Mendelian sampling covariability of marker effects and genetic values
- Sarah Bonk^{1},
- Manuela Reichelt^{1},
- Friedrich Teuscher^{1},
- Dierck Segelke^{2} and
- Norbert Reinsch^{1}Email author
Received: 10 June 2015
Accepted: 7 April 2016
Published: 23 April 2016
Abstract
Background
Measures of the expected genetic variability among full-sibs are of practical relevance, such as in the context of mating decisions. An important application field in animal and plant breeding is the selection and allocation of mates when large or small amounts of genetic variability among offspring are desired, depending on user-specific goals. Estimates of the Mendelian sampling variance can be obtained by simulating gametes from parents with known diplotypes. Knowledge of recombination rates and additive marker effects is also required. In this study, we aimed at developing an exact method that can account for both additive and dominance effects.
Results
We derived parent-specific covariance matrices that exactly quantify the within-family (co-)variability of additive and dominance marker effects. These matrices incorporate prior knowledge of the parental diplotypes and recombination rates. When combined with additive marker effects, they allow the exact derivation of the Mendelian sampling (co-)variances of (estimated) breeding values for several traits, as well for the aggregate genotype. A comparative analysis demonstrated good average agreement between the exact values and the simulation results for a practical dataset (74,353 German Holstein cattle).
Conclusions
The newly derived method is suitable for calculating the exact amount of intra-family variation of the estimated breeding values and genetic values (comprising additive and dominance effects).
Background
The degree of genetic variability among full-sibs is known as Mendelian sampling variance. This variability is due to the inheritance of random samples of alleles from both parents. For a quantitative trait, the amount of this variability depends on the parental degree of heterozygosity, \(1-F_{{\mars}}\), where \(F_{{\mars}}\) (\(F_{{\venus}}\)) is the inbreeding coefficient of an individual’s sire (dam), which is derived from the pedigree. Under additivity and with unlinked loci, the Mendelian sampling variance is the sum of two parental contributions, \(\frac{1}{4}\sigma ^2_a (1-F_{{\mars}})+ \frac{1}{4}\sigma ^2_a (1-F_{{\venus}})\), where \(\sigma ^2_a\) is the additive genetic variance [1]. The latter expression is of general importance in quantitative genetics, especially in the context of estimating genetic parameters and in genetic evaluations. In certain models (e.g. [2, 3]), it is used explicitly for the relative weighting of observations from progeny of inbred versus non-inbred parents. Moreover, the inverse Mendelian sampling variance plays a pivotal role in direct inversion of the numerator relationship matrix [4].
New methods to track Mendelian sampling variance are based on the availability of phased genotypes (diplotypes) at genetic markers across the genome as a byproduct of genomic selection (e.g. [5, 6]). Single nucleotide polymorphism (SNP) diplotypes of parents differ in terms of three features: the degree of heterozygosity, the genotypes at homozygous loci, and the linkage phase between loci. All of these features have consequences for the variability of gametes that are generated by a particular individual, and thus for the variance among the progeny in a family. A small within-family genetic variation contributes to phenotypic uniformity, which is desired e.g. for birth-weight of piglets (e.g. [7]), while a large Mendelian sampling variability may increase selection opportunity between sibs [6]. When phased genotypes are available, it is possible to simulate a large sample of the population of gametes of a selection candidate by considering recombination within chromosomes, as was demonstrated in a study on 58,035 Holsteins [5]. However, to the best of our knowledge, exact formulae for calculating the Mendelian sampling variance from phased genotypes have not been reported previously.
In this study, we provide the requisite formulae for the exact calculation of within-family genetic variation. The within-family covariance matrix between the additive and dominance effects of all markers can be derived exactly from phased SNP-genotypes and the known genetic distances between markers. Conversion to the within-family variance of (estimated) additive and dominance values for a trait is then achieved via the (estimated) additive and dominance marker effects. We provide a comparison between the results obtained by simulations and the exact method, as well as a brief discussion of the application of this method to the allocation of mates.
Methods
In the following, a breeding population is assumed, where phased SNP-genotypes are available for all potential mating partners, as well as estimates of the SNP-effects for all traits. Furthermore, it is assumed that the genetic distances between all markers are known in terms of their recombination rates, which are summarized in a comprehensive genetic map for all SNPs.
Subsequently we demonstrate that this additive property of the covariance matrix vanishes when dominance marker effects are included.
Definitions and the case of pure additivity
Ten classes of parental diplotypes with different two-locus genotypes and distributions of produced gametes
Parental diplotype | Genotype indicators | Probabilities of gametes | Characterizing parameters | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(c_{a,i}\) | \(c_{a,j}\) | \(c_{d,i}\) | \(c_{d,j}\) | \(p_{A-A}\) | \(p_{A-B}\) | \(p_{B-A}\) | \(p_{B-B}\) | \(D_{i,j}\) | \(p_{i}\) | \(p_{j}\) | |
\(\begin{array}{c} A-A \\ A-A \end{array}\) | 1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
\(\begin{array}{c} A-A \\ A-B \end{array}/\begin{array}{c} A-B \\ A-A \end{array}\) | 1 | 0 | −1 | 1 | \(\frac{1}{2}\) | \(\frac{1}{2}\) | 0 | 0 | 0 | 1 | \(\frac{1}{2}\) |
\(\begin{array}{c} A-A \\ B-A \end{array}/\begin{array}{c} B-A \\ A-A \end{array}\) | 0 | 1 | 1 | −1 | \(\frac{1}{2}\) | 0 | \(\frac{1}{2}\) | 0 | 0 | \(\frac{1}{2}\) | 1 |
\(\begin{array}{c} A-B \\ A-B \end{array}\) | 1 | −1 | −1 | −1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
\(\begin{array}{c} B-A \\ B-A \end{array}\) | −1 | 1 | −1 | −1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
\(\begin{array}{c} A-A \\ B-B \end{array}/\begin{array}{c} B-B \\ A-A \end{array}\) | 0 | 0 | 1 | 1 | \(\frac{1-\theta _{i,j}}{2}\) | \(\frac{\theta _{i,j}}{2}\) | \(\frac{\theta _{i,j}}{2}\) | \(\frac{1-\theta _{i,j}}{2}\) | \(\frac{1-2\theta _{i,j}}{4}\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) |
\(\begin{array}{c} A-B \\ B-A \end{array}/\begin{array}{c} B-A \\ A-B \end{array}\) | 0 | 0 | 1 | 1 | \(\frac{\theta _{i,j}}{2}\) | \(\frac{1-\theta _{i,j}}{2}\) | \(\frac{1-\theta _{i,j}}{2}\) | \(\frac{\theta _{i,j}}{2}\) | \(-\frac{1-2\theta _{i,j}}{4}\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) |
\(\begin{array}{c} A-B \\ B-B \end{array}/\begin{array}{c} B-B \\ A-B \end{array}\) | 0 | −1 | 1 | −1 | 0 | \(\frac{1}{2}\) | 0 | \(\frac{1}{2}\) | 0 | \(\frac{1}{2}\) | 0 |
\(\begin{array}{c} B-A \\ B-B \end{array}/\begin{array}{c} B-B \\ B-A \end{array}\) | −1 | 0 | −1 | 1 | 0 | 0 | \(\frac{1}{2}\) | \(\frac{1}{2}\) | 0 | 0 | \(\frac{1}{2}\) |
\(\begin{array}{c} B-B \\ B-B \end{array}\) | −1 | −1 | −1 | −1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
In Table 1 the probability of the appearance of allele A at the ith locus is denoted by \(p_i\). Note that this definition applies to the diplotypes as well as to the gametes of the parent. The values for \(p_i\) and \(p_j\) are given in columns 11 and 12 of Table 1. All entries in Table 1 apply to both the sire (upper index \({\mars}\)) and the dam (upper index \({\venus}\)).
Two-locus genotype probabilities in a full-sib family
L1 | L2 | Diplotypes | Probabilities |
---|---|---|---|
BB | BB | \(\begin{array}{c} B-B \\ B-B \end{array}\) | \(p^{{\mars}}_{B-B}p^{{\venus}}_{B-B}\) |
AB/BA | \(\begin{array}{c} B-A \\ B-B \end{array}/\begin{array}{c} B-B \\ B-A \end{array}\) | \(p^{{\mars}}_{B-A}p^{{\venus}}_{B-B}+p^{{\mars}}_{B-B}p^{{\venus}}_{B-A}\) | |
AA | \(\begin{array}{c} B-A \\ B-A \end{array}\) | \(p^{{\mars}}_{B-A}p^{{\venus}}_{B-A}\) | |
AB/BA | BB | \(\begin{array}{c} A-B \\ B-B \end{array}/\begin{array}{c} B-B \\ A-B \end{array}\) | \(p^{{\mars}}_{A-B}p^{{\venus}}_{B-B}+p^{{\mars}}_{B-B}p^{{\venus}}_{A-B}\) |
AB/BA | \(\begin{array}{c} A-A \\ B-B \end{array}/\, \begin{array}{c} B-B \\ A-A \end{array}/\, \begin{array}{c} A-B \\ \, B-A \end{array}/\,\begin{array}{c} B-A \\ \, A-B \end{array}\) | \(p^{{\mars}}_{A-A}p^{{\venus}}_{B-B}+p^{{\mars}}_{B-B}p^{{\venus}}_{A-A}+p^{{\mars}}_{A-B}p^{{\venus}}_{B-A}+p^{{\mars}}_{B-A}p^{{\venus}}_{A-B}\) | |
AA | \(\begin{array}{c} A-A \\ B-A \end{array}/\begin{array}{c} B-A \\ A-A \end{array}\) | \(p^{{\mars}}_{A-A}p^{{\venus}}_{B-A}+p^{{\mars}}_{B-A}p^{{\venus}}_{A-A}\) | |
AA | BB | \(\begin{array}{c} A-B \\ A-B \end{array}\) | \(p^{{\mars}}_{A-B}p^{{\venus}}_{A-B}\) |
AB/BA | \(\begin{array}{c} A-A \\ A-B \end{array}/\begin{array}{c} A-B \\ A-A \end{array}\) | \(p^{{\mars}}_{A-A}p^{{\venus}}_{A-B}+p^{{\mars}}_{A-B}p^{{\venus}}_{A-A}\) | |
AA | \(\begin{array}{c} A-A \\ A-A \end{array}\) | \(p^{{\mars}}_{A-A}p^{{\venus}}_{A-A}\) |
Two-locus genotype probabilities as functions of characteristic parameters
L1 | L2 | Probabilities |
---|---|---|
BB | BB | \([D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i)(1- p^{{\mars}}_j)][D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i)(1- p^{{\venus}}_j)]\) |
AB/BA | \([-D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i) p^{{\mars}}_j] [D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i)(1- p^{{\venus}}_j)]+[D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i)(1- p^{{\mars}}_j)][-D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i) p^{{\venus}}_j]\) | |
AA | \([-D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i) p^{{\mars}}_j][-D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i) p^{{\venus}}_j]\) | |
AB/BA | BB | \([-D^{{\mars}}_{i,j}+p^{{\mars}}_i (1-p^{{\mars}}_j)][D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i)(1- p^{{\venus}}_j)]+[D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i)(1- p^{{\mars}}_j)][-D^{{\venus}}_{i,j}+p^{{\venus}}_i (1-p^{{\venus}}_j)]\) |
AB/BA | \(\begin{aligned}&[D^{{\mars}}_{i,j}+p^{{\mars}}_i p^{{\mars}}_j][D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i)(1- p^{{\venus}}_j)]+[D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i)(1- p^{{\mars}}_j)][D^{{\venus}}_{i,j}+p^{{\venus}}_i p^{{\venus}}_j]\\&\quad+[-D^{{\mars}}_{i,j}+p^{{\mars}}_i (1-p^{{\mars}}_j)][-D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i) p^{{\venus}}_j]+[-D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i) p^{{\mars}}_j][-D^{{\venus}}_{i,j}+p^{{\venus}}_i (1-p^{{\venus}}_j)]\end{aligned}\) | |
AA | \([D^{{\mars}}_{i,j}+p^{{\mars}}_i p^{{\mars}}_j][-D^{{\venus}}_{i,j}+(1-p^{{\venus}}_i) p^{{\venus}}_j]+[-D^{{\mars}}_{i,j}+(1-p^{{\mars}}_i) p^{{\mars}}_j][D^{{\venus}}_{i,j}+p^{{\venus}}_i p^{{\venus}}_j]\) | |
AA | BB | \([-D^{{\mars}}_{i,j}+p^{{\mars}}_i (1-p^{{\mars}}_j)][-D^{{\venus}}_{i,j}+p^{{\venus}}_i (1-p^{{\venus}}_j)]\) |
AB/BA | \([D^{{\mars}}_{i,j}+p^{{\mars}}_i p^{{\mars}}_j][-D^{{\venus}}_{i,j}+p^{{\venus}}_i (1-p^{{\venus}}_j)]+[-D^{{\mars}}_{i,j}+p^{{\mars}}_i (1-p^{{\mars}}_j)][D^{{\venus}}_{i,j}+p^{{\venus}}_i p^{{\venus}}_j]\) | |
AA | \([D^{{\mars}}_{i,j}+p^{{\mars}}_i p^{{\mars}}_j][D^{{\venus}}_{i,j}+p^{{\venus}}_i p^{{\venus}}_j]\) |
The variance \(\text {var}(c_{a,i})\) becomes zero at loci for which both parents are homozygous. The corresponding rows and columns of the covariance matrix only contain zeroes, which causes a rank deficiency. The corresponding diagonal and off-diagonal elements in the correlation matrix R are defined as zero (although they are not defined in a strictly mathematical sense) for our purposes in order to maintain rank equality between \({\varvec{\Omega }}\) and R.
Joint additive and dominance genetic effects
Mendelian covariance with multiple traits
This quantity has a pivotal role in mating decisions because the total breeding value (defined as the linear combination of single breeding values \({\mathbf {f}}^{\prime }{\mathbf {t}}\), \({\mathbf {t}}=(t_1,\ldots ,t_N)\)) is the most important criterion for selection.
Practical application
We compared the exact method with a recently published simulation approach [5]. The Mendelian sampling variance of gametes was calculated with both methods for each animal from a dataset that included the diplotypes of 74,353 male and female German Holstein cattle. Identical sets of recombination rates and estimates of additive marker effects were used for both methods. These parameters were derived from routine genomic evaluation data. This comparison was done for four traits: fat yield (FKG), protein yield (PKG), somatic cell score (SCS), and the direct genetic effect on stillbirth (SBd).
The Mendelian covariances between traits were also obtained using the exact method by applying Eq. 29. Furthermore, the four traits were combined with weights of 0.1, 0.4, 0.375, and 0.175 for the traits FKG, PKG, SCS, and SBd, respectively, and the (gametic) Mendelian variances were computed for this aggregate genotype. The covariances and aggregate genotypes were not implemented in the simulation method, so no comparisons could be made with the simulation method for these quantities.
Results
The coefficient of variation of the Mendelian variances of the aggregate genotype was 18.7 %, which was similar to the coefficient of variation of the Mendelian variances of PKG (19.0 %), SCS (20.2 %), and SBd (21.5 %), but somewhat smaller than that of FKG (30.3 %).
Discussion
Fields of application and computational aspects
In general, the method described in this study can be applied to all diploid animals and plants. Of course, all relevant input parameters must be known, such as the marker maps, marker effects, and phased genotypes. Crosses of double haploid (i.e. fully inbred) lines occur as parents in breeding programs for plant species such as e.g. maize. An advantage of such parents is that they provide reliable diplotypes because of the complete homozygosity of genotyped grandparents, whereas the derivation of diplotypes is prone to some degree of phasing error in non-inbred populations [10]. In cases where the phase of some SNPs is only known at a probabilistic level, it may be an option to average the Mendelian sampling variances over all possible linkage phases. Simplifications may be possible, e.g. by taking only the most probable diplotypes into account. However, we did not investigate this question in detail.
For humans and mice, it has been found that marker maps generally differ for male and female parents [11]. All the covariances can be adjusted easily for sex-specific recombination rates, which is achieved most easily in the pure additive case by applying the male and female recombination rates to set up \({\varvec{\Omega }}^{{\mars}}\) and \({\varvec{\Omega }}^{{\venus}}\). In the general case, the LD measures for both the paternal and maternal gametes \(D^{{\mars}}\) and \(D^{{\venus}}\) must be adjusted in order to obtain the adjusted covariances.
The sex of the full sibs matters for the inclusion of sex chromosomes because when all considered progeny are female, the sire can be treated as homozygous at all X-chromosomal loci and the calculation can proceed as usual. When the focus is on male progeny, such as young bulls obtained from elite matings, there is no X-chromosomal paternal contribution to the Mendelian sampling variance. Of course, dominance has no effect on the X-chromosomal Mendelian variance in males, unlike for females.
From a computational viewpoint, the purely additive case is most convenient because the parental contributions to the Mendelian variances and covariances can be calculated for a large list of potential parents and all traits by setting up the parent-specific covariance matrix. Subsequently, the parental contributions only have to be added to the total within-family variance for each mating considered. The computational time required by the exact method was roughly the same as that for the simulation approach. However, the computational demand would increase for the latter case if the Monte Carlo error needs to be reduced further.
Population-averaged Mendelian sampling variances for single traits were previously derived from large numbers of phased genotypes and available estimates of additive marker effects by Cole and VanRaden [6]. Neither simulation of gametes nor covariance matrices were used in this study since loci on the same chromosome were either assumed to be perfectly linked or fully independent. Consequently, their respective results can only be interpreted as upper and lower limits. In another study, Segelke et al. [5] took recombination within chromosomes into account by simulating gametes of individuals with known diplotypes. Parental contributions to the within-family additive genetic variance were expressed as standard deviations of gamete breeding values in a family-specific manner.
Consideration of the aggregate genotype calls for a full Mendelian sampling covariance matrix across traits, which, in the additive case, can also be derived by simulation, but this has not yet been reported in the literature. This requires that genomic breeding values are estimated for each trait of interest and each single simulated gamete and then averages of squares and cross-products are calculated over gametes. If dominance effects are to be included, pairs of paternal and maternal gametes have to be simulated. The simulation-inherent Monte Carlo errors of all single-trait variances and all pair-wise covariances will, of course, propagate and induce a joint Monte Carlo error of the resulting variability in the aggregate genotype.
From a producer’s perspective, phenotypic uniformity of a population of plants or animals is desirable because it facilitates management. Matings with high additive genetic merit and low within-family genetic variance [6] may be attractive to achieve that goal. Dominance—if of some importance for the traits under consideration—could be included for the same purpose. Breeding organizations, in contrast, are probably more interested in offspring with exceptionally high breeding values [6], since e.g. in dairy cattle, semen prices are non-linearly related to the genetic merit of bulls. For a particular mating, the probability that the estimated breeding value of offspring will be greater than a certain threshold can be determined from a normal distribution with family-specific mean and variance. The opportunity to breed the desired animals of top genetic merit can then be maximized by choosing the matings with the highest probabilities among all possible matings, possibly by taking some constraints such as inbreeding into account.
The average observed degree of homozygosity in the German Holstein dataset was 65.3 %, with a range from 25.2 to 88.0 %. These high degrees of homozygosity were exploited for computational speed by deleting the rows and columns for homozygous markers from the covariance matrix and the respective marker effects from \({\hat{\mathbf{m}}}\). Therefore, the dimensions of the remaining vector of marker effects and the remaining covariance matrix were reduced greatly, leading to considerable computational time savings. Note that both parents had to be homozygous in the dominance case in order to reduce matrix dimension in a similar way. Clearly, computational time can be decreased by implementing parallel calculation of individuals and chromosomes. However, in the presence of dominance, each considered mating must be computed with its own covariance matrix, and thus only matings can be parallelized (the chromosomes are unaffected).
Choosing alternative genotype indicators
The formulae for the covariances and correlations are functions of the chosen genotype indicators, for which different options exist [12]. In the present study, we used (1, 0, −1) (Eq. 3) for additive effects and (−1, 1,−1) (Eq. 4) for dominance effects, but other possible indicators include (0, 1, 2) for additive effects and (0, 1, 0) or \(\left( -\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right)\) for dominance effects. All these indicators can be transformed into each other by a shift and/or a multiplication by a constant. Simply shifting the indicators does not influence the (co)variance, so it is also possible to use the formulae described in this study when additive marker effects have been estimated via the (0, 1, 2) coding.
The allele frequencies u and v, which are required for the transformation, are already available because they are required for routine estimation.
Conclusions
In this study, we proposed a new method for the exact calculation of Mendelian sampling (co-)variances based on knowledge of phased marker genotypes and marker effect estimates and we derived all the requisite formulae. The method considers inbreeding but also the absolute level of homozygosity, as indicated by the marker genotypes, while it also considers the linkage phase of the markers in both parents.
We demonstrated the applicability of our method by comparing its results with results produced by an established simulation method using a large dairy cattle dataset. We found that both approaches agreed within the range of the Monte Carlo error, which is inherent in the simulation, but which can be fully avoided because the derived covariance matrices represent an infinitely large number of progeny.
Declarations
Authors' contributions
SB and FT derived the formulae for all the covariances, with the help of NR. FT introduced the use of LD parameters and their clear representation. The theory was implemented in Fortran programs by MR and SB, with some help from NR. DS ran the simulations and the exact calculations at VIT, while SB performed the comparative analysis. The manuscript was drafted by SB, with contributions from all the coauthors. NR was responsible for the general concept and supervised all of the work. All authors read and approved the final manuscript.
Acknowledgements
We thank Friedrich Reinhard (VIT Verden) for fruitful discussions. We thank three anonymous reviewers for their valuable comments. Financial support for SB and MR from BMBF via Phänomics under Grant 0315536G is gratefully acknowledged. The publication of this article was funded by the Open Access fund of the Leibniz Association.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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