Bayesian analysis of genetic architecture of quantitative trait using data of crosses of multiple inbred lines

Abstract

Using the data of crosses of multiple of inbred lines for mapping QTL can increase QTL detecting power compared with only cross of two inbred lines. Although many fixed-effect model methods have been proposed to analyze such data, they are largely based on one-QTL model or main effect model, and the interaction effects between QTL are always neglected. However, effectively separating the interaction effects from the residual error can increase the statistical power. In this article, we both extended the novel Bayesian model selection method and Bayesian shrinkage estimation approaches to multiple inbred line crosses. With two extensions, interacting QTL are effectively detected with high solution; in addition, the posterior variances for both main effects and interaction effects are also subjected to full Bayesian estimate, which is more optimal than two step approach involved in maximum-likelihood. A series of simulation experiments have been conducted to demonstrate the performance of the methods. The computer program written in FORTRAN language is freely available on request.

Appendix: conditional posterior distributions

Conditional posterior distribution of model effects

The conditional posterior distributions of all model effects are normal. For popular mean u, the conditional posterior distribution of it follows

$$ u\sim N\left( {\left( {1/nm} \right)\sum_{i=1,j=1}^{n,m} {(y_{ij} -g_{ij} ),\left( {1/mn} \right)\sigma _e^2 } } \right). $$

(A1)

If the number of individuals in each cross is different, n will be substituted by n _j , and the rules holds in all steps. For cross specific effect, the conditional posterior distribution of it is

$$ c_j \sim N\left( {\sum_{i=1}^n {(y_{ij} -g_{ij} +c_j )/n,\left( {1/mn} \right)\sigma _e^2 },\;\;\left( {1/mn} \right)\sigma _e^2 } \right), $$

(A2)

for j = 1,...,m. The conditional posterior distribution for main effect b _jq is

$$ b_{jq} \sim N\left( {\left(\sigma _e^2 \sigma _q^{-2} +\sum_{i=1}^n {x_{ij,q}^2 } \right)^{-1}\sum_{i=1}^n {x_{ij,q} (y_{ij} +\gamma _q x_{ij,q} b_{jq} -g_{ij} )},\;\;\left(\sigma _e^2 \sigma _q^{-2} +\sum_{i=1}^n {x_{ij,q}^2 }\right )^{-1}\sigma _e^2 } \right), $$

(A3)

q = 1,...,K and j = 1,...,m, and for interaction effect w _{j,q_1 q_2} is

$$ \begin{aligned} w_{j,q_1 q_2}&\sim N\left(\left(\sigma _e^2 \sigma _{q_1 q_2 }^{-2} +\sum_{i=1}^n {x_{ij,q_1 q_2 }^2 } \right)^{-1}\sum_{i=1}^n {x_{ij,q_1 q_2 } (y_{ij} +\gamma _{q_1 q_2 } x_{ij,q_1 q_2 } w_{j,q_1 q_2 } -g_{ij})},\right.\\ &\left.\left(\sigma_e^2 \sigma_{q_1 q_2 }^{-2} +\sum_{i=1}^n{x_{ij,q_1 q_2}^2}\right)^{-1}\sigma_e^2\right),\\ \end{aligned} $$

(A4)

for q ₁ = 1,...,K, q ₂ < q ₁ and j = 1,...,m.

Conditional posterior distribution of residual error σ ² _e

For the residual variance σ ² _e , the conditional posterior distribution is a scaled inverse chi-square, $\sigma _e^2 \sim \chi ^{-2}(mn,SS_e ),$ where,

$$ SS_e =\sum_{i=1,j=1}^{n,m} {(y_{ij} -g_{ij} } )^{2}. $$

(A5)

Conditional posterior distribution of QTL genotypes

The conditional posterior distribution of QTL genotypes is

$$ p(x_{ij,q}=z\left| {y_{ij},}\right.m_j,m_{j+1},\lambda_j,\varvec{\gamma},x_{ij\cdot}, \varvec{\theta}) =\frac{p(y_{ij} \left|{\varvec{\gamma},x_{ij(-q)} },\right. x_{ij,q} =z,\varvec{\theta})\cdot p(x_{ij,q} =z\left| {\lambda _j,} \right. m_j,m_{j+1})}{\sum_{x_{ij,q}\in\{-1,1\}}p(y_{ij} \left| {\varvec{\gamma},x_{ij(-q)}},\right. x_{ij,q},\varvec{\theta})\cdot p(x_{ij,q}\left|{\lambda_j,} \right.m_j,m_{j+1})} $$

(A6)

where, z = −1 or 1, indicates two different genotypes of QTL.