A framework for detecting and characterizing genetic background-dependent imprinting effects

Abstract

Genomic imprinting, where the effects of alleles depend on their parent-of-origin, can be an important component of the genetic architecture of complex traits. Although there has been a rapidly increasing number of studies of genetic architecture that have examined imprinting effects, none have examined whether imprinting effects depend on genetic background. Such effects are critical for the evolution of genomic imprinting because they allow the imprinting state of a locus to evolve as a function of genetic background. Here we develop a two-locus model of epistasis that includes epistatic interactions involving imprinting effects and apply this model to scan the mouse genome for loci that modulate the imprinting effects of quantitative trait loci (QTL). The inclusion of imprinting leads to nine orthogonal forms of epistasis, five of which do not appear in the usual two-locus decomposition of epistasis. Each form represents a change in the imprinting status of one locus across different classes of genotypes at the other locus. Our genome scan identified two different locus pairs that show complex patterns of epistasis, where the imprinting effect at one locus changes across genetic backgrounds at the other locus. Thus, our model provides a framework for the detection of genetic background-dependent imprinting effects that should provide insights into the background dependence and evolution of genomic imprinting. Our application of the model to a genome scan supports this assertion by identifying pairs of loci that show reciprocal changes in their imprinting status as the background provided by the other locus changes.

Appendices

Appendix 1: model details

The order of the vector of two-locus genotypic values (G _AB) is constructed using the Kronecker product of the two single-locus genotypic value vectors, where G _AB = G _B ⊗ G _A (cf. Appendix 1 in Álvarez-Castro and Carlborg 2007). The Kronecker product gives the order of the genotypic values, not the actual products (i.e., multiplication) of values. Since each single-locus genotypic value has two subscripts corresponding to the maternally and paternally inherited alleles at the locus, the two-locus genotypic values simply contain the combination of subscripts taken from the two single-locus genotypic values, with the A locus subscripts appearing first. Thus, in the resulting vector of two-locus genotypic values, G _ijkl, the first two subscripts refer to the two A locus alleles (i = paternally inherited allele, j = maternally inherited allele) and the last two refer to the B locus alleles (k = maternally inherited allele, l = paternally inherited allele). This can be visualized as

$$ \left[ {\begin{array}{*{20}c} {G_{B11} } \\ {G_{B12} } \\ {G_{B21} } \\ {G_{B22} } \\ \end{array} } \right] \otimes \left[ {\begin{array}{*{20}c} {G_{A11} } \\ {G_{A12} } \\ {G_{A21} } \\ {G_{A22} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {G_{B11} \cdot G_{A11} } \\ {G_{B11} \cdot G_{A12} } \\ {G_{B11} \cdot G_{A21} } \\ {G_{B11} \cdot G_{A22} } \\ {G_{B12} \cdot G_{A11} } \\ {G_{B12} \cdot G_{A12} } \\ {G_{B12} \cdot G_{A21} } \\ {G_{B12} \cdot G_{A22} } \\ {G_{B21} \cdot G_{A11} } \\ {G_{B21} \cdot G_{A12} } \\ {G_{B21} \cdot G_{A21} } \\ {G_{B21} \cdot G_{A22} } \\ {G_{B22} \cdot G_{A11} } \\ {G_{B22} \cdot G_{A12} } \\ {G_{B22} \cdot G_{A21} } \\ {G_{B22} \cdot G_{A22} } \\ \end{array} } \right] \to {\mathbf{G}}_{AB} = \left[ {\begin{array}{*{20}c} {G_{1111} } \\ {G_{1211} } \\ {G_{2111} } \\ {G_{2211} } \\ {G_{1112} } \\ {G_{1212} } \\ {G_{2112} } \\ {G_{2212} } \\ {G_{1121} } \\ {G_{1221} } \\ {G_{2121} } \\ {G_{2221} } \\ {G_{1122} } \\ {G_{1222} } \\ {G_{22122} } \\ {G_{2222} } \\ \end{array} } \right] $$

Like the matrix of two-locus genotypic values, the structure of the vector of two-locus genetic effects (E _AB) is defined by the Kronecker product of the two single-locus vectors (E _A and E _B), E _AB = E _B ⊗ E _A, but with the reference point in each vector first replaced with a 1 (Álvarez-Castro and Carlborg 2007). The products of single-locus effects in the resulting vector are replaced with corresponding epistasis terms (e.g., a _A a _B would be replaced with aa, which corresponds to the interaction between additive effects of the two loci, see below), resulting in a vector that contains a total of 16 terms, 9 of which correspond to epistasis between the two loci:

$$ \left[ {\begin{array}{*{20}c} 1 \\ {a_{B} } \\ {d_{B} } \\ {i_{B} } \\ \end{array} } \right] \otimes \left[ {\begin{array}{*{20}c} 1 \\ {a_{A} } \\ {d_{A} } \\ {i_{A} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 \\ {a_{A} } \\ {d_{A} } \\ {i_{A} } \\ {a_{B} } \\ {a_{B} a_{A} } \\ {a_{B} d_{A} } \\ {a_{B} i_{A} } \\ {d_{B} } \\ {d_{B} a_{A} } \\ {d_{B} d_{A} } \\ {d_{B} i_{A} } \\ {i_{B} } \\ {i_{B} a_{A} } \\ {i_{B} d_{A} } \\ {i_{B} i_{A} } \\ \end{array} } \right] \to {\mathbf{\rm E}}_{AB} = \left[ {\begin{array}{*{20}c} R \\ {a_{A} } \\ {d_{A} } \\ {i_{A} } \\ {a_{B} } \\ {aa} \\ {da} \\ {ia} \\ {d_{B} } \\ {ad} \\ {dd} \\ {id} \\ {i_{B} } \\ {ai} \\ {di} \\ {ii} \\ \end{array} } \right] $$

Appendix 2: extension of the model to arbitrary allele frequencies

In this appendix we extend the model to populations with arbitrary allele frequencies that are in Hardy–Weinberg equilibrium. This approach follows the general two-allele model presented in Zeng et al. (2005), which has been called the G2A model by Álvarez-Castro and Carlborg (2007). It can be extended to any population using the framework in Álvarez-Castro and Carlborg (2007). We do not present a full description of the G2A model since a full description and justification is presented by Zeng et al. (2005). The G2A model differs from the model presented in the text above in that the values of the genotypic index variables (coefficients) depend on allele frequencies at the loci (to maintain orthogonality across allele frequencies). When the two alleles are at equal frequency, the G2A model reduces to the Cockerham model presented in the text.

Because the genotypic index variables depend on allele frequency, we denote the frequency of the A ₁ allele as p ₁ and the frequency of the A ₂ allele as p ₂. Under the G2A model the additive and dominance effect design variables, w _A and v _A, respectively are defined as (extended to consider the two heterozygotes separately)

$$ w_{A} = \left\{ \begin{gathered} 2p_{2} {\text{ for }}A_{ 1} A_{ 1} \hfill \\ (p_{2} - p_{1} ){\text{ for }}A_{ 1} A_{ 2} \hfill \\ (p_{2} - p_{1} ){\text{ for }}A_{ 2} A_{ 1} \hfill \\ - 2p_{1} {\text{ for }}A_{ 2} A_{ 2} \hfill \\ \end{gathered} \right.,\quad {\text{and}} \quad v_{A} = \left\{ {\begin{array}{*{20}c} \hfill { - 2p_{2}^{2} {\text{ for }}A_{ 1} A_{ 1} } \\ \hfill {2p_{1} p_{2} {\text{ for }}A_{ 1} A_{ 2} } \\ \hfill {2p_{1} p_{2} {\text{ for }}A_{ 2} A_{ 1} } \\ \hfill { - 2p_{1}^{2} {\text{ for }}A_{ 2} A_{ 2} } \\ \end{array} } \right. $$

(14)

The inclusion of the imprinting genotypic index variable in the G2A model is straightforward because the two heterozygotes are always at equal frequency at Hardy-Weinberg equilibrium. As a result, the value of the imprinting genotypic index variable u _A does not depend on allele frequencies and therefore is orthogonal from the additive and the dominance genotypic index variables at all frequencies. The value of the genotypic index variable for the G2A model matches that given in Eq. 3. This makes the genetic-effect design matrix for the G2A model:

$$ {\mathbf{S}}_{A} = \left[ {\begin{array}{llll} 1 & {2p_{2} } & { - 2p_{2}^{2} } & 0 \\ 1 & {\left( {p_{2} - p_{1} } \right)} & {2p_{1} p_{2} } & 1 \\ 1 & {\left( {p_{2} - p_{1} } \right)} & {2p_{1} p_{2} } & { - 1} \\ 1 & { - 2p_{1} } & { - 2p_{1}^{2} } & 0 \\ \end{array} } \right] $$

(15)

and the corresponding vector of genetic effects:

$$ {\mathbf{E}}_{A} = \left[ {\begin{array}{r} R \\ {a_{A} } \\ {d_{A} } \\ {i_{A} } \\ \end{array} } \right] = {\mathbf{S}}_{A}^{ - 1} {\mathbf{G}}_{A} = \left[ {\begin{array}{rrrr} {p_{1}^{2} } & {p_{1} p_{2} } & {p_{1} p_{2} } & {p_{2}^{2} } \\ {p_{1} } & {{\tfrac{1}{2}}\left( {p_{2} - p_{1} } \right)} & {{\tfrac{1}{2}}\left( {p_{2} - p_{1} } \right)} & { - p_{2} } \\ { - {\tfrac{1}{2}}} & {{\tfrac{1}{2}}} & {{\tfrac{1}{2}}} & { - {\tfrac{1}{2}}} \\ 0 & { - {\tfrac{1}{2}}} & { - {\tfrac{1}{2}}} & 0 \\ \end{array} } \right]\left[ {\begin{array}{r} {G_{A11} } \\ {G_{A12} } \\ {G_{A21} } \\ {G_{A22} } \\ \end{array} } \right] $$

(16)

The reference point for the model remains the mean of the genotypic values but is now weighted by their frequency. In the G2A model, the measure of the additive effect a _A is frequency-dependent and now represents the average effect of an allele substitution rather than the typical frequency-independent additive effect or additive genotypic value. The dominance and imprinting effects, however, are independent of allele frequencies for the single-locus model (but not for multiple loci Zeng et al. 2005).

The two-locus extension of the G2A model for ordered genotypes follows the extension of the single-locus Cockerham model to the two-locus case. The structure of the two-locus vectors of genotypic values (G _AB) and genetic effects (E _AB) match those in Appendix 1. To extend the model to two loci, one-first needs to define the allele frequencies at the B locus; therefore, we denote the frequency of the B ₁ allele as q ₁ and the frequency of the B ₂ allele as q ₂ = 1 − q ₁. Using these allele frequencies, a genetic-effect design matrix for locus B can be constructed following the structure in Eq. 15:

$$ {\mathbf{S}}_{B} = \left[ {\begin{array}{rrrr} 1 & {2q_{2} } & { - 2q_{2}^{2} } & 0 \\ 1 & {\left( {q_{2} - q_{1} } \right)} & {2q_{1} q_{2} } & 1 \\ 1 & {\left( {q_{2} - q_{1} } \right)} & {2q_{1} q_{2} } & { - 1} \\ 1 & { - 2q_{1} } & { - 2q_{1}^{2} } & 0 \\ \end{array} } \right] $$

(17)

As in the Cockerham model presented in the text, the two single-locus genetic-effect design matrices (Eqs. 15 and 17) can be used to derive the genotypic index values for the two-locus model as S _AB = S _B ⊗ S _A. The resulting matrix, which is given in Supplementary Table 2, is analogous to that presented in Table 1 of Zeng et al. (2005), but with the additional dimensions added for the ordered genotypes. As in the model of unordered genotypes from Zeng et al. (2005), the values of all coefficients involving additive and dominance effects (including all epistatic terms involving interactions between these effects) depend on allele frequencies. However, coefficients for the imprinting effects of both loci (i _A and i _B) and their interaction (ii) do not depend on allele frequencies. The inverse of the two-locus genetic-effect design matrix defines the two-locus genetic effects in terms of two-locus genotypic values. The inverse of the two-locus genetic-effect design matrix, ${\mathbf{S}}_{AB}^{ - 1}$, which gives the definitions of the two-locus genetic effects in terms of the genotypic values, is given in Supplementary Table 3.