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
\[\theta_{H}=2H\ -\ 1\]
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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).
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