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A complete classification of epistatic two-locus models.

Hallgrímsdóttir IB, Yuster DS - BMC Genet. (2008)

Bottom Line: Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for.The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms.We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, UK. ingileif@stats.ox.ac.uk

ABSTRACT

Background: The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results: We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion: The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

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Related in: MedlinePlus

Example. Example of observed epistasis. A visual representation of the 30 trait/QTL pairs. The phenotype average for each genotype is given by the heights of the bars, the corresponding shape is also given, and the trait (A-E) and QTL pair (1–21) listed. Under each panel we list the cluster it falls into (1–4) and the group given by [24] (A, B, H, L, U).
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Figure 6: Example. Example of observed epistasis. A visual representation of the 30 trait/QTL pairs. The phenotype average for each genotype is given by the heights of the bars, the corresponding shape is also given, and the trait (A-E) and QTL pair (1–21) listed. Under each panel we list the cluster it falls into (1–4) and the group given by [24] (A, B, H, L, U).

Mentions: We computed the shapes of the 30 observations and found that 23 of the 387 shapes occurred, or 16 out of 69 up to symmetry. The data are shown in Figure 6. For each observation we show a bar-chart of the phenotype means and the corresponding shape. The point in the upper left corner of the shape corresponds to the genotype BB/BB, and the point in the lower right corner corresponds to LL/LL. Although in most applications one would consider the two alleles at a locus to be interchangeable we do not here, since they come from different chicken lines. To group together observations with similar genetic effects we clustered the shapes based on the pair-wise distances between them, using complete linkage hierarchical clustering. There are four main clusters in the resulting dendrogram (not shown). Under each panel in Figure 6 we list which cluster it falls into, and in parentheses we list which group it belongs to according to [24]. The observations are ordered based on the hierarchical clustering with observations in the same cluster listed together and observations within each cluster listed according to the distance between them, as far as possible. For four observations we switched the order of the first and second locus, compared to the order in [24], in order to minimize the distance to the closest observation. Within a cluster, the distance between the shapes in side-by-side panels is typically one but occasionally two. Many of the observed shapes are adjacent to more than one other shape, so two shapes that are not adjacent in Figure 6 may still be close. Consider the last row in Figure 6. In all five panels the values of the genotypes BB/BL, BL/BB and LL/LL dominate the shape, resulting in a central triangular plane. The value at BB/BB varies considerably but does not affect the shape. The shape that each of the observations fall into is, however, affected by the values of BL/LL and LL/BL. When they are relatively high an additional partition is added in the shape. Recall from the previous section that this shape is observed when there is both additive × dominance and dominance × additive interaction in the data. The shapes in the second-to-last row indicate strong dominance × dominance interaction (compare to the shape given in Figure 5). In the last two observations in the row, dom × dom is the strongest interaction, whereas the first three also show strong add × dom interaction.


A complete classification of epistatic two-locus models.

Hallgrímsdóttir IB, Yuster DS - BMC Genet. (2008)

Example. Example of observed epistasis. A visual representation of the 30 trait/QTL pairs. The phenotype average for each genotype is given by the heights of the bars, the corresponding shape is also given, and the trait (A-E) and QTL pair (1–21) listed. Under each panel we list the cluster it falls into (1–4) and the group given by [24] (A, B, H, L, U).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2289835&req=5

Figure 6: Example. Example of observed epistasis. A visual representation of the 30 trait/QTL pairs. The phenotype average for each genotype is given by the heights of the bars, the corresponding shape is also given, and the trait (A-E) and QTL pair (1–21) listed. Under each panel we list the cluster it falls into (1–4) and the group given by [24] (A, B, H, L, U).
Mentions: We computed the shapes of the 30 observations and found that 23 of the 387 shapes occurred, or 16 out of 69 up to symmetry. The data are shown in Figure 6. For each observation we show a bar-chart of the phenotype means and the corresponding shape. The point in the upper left corner of the shape corresponds to the genotype BB/BB, and the point in the lower right corner corresponds to LL/LL. Although in most applications one would consider the two alleles at a locus to be interchangeable we do not here, since they come from different chicken lines. To group together observations with similar genetic effects we clustered the shapes based on the pair-wise distances between them, using complete linkage hierarchical clustering. There are four main clusters in the resulting dendrogram (not shown). Under each panel in Figure 6 we list which cluster it falls into, and in parentheses we list which group it belongs to according to [24]. The observations are ordered based on the hierarchical clustering with observations in the same cluster listed together and observations within each cluster listed according to the distance between them, as far as possible. For four observations we switched the order of the first and second locus, compared to the order in [24], in order to minimize the distance to the closest observation. Within a cluster, the distance between the shapes in side-by-side panels is typically one but occasionally two. Many of the observed shapes are adjacent to more than one other shape, so two shapes that are not adjacent in Figure 6 may still be close. Consider the last row in Figure 6. In all five panels the values of the genotypes BB/BL, BL/BB and LL/LL dominate the shape, resulting in a central triangular plane. The value at BB/BB varies considerably but does not affect the shape. The shape that each of the observations fall into is, however, affected by the values of BL/LL and LL/BL. When they are relatively high an additional partition is added in the shape. Recall from the previous section that this shape is observed when there is both additive × dominance and dominance × additive interaction in the data. The shapes in the second-to-last row indicate strong dominance × dominance interaction (compare to the shape given in Figure 5). In the last two observations in the row, dom × dom is the strongest interaction, whereas the first three also show strong add × dom interaction.

Bottom Line: Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for.The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms.We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, UK. ingileif@stats.ox.ac.uk

ABSTRACT

Background: The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results: We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion: The circuits provide information on epistasis beyond that contained in the additive x additive, additive x dominance, and dominance x dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

Show MeSH
Related in: MedlinePlus