## Abstract

Genomic imprinting is interpreted as a phenomenon, in which some genes inherited from one parent are not completely expressed due to modification of the genome caused during gametogenesis. Subsequently, the expression level of an allele at the imprinted gene is changed dependent on the parental origin, which is referred to as the parent-of-origin effect. In livestock, some QTL for reproductive performance and meat productivity have been reported to be imprinted. So far, methods detecting imprinted QTL have been proposed on the basis of interval mapping, where only a single QTL was tested at a time. In this study, we developed a Bayesian method for simultaneously mapping multiple QTL, allowing the inference about expression modes of QTL in an outbred F_{2} family. The inference about whether a QTL is Mendelian or imprinted was made using Markov chain Monte Carlo estimation by comparing the goodness-of-fits between models, assuming the presence and the absence of parent-of-origin effect at a QTL. We showed by the analyses of simulated data sets that the Bayesian method can effectively detect both Mendelian QTL and imprinted QTL.

GENOMIC imprinting is interpreted as a phenomenon, in which some genes inherited from one parent are not completely expressed due to modification of the genome caused during gametogenesis. Subsequently, expression levels of alleles at imprinted genes are dependent on the parent from which they are inherited and the identical allele shows different effects corresponding to the parental origins, which are called the parent-of-origin effects. When one of the paternal allele and the maternal allele at a gene is exclusively expressed and the other is completely silenced, this feature of imprinting is referred to as complete imprinting and the gene can be said to be completely imprinted. On the other hand, when both the paternal allele and the maternal allele at a gene are expressed at different levels depending on the parental origin, this feature of imprinting is referred to as partial imprinting and the gene can be said to be partially imprinted.

It has been shown that some genes and traits were influenced by genomic imprinting in several organisms. Especially in livestock such as pigs, some QTL affecting traits of economical importance, including reproduction and meat productivity, have been reported to be imprinted with allelic effect changed by the parental origin in several researches (Knott *et al*. 1998; Jeon *et al*. 1999; Nezer *et al*. 1999; De Koning *et al*. 2000; Tuiskula-Haavisto *et al*. 2004), where interval mapping methods were used in genome scans for imprinted QTL in F_{2} families derived from crossing two outbred lines. These interval-mapping methods for imprinted QTL were based on a regression model, where phenotypic values of a trait in F_{2} individuals are regressed on the line origin of alleles as well as on the parental origin of alleles that can be traced back from the F_{2} individuals to their F_{1} parents at a putative QTL in outbred F_{2} populations. Recently, interval mapping for exploring imprinted QTL in inbred F_{2} populations, derived from crosses between two inbred lines, was also devised by Cui *et al*. (2006) on the basis of the maximum-likelihood method, where two heterozygous QTL genotypes, say, *Q*_{1}*Q*_{2} and *Q*_{2}*Q*_{1}, with *Q*_{1} and *Q*_{2} being two alleles at a QTL and the first and second alleles indicating paternal and maternal ones, respectively, can be discriminated with the EM algorithm using phenotypic values even in inbred F_{2} populations.

In the report by De Koning *et al*. (2002), the statistical models of interval mapping for imprinted QTL in outbred F_{2} families were reviewed and the efficiencies of the models in QTL detection were investigated using simulation experiments. They proposed a two-step statistical procedure to identify the expression mode for a detected QTL as follows: (A) test of H_{0}: Mendelian QTL model *vs.* H_{1}: full imprinted QTL model that assumes the parent-of-origin effects of QTL and thus includes both cases of partial imprinting and complete imprinting and (B) test of H_{0}: completely imprinted QTL model *vs.* H_{1}: Mendelian QTL model. Tuiskula-Haavisto *et al*. (2004) applied a similar statistical procedure to detection of parent-of-origin effects at QTL in a chicken F_{2} population.

Since the model for QTL expression is fixed as a Mendelian QTL model or an imprinted QTL model in the analyses with interval mapping, multiple steps are required to identify the expression mode of QTL, as seen in De Koning *et al*. (2002) and Tuiskula-Haavisto *et al*. (2004), which might cause a problem in determining thresholds due to multiple testing. Therefore, methods to simultaneously map multiple QTL are more suitable for detecting imprinted QTL and discriminating between Mendelian and imprinting expressions of QTL.

In this article, we developed a Bayesian method using a Markov chain Monte Carlo (MCMC) algorithm that enabled us to map multiple QTL simultaneously and to infer the expression mode for each detected QTL in F_{2} families derived from crossing two outbred lines. We utilized a reversible-jump MCMC (RJ–MCMC) procedure in moving across competing models corresponding to several expression modes of QTL, including Mendelian expression and imprinting expressions that were furthermore classified into partial imprinting and complete imprinting with exclusive paternal or maternal expression. Although several Bayesian methods have been devised for QTL mapping in experimental populations such as F_{2} families (Satagopan *et al*. 1996; Sillanpää and Arjas 1998; Hayashi and Awata 2005) and actual populations of more complicated structure (Heath 1997; Uimari and Hoeschele 1997; Bink and Van Arendonk 1999; Uimari and Sillanpää 2001; Yi and Xu 2001), there have been no Bayesian methods incorporating the procedure of inference for the expression mode of QTL. The power to detect both Mendelian QTL and imprinted QTL and the accuracy in the inference for the expression mode of QTL were evaluated using simulation experiments for the developed Bayesian method.

## PREMISE FOR ANALYSES OF IMPRINTED QTL

#### Outbred F_{2} family:

In this study, we consider detection of QTL in a three-generation family derived from a cross between two outbred lines, denoted by P_{1} and P_{2}, in the F_{0} generation. The family consists of the two grandparental lines, F_{1} parents, and F_{2} individuals. We assume that P_{1} and P_{2} may segregate at the marker loci, but are fixed for alternative alleles at QTL that are indicated by *Q*_{1} and *Q*_{2} for P_{1} and P_{2}, respectively. Moreover, we assumed that the marker information was sufficient to trace the parental origins of QTL alleles back from each F_{2} individual to its parents in the F_{1} generation and consequently two heterozygote *Q*_{1}*Q*_{2} and *Q*_{2}*Q*_{1} are discriminated in F_{2} individuals, where the first allele and the second allele indicate the paternally inherited allele and the maternally inherited allele, respectively.

#### Expression modes of QTL:

Expression modes of QTL are here classified into Mendelian expression (*E*_{1}), partial imprinting (*E*_{2}), where both the paternal allele and the maternal allele are expressed, but at different levels depending on their parental origin, and complete imprinting, which is furthermore divided into exclusive paternal expression (*E*_{3}) and exclusive maternal expression (*E*_{4}). Under these four modes, *E*_{1}, *E*_{2}, *E*_{3}, and *E*_{4}, the genetic contribution of QTL, *g*, was differently modeled as follows. In the expression mode of *E*_{1}, we write *g* = 2*a*, *d*, and −2*a* for QTL genotypes *Q*_{1}*Q*_{1}, *Q*_{1}*Q*_{2} (*Q*_{2}*Q*_{1}), and *Q*_{2}*Q*_{2}, respectively, where 2*a* and *d* are regarded as the additive effect and the dominance effect of QTL and there is no difference in genetic contributions between two heterozygotes, *Q*_{1}*Q*_{2} and *Q*_{2}*Q*_{1}. In the expression mode of *E*_{2}, denoting the paternally inherited QTL effect by *a*_{pat} and the maternally inherited QTL effect by *a*_{mat}, we can express the genetic contribution of QTL as *g* = *a*_{pat} + *a*_{mat}, *a*_{pat} − *a*_{mat} + *d*, −*a*_{pat} + *a*_{mat} + *d*, and −*a*_{pat} − *a*_{mat}, corresponding to QTL genotypes *Q*_{1}*Q*_{1}, *Q*_{1}*Q*_{2}, *Q*_{2}*Q*_{1}, and *Q*_{2}*Q*_{2}, respectively (De Koning *et al*. 2000, 2002). In the case of complete imprinting, we obtain *g* = *a*_{pat} for *Q*_{1}*Q*_{1} and *Q*_{1}*Q*_{2} and *g* = −*a*_{pat} for *Q*_{2}*Q*_{1} and *Q*_{2}*Q*_{2} under the mode *E*_{3} by taking *a*_{mat} = *d* = 0 in *E*_{2} and *g* = *a*_{mat} for *Q*_{1}*Q*_{1} and *Q*_{2}*Q*_{1} and *g* = −*a*_{mat} for *Q*_{1}*Q*_{2} and *Q*_{2}*Q*_{2} under the mode *E*_{4} (De Koning *et al*. 2000, 2002). Different notations were used in Cui *et al*. (2006) for the genetic contribution of imprinted QTL.

## ESTIMATION PROCEDURE IN BAYESIAN QTL MAPPING

#### Statistical model for Mendelian and imprinted QTL:

We assume that observations of the phenotype of a trait, **y**, are available for F_{2} individuals of size *n* as well as marker information, **M**, including genotypic data at markers for F_{0}, F_{1}, and F_{2} individuals and the linkage map of markers. We can apply a linear model for **y**,(1)where **X** is a known design matrix for a vector of nongenetic effect **b** (including the overall mean), *N* is the number of QTL affecting a trait, **g**_{q} (*q* = 1, 2, … , *N*) is an *n* × 1 vector for the genetic contribution of the *q*th QTL in F_{2} individuals, and **e** is the vector of residual (environmental) effect following an *n*-variate normal distribution with mean vector **0** and covariance matrix σ_{e}^{2}**I**_{n} (**I**_{n} is the *n* × *n* identity matrix). The *i*th element of **g**_{q}, denoted as *g _{qi}*, indicates the genetic contribution of the

*q*th QTL for the

*i*th F

_{2}individual. As described in the previous section,

*g*is expressed in different forms corresponding to the QTL genotype of the individual and the expression mode of the QTL. For example, when the QTL genotype of the individual is

_{qi}*Q*

_{1}

*Q*

_{2}, we write the genetic effects at the QTL as

*g*=

_{qi}*d*, +

_{q}*d*, , or , corresponding to the expression mode

_{q}*E*

_{1},

*E*

_{2},

*E*

_{3}, or

*E*

_{4}, respectively, with subscript

*q*indicating the

*q*th QTL.

#### Prior and posterior distributions for QTL parameters:

We denote the expression mode of *N* QTL by **S** = (*S*_{1}, *S*_{2},…, *S _{N}*), where

*S*is the expression mode of the

_{q}*q*th QTL taking

*E*

_{1},

*E*

_{2},

*E*

_{3}, or

*E*

_{4}, and genetic effects of the

*q*th QTL by

**a**

_{q}, the components of which are differently specified as

**a**

_{q}= (

*a*,

_{q}*d*), (,

_{q}*d*), (), and () depending on

_{q}*S*=

_{q}*E*

_{1},

*E*

_{2},

*E*

_{3}, and

*E*

_{4}, respectively. Genetic effects for all QTL are collectively written by

**a**= (

**a**

_{1},

**a**

_{2},…,

**a**

_{q}). Let λ

*denote the location of the*

_{q}*q*th QTL and

**λ**= (λ

_{1}, λ

_{2},…, λ

*) denote the locations of all QTL. The genotypes at all QTL are written as*

_{N}**G**

_{i}=

**(**

*G*

_{1i},

*G*

_{2i},…,

*G*) for the

_{Ni}*i*th F

_{2}individual and the QTL genotypes of all F

_{2}individuals are given as

**G**=

**(G**

_{1},

**G**

_{2},…,

**G**

_{n}), which are sampled on the basis of the marker information

**M**.

The model parameters and QTL genotypes to be inferred via the MCMC algorithm are written as **θ** = (*N*, **a**, **λ**, **S**, **G**, **b**, σ_{e}^{2}). The joint prior density function for **θ** can be calculated as the product of the prior density for each of components:(2)

The prior distribution of the number of QTL, *p*(*N*), is assumed to be a truncated Poisson distribution with mean μ and a predetermined maximum number (Jannink and Fernando 2004; Sillanpää *et al*. 2004). The probability distribution of QTL genotypes of F_{2} individuals, *p*(**G** | **M**, **λ**), is given as a product of the probabilities for genotypes at each QTL in each F_{2} individual, *p*(*G _{qi}* |

**M**,

**λ**) (

*q*= 1, 2,…,

*N*;

*I*= 1, 2,…,

*n*), which is obtained following the method proposed by Haley

*et al*. (1994). In brief, the probabilities of

*G*being

_{qi}*Q*

_{1}

*Q*

_{1},

*Q*

_{1}

*Q*

_{2},

*Q*

_{2}

*Q*

_{1}, and

*Q*

_{2}

*Q*

_{2}, which are denoted as

*p*

_{11qi},

*p*

_{12qi},

*p*

_{21qi}, and

*p*

_{22qi}, respectively, can be calculated on the basis of marker genotypes of the individual and using the recombination frequencies between QTL and markers. Prior distributions of other parameters might be given as uniform distributions or some other distributions according to the prior information available.

Given the number of QTL, *N*, QTL expression modes, **S**, and QTL genotypes, **G**, the distribution of **y**, *p*(**y** | *N*, **S**, **G**, **a**, **b**, σ_{e}^{2}), is written as a normal distribution from (1) as follows:(3)

Bayesian estimation of **θ** is carried out on the basis of the posterior distribution *p*(**θ** | **y**, **M**), which is expressed as(4)MCMC sampling is used for Bayesian inference about each of the components of **θ**. The details of MCMC sampling adopted in this study are described in the appendix.

## SIMULATION EXPERIMENTS

We evaluated the efficiency in detecting QTL and the accuracy of the inference about the expression mode of QTL using the proposed Bayesian method by analyzing simulated data sets in F_{2} populations derived from a cross between two founder outbred lines, P_{1} and P_{2}. Two scenarios, referred to as case I and case II hereafter, were considered in simulation experiments. In case I, we evaluated the efficiency of the Bayesian method in detecting a single QTL under different expression modes. In case II, we investigated the capability of the Bayesian method to resolve multiple QTL with different expression modes. In both case I and case II, the performances of the Bayesian method were assessed in comparison with the interval-mapping procedure for detecting imprinted QTL.

#### Evaluation of efficiencies in detecting a single QTL:

In case I, we evaluated the performance of the developed Bayesian method in detecting a single QTL, with Mendelian expression mode (*E*_{1}) and completely imprinting modes (*E*_{3} and *E*_{4}), on a chromosome when P_{1} and P_{2} were either fixed for alternative QTL alleles or segregating at frequencies of 0.80 and 0.20 for the positive allele at biallelic QTL in P_{1} and P_{2}, respectively. The structures of a simulated F_{2} family and a simulated genome were almost the same as those of De Koning *et al.* (2002), which are summarized as follows. We generated F_{1} individuals by random mating of 20 males in P_{1} and 80 females in P_{2} (4 females per male), where 20 F_{1} males and 80 F_{1} females were obtained. They were randomly mated to produce 400 F_{2} offspring (5 offspring per female) with equal sex ratio. A biallelic QTL with expression mode *E*_{1}, *E*_{3}, or *E*_{4} was simulated at 46 cM on a chromosome of 100 cM length, on which 11 markers were located every 10 cM. We simulated eight alleles for every marker, with four line-specific alleles segregating at equal frequencies in P_{1} and P_{2}. The phenotype of each F_{2} individual was determined by the sum of an effect of QTL located on a chromosome, 10 unlinked QTL effects, each with an additive effect of 0.25 and segregating at a frequency of 0.5 in both P_{1} and P_{2}, and an environmental effect sampled from a normal distribution with mean of 0 and a variance of 0.47. QTL effects were given as 0.25 and 0.5. The proportion of total variance explained by the simulated QTL, denoted by , ranged from 0.04 (2*a* = 0.25, *d* = 0.0) to 0.25 (*a*_{pat} = 0.5 or *a*_{mat} = 0.5).

We simulated 200 replicates of data, each of which included marker genotypes for all individuals and sex records and phenotypes for F_{2} individuals, for each case of variable QTL effects under different expression modes when founder lines were fixed with alternative alleles or segregating at QTL. We first generated 200 replicates of data including the information for genotypes of markers and a simulated QTL, sex records, genetic effects at 10 unlinked loci, and environmental effects for individuals prior to the generation of phenotypic values. Then we obtained the data sets of phenotypic values for F_{2} individuals by assigning different expression modes and corresponding genetic effects to the simulated QTL on the basis of the information generated beforehand. Therefore, the same 200 replicates of data sets including marker and QTL genotypes, sex records, genetic effects at 10 unlinked loci, and environmental effects were commonly used for generating phenotypic values for variable genetic effects of QTL under different expression modes. The data of marker genotypes, sex records, and phenotypes were only subject to the analysis, where the model including sex effect was applied although no sex effect was simulated.

Each of 200 replicates was analyzed by the Bayesian method to evaluate the power to detect a simulated QTL and the accuracy of inference about the QTL expression mode. For the prior distribution of the QTL number, *p*(*N*), a Poisson distribution with mean 2 was assumed. We assumed that prior probabilities of expression modes for a QTL were *p*(*S* = *E*_{1}) = 0.6, *p*(*S* = *E*_{2}) = 0.2, *p*(*S* = *E*_{3}) = 0.1, and *p*(*S* = *E*_{4}) = 0.1. The priors of the other parameters, *p*(**a** | *N*, **S**), *p*(**λ** | *N*), *p*(**b**), and *p*(σ_{e}^{2}), were assumed to be uniform distributions over ranges of possible values of parameters, where fixed effects **b** included a general mean and sex effect. For the proposal probabilities of QTL expression modes, which are transition probabilities between QTL expression modes used in the update of **S**, we assigned the values of δ = 0.2, κ = 0.6, ω = 0.1, η = 0.1, ζ = 0.1 (for δ, κ, ω, η, ζ, see Table 1).

In case I, we performed 50,000 MCMC cycles with the last 40,000 cycles used for sampling the values of parameters every 10 cycles in the Bayesian analysis. Therefore total number of samples kept was 4000. We made some posterior inference on the basis of the posterior QTL intensity (QI) (Sillanpää and Arjas 1998), which was defined as the relative frequency of the cycles in which a QTL was detected in each bin with length of 1 cM along a chromosome. For the judgment of whether or not QTL detection is successful in each analysis, the total QI summed over all bins of a chromosome was used as the criterion. This summed QI (SQI) can be regarded as the strength of evidence for linkage of QTL to the chromosome. The threshold values for SQI were determined with the analyses of another 200 replicates of data sets generated under the same condition as described above except that there were no QTL segregating between P_{1} and P_{2} on the chromosome. We evaluated the variation of SQI calculated on a whole chromosomal region (100 bins) through analyses of the 200 null data sets, which included phenotypes determined by effects of 10 unlinked loci and an environmental effect in the absence of QTL on the chromosome, but provided the same data for marker genotypes and sex records of all individuals as the original data sets generated under the presence of a QTL on the chromosome. As a result, we obtained threshold values of SQI of 0.66, corresponding to a type I error of 0.05. This means that the times of SQI exceeding a value of 0.66 by chance are expected to be ≤10 in the analyses of 200 null data sets.

Accordingly, we regarded the detection of QTL as successful when SQI exceeded the specified threshold value. Although there was the possibility that more than one QTL were simultaneously fitted in the model with the Bayesian method, we treated such multiple QTL fitted as an identical QTL. When a QTL was detected, the location of each QTL was estimated as a posterior mean of the positions of fitted QTL. We considered the posterior probabilities for expression modes of QTL to infer the QTL expression modes. The expression mode having the highest posterior probability was regarded as the most probable expression mode for a detected QTL in the analysis of each data set. The posterior distributions for genetic effects of detected QTL were obtained, corresponding to each of the expression modes. Therefore, the QTL effects were estimated under different expression modes.

To compare the efficiencies of detecting QTL and the accuracies of inference about the expression mode of QTL between the Bayesian method and conventional methods, we also analyzed the same data sets with the interval mapping method for imprinted QTL (De Koning *et al*. 2002). In the analyses with interval mapping, the detection of QTL was attempted on the basis of an *F*-test of a model assuming no QTL (H_{0}: null model) *vs.* a model assuming the presence of QTL with expression mode *E*_{2} (H_{1}: model *E*_{2}) that was regarded as a full model. This test was denoted by T1. Moreover, expression modes of the detected QTL were inferred by carrying out an additional two hypothesis tests, T2 and T3, where T2 is an *F*-test of the model *E*_{1} (H_{0}) *vs.* the model *E*_{2} (H_{1}) and T3 is another *F*-test of either the model *E*_{3} or the model *E*_{4} (H_{0}) *vs.* the model *E*_{2} (H_{1}). Accordingly, the interval-mapping (IM) procedure for identifying imprinted QTL consisted of three tests, T1, T2, and T3. The threshold of *F*-ratios in T1 with a chromosomewise significance level of *P* = 0.05 was obtained as 4.12 by the analyses of the 200 null data sets generated assuming no QTL. Two statistical tests to infer QTL expression modes, T2 and T3, were carried out on the position of detected QTL with the highest *F*-ratios in T1; thus, nominal threshold values with a pointwise significance level of *P* = 0.05 were adopted in T2 and T3. The threshold values of T2 and T3 were obtained as 3.87 and 3.02, respectively, which were the 0.95th quantiles of the corresponding *F* distributions. The interval-mapping procedure for discriminating QTL expression modes described here was similar to that adopted by Tuiskla-Haavisto *et al*. (2004), where the *F*-test of either the model *E*_{3} or the model *E*_{4} *vs.* the null model was first performed to detect QTL indicating the imprinting expression and then T2 and T3 were considered at the position of the QTL to confirm the expression mode.

The detection of simulated QTL was regarded as successful in IM when T1 was significant at any position on a chromosome. The expression mode of QTL detected in T1 was inferred by additional tests T2 and T3 as follows: QTL expression was inferred as *E*_{1} when T2 was nonsignificant, as *E*_{2} when both T2 and T3 were significant, and as *E*_{3} or *E*_{4} when T2 was significant but T3 was nonsignificant. The estimates for genetic effects of QTL were obtained for each of the QTL expression modes at the position of detected QTL with the highest *F*-ratio in T1, which were least-squares estimates of models corresponding to QTL expression modes.

#### Evaluation of efficiencies in resolving multiple QTL:

In case II of the simulation experiments, our objective was to evaluate the efficiencies of the Bayesian method in resolving multiple QTL of different expression modes. We considered a genome consisting of two chromosomes, denoted by LG1 and LG2, each of length 100 cM, on which 11 markers were located every 10 cM for each. We located three biallelic QTL, denoted by QTL1, QTL2, and QTL3, on LG1 and LG2, where QTL1 and QTL2 were linked and located at 25 and 75 cM, respectively, on LG1 and QTL3 was at 65 cM on LG2. We assumed that QTL1, QTL2, and QTL3 were exclusively maternal (*E*_{4}) with a genetic effect of , Mendelian (*E*_{1}) with 2*a*_{2} = 0.5 and *d*_{2} = 0.0, and exclusively paternal (*E*_{3}) with , respectively. We assumed that P_{1} and P_{2} were fixed with alternative alleles at all three QTL. For the structure of the F_{2} family, the generation of marker genotypes, and the effects of unlinked QTL and environmental effect, the same settings as in case I were used. The proportions of total variance explained by QTL1, QTL2, and QTL3 were calculated as , 0.131, and 0.042, respectively, taking the covariance between the effects of QTL1 and QTL2 into consideration.

We generated 100 replicates of data sets in case II. Each data set was analyzed by the Bayesian method to evaluate the power to detect each QTL and the accuracy of inference about QTL expression modes. We performed 100,000 MCMC cycles with the last 80,000 cycles used for sampling the values of parameters every 20 cycles (4000 cycles available for sampling) for the Bayesian estimation. We used SQI obtained on the QTL region as the criterion for successful QTL detection as in case I. The threshold values for the SQIs were determined with the analyses of another 100 data sets generated under the condition that there are no QTL on LG1 and LG2 segregating between P_{1} and P_{2}. Since two chromosomes were considered in case II, the distribution of the maximum value of two SQIs, each obtained in each chromosome, was investigated with 100 null data sets to determine the threshold value corresponding to the prespecified genomewide significance level. We obtained threshold values of SQIs of 0.69, corresponding to genomewide type I error of 0.05. The successful detection of QTL3 was determined on the basis of SQI on LG2, which is QIs summed over all bins of 1 cM length on LG2. The detection of QTL1 and QTL2 located on LG1 was regarded as successful when the SQI calculated in the two respective QTL regions, *i.e.*, [0 cM, 50 cM] and (50 cM, 100 cM] on LG1, exceeded the threshold value of 0.69. Moreover, the capability of simultaneously detecting both QTL1 and QTL2 was evaluated by the fraction of the iterations at which the two QTL were fitted at the same time, in all MCMC iterations. The locations, expression modes, and genetic effects of QTL were estimated as the means of the posterior distributions of the parameters, as described in case I. For discriminating QTL1 and QTL2, the posterior distributions were considered on the two regions, [0 cM, 50 cM] and (50 cM, 100 cM] of LG1, respectively.

We analyzed the same data sets with the IM procedure, consisting of three statistical tests, T1, T2, and T3, in the same way as adopted in case I. The threshold value with genomewide significance level of *P* = 0.05 in T1 was obtained as 4.75 by the analyses of the 100 null data sets generated in the absence of QTL. For T2 and T3, the threshold values were determined as in case I. The detection of QTL3 was regarded as successful when T1 was significant with a significance level of *P* = 0.05 at any point on LG2. For QTL1 and QTL2, we decided that the two QTL were simultaneously detected when the *F*-ratio showed two peaks exceeding the threshold value in the respective regions, [0 cM, 50cM] and (50cM, 100cM] of LG1, and decreased below the threshold value between the two peaks in T1. Only one of QTL1 and QTL2 was judged to be detected, when the *F*-ratio was significant in either region, [0 cM, 50 cM] or (50 cM, 100 cM] of LG1, or when the *F*-ratio was significant in both of the regions but the criterion for simultaneous detection of QTL1 and QTL2, as described above, was not fulfilled. In the latter case, either one of QTL1 or QTL2 was determined to be detected according to the position of the maximum *F*-ratio observed. The estimated location of each detected QTL was obtained as the position showing the peak of the *F*-ratio. The expression mode of QTL successfully detected in T1 was inferred by T2 and T3 performed at the estimated QTL position. The estimates of QTL effects were obtained under each expression mode as in case I.

## RESULTS OF SIMULATIONS

#### Results of a single-QTL detection:

The power of detecting a single QTL and the estimates of the location, effects, and expression mode of a QTL obtained by analyses of 200 data sets in case I with the Bayesian method and the IM method are shown in Table 2 for some simulated QTL. As the same simulation settings as those in De Koning *et al.* (2002) were used in case I, the results obtained by the analyses with IM were comparable to the results reported by them. The performance of the Bayesian method in case I, which was similar to that of IM as shown in Table 2, is summarized as follows.

When founder lines were fixed for different QTL alleles, the power of the Bayesian method in detecting a QTL was quite high. Even for small additive QTL with effect of 2*a* = 0.25, detection was successful in 83% of the replicates, where a QTL was correctly inferred as being Mendelian in 94.6% (78.5/83 = 0.946, see Table 2) of the replicates with successful detection. The power was increased to 95.5% for dominant QTL with effects of 2*a* = *d* = 0.25 and attained 100% for additive QTL with effect of 2*a* = 0.5, where a detected QTL was accurately inferred as being a Mendelian QTL in most of the replicates with successful detection (Table 2). When founder lines were fixed with alternative QTL alleles, the Bayesian method also successfully detected an imprinted QTL in most of the replicates, where the powers of detecting QTL were 99.5 and 99% of the replicates under expression modes *E*_{3} (exclusively paternal expression) and *E*_{4} (exclusively maternal expression) with effects of 0.25, respectively. The proportions of accurate inference about QTL expression modes were 89.4 and 92.4% for such QTL under *E*_{3} and *E*_{4}, respectively.

When founder lines were segregating for the positive allele of QTL with frequencies of 0.80 and 0.20 in P_{1} and P_{2}, respectively, the powers of the Bayesian method in detecting QTL were decreased for both Mendelian and imprinted QTL. The decrease in powers for Mendelian QTL with the smaller effects (2*a* = 0.25 or 2*a* = *d* = 0.25) was remarkable, where the segregation at QTL in founder lines reduced the powers to 43.5 and 48% for additive and dominant QTL, respectively (Table 2). Spurious imprinting was inferred more frequently. In the additive QTL with 2*a* = 0.25, the QTL was inferred as being imprinted in 19.6% of the replicates detecting QTL. For the imprinted QTL with smaller effects (*a*_{pat} = 0.25 or *a*_{mat} = 0.25), the power was also much reduced and the inference about QTL expression mode was less accurate. However, the reduction in the power due to the QTL segregation within founder lines was less for the Bayesian method than IM for small QTL with effects of 0.25 (Table 2).

#### Results of multiple-QTL resolution:

The power of detecting each of three QTL and the accuracy in estimation of the location and expression mode of each QTL obtained by analyses of 100 data sets using the Bayesian method and IM are shown in Table 3.

The Bayesian methods showed higher efficiency in resolving two linked QTL, QTL1 and QTL2, than IM, where the estimates of effects and positions of the two QTL were unbiased and the QTL expression modes were accurately inferred with the Bayesian method although IM often provided biased estimates for effects and the incorrect inference about QTL expression modes for the two linked QTL. For QTL1 with smaller effect (), which was located near QTL2 with greater effects (), the Bayesian method showed much higher power than IM, where the powers of the Bayesian method and IM were 80 and 57%, respectively. We considered the fractions of the iterations, which accepted the model simultaneously fitting QTL1 and QTL2, in all MCMC iterations for evaluating the capability of the Bayesian method to resolve QTL1 and QTL2. The fractions of the iterations with simultaneous detection of QTL1 and QTL2 were averaged over 100 replicates as 0.82. For QTL3 with the smallest effects (), the Bayesian method showed low power of 71%, but still greater than IM with power of 63%.

Considering the proportions of the replicates correctly inferring the expression mode for each QTL in the replicates with QTL detection, which were regarded as the conditional efficiency of the inference for QTL expression mode given the QTL was detected, the Bayesian method showed such proportions higher than IM. These proportions were 0.98, 0.96, and 0.86 with the Bayesian method for QTL1, QTL2, and QTL3, respectively, while the corresponding proportions with IM were 0.35, 0.68, and 0.75, respectively.

The posterior probabilities for *N*, the number of QTL fitted in the model with the Bayesian methods, were investigated in the analysis of each data set and averaged over 100 replicates. The averaged posterior probabilities for *N* are shown in Table 4.

## DISCUSSION

#### Criterion for QTL detection with the Bayesian method:

The results of a Bayesian QTL analysis are provided as the form of the posterior distributions of parameters including the number of QTL affecting a trait, genetic effects and locations of QTL, and other QTL characteristics such as the QTL expression mode that is of main concern in this study. For the determination of the presence of QTL on the chromosomal region, the posterior probability of QTL linkage to that region, which is an estimate of the posterior distribution for QTL locations, is of greatest interest. The fraction of MCMC iterations in which at least one QTL is located at each bin with some length is defined as the posterior QTL intensity of that bin, denoted by QI, and used as an estimate of the posterior QTL locations (Sillanpää and Arjas 1998, 1999). However, QI is much affected by the length of the bin and will be lower for a bin with smaller length. Thus, QI summed over all bins on a chromosomal region, which is referred to as SQI, was used as a criterion for the determination of the presence of QTL linked to the region in the Bayesian analysis of this study. The threshold of SQI was obtained by the empirical distribution of SQI calculated on each chromosome in the absence of QTL, where null data with no simulated QTL were generated in the simulation experiments. For the threshold of SQI corresponding to the genomewide significance level, the empirical null distribution of maximum SQIs over a whole genome consisting of multiple chromosomes was considered as in the case of the determination of thresholds for a conventional LOD score and *F*-ratio calculated in the interval mapping. When SQI exceeded a threshold with a genomewide significance level of *P* = 0.05 on some regions, we determined the presence of QTL on such regions.

We compared the performance between the developed Bayesian method and the IM procedure in the simulation experiments. Although the strict comparison between the Bayesian method and conventional methods is difficult, the comparison based on simulated data is reasonable by controlling the type-I error rate for QTL detection with the Bayesian methods and IM based on the thresholds for SQI and the *F*-ratio. Therefore, the powers of QTL detection and the proportions of the replicates with accurate inference of QTL expression mode obtained in the simulation study as shown in Tables 2 and 3 could be utilized for suitable evaluation of the performance of the Bayesian method in comparison with IM. The problem of the interpretation for QTL detection with the Bayesian method was reviewed by Wijsman and Yu (2004).

#### Detection of QTL in the simulation study:

In case I of the simulation experiments, where a single biallelic QTL was located on a chromosome, the power of QTL detection was expected to be similar in the Bayesian method and IM since the advantage of simultaneously fitting multiple QTL in the Bayesian method was not taken for the case of a single QTL. As shown in Table 2, comparable powers of detecting QTL were obtained in case I for the Bayesian method and IM when founder lines were fixed with alternative QTL alleles. The similar powers between the Bayesian method and IM indicated that the comparison of the two methods under the control of type-I error rate was suitable.

In case II where we assumed that three QTL were located on two chromosomes, the Bayesian method showed much higher efficiencies in detecting each QTL and in estimation of QTL effects and expression modes than IM. For the detection of QTL1 and QTL2 with the Bayesian method, SQIs obtained on the partial region of 50 cM length were compared with a threshold corresponding to genomewide type I error of 0.05, which was obtained by considering the empirical variation of maximum values of SQIs calculated on whole-chromosomal regions in the absence of QTL. Accordingly the criterion for the detection of QTL1 and QTL2 might be too strict for the Bayesian method. The averaged posterior probabilities of *N* shown in Table 4 could be used for evaluation of the efficiency in successfully detecting all simulated QTL with the Bayesian method. The probability of *N* ≥ 3, which was considered to indicate such power of detecting all QTL, was 0.681.

#### Prior of QTL expression mode:

In this study, the prior probabilities of QTL expression modes were given as 0.6, 0.2, 0.1, and 0.1 for *E*_{1}, *E*_{2}, *E*_{3}, and *E*_{4}, since it is considered that the number of genes with Mendelian expression is generally larger than the number of genes imprinted in a whole genome of organisms. For the probabilities of transition between models of expression modes, we assigned the fixed values of δ = 0.2, κ = 0.6, ω = 0.1, η = 0.1, and ζ = 0.1 in the analysis of this study, following the same prior information for the possible expression mode of genes. In these settings of the prior and transition probabilities for QTL expression modes, the Mendelian expression model was more frequently considered for model fitting in MCMC iterations than other expression models; accordingly, the power and the accuracy of inferring expression modes for imprinted QTL might be reduced. Therefore, we performed additional simulation experiments, where data sets in case II were reanalyzed by the Bayesian method with the prior and transition probabilities for expression modes modified as uniform by assigning the prior probability of 0.25 to each expression mode and giving the values as δ = 0.5, κ = 0.25, ω = 0.25, η = 0.5, and ζ = 0.5 for the transition probabilities among expression modes. The result of this additional analysis is shown in Table 5. In a comparison of Table 5 with Table 3, the power of detecting imprinted QTL (QTL1 and QTL3) was enhanced, while the power for Mendelian QTL (QTL2) was unchanged but it was more frequently misidentified as partially imprinted QTL (*E*_{2}), by modifying the prior and transition probabilities of expression modes as uniform. Since the model of *E*_{2} is a full model including *E*_{1}, *E*_{3}, and *E*_{4} as submodels, the frequency of misidentification for QTL with *E*_{1}, *E*_{3}, or *E*_{4} as QTL with *E*_{2} depends on the prior and transition probabilities for expression modes.

From the result of this additional analysis, it was shown that we can enhance the power of detecting the QTL and the accuracy of inference for expression mode of the QTL by arranging the values of these parameters so as to increase the chance that the model of correct expression mode for a QTL is tested for model fitting. However, true expression modes of QTL are usually unknown prior to the analysis and multiple QTL with different expression modes might be involved in the actual analysis. More suitable settings for the values of these parameters would require further investigation. In the actual analysis, it would be either suitable to adopt the prior and transition probabilities for expression modes with larger probability for *E*_{1} than for other expression modes such that the Mendelian expression model is tested more frequently than imprinting expression models on the basis of the general information of a smaller number of genes with imprinting expression or suitable to apply the uniform prior and transition probabilities for expression modes.

#### Conclusion:

It has been known in mammals that the imprinted genes are not distributed uniformly through the genome, but cluster together (Reik and Walter 2001). Therefore, there is the possibility that multiple linked QTL with different expression modes affect a trait. The development of useful statistical tools for resolving such linked QTL would be desired. The Bayesian statistical framework is more suitable than interval-mapping methods for addressing the problem of separately detecting each of multiple QTL with different expression modes. Bayesian inferences can provide a flexible and useful platform for investigation of genetic structure underlying a phenotype of a trait, which could be more clearly elucidated with the methods for simultaneously detecting multiple QTL affecting the trait. The Bayesian method incorporating variable models for QTL expression, as proposed in this study, could contribute to the exploration of cryptic imprinted QTL that have remained undetected so far with conventional methods.

## APPENDIX: MCMC SAMPLING TO ESTIMATE QTL PARAMETERS

The MCMC cycle to estimate parameters **θ** in the model consists of the following steps: (a) updating the effects at each QTL **a** = (**a**_{1}, **a**_{2},…, **a**_{N}), (b) updating the fixed effects **b** and residual variance σ_{e}^{2}, (c) updating QTL locations **λ** = (λ_{1}, λ_{2},…, λ* _{N}*), (d) updating QTL genotypes

**G**for each F

_{2}individual, (e) updating the QTL expression mode

**S**= (

*S*

_{1},

*S*

_{2},…,

*S*), and (f) adding one new QTL to the model or removing one existing QTL from the model.

_{N}The QTL effects **a** are updated locus by locus and component by component via Gibbs sampling. The full conditional distribution of **a**_{q} (*q* = 1, 2,…, *N*) given other parameters and **y** is easily constructed by (3) assuming suitable prior distributions. For instance, in the case of *S _{q}* =

*E*

_{3}(exclusive paternal expression) and assuming a uniform prior distribution,

**a**

_{q}= () is normally distributed with mean

**z**

_{q}′(

**y**−

**Xb −**)/

*n*and variance σ

_{e}^{2}/

*n*, where

**z**

_{q}′ = (

*z*

_{q}_{1},

*z*

_{q}_{2},…,

*z*) are indicator variables for genotypes at the

_{qn}*q*th QTL of F

_{2}individuals taking values

*z*= 1 for the genotypes

_{qi}*Q*

_{1}

*Q*

_{1}and

*Q*

_{1}

*Q*

_{2}and −1 for

*Q*

_{2}

*Q*

_{1}and

*Q*

_{2}

*Q*

_{2}.

An update of the fixed effects **b** and residual variance σ_{e}^{2} is also performed with Gibbs sampling. The full conditional distribution of **b** is normal and that of σ_{e}^{2} is inverted chi square, when conjugate prior distributions are adopted for the parameters.

Moreover, Gibbs sampling is applicable to an update of QTL genotypes **G** of F_{2} individuals. The full conditional probability for the genotype at the *q*th QTL for the *i*th F_{2} individual, *G _{qi}*, given other parameters and variables is easily obtained. For example, in the case of

*S*=

_{q}*E*

_{4}(exclusive maternal expression), the mean of

*y*is written as μ

_{i}_{1}= for

*G*=

_{qi}*Q*

_{1}

*Q*

_{1}and

*Q*

_{2}

*Q*

_{1}or μ

_{2}= for

*G*=

_{qi}*Q*

_{1}

*Q*

_{2}and

*Q*

_{2}

*Q*

_{2}, where

*x*is the

_{ij}*i*,

*j*th component of design matrix

**X**and

*b*is the

_{j}*j*th component of fixed effects

**b**. Therefore, in the case of

*S*=

_{q}*E*

_{4}, the full conditional probability that

*G*=

_{qi}*Q*(

_{k}Q_{l}*k*,

*l*= 1, 2) can be written aswhere ϕ(

*y*| μ, σ

^{2}) denotes the normal density function with mean μ and variance σ

^{2}. The QTL genotype

*G*is updated by sampling a new genotype from this probability distribution.

_{qi}The update of other parameters (steps c, e, and f) is carried out by a Metropolis–Hastings algorithm (Metropolis *et al*. 1953; Hastings 1970) as described below.

#### Updating QTL locations and genotypes:

The locations of QTL, **λ**, are updated for one QTL at a time. For the *q*th QTL, new proposal values of λ* _{q}*, denoted by λ

**, are sampled from a symmetric uniform distribution centered at the previous values of λ*

_{q}*. Accordingly, a new genotype at the*

_{q}*q*th QTL,

**G**

_{q}*, is proposed for each F

_{2}individual corresponding to the proposed QTL position. We accept λ

** with probabilitywhere*

_{q}**λ*** and

**G*** are proposal values of

**λ**and

**G**with the components corresponding to the

*q*th QTL replaced by new values.

#### Updating QTL expression mode:

The QTL expression modes **S** are updated QTL by QTL via a RJ–MCMC algorithm (Green 1995). In the update of **S**, for the present mode of the *q*th QTL, *S _{q}*, the new mode

*S** is proposed with probability distribution

_{q}*q*(

*S** |

_{q}*S*). We chose

_{q}*q*(

*S** |

_{q}*S*) as shown in Table 1, considering inclusion relation between models corresponding to

_{q}*E*

_{1},

*E*

_{2},

*E*

_{3}, and

*E*

_{4}, where δ, κ, ω, η, and ζ are fixed values between 0 and 1, which are prespecified according to prior information available. The model with

*E*

_{2}is regarded as a saturated model and the models with

*E*

_{1},

*E*

_{3}, and

*E*

_{4}are as submodels of

*E*

_{2}. There is no inclusion relation between models with

*E*

_{1},

*E*

_{3}, and

*E*

_{4}. Accordingly, we assumed that the probabilities of direct transition between models with

*E*

_{1},

*E*

_{3}, and

*E*

_{4}are zero (Table 1);

*i.e*.,

*q*(

*S** =

_{q}*E*

_{3}|

*S*=

_{q}*E*

_{1}) =

*q*(

*S** =

_{q}*E*

_{4}|

*S*=

_{q}*E*

_{1}) =

*q*(

*S** =

_{q}*E*

_{1}|

*S*=

_{q}*E*

_{3}) =

*q*(

*S** =

_{q}*E*

_{4}|

*S*=

_{q}*E*

_{3}) =

*q*(

*S** =

_{q}*E*

_{1}|

*S*=

_{q}*E*

_{4}) =

*q*(

*S** =

_{q}*E*

_{3}|

*S*=

_{q}*E*

_{4}) = 0. The dimensionality of

**a**

_{q}is changed along with the update of

*S*. Denoting the proposed new values of

_{q}**a**

_{q}by

**a**

_{q}*, the components of

**a**

_{q}* are obtained by a transformation of the components of

**a**

_{q}as follows. For

*S*=

_{q}*E*

_{1}and

*S** =

_{q}*E*

_{2},

**a**

_{q}and

**a**

_{q}* are written as

**a**

_{q}= (

*a*,

_{q}*d*) and

_{q}**a**

_{q}* = (, where we obtain

**a**

_{q}* from

**a**

_{q}using a transformationwith

*u*being a random variable used for dimension matching and sampled from a suitable probability distribution such as a uniform distribution. In the transformation, we can easily show that the Jacobian is |∂

**a**

_{q}*/∂(

**a**

_{q},

*u*)| = 1. Other forms of transformation with Jacobian = 1 might be possible. For other combinations of

*S*and

_{q}*S**, a suitable transformation between

_{q}**a**

_{q}and

**a**

_{q}* would be found without difficulty. Using a transformation with the Jacobian = 1, the probability of accepting the proposed value

*S** is expressed aswhere

_{q}**S*** and

**a*** are

**S**and

**a**with the

*q*th components replaced by

*S** and

_{q}**a**

_{q}*, respectively.

#### Updating QTL number:

In the update of QTL number, the RJ–MCMC algorithm is also used. The number of QTL, *N*, is updated by adding one new QTL to a model with probability *p*_{a} or deleting one existing QTL from a model with probability *p*_{d} in the way described in Jannink and Fernando (2004) and Sillanpää *et al*. (2004). For a proposed QTL number *N**, there are three possible values: *N** = *N* + 1, *N** = *N* − 1, and *N** = *N* with probabilities *p*_{a}, *p*_{d}, and 1 − *p*_{a} − *p*_{d}.

When attempting to add one new QTL, first the location of the QTL λ* _{N*}* is sampled from a uniform distribution over a whole-genome region and one of the expression modes,

*E*

_{1},

*E*

_{2},

*E*

_{3}, and

*E*

_{4}, is randomly assigned to the new QTL following the prior probabilities of the expression modes. Next, the QTL effects are determined from uniform distributions and the genotypes of the new QTL for F

_{2}individuals are obtained by sampling values from

*p*(

**G**|

**M**,

**λ**). One new QTL is accepted with probabilitywhere

**S***,

**G***, and

**a*** indicate

**S**,

**G**, and

**a**with the

*N*+ 1 elements added corresponding to the new QTL.

For deleting one existing QTL, a random choice is made among the existing QTL. The chosen QTL is then proposed to delete from the model. If the *q*th QTL is proposed to delete, the probability for the deletion iswhere **S***, **G***, and **a*** correspond to **S**, **G**, and **a** except that the items corresponding to the *q*th QTL are deleted.

## Footnotes

Communicating editor: C. Haley

- Received September 4, 2007.
- Accepted November 2, 2007.

- Copyright © 2008 by the Genetics Society of America