Genome-based prediction of testcross values in maize (original) (raw)

Abstract

This is the first large-scale experimental study on genome-based prediction of testcross values in an advanced cycle breeding population of maize. The study comprised testcross progenies of 1,380 doubled haploid lines of maize derived from 36 crosses and phenotyped for grain yield and grain dry matter content in seven locations. The lines were genotyped with 1,152 single nucleotide polymorphism markers. Pedigree data were available for three generations. We used best linear unbiased prediction and stratified cross-validation to evaluate the performance of prediction models differing in the modeling of relatedness between inbred lines and in the calculation of genome-based coefficients of similarity. The choice of similarity coefficient did not affect prediction accuracies. Models including genomic information yielded significantly higher prediction accuracies than the model based on pedigree information alone. Average prediction accuracies based on genomic data were high even for a complex trait like grain yield (0.72–0.74) when the cross-validation scheme allowed for a high degree of relatedness between the estimation and the test set. When predictions were performed across distantly related families, prediction accuracies decreased significantly (0.47–0.48). Prediction accuracies decreased with decreasing sample size but were still high when the population size was halved (0.67–0.69). The results from this study are encouraging with respect to genome-based prediction of the genetic value of untested lines in advanced cycle breeding populations and the implementation of genomic selection in the breeding process.

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Acknowledgments

This research was funded by the German Federal Ministry of Education and Research (BMBF) within the AgroClustEr “Synbreed—Synergistic plant and animal breeding” (FKZ: 0315528A).

Author information

Authors and Affiliations

  1. Plant Breeding, Technische Universität München, Emil-Ramann-Straße 4, 85354, Freising, Germany
    Theresa Albrecht, Valentin Wimmer, Hans-Jürgen Auinger & Chris-Carolin Schön
  2. Department of Animal Sciences, Georg-August-Universität Göttingen, 37075, Göttingen, Germany
    Malena Erbe & Henner Simianer
  3. KWS SAAT AG, 37555, Einbeck, Germany
    Carsten Knaak & Milena Ouzunova

Authors

  1. Theresa Albrecht
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  2. Valentin Wimmer
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  3. Hans-Jürgen Auinger
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  4. Malena Erbe
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  5. Carsten Knaak
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  6. Milena Ouzunova
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  7. Henner Simianer
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  8. Chris-Carolin Schön
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Corresponding author

Correspondence toChris-Carolin Schön.

Additional information

Communicated by J. Snape.

T. Albrecht and V. Wimmer contributed equally to this work.

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122_2011_1587_MOESM1_ESM.doc

Supplemental Fig. 1 Linkage disequilibrium between pairs of marker loci within the same linkage group as a function of physical distance on the reference map for 31 parental inbred lines (DOC 540 kb)

Appendix

Appendix

In this Appendix we show the functional dependency of the genomic variance components pertaining to Model 2, Model SM and the random regression model suggested by Meuwissen et al. (2001) denoted Model RR.

All three models can be seen as special cases of a more general model

mathbfy=Xvarvecupbeta+(W−Q)m+e{\mathbf{y = X{\varvec{\upbeta}} + (W - Q)m + e}}mathbfy=Xvarvecupbeta+(W−Q)m+e

where \( {\varvec{\upbeta}} \) is a vector of fixed effects and X is a design matrix assigning fixed effects to the phenotypes**, W** is an N × M design matrix assigning M SNP marker genotypes coded 0 or 2 to N phenotypes. Random marker effects in vector m are assumed to follow a normal distribution with \( {\mathbf {m}} \sim N(0,{\mathbf {I}}\sigma_{m}^{2} ) \), where I is an identity matrix and \( \sigma_{m}^{2} \) denotes the proportion of the testcross variance contributed by each individual SNP marker. Q is an N × M matrix composed of M uniform column vectors \( {\mathbf{Q}} = \left\{ {q_{1} {\bf c},q_{2} {\bf c}, \cdots ,q_{M} {\bf c}} \right\} \) where \( q_{m} \) is a scalar correction term for marker m and \( {\mathbf {c}} \) is a column vector of length N containing only 1s.

The estimated SNP effects and the corresponding variance component \( \sigma_{m}^{2} \) in the general model are unaffected by the choice of Q, since subtracting the correction terms shifts the intercept, but does not change the slope of the regression of the phenotype on each SNP (Habier et al. 2007; Piepho 2009). For Model RR, Q is the null matrix, but the same variance components as estimated from Model RR will be received with any other Q.

For Model 2 the kinship between lines i and j is modeled by the matrix U calculated as

mathbfU=frac(mathbfW−mathbfP)(mathbfW−mathbfP)′8sumnolimitsm=1Mpm(1−pm){\mathbf{U}} = {\frac{{({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'}}{{8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )} }}}mathbfU=frac(mathbfW−mathbfP)(mathbfW−mathbfP)′8sumnolimitsm=1Mpm(1−pm)

and the variance–covariance matrix of the phenotype vector y can be written as

mathbfVtextModel2=frac(mathbfW−mathbfP)(mathbfW−mathbfP)′8sumnolimitsm=1Mpm(1−pm)sigmau2+mathbfIsigma2{\mathbf{V}}_{\text{Model2}} = {\frac{{({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'}}{{8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )} }}}\sigma_{u}^{2} + {\mathbf{I}}\sigma_{{}}^{2}mathbfVtextModel2=frac(mathbfW−mathbfP)(mathbfW−mathbfP)′8sumnolimitsm=1Mpm(1−pm)sigmau2+mathbfIsigma2

Using \( {\mathbf{Q}} = {\mathbf{P}} \) in the generalized model, the variance–covariance matrix of the phenotype vector y is

mathbfVrmP=(mathbfW−mathbfP)(mathbfW−mathbfP)′sigmam2+mathbfIsigma2{\mathbf{V}}_{{\rm P}} = ({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'\sigma_{m}^{2} + {\mathbf{I}}\sigma_{{}}^{2}mathbfVrmP=(mathbfW−mathbfP)(mathbfW−mathbfP)′sigmam2+mathbfIsigma2

Hence, \( {\mathbf{V}}_{\text{Model2}} = {\mathbf{V}}_{P} \) if

sigmau2=sigmam2times8sumnolimitsm=1Mpm(1−pm)\sigma_{u}^{2} = \sigma_{m}^{2} \times 8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )}sigmau2=sigmam2times8sumnolimitsm=1Mpm(1−pm)

In Model SM, the matrix S can be written as

mathbfS=frac(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′2M(1−smin)+fracM−2Msmin2M−2MsminmathbfJNtimesN{\mathbf{S}} = {\frac{{({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'}}{{2M(1 - s_{\min } )}}} + {\frac{{M - 2Ms_{\min } }}{{2M - 2Ms_{\min } }}}{\mathbf{J}}_{N \times N}mathbfS=frac(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′2M(1−smin)+fracM−2Msmin2M−2MsminmathbfJNtimesN

The second term reflects a constant, which is added to all elements of the matrix. This is equivalent to a constant random block effect and thus fully confounded with the fixed intercept (Piepho et al. 2008; Williams et al. 2006). Hence, ignoring the second term will not affect the estimated variance components.

The numerator of the first term of S is a special case of the numerator of the generalized model with \( {\mathbf{Q}} = {\mathbf{J}}_{N \times M} \), i.e. assuming \( q_{m} = 1 \) for all loci, which is equivalent to the assumption of allele frequency \( p_{m} = 0.5 \) for all SNPs in Model 2. Using the same argument as above,

mathbfVtextSM=frac(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′2M(1−smin)sigmas2+mathbfIsigma2{\mathbf{V}}_{\text{SM}} = {\frac{{({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'}}{{2M(1 - s_{\min } )}}}\sigma_{s}^{2} + {\mathbf{I}}\sigma^{2}mathbfVtextSM=frac(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′2M(1−smin)sigmas2+mathbfIsigma2

For the special case \( {\mathbf{Q}} = {\mathbf{J}}_{N \times M} \) in the general model, the variance–covariance matrix of the phenotype vector y is

mathbfVJ=(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′sigmam2+mathbfIsigma2{\mathbf{V}}_{J} = ({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'\sigma_{m}^{2} + {\mathbf{I}}\sigma^{2}mathbfVJ=(mathbfW−mathbfJNtimesM)(mathbfW−mathbfJNtimesM)′sigmam2+mathbfIsigma2

and \( {\mathbf{V}}_{\text{SM}} = {\mathbf{V}}_{J} \) if

sigmas2=sigmam2times2M(1−smin).\sigma_{s}^{2} = \sigma_{m}^{2} \times 2M(1 - s_{\min } ).sigmas2=sigmam2times2M(1−smin).

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Albrecht, T., Wimmer, V., Auinger, HJ. et al. Genome-based prediction of testcross values in maize.Theor Appl Genet 123, 339–350 (2011). https://doi.org/10.1007/s00122-011-1587-7

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