The main occupation o f the theory is to predict breeding values o f individuals, estimate heritabilities o f traits, and to predict selection responses. 1,2 Its main methodological tools are matrix algebra, linear models and variance component partitioning. The phenotypic variance (Vp) o f a given metric trait is often in textbooks split into additive genetic variance ( Va) , domi= nance genetic variance (Vo), epistatic or interaction variance (Vj) and environmental variance (V f). Conceptually, modern quantitative genetic theory has not in principle moved beyond what may be developed from a single locus model with two alleles, one dominant and one

recessive, in a random m ating popu lation . 1,5 Th e traditional way o f developing this mathematical-statistical machinery is to start from the concepts o f additive and dominant gene actions, introduce a linear approximation in the form o f a least-squares regression o f genotypic value on gene content in the single-locus case, define the statistically motivated terms average effect (or breeding value; A) and dominance deviation (D) (Fig. la), and from this develop expressions for the variances o f A (Va) and D (Vp), and heritability o f the trait, as a function o f allele frequency and the degree o f dominant gene action (d) (Fig. lb ). This in turn allows establishment o f the highly instrumental allele frequency-independent relationship between the performance covariance o f relatives and the additive variance Va (COV(Offspring Par= ent) =1/2 Va) (see ref. 2 for a systematic presentation). By assuming random mating and inde= pendent segregation o f loci over several generations, the single-locus results are valid for the multilocus case without any further theoretical development. From this foundation, expres= sions for epistatic variance (Vj) can also be developed.2,6,7

In a keynote address for a 1959 Cold Spring Harbor Symposium, Ernst Mayr criticized quantitative genetics models as grossly oversimplified “beanbag genetics” , arguing that they naively treated each individual gene as an independent unit (see ref. 8 for a detailed historical exposition). Even though this is not stricdy true, it is fair to say that the theory, as it is used within production biology and evolutionary biology, is to a large degree based on the concep= tion that a very large number o f alleles are responsible for the observed genetic variation o f a trait (the infinitesimal model), and that these alleles behave as discrete elements whose effects can be summed within and between loci. In fact, as long as the effect from each gene is pre= sumed to be small it does not really matter in operational terms whether interactions within and between some o f these genes are acknowledged or not. For this reason I think that as long as quantitative genetics theory is based on the infinitesimal model as such, it qualifies to be characterized as beanbag genetics despite that Mayr s reasons for coining the term may be questioned. (O. Kempthorne, one o f the founders o f quantitative genetics, has in fact also made use o f the term to characterize the theory) .9