types of data, but a positive reading shows that the method is robust to departures from the true . However, it will be exciting to ascertain whether can be estimated in carefully designed experiments using the approach developed here. It is also clear that the theory presented here can be applied to model imprinting with only a few changes, e.g., setting and for male imprinting.

In conclusion, our work adds to previous methods of QTL analysis for sex chromosomes in three main aspects: first, a model for dosage compensation is used such that both the pseudoautosomal and the nonhomologous region are studied using a coherent statistical modeling; second, a series of increasingly sophisticated genetic models have been applied, showing that model choice is a critical aspect of QTL detection and is specially relevant for sex chromosome analysis; and third, it uses all available pedigree information to compute the IBD probabilities conditional on marker information. We have illustrated the theoretical approach with an analysis of original pig real data, and we have explored a variety of models. From this analysis, we can conclude that the distal part of the differential region contains one (or more) QTL affecting IMF and, perhaps, the a* color component. As far as genetic action is concerned, the analyses with these traits, as well as other traits analyzed (our unpublished results), all lead to the conclusion that the genetic variance within the Iberian line used was zero. This agrees with expectations due to the small number of Iberian founders, together with the high relationship coefficient between the Guardyerbas boars, 0.75 on average. It should be recalled that all Guadyerbas individuals are highly related; their inbreeding coefficient is >0.3 on average (RODRIGANEZ et al. 1997 ). In addition, there is evidence for overdominance (IMF) and that genetic segregation might exist within Landrace founders (a* color component). Finally, and given that the power to detect sex-linked QTL is lower than for autosomes, it would be most interesting to carry out a joint analysis like that of porcine SSC4 (WALLING et al. 2000 ) using our approach. If more than two founder breeds are involved, the theory can be extended; e.g., (6a) can be replaced by and similarly for the remaining equations.

	ACKNOWLEDGMENTS

We are specially grateful to Pere Borràs and Eva Ramells together with all the personnel in Nova Genètica and Copaga's slaughterhouse for their cooperation. We thank Romain Pique for assistance in analyzing the data. We are also grateful to L. Silió, M. C. Rodríguez, and M. A. Toro for comments. The microsatellite primers were kindly provided by M. F. Rothschild. The Iberian boars were a gift from the SIA "El Dehesón del Encinar" (Toledo, Spain) that belongs to the Junta de Castilla-La Mancha. Pure Landrace animals and the rest of the pedigree were produced in Nova Genètica (Lleida, Spain). Alex Clop acknowledges a predoctoral fellowship from Universitat Autònoma de Barcelona. Work was funded by Comisión Interministerial de Ciencia y Tecnología grants AGF96-2510-C05 and AGF99-0284-C02 (Spain).

Manuscript received November 6, 2001; Accepted for publication May 6, 2002.

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