Computation of the inverse additive relationship matrix for autopolyploid and multiple-ploidy populations.

KEY MESSAGE: Rules to generate the inverse additive relationship matrix (A -1 ) are defined to enable the adoption restricted maximum likelihood (REML) and best linear unbiased prediction (BLUP) in autopolyploid populations with multiple ploidy levels. Many important agronomic, horticultural, ornamental, forestry, and aquaculture species are autopolyploids. However, the adoption of restricted maximum likelihood (REML), for estimating co/variance components, and best linear unbiased prediction (BLUP), for predicting breeding values, has been hampered in autopolyploid breeding by the absence of an appropriate means of generating the inverse additive relationship matrix (A -1 ). This paper defines rules to generate the A -1 of autopolyploid populations comprised of individuals of the same or different ploidy-levels, including populations exhibiting (1) odd-numbered ploidy levels (e.g. triploids), (2) sex-based differences in the probability that gametic genes are identical by descent and (3) somatic chromosome doubling. Inbreeding, due to double reduction, in autopolyploid founders in the absence of mating among relatives is also accounted for. A previously defined approach is modified, whereby rules are initially defined to build an inverse matrix of kinship coefficients (K -1 ), which is then used to generate A -1 . An R package (polyAinv; https://github.com/mghamilton/polyAinv ) to implement these rules has been developed and examples of analyses provided. The adoption of REML and BLUP methods made possible by these new rules has the potential to provide further insights into the quantitative genetic architecture of autopolyploid and multiple-ploidy populations, improve estimates of breeding values, and increase genetic gains made through recurrent selection.

Genetic control of interactions among individuals: contrasting outcomes of indirect genetic effects arising from neighbour disease infection and competition in a forest tree.

Indirect genetic effects (IGEs) are heritable effects of individuals on trait values of their conspecifics. IGEs may substantially affect response to selection, but empirical studies on IGEs are sparse and their magnitude and correlation with direct genetic effects are largely unknown in plants. Here we used linear mixed models to estimate genetic (co)variances attributable to direct and indirect effects for growth and foliar disease damage in a large pedigreed population of Eucalyptus globulus. We found significant IGEs for growth and disease damage, which increased with age for growth. The correlation between direct and indirect genetic effects was highly negative for growth, but highly positive for disease damage, consistent with neighbour competition and infection, respectively. IGEs increased heritable variation by 71% for disease damage, but reduced heritable variation by 85% for growth, leaving nonsignificant heritable variation for later age growth. Thus, IGEs are likely to prevent response to selection in growth, despite a considerable ordinary heritability. IGEs change our perspective on the genetic architecture and potential response to selection. Depending on the correlation between direct and indirect genetic effects, IGEs may enhance or diminish the response to natural or artificial selection compared with that predicted from ordinary heritability.

Use of the numerator relationship matrix in genetic analysis of autopolyploid species.

Mixed models incorporating the inverse of a numerator relationship matrix (NRM) are widely used to estimate genetic parameters and to predict breeding values in animal breeding. A simple and quick method to directly calculate the inverse of the NRM has been historically developed for diploid animal species. Mixed models are less used in plant breeding partly because the existing method for diploids is not applicable to autopolyploid species. This is because of the phenomenon of double reduction and the possibility that gametes carry alleles which are identical by descent. This paper generalises the NRM and its inverse for autopolyploid species, so it can be easily incorporated into their genetic analysis. The technique proposed is to first calculate the kinship coefficient matrix and its inverse as a precursor to calculating the NRM and its inverse. This allows the NRM to be calculated for populations containing individuals of mixed ploidy levels. This generalization can also accommodate uncertain parentage by generating the "average" relationship matrix. The possibility that non-inbred parents can produce inbred progeny (double reduction) is also discussed. Rules are outlined that are applicable for any level of ploidy. Examples of use of the matrix are provided using simulated pedigrees.