Association mapping in multiple segregating populations of sugar beet (Beta vulgaris L.)

Abstract

Association mapping in multiple segregating populations (AMMSP) combines high power to detect QTL in genome-wide approaches of linkage mapping with high mapping resolution of association mapping. The main objectives of this study were to (1) examine the applicability of AMMSP in a plant breeding context based on segregating populations of various size of sugar beet (Beta vulgaris L.), (2) compare different biometric approaches for AMMSP, and (3) detect markers with significant main effect across locations for nine traits in sugar beet. We used 768 F _n (n = 2, 3, 4) sugar beet genotypes which were randomly derived from 19 crosses among diploid elite sugar beet clones. For all nine traits, the genotypic and genotype × location interaction variances were highly significant (P < 0.01). Using a one-step AMMSP approach, the total number of significant (P < 0.05) marker-phenotype associations was 44. The identification of genome regions associated with the traits under consideration indicated that not only segregating populations derived from crosses of parental genotypes in a systematic manner could be used for AMMSP but also populations routinely derived in plant breeding programs from multiple, related crosses. Furthermore, our results suggest that data sets, whose size does not permit analysis by the one-step AMMSP approach, might be analyzed using the two-step approach based on adjusted entry means for each location without losing too much power for detection of marker-phenotype associations.

Appendix

Statistical models with random effects for genotypes can be used in AMMSP approaches to model the relatedness among genotypes. In this context it is assumed that

$$ \hbox{Var}({{\mathbf{g}}}) \sim {\bf A}\sigma_g^2, $$

where g is the vector containing the genotypic effects of all entries and A is the numerator relationship matrix. However, in AMMSP studies in which marker data are not available for all entries, separate genotypic effects have to be assumed for entries with (g ₁) and without (g ₂) available marker data:

$$ {{\mathbf{g}}} = \left( \begin{array}{l} {{\mathbf{g}}}_1\\ {{\mathbf{g}}}_2\\ \end{array} \right) \sim {\rm MVN}\left[\left( \begin{array}{l} {{\mathbf{0}}}\\ {{\mathbf{0}}}\\ \end{array} \right), \left( \begin{array}{ll} {{\mathbf{A}}}_{11}&{{\mathbf{A}}}_{12}\\ {{\mathbf{A}}}_{21}&{{\mathbf{A}}}_{22}\\ \end{array} \right) \sigma_{g}^2 \right]. $$

Assume g ₁ = f ₁ + h ₁, where f ₁ is the vector of genotypic effects of the marker locus under consideration, then:

$$\left( \begin{array}{l} {{\mathbf{f}}}_1\\ {{\mathbf{h}}}_1\\ {{\mathbf{g}}}_2\\ \end{array} \right) \sim {\rm MVN} \left[\left( \begin{array}{l} {{\mathbf{0}}}\\ {{\mathbf{0}}}\\ {{\mathbf{0}}}\\ \end{array} \right), \left( \begin{array}{lll} p_g{{\mathbf{A}}}_{11}&{{\mathbf{0}}}&p_g{{\mathbf{A}}}_{12}\\ {{\mathbf{0}}}&(1-p_g){{\mathbf{A}}}_{11}&(1-p_g){{\mathbf{A}}}_{12}\\ p_g{{\mathbf{A}}}_{21}&(1-p_g){{\mathbf{A}}}_{21}& {{\mathbf{A}}}_{22}\\ \end{array} \right) \sigma_{g}^2= \left( \begin{array}{ll} {{\mathbf{G}}}&{{\mathbf{H}}}\\ {{\mathbf{H}}}^{\prime}&{{\mathbf{K}}} \end{array} \right) \sigma_{g}^2 \right], $$

where p _g is the proportion explained genotypic variance of the marker locus under consideration, G = p _g A ₁₁, H = (0 p _g A ₁₂), and

$$ {{\mathbf{K}}} = \left( \begin{array}{ll} (1-p_g){{\mathbf{A}}}_{11}&(1-p_g){{\mathbf{A}}}_{12}\\ (1-p_g){{\mathbf{A}}}_{12}&{{\mathbf{A}}}_{22}\\ \end{array} \right). $$

Based on the formulas for the conditional multivariate normal distribution (Searle et al. 1992), it can be concluded that conditioning on f ₁ leads to:

$$ ({{\mathbf{h}}}_1,{{\mathbf{g}}}_1|{{\mathbf{f}}}_1) \sim \left[ ({{\mathbf{H}}}^\prime{{\mathbf{G}}}^{-1}{{\mathbf{f}}}_1), ({{\mathbf{K}}}-{{\mathbf{H}}}^\prime{{\mathbf{G}}}^{-1}{{\mathbf{H}}}) \sigma_g^2\right], $$

where

$${{\mathbf{H}}}^\prime{{\mathbf{G}}}^{-1}{{\mathbf{f}}}_1= \left( {{\mathbf{0}}}\\ {{\mathbf{A}}}_{21}{{\mathbf{A}}}_{11}^{-1}{{\mathbf{f}}}_1\\ \right) $$

and

$$ \begin{aligned} {{\mathbf{K}}} - {{\mathbf{H}}}^\prime{{\mathbf{G}}}^{-1}{{\mathbf{H}}}&= \left( \begin{array}{ll} (1-p_g){{\mathbf{A}}}_{11}&(1-p_g){{\mathbf{A}}}_{12}\\ (1-p_g){{\mathbf{A}}}_{12}&{{\mathbf{A}}}_{22}\\ \end{array} \right) - \left( \begin{array}{ll} {{\mathbf{0}}}&{{\mathbf{0}}}\\ {{\mathbf{0}}}&p_g{{\mathbf{A}}}_{21}{{\mathbf{A}}}_{11}^{-1}{{\mathbf{A}}}_{12}\\ \end{array} \right)\\ &= \left( \begin{array}{ll} (1-p_g){{\mathbf{A}}}_{11}&(1-p_g){{\mathbf{A}}}_{12}\\ (1-p_g){{\mathbf{A}}}_{12}&{{\mathbf{A}}}_{22}-p_g{{\mathbf{A}}}_{21}{{\mathbf{A}}}_{11}^{-1}{{\mathbf{A}}}_{12} \\ \end{array} \right). \end{aligned} $$

Under the assumption that conditioning on a marker locus is equivalent to conditioning on f ₁, this result suggests that in the conditional density of all entries, the expected value of the entries without available marker data is shifted by conditioning on the marker data of the entries with available marker data. This problem can be overcome by replacing f ₁ by a regression approach on the marker $({\bf f}_1={\bf X}_1{\varvec{\alpha}}),$ which leads to:

$$ \left({{\mathbf{h}}}_1,{{\mathbf{g}}}_2|{{\mathbf{f}}}_1\approx{{\rm Marker}} \right) \sim {\rm MVN}\left[\left( \begin{array}{l} {{\mathbf{0}}}\\ {{\mathbf{A}}}_{21}{{\mathbf{A}}}_{11}^{-1}{{\mathbf{X}}}_1 {\varvec{\alpha}}\\ \end{array} \right), \left({{\mathbf{K}}} - {{\mathbf{H}}}^\prime {{\mathbf{G}}}^{-1}{{\mathbf{H}}} \right) \sigma_{\check{g}}^2 \right] $$

and

$$ \left({{\mathbf{g}}}_1, {{\mathbf{g}}}_2|{{\mathbf{f}}}_1\approx {\rm Marker} \right) \sim {\rm MVN} \left[\left( \begin{array}{l} {{\mathbf{X}}}_1{\varvec{\alpha}}\\ {{\mathbf{X}}}_2 {\varvec{\alpha}}\\ \end{array} \right), \left({{\mathbf{K}}} - {{\mathbf{H}}}^\prime {{\mathbf{G}}}^{-1}{{\mathbf{H}}} \right) \sigma_{\check{g}}^2 \right], $$

where X ₂ = A ₂₁ A ⁻¹₁₁ X ₁ and $\sigma_{\check{g}}^2 =(1-p_g)\sigma_{g}^2.$