Box 1. Hybrid genetic effects Definition of heterosis Within recent ecological and evolutionary studies, the term “heterosis” is mostly used to imply beneficial outcomes of outcrossing, analogous to the case of increased yield in crops. However, definitions vary across the literature regarding direction and degree of deviation from the mid- or maximum parental value (e.g. Hayes 1946, Stern 1948). In line with our proposed goal of bridging the research agendas on positive and negative hybrid fitness effects, we use “heterosis” to mean a deviation from the mid-parental value (“mid-parent heterosis”) in either direction. Whenever relevant, we further qualify heterosis as mid- vs. best-parent heterosis, indicating the mid- or maximal parental value for reference, or positive vs. negative heterosis, indicating deviations above or below the expected value, respectively. Predicting outcomes of outbreeding Line-cross theory was developed to predict outcomes of outbreeding between inbred (“pure”) parental lines as the result of underlying genetic effects, with the ambition to judiciously cross lines to maximise key traits of economic value in hybrid offspring (Cockerham 1954, Lynch 1991, Lynch & Walsh 1998, Mather & Jinks 1982, Zeng et al. 2005). Line-cross theory predicts outcomes of outbreeding from composite genetic effects (i.e. overall genetic effect across all genes considered), including additive (α1), dominance (δ1) and epistatic genetic effects (αq and δq indicate additive-by-additive and dominance-by-dominance epistatic interactions between loci, respectively, and αqδq indicates additive-by-dominance epistatic interactions; q indicates the number of loci and q>1). Coefficients for each of these composite genetic effects are derived for each parental and filial generation, taking the mean phenotype of a specific generation as the point of reference. We focus on the F2-metric model (Cockerham 1954), which takes the mean trait value for the F2s (μ0) as the reference (but see e.g. Mather & Jinks 1982 for an alternative point of reference). The coefficients for each filial generation can be estimated from the hybridity (θH) and source (θS) indices which, in turn, are estimated from the proportion of homozygote (i.e. fixed) divergent sites in each of the parental lines (p1 for P1 and p2 for P2), out of a total difference of “d” substitutions between the two parental populations (Lynch 1991). The mean trait value (μ) across parental and filial generations is, then: \[\mu\ =\ 1\ast\mu_{0}\ +\ \theta_{S}\ast\alpha_{1}\ +\ \theta_{H}\ast\delta_{1}\ +\ \theta_{S}^{2}\ast\alpha_{2}\ +\ \theta_{S}\theta_{H}\ast\alpha_{1}\delta_{1}\ +\ \theta_{H}^{2}\ast\delta_{2}\ \] Where: \[\theta_{S}=2S\ -\ 1\] and \[\theta_{H}=2H\ -\ 1\]
Following Lynch’s (1991) notations, where pm and pf indicate the expected fraction of P1 alleles in the sire and dam, respectively, S and H are calculated as: \[S=\ \frac{\left(p_{m}\ +\ p_{f}\right)}{2}\] and \[H=\ p_{f}\left(1-p_{m}\right)+p_{m}\left(1-p_{f}\right)\] Therefore, S indicates the expected fraction of P1 alleles in an offspring, while H indicates the probability of heterozygosity (i.e. one P1 allele and one P2 allele at a locus). θS ranges from +1 when all alleles derive from P1 (S=1) to -1 when all alleles derive from P2 (S=0). θH ranges from -1 when individuals have only P1 or only P2 alleles (H=0) to +1 when individuals are outbred at all loci (H=1; e.g. F1s). This equation predicts the mean phenotype of the filial generations (Fig. 3), under the assumptions of population allele frequencies of 0.5, with Hardy-Weinberg and linkage equilibrium (Cockerham 1954, Lynch & Walsh 1998; see Zeng et al. 2005 and citing literature for generalizations independent of allele frequencies and linkage disequilibrium).