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Genomic models for dominance effects

Model

Given SNP marker data, nonadditive genetic effects can be directly integrated into a mixed-model framework. There are several ways to model nonadditive effects. The textbook deals with a simple model: the GBLUP analogue. We can construct a separate “relationship matrix” for each nonadditive effect and use these matrices in the mixed model.

The dominance effect is the most important nonadditive effect in quantitative genetic studies. A typical mixed model for the dominance effect is as follows (Equation 13.5). \[ \mathbf{y} = \mathbf{Xb} + \mathbf{T}\mathbf{u}_a + \mathbf{V}\mathbf{u}_d + \mathbf{e} \] This is an extension of the additive model (breeding-value model): \(\mathbf{T}\) (often written as \(\mathbf{Z}\)) is a centered marker matrix associated with the additive genetic effect \(\mathbf{u}_a\), and \(\mathbf{V}\) is another centered marker matrix associated with the dominance effect \(\mathbf{u}_d\). The additive and dominance relationship matrices are defined as follows: \[ \mathrm{var}(\mathbf{u}_a) = \mathbf{T}\mathbf{T}'/k_a = \mathbf{G}\sigma_a^2 \quad \text{and} \quad \mathrm{var}(\mathbf{u}_d) = \mathbf{V}\mathbf{V}'/k_d = \mathbf{D}\sigma_d^2 \] where \(k_a\) and \(k_d\) are functions of the allele frequencies across all marker loci for each effect, \(\mathbf{G}\) (\(\mathbf{D}\)) is the additive (dominance) genomic relationship matrix, and \(\sigma_a^2\) and \(\sigma_d^2\) are the additive genetic and dominance variances, respectively. Given the variance-covariance structure, we can construct a system of mixed-model equations and predict the effects as solutions of those equations. This section uses the widely accepted GBLUP dominance model.

BLUPF90+ does not have a dedicated option for creating nonadditive relationship matrices, unlike \(\mathbf{G}\) for additive effects. The user should prepare the relationship matrix and plug it into the equations, as in the previous sections.

Files

The data file (mr133.dat) contains five columns.

1. Animal
2. Sire ID
3. Dam ID
4. Pen
5. Phenotype (WWT)

The pedigree file (mr133.ped) has 15 animals.

BLUPF90+ creates the genomic relationship matrix from the marker (mr133.gen) and cross-reference (mr133.xrf) files, but the inverse of the dominance relationship matrix is read from the file (mr133.Di). The parameter file (mr133.par) is as follows.

DATAFILE
mr133.dat
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
3
OBSERVATION(S)
5
WEIGHT(S)

EFFECTS:
4  2 cross
1 15 cross
1 15 cross
RANDOM_RESIDUAL VALUES
120.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
mr133.ped
(CO)VARIANCES
90.0
RANDOM_GROUP
3
RANDOM_TYPE
user_file
FILE
mr133.Di
(CO)VARIANCES
80.0
OPTION solv_method FSPAK
OPTION SNP_file mr133.gen mr133.xrf
OPTION no_quality_control
OPTION AlphaBeta 1.00 0.00
OPTION GammaDelta 0.01 0.00
OPTION tunedG 0

The setup of the additive genomic relationship matrix is the same as that shown in the GBLUP section.

Solutions

The solutions are identical to those shown in the textbook (pp. 233–234).

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