J. Jesus Céron-Rojas

Genomic prediction in CIMMYT maize and wheat breeding programs.

Genomic selection (GS) has been implemented in animal and plant species, and is regarded as a useful tool for accelerating genetic gains. Varying levels of genomic prediction accuracy have been obtained in plants, depending on the prediction problem assessed and on several other factors, such as trait heritability, the relationship between the individuals to be predicted and those used to train the models for prediction, number of markers, sample size and genotype × environment interaction (GE). The main objective of this article is to describe the results of genomic prediction in International Maize and Wheat Improvement Center’s (CIMMYT’s) maize and wheat breeding programs, from the initial assessment of the predictive ability of different models using pedigree and marker information to the present, when methods for implementing GS in practical global maize and wheat breeding programs are being studied and investigated. Results show that pedigree (population structure) accounts for a sizeable proportion of the prediction accuracy when a global population is the prediction problem to be assessed. However, when the prediction uses unrelated populations to train the prediction equations, prediction accuracy becomes negligible. When genomic prediction includes modeling GE, an increase in prediction accuracy can be achieved by borrowing information from correlated environments. Several questions on how to incorporate GS into CIMMYT’s maize and wheat programs remain unanswered and subject to further investigation, for example, prediction within and between related bi-parental crosses. Further research on the quantification of breeding value components for GS in plant breeding populations is required.

A Restricted Selection Index Method Based on Eigenanalysis

Selection indices, used in animal and plant breeding to select the best individuals for the next breeding cycle, are based on phenotypic observations of traits recorded in candidate individuals. The restrictive selection index (RSI) facilitates maximizing the genetic progress of some characters, while leaving others unchanged. Recently a selection index (SI) was proposed based on the eigen analysis method (ESIM), in which the first eigenvector (from the largest eigenvalue) is used as the SI criterion, and its elements determine the proportion of the trait that contributes to the SI. However, the current ESIM, which has two main limitations, is based on the assumption that the vector of coefficients of the index is equal to the genotypic variance-covariance matrix among the traits multiplied by the vector of economic weights, and does not allow one to restrict the number of traits. In this study, we develop a more general ESIM that has two main features, namely, it makes no assumption concerning the coefficients of the index and it can be generalized to a restrictive ESIM (RESIM). We use two datasets to illustrate the theoretical results and practical use of ESIM and RESIM, and to compare them with standard unrestrictive and restrictive selection indices. The main advantages of RESIM over traditional unrestrictive and restrictive SIs are that its statistical sampling properties are known; its selection responses are equal to or greater than those estimated from the traditional restrictive SI; and it does not require economic weights and thus can be used in practical applications when all or some of the traits need to be improved simultaneously (traditional SIs cannot improve several traits simultaneously if economic weights are not available). Finally, we prove that the coefficients of the traditional RSI belong to the space generated by the eigenvectors of RESIM.

A Molecular Selection Index Method Based on Eigenanalysis

The traditional molecular selection index (MSI) employed in marker-assisted selection maximizes the selection response by combining information on molecular markers linked to quantitative trait loci (QTL) and phenotypic values of the traits of the individuals of interest. This study proposes an MSI based on an eigenanalysis method (molecular eigen selection index method, MESIM), where the first eigenvector is used as a selection index criterion, and its elements determine the proportion of the traits contribution to the selection index. This article develops the theoretical framework of MESIM. Simulation results show that the genotypic means and the expected selection response from MESIM for each trait are equal to or greater than those from the traditional MSI. When several traits are simultaneously selected, MESIM performs well for traits with relatively low heritability. The main advantages of MESIM over the traditional molecular selection index are that its statistical sampling properties are known and that it does not require economic weights and thus can be used in practical applications when all or some of the traits need to be improved simultaneously.

A Genomic Selection Index Applied to Simulated and Real Data.

A genomic selection index (GSI) is a linear combination of genomic estimated breeding values that uses genomic markers to predict the net genetic merit and select parents from a nonphenotyped testing population. Some authors have proposed a GSI; however, they have not used simulated or real data to validate the GSI theory and have not explained how to estimate the GSI selection response and the GSI expected genetic gain per selection cycle for the unobserved traits after the first selection cycle to obtain information about the genetic gains in each subsequent selection cycle. In this paper, we develop the theory of a GSI and apply it to two simulated and four real data sets with four traits. Also, we numerically compare its efficiency with that of the phenotypic selection index (PSI) by using the ratio of the GSI response over the PSI response, and the PSI and GSI expected genetic gain per selection cycle for observed and unobserved traits, respectively. In addition, we used the Technow inequality to compare GSI vs. PSI efficiency. Results from the simulated data were confirmed by the real data, indicating that GSI was more efficient than PSI per unit of time.

Rindsel: an R package for phenotypic and molecular selection indices used in plant breeding.

Selection indices are estimates of the net genetic merit of the individual candidates for selection and are calculated based on phenotyping and molecular marker information collected on plants under selection in a breeding program. They reflect the breeding value of the plants and help breeders to choose the best ones for next generation. Rindsel is an R package that calculates phenotypic and molecular selection indices.

Linear Selection Indices in Modern Plant Breeding

We describe the main characteristics of two approaches to the linear selection indices theory. The first approach is called standard linear selection indices whereas the second of them is called eigen selection index methods. In the first approach, the economic weights are fixed and known, whereas in the second approach the economic weights are fixed but unknown. This is the main difference between both approaches and implies that the eigen selection index methods include to the standard linear selection indices because they do not require that the economic weights be known. Both types of indices predict the net genetic merit and maximize the selection response, and they give the breeder an objective criterion to select individuals as parents for the next selection cycle. In addition, in the prediction they can use phenotypic, markers, and genomic information. In both approaches, the indices can be unrestricted, null restricted or predetermined proportional gains and can be used in the context of single-stage or multistage breeding selection schemes. We describe the main characteristics of the two approaches to the linear selection indices theory and we finish this chapter describing the Lagrange multiplier method, which is the main tool to maximize the selection index responses. Linear selection indices that assume that economic weights are fixed and known to predict the net genetic merit are based on the linear selection index theory originally developed by Smith (1936), Hazel and Lush (1942), and Hazel (1943). They are called standard linear selection indices in this introduction. Linear selection indices that assume that economic weights are fixed but unknown are based on the linear selection index theory developed by Cerón-Rojas et al. (2008a, 2016) and are called Eigen selection index methods. The Eigen selection index methods include the standard linear selection indices as a particular case because they do not require the economic weights to be known. To understand the Eigen selection index methods theory, the point is to see that this is an application of the canonical correlation theory to the standard linear selection index context. The multistage linear selection index theory will be described only in the context of the standard linear selection indices. As we shall see, there are three main types of LSI: phenotypic, marker, and genomic. Each can be unrestricted, null restricted or

Linear Marker and Genome-Wide Selection Indices

There are two main linear marker selection indices employed in marker-assisted selection (MAS) to predict the net genetic merit and to select individual candidates as parents for the next generation: the linear marker selection index (LMSI) and the genome-wide LMSI (GW-LMSI). Both indices maximize the selection response, the expected genetic gain per trait, and the correlation with the net genetic merit; however, applying the LMSI in plant or animal breeding requires genotyping the candidates for selection; performing a linear regression of phenotypic values on the coded values of the markers such that the selected markers are statistically linked to quantitative trait loci that explain most of the variability in the regression model; constructing the marker score, and combining the marker score with phenotypic information to predict and rank the net genetic merit of the candidates for selection. On the other hand, the GW-LMSI is a single-stage procedure that treats information at each individual marker as a separate trait. Thus, all marker information can be entered together with phenotypic information into the GW-LMSI, which is then used to predict the net genetic merit and select candidates. We describe the LMSI and GW-LMSI theory and show that both indices are direct applications of the linear phenotypic selection index theory to MAS. Using real and simulated data we validated the theory of both indices.

Linear Molecular and Genomic Eigen Selection Index Methods

The three main linear phenotypic eigen selection index methods are the eigen selection index method (ESIM), the restricted ESIM (RESIM) and the predetermined proportional gain ESIM (PPG-ESIM). The ESIM is an unrestricted index, but the RESIM and PPG-ESIM allow null and predetermined restrictions respectively to be imposed on the expected genetic gains of some traits, whereas the rest remain without any restrictions. These indices are based on the canonical correlation, on the singular value decomposition, and on the linear phenotypic selection indices theory. We extended the ESIM theory to the molecular-assisted and genomic selection context to develop a molecular ESIM (MESIM), a genomic ESIM (GESIM), and a genome-wide ESIM (GW-ESIM). Also, we extend the RESIM and PPG-ESIM theory to the restricted genomic ESIM (RGESIM), and to the predetermined proportional gain genomic ESIM (PPG-GESIM) respectively. The latter five indices use marker and phenotypic information jointly to predict the net genetic merit of the candidates for selection, but although MESIM uses only statistically significant markers linked to quantitative trait loci, the GW-ESIM uses all genome markers and phenotypic information and the GESIM, RGESIM, and PPG-GESIM use the genomic estimated breeding values and the phenotypic values to predict the net genetic merit. Using real and simulated data, we validated the theoretical results of all five indices.

The Linear Phenotypic Selection Index Theory

The main distinction in the linear phenotypic selection index (LPSI) theory is between the net genetic merit and the LPSI. The net genetic merit is a linear combination of the true unobservable breeding values of the traits weighted by their respective economic values, whereas the LPSI is a linear combination of several observable and optimally weighted phenotypic trait values. It is assumed that the net genetic merit and the LPSI have bivariate normal distribution; thus, the regression of the net genetic merit on the LPSI is linear. The aims of the LPSI theory are to predict the net genetic merit, maximize the selection response and the expected genetic gains per trait (or multi-trait selection response), and provide the breeder with an objective rule for evaluating and selecting parents for the next selection cycle based on several traits. The selection response is the mean of the progeny of the selected parents, whereas the expected genetic gain per trait, or multi-trait selection response, is the population means of each trait under selection of the progeny of the selected parents. The LPSI allows extra merit in one trait to offset slight defects in another; thus, with its use, individuals with very high merit in one trait are saved for breeding even when they are slightly inferior in other traits. This chapter describes the LPSI theory and practice. We illustrate the theoretical results of the LPSI using real and simulated data. We end this chapter with a brief description of the quadratic selection index and its relationship with the LPSI.

Constrained Linear Phenotypic Selection Indices

The linear phenotypic selection index (LPSI), the null restricted LPSI (RLPSI), and the predetermined proportional gains LPSI (PPG-LPSI) are the main phenotypic selection indices used to predict the net genetic merit and select parents for the next selection cycle. The LPSI is an unrestricted index, whereas the RLPSI and the PPG-LPSI allow restrictions equal to zero and predetermined proportional gain restrictions respectively to be imposed on the expected genetic gain values of the trait to make some traits change their mean values based on a predetermined level while the rest of the trait means remain without restrictions. One additional restricted index is the desired gains LPSI (DG-LPSI), which does not require economic weights and, in a similar manner to the PPG-LPSI, allows restrictions to be imposed on the expected genetic gain values of the trait to make some traits change their mean values based on a predetermined level. The aims of RLPSI and PPG-LPSI are to maximize the selection response, the expected genetic gains per trait, and provide the breeder with an objective rule for evaluating and selecting parents for the next selection cycle based on several traits. This chapter describes the theory and practice of the RLPSI, PPG-LPSI, and DG-LPSI. We show that the PPG-LPSI is the most general index and includes the LPSI and the RLPSI as particular cases. Finally, we describe the DG-LPSI as a modification of the PPG-LPSI. We illustrate the theoretical results of all the indices using real and simulated data.