Genetic control of the performance of maize hybrids using complex pedigrees and microsatellite markers

Abstract

This study seeks to quantify the importance of epistatic effects on the heterosis of maize using complex pedigrees in a single model of the so-called bi-, tri-, and tetra-alleles in an analysis with and without information from microsatellite markers. To this end, 51 inbred lines were sampled from different backgrounds, obtaining 6 double cross hybrids, 14 triple cross hybrids, and 58 single cross hybrids, for a total of 78 hybrids. Seventy-nine microsatellite markers were used in the genoty** of the 51 lines. These markers were distributed throughout the 10 linkage groups in maize. This information was used to construct an information matrix on kinship. The mixed models and restricted maximum likelihood approaches were used to estimate additive, dominant and epistatic effects. It was observed that the dominant by dominant epistasis was the most important effect related to genetic control of the heterosis in maize. Also, our study demonstrated that it is possible to exploit a large amount of information when we jointly analyze simple, double, and three-way cross hybrids under the same model. Using this approach, it is possible to dissect heterosis into several components and to adopt the best crossbreeding strategy based on the importance of each component. Additionally, it was possible to verify that the use of molecular markers improves the accuracy of calculating the epistatic and dominance effects. Thus, using the current state-of-art in quantitative genetics and statistical methods the concept of crossbreeding can be expanded to frontiers that are far beyond the traditional general and specific combining ability.

Appendix

Using the genetic model it is possible to recovery the inter-genotypic information in disconnected blocks. For instance, to consider four hybrids (two single cross, one double cross and three-way cross) allocated in two different blocks. In the block 1 we have allocated the hybrids 1x2 and 1x3. On the other hand, the double cross hybrid (1x2) x (3x4) and the three-way cross (1x3)x2 were allocated in the block 2. Using traditional analysis (ANOVA), the contrasts among these hybrids are not estimable and the treatments, in theory, are not comparable. However, using Bayesian approach or likelihood inference by mixed models as used in this work, we can deal with this problem using a genetic model under complex pedigree (Bueno Filho and Gilmour 2003). Thus, we have four parental involved in the crosses instead four hybrids and six possible dominant effects to be estimated. The design matrix for blocks, additive and dominant effects are given below. In order to simplify the epistatic effects were omitted. Details about X, Z1 and Z2 are given in the model (9).

X		Z1				Z2
β1	β2	a1	a2	a3	a4	d12	d13	d14	d23	d24	d34
1	0	1	1	0	0	1	0	0	0	0	0
1	0	0	0	1	1	0	0	0	1	0	0
0	1	0.5	0.5	0.5	0.5	0	0.25	0.25	0	0.25	0.25
0	1	0.5	1	0.5	0	0.5	0	0	0.5	0	0

It is evident that instead to use the hybrids model and using the pedigree model we can link all genetic effects distributed in all blocks (Bueno Filho and Gilmour 2003). Concatenating Z₁ and Z₂ and assuming θ as the vector for random effects in the model (9), the solution for θ is given by:

$$ \begin{gathered} \hat{\theta } = \left( {Z^{T} Z + G^{ - 1} } \right)^{ - 1} Z^{T} (y - X\beta ) \hfill \\ \hat{\theta }\left( {Z^{T} Z + G^{ - 1} } \right) = Z^{T} \left( {y - X\left( {X^{T} X} \right)^{ - 1} X^{T} (y - Z\theta )} \right) \hfill \\ \hat{\theta }\left( {Z^{T} Z + G^{ - 1} + Z^{T} X\left( {X^{T} X} \right)^{ - 1} X^{T} Z} \right) = Z^{T} (y - X\beta ^{*}) \hfill \\ \end{gathered} $$

where β* is the OLS estimator for blocks effects given by $\left( {X^{T} X} \right)^{ - 1} X^{T} y.$ In the mixed models equation the matrices Z ^TX or $X^{T} Z$ are matrices of incidence of the genetics effects in each block whose result is (in the current example) given by:

X TZ1				X TZ2
1	1	1	1	1	0	0	1	0	0
1	1.5	1	0.5	0.5	0.25	0.25	0.5	0.25	0.25

In this matrix we can observe that all additive effects are present in both blocks. However, only two dominant effects are present in the block 1. This matrix can be renamed as N, and the matrix $\left( {X^{T} X} \right)$ as K—the block size matrix. Using this new notation the solution for random effects is given by:

$$ \hat{\theta }\left( {Z^{T} Z + G^{ - 1} + N^{T} K^{ - 1} N} \right) = Z^{T} (y - X\beta ^{*}) $$

Whose matrix $N^{T} K^{ - 1} N$ is given by:

1	1.25	1	0.75	0.75	0.125	0.125	0.75	0.125	0.125
1.25	1.625	1.25	0.875	0.875	0.1875	0.1875	0.875	0.1875	0.1875
1	1.25	1	0.75	0.75	0.125	0.125	0.75	0.125	0.125
0.75	0.875	0.75	0.625	0.625	0.0625	0.0625	0.625	0.0625	0.0625
0.75	0.875	0.75	0.625	0.625	0.0625	0.0625	0.625	0.0625	0.0625
0.125	0.1875	0.125	0.0625	0.0625	0.03125	0.03125	0.0625	0.03125	0.03125
0.125	0.1875	0.125	0.0625	0.0625	0.03125	0.03125	0.0625	0.03125	0.03125
0.75	0.875	0.75	0.625	0.625	0.0625	0.0625	0.625	0.0625	0.0625
0.125	0.1875	0.125	0.0625	0.0625	0.03125	0.03125	0.0625	0.03125	0.03125
0.125	0.1875	0.125	0.0625	0.0625	0.03125	0.03125	0.0625	0.03125	0.03125

If so, we can recover the inter-genotypic information in different hybrids randomized in disconnected blocks making all genetics effects comparable by use of the complete matrix $N^{T} K^{ - 1} N.$ While in traditional analysis of incomplete block design the recovery of (inter-block) information is obtained by taking genotypes as fixed and blocks as random, in this work we used the blocks as fixed and genotypes as random making all genetics effects comparable and linked as previously showed by Bueno Filho and Gilmour (2003).