MIM: an indirect method to assess inbreeding and coancestry in large incomplete pedigrees of selected dairy cattle

Estimating genomic relationships of metafounders across and within breeds using maximum likelihood, pseudo-expectation-maximization maximum likelihood and increase of relationships

BACKGROUND

The theory of "metafounders" proposes a unified framework for relationships across base populations within breeds (e.g. unknown parent groups), and base populations across breeds (crosses) together with a sensible compatibility with genomic relationships. Considering metafounders might be advantageous in pedigree best linear unbiased prediction (BLUP) or single-step genomic BLUP. Existing methods to estimate relationships across metafounders Γ are not well adapted to highly unbalanced data, genotyped individuals far from base populations, or many unknown parent groups (within breed per year of birth).

METHODS

We derive likelihood methods to estimate Γ . For a single metafounder, summary statistics of pedigree and genomic relationships allow deriving a cubic equation with the real root being the maximum likelihood (ML) estimate of Γ . This equation is tested with Lacaune sheep data. For several metafounders, we split the first derivative of the complete likelihood in a term related to Γ , and a second term related to Mendelian sampling variances. Approximating the first derivative by its first term results in a pseudo-EM algorithm that iteratively updates the estimate of Γ by the corresponding block of the H-matrix. The method extends to complex situations with groups defined by year of birth, modelling the increase of Γ using estimates of the rate of increase of inbreeding ( Δ F ), resulting in an expanded Γ and in a pseudo-EM+ Δ F algorithm. We compare these methods with the generalized least squares (GLS) method using simulated data: complex crosses of two breeds in equal or unsymmetrical proportions; and in two breeds, with 10 groups per year of birth within breed. We simulate genotyping in all generations or in the last ones.

RESULTS

For a single metafounder, the ML estimates of the Lacaune data corresponded to the maximum. For simulated data, when genotypes were spread across all generations, both GLS and pseudo-EM(+ Δ F ) methods were accurate. With genotypes only available in the most recent generations, the GLS method was biased, whereas the pseudo-EM(+ Δ F ) approach yielded more accurate and unbiased estimates.

CONCLUSIONS

We derived ML, pseudo-EM and pseudo-EM+ Δ F methods to estimate Γ in many realistic settings. Estimates are accurate in real and simulated data and have a low computational cost.

From clinics to (cow)mics: a reproductive journey

This manuscript describes the different topics I have been involved in the fields of reproductive physiology and embryo biotechnologies with attempts to address practical issues raised mainly by the breeding industry. The journey started with phenotyping work in the field of reproductive physio-pathology. Other issues were related to the optimization of reproductive biotechnologies to favorize genetic selection. The implementation of genomic selection raised opportunities to develop the use embryo biotechnologies and showed the interest of combining them in the case of embryo genotyping. There is still a need to refine phenotyping for reproductive traits especially for the identification of markers of uterine dysfunction. It is believed that new knowledge generated by combining different molecular approaches will be the source of applications that may benefit AI practice and embryo technologies.

Ancestral Relationships Using Metafounders: Finite Ancestral Populations and Across Population Relationships

Recent use of genomic (marker-based) relationships shows that relationships exist within and across base population (breeds or lines). However, current treatment of pedigree relationships is unable to consider relationships within or across base populations, although such relationships must exist due to finite size of the ancestral population and connections between populations. This complicates the conciliation of both approaches and, in particular, combining pedigree with genomic relationships. We present a coherent theoretical framework to consider base population in pedigree relationships. We suggest a conceptual framework that considers each ancestral population as a finite-sized pool of gametes. This generates across-individual relationships and contrasts with the classical view which each population is considered as an infinite, unrelated pool. Several ancestral populations may be connected and therefore related. Each ancestral population can be represented as a "metafounder," a pseudo-individual included as founder of the pedigree and similar to an "unknown parent group." Metafounders have self- and across relationships according to a set of parameters, which measure ancestral relationships, i.e., homozygozities within populations and relationships across populations. These parameters can be estimated from existing pedigree and marker genotypes using maximum likelihood or a method based on summary statistics, for arbitrarily complex pedigrees. Equivalences of genetic variance and variance components between the classical and this new parameterization are shown. Segregation variance on crosses of populations is modeled. Efficient algorithms for computation of relationship matrices, their inverses, and inbreeding coefficients are presented. Use of metafounders leads to compatibility of genomic and pedigree relationship matrices and to simple computing algorithms. Examples and code are given.

Variance and covariance of actual relationships between relatives at one locus

The relationship between pairs of individuals is an important topic in many areas of population and quantitative genetics. It is usually measured as the proportion of thegenome identical by descent shared by the pair and it can be inferred from pedigree information. But there is a variance in actual relationships as a consequence of Mendelian sampling, whose general formula has not been developed. The goal of this work is to develop this general formula for the one-locus situation,. We provide simple expressions for the variances and covariances of all actual relationships in an arbitrary complex pedigree. The proposed method relies on the use of the nine identity coefficients and the generalized relationship coefficients; formulas have been checked by computer simulation. Finally two examples for a short pedigree of dogs and a long pedigree of sheep are given.

Compatibility of pedigree-based and marker-based relationship matrices for single-step genetic evaluation

BACKGROUND

Single-step methods provide a coherent and conceptually simple approach to incorporate genomic information into genetic evaluations. An issue with single-step methods is compatibility between the marker-based relationship matrix for genotyped animals and the pedigree-based relationship matrix. Therefore, it is necessary to adjust the marker-based relationship matrix to the pedigree-based relationship matrix. Moreover, with data from routine evaluations, this adjustment should in principle be based on both observed marker genotypes and observed phenotypes, but until now this has been overlooked. In this paper, I propose a new method to address this issue by 1) adjusting the pedigree-based relationship matrix to be compatible with the marker-based relationship matrix instead of the reverse and 2) extending the single-step genetic evaluation using a joint likelihood of observed phenotypes and observed marker genotypes. The performance of this method is then evaluated using two simulated datasets.

RESULTS

The method derived here is a single-step method in which the marker-based relationship matrix is constructed assuming all allele frequencies equal to 0.5 and the pedigree-based relationship matrix is constructed using the unusual assumption that animals in the base population are related and inbred with a relationship coefficient γ and an inbreeding coefficient γ / 2. Taken together, this γ parameter and a parameter that scales the marker-based relationship matrix can handle the issue of compatibility between marker-based and pedigree-based relationship matrices. The full log-likelihood function used for parameter inference contains two terms. The first term is the REML-log-likelihood for the phenotypes conditional on the observed marker genotypes, whereas the second term is the log-likelihood for the observed marker genotypes. Analyses of the two simulated datasets with this new method showed that 1) the parameters involved in adjusting marker-based and pedigree-based relationship matrices can depend on both observed phenotypes and observed marker genotypes and 2) a strong association between these two parameters exists. Finally, this method performed at least as well as a method based on adjusting the marker-based relationship matrix.

CONCLUSIONS

Using the full log-likelihood and adjusting the pedigree-based relationship matrix to be compatible with the marker-based relationship matrix provides a new and interesting approach to handle the issue of compatibility between the two matrices in single-step genetic evaluation.

Optimized diffusion of buck semen for saving genetic variability in selected dairy goat populations

Background

Current research on quantitative genetics has provided efficient guidelines for the sustainable management of selected populations: genetic gain is maximized while the loss of genetic diversity is maintained at a reasonable rate. However, actual selection schemes are complex, especially for large domestic species, and they have to take into account many operational constraints. This paper deals with the actual selection of dairy goats where the challenge is to optimize diffusion of buck semen on the field. Three objectives are considered simultaneously: i) natural service buck replacement (NSR); ii) goat replacement (GR); iii) semen distribution of young bucks to be progeny-tested. An appropriate optimization method is developed, which involves five analytical steps. Solutions are obtained by simulated annealing and the corresponding algorithms are presented in detail.

Results

The whole procedure was tested on two French goat populations (Alpine and Saanen breeds) and the results presented in the abstract were based on the average of the two breeds. The procedure induced an immediate acceleration of genetic gain in comparison with the current annual genetic gain (0.15 genetic standard deviation unit), as shown by two facts. First, the genetic level of replacement natural service (NS) bucks was predicted, 1.5 years ahead at the moment of reproduction, to be equivalent to that of the progeny-tested bucks in service, born from the current breeding scheme. Second, the genetic level of replacement goats was much higher than that of their dams (0.86 unit), which represented 6 years of selection, although dams were only 3 years older than their replacement daughters. This improved genetic gain could be achieved while decreasing inbreeding coefficients substantially. Inbreeding coefficients (%) of NS bucks was lower than that of the progeny-tested bucks (-0.17). Goats were also less inbred than their dams (-0.67).

Conclusions

It was possible to account for complex operational constraints while developing goat selection schemes, both efficient and sustainable. Therefore, the recommended selection and mating decisions might receive attention from goat breeders using both AI and NS.