Mapping interacting QTL for count phenotypes using hierarchical Poisson and binomial models: an application to reproductive traits in mice

Abstract

We proposed hierarchical Poisson and binomial models for mapping multiple interacting quantitative trait loci (QTLs) for count traits in experimental crosses. We applied our methods to two counted reproductive traits, live fetuses (LF) and dead fetuses (DF) at 17 days gestation, in an F2 female mouse population. We treated observed number of corpora lutea (ovulation rate) as the baseline and the total trials in our Poisson and binomial models, respectively. We detected more than 10 QTLs for LF and DF, most having epistatic and pleiotropic effects. The epistatic effects were larger, involved more QTLs, and explained a larger proportion of phenotypic variance than the main effects. Our analyses revealed a complex network of multiple interacting QTLs for the reproductive traits, and increase our understanding of the genetic architecture of reproductive characters. The proposed statistical models and methods provide valuable tools for detecting multiple interacting QTLs for complex count phenotypes.

Figure 1

Histograms of numbers of live fetuses (LF), dead fetuses (DF), and ovulation rate (OR) (also called corpora lutea).

Figure 2

Poisson main-effect (left) and epistatic (right) models for live fetuses (LF): estimated effects with ± 1 standard errors (dots and short lines), p-values (rescaled as −log₁₀ p/10) (triangles), and deviance and AIC. The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.

Figure 3

Poisson main-effect model without model search for live fetuses (LF): estimated effects with ± 1 standard errors (dots and short lines), and p-values (rescaled as −log₁₀ p/10) (triangles). The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.

Figure 4

Binomial main-effect (left) and epistatic (right) models for live fetuses (LF): estimated effects with ± 1 standard errors (dots and short lines), p-values (rescaled as −log₁₀ p/10) (triangles), and deviance and AIC. The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.

Figure 5

Poisson main-effect (left) and epistatic (right) models for dead fetuses (DF): estimated effects with ± 1 standard errors (dots and short lines), p-values (rescaled as −log₁₀ p/10) (triangles), and deviance and AIC. The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.

Figure 6

Binomial main-effect (left) and epistatic (right) models for dead fetuses (DF): estimated effects with ± 1 standard errors (dots and short lines), p-values (rescaled as −log₁₀ p/10) (triangles), and deviance and AIC. The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.

Figure 7

Treating discrete live fetuses (LF) and dead fetuses (DF) as normally continuous traits. a. main-effect model for LF; b. epistatic model for LF; c. main-effect model for DF; d. epistatic model for DF; estimated effects with ± 1 standard errors (dots and short lines), p-values (rescaled as −log₁₀ p/10) (triangles), and deviance and AIC. The notation for additive effect (C@h)a dominance effect (C@h)d, indicates chromosome C and position h cM. The term X₁:X₂ represents interaction between X₁ and X₂. The two gray lines indicate the p-values of 0.05 and 0.001, respectively.