Field experiments
The experiments were carried out in the State of Minas Gerais, Brazil, on the experimental fields of Agricultural Research Institute of Minas Gerais State (EPAMIG) in the cities of Leopoldina (21° 31' 48.01 '' S, 42° 38' 24'' W), Lambari (21° 58' 11.24'' S, 45° 20' 59.6'' W) and Janaúba (15° 48' 77'' S, 43° 17' 59.09'' W).
Twenty-five rice (Oryza sativa L) genotypes belonging to the flood-irrigated rice breeding program of the state of Minas Gerais were evaluated, and five of these genotypes were used as experimental controls (Rubelita, Seleta, Ourominas, Predileta, and Rio Grande). These genotypes were evaluated in comparative trials after multiple generations of selection, and in addition, they are known for their high yield, uniform growth rate and plant growth, resistance to major diseases, and for their excellent grain quality. The traits evaluated were grain yields (GY, Kg ha− 1), grain length (GL, mm), grain width (GW, mm) and grain thickness (GT, mm), grain length, and grain width and weight of 100 grains (GWH, g) in the agricultural year 2016/2017.
The design used in all experiments was a randomized complete block design with three replications. The experiments were conducted in floodplain soils with continuous flood irrigation. Management practices were carried out according to recommendations for flood-irrigated rice in the relevant regions (Soares et al., 2005).
The useful plot area consisted of 4 m central of three internal rows (4 m x 0.9 m, 3.60 m2 total). The soil preparation was carried out by plowing and harrowing around 30 days before sowing and harrowing on the eve of the installation of the tests. For planting fertilization, a mixture of 100 kg ha− 1 of ammonium sulfate, 300 kg ha− 1 of simple superphosphate, and 100 kg ha− 1 of potassium chloride was used, applied to the plot, and incorporated into the soil before planting. The fertilization in the top dressing was carried out approximately 60 days after the installation of the experiments, with 200 kg ha− 1 of ammonium sulfate. The weeds were controlled with the use of herbicides and manual weeding. Sowing was carried out in the planting line with a density of 300 seeds m− 2. The irrigation started around 10–15 days after the implantation of the experiments, and the water was only removed close to the maturation of the material later. The harvest was carried out when the grains reached a humidity of 20–22%. Grain production data were obtained by weighing all grains harvested in the useful plot, after cleaning and uniform drying in the sun, until they reached a humidity of 13%.
Biometric Analysis
The measured traits were analyzed using the univariate model and the multi-trait model through the Bayesian approach of Markov Chain Monte Carlo (MCMC).
The multi-trait model was given by:
$$y=X\beta +Zg+ \epsilon$$
where y is the vector of phenotypic data, and the conditional distribution was given by: y| \(\beta\), g, i, G, R ~ N (X\(\beta\) + Zg, R⊗I), G is the matrix of genotypic covariance, R is the matrix of residual covariance. I is an identity matrix, \(\beta\) is vector of systematic effects (genotypes mean and replication effects), assumed as \(\beta\) ~ N (\(\beta\), Σ\(\beta\)⊗I). g is the vector of genotype effects, assumed as g|\(\text{G}\), ~ N (0, G⊗I). e is the vector of residuals, assumed as e |\(R\), ~ N (0, R⊗I). The uppercase bold letters X and Z refer to the incidence matrices for the effects \(\beta\) and g, respectively.
We assume that G and R follow an inverted Wishart distribution WI (v, V), with hyperparameters v and V (Sorensen and Gianola, 2002). Hyperparameters for all prior distributions have been selected to provide non-informative or flat prior distributions. For the systematic effect (β), a pre-uniform distribution was assigned. In addition, the parameters β, g, G, R were estimated following the set posterior distribution: P(β, g, G, R |y) α P(y | β, g, G, R )× P(β, g, G, R).
For the model, the package was used “MCMCglmm” (Hadfield et al., 2010) of the R software (R Development Core Team, 2020). A total of 10,000,000 samples were generated and assumed a burn-in period and thin range of 500,000 and 10 iterations, respectively, resulting in a final total of 50,000 samples. The convergence of the MCMC was verified by the criterion of Geweke et al. (1992), carried out in two R software packages: "boa" (Smith et al., 2007) and "CODA" (Convergence Diagnosis and Output Analysis) (Plummer et al., 2006).
The model was compared using the deviation information criterion (DIC) proposed by Spiegelhalter et al. (2002):
$$DIC=D\left(\stackrel{-}{\theta }\right)+{2p}_{D}$$
where \(D\left(\stackrel{-}{\theta }\right)\)is a point estimate of the deviance obtained by replacing the parameters with their posterior means estimates in the likelihood function and \({p}_{D}\) is the effective number of model parameters. Models with a lower DIC should be preferred over models with a higher DIC.
The components of variance, broad-sense heritability, and genotypic correlation coefficients between traits and breeding values were calculated from the posterior distribution. The package “boa” (Smith et al., 2007) R software was used to calculate the intervals of higher posterior density (HPD) for all parameters. A posteriori estimates for broad-sense heritability (\({h}^{2}\)) of the six traits for each iteration were calculated from the later samples of the variance components obtained by the multivariate model, using the expression:
$${h}^{2\left(i\right)}= \frac{{\sigma }_{g}^{2\left(i\right)}}{\left({\sigma }_{g}^{2\left(i\right)}+ {\sigma }_{r}^{2\left(i\right)}+ {\sigma }_{\epsilon }^{2\left(i\right)}\right)}$$
where: \({\sigma }_{g}^{2\left(i\right)}, {\sigma }_{r}^{2\left(i\right)},\)and \({\sigma }_{\epsilon }^{2\left(i\right)}\) are the genetic, replication, and residual variations for each iteration, respectively.
For the multi-trait model, the genetic correlation coefficients between the pairs of traits in each environment were obtained, as suggested by Piepho et al. (2018), using the expression below for all models:
$${\rho }_{l\left(\text{1,2}\right)}= \frac{{\sigma }_{gl\left(\text{1,2}\right)}}{\sqrt{{\sigma }_{gl\left(1\right)}^{2}{\sigma }_{gl\left(2\right)}^{2}}}$$
where \({\widehat{\sigma }}_{gl}^{2}\) represents the genetic variance of the evaluated trait and \({\sigma }_{gl\left(\text{1,2}\right)}\) represents the genetic covariance between pairs of traits.
Genetic selection based on selection index
The multi-trait index based on factor analysis and genotype-ideotype distance (FAI-BLUP) (Rocha et al., 2018) was used to identify superior genotypes to be selected in the flood-irrigated rice breeding program.
$${P}_{ij}= \frac{\frac{1}{{d}_{ij}}}{\sum _{i=1;j=1}^{i=n;j=m}\frac{1}{{d}_{ij}}}$$
,
where, \({P}_{ij}\): probability of the ith genotype (i = 1, 2, ..., n) to be similar to the jth ideotype (j = 1, 2, ..., m);\({d}_{ij}\): genotype-ideotype distance from ith genotype to jth ideotype – based on standardized mean distance.
Selection gains (SG) were obtained directly from the FIA-BLUP result considering four different selection intensities: 12%, 20%, 40%, and 60%, which referred to the selection of 3, 5, 10, and 15 genotypes, respectively, as follows:
$$SG\left(\%\right)=\left(\frac{{X}_{s}-{X}_{0} }{{X}_{0}}\right)$$
,
where \({X}_{s}\)is the overall mean of the estimated breeding values of the selected genotypes, and \({X}_{0}\) is the general population average.