156 Utilisation and conservation of farm animal genetic resources
John Woolliams
the breeding value is an aggregation of the individual’s Mendelian term and those of its ancestors, so substituting A for a in Equation 7.2 would result in double counting.
Grundy et al. 1998 predict that as a consequence of Equations 7.1 and 7.2, breeding
schemes optimised to maximise gain for the same rate of inbreeding should allocate long-term contributions of individuals in relation to their estimated Mendelian
sampling term, a prediction conirmed by Avendaño et al. 2004, as described below.
Consequently the target contribution will change over time partly because estimates of genetic merit change, with errors reducing in magnitude as more information becomes
available over time. his is not the only source of uncertainty in desired contribution: even if the breeding value of all individuals is always known with full accuracy, the desired
contribution of an individual parent will change as the genetic values of the ofspring become known, since their contributions cannot be determined independently without
changing the long-term contribution of the parent remember
r
parent
= ½ Σ r
ofspring
.
4.1. Optimum contributions: The problem
his is commonly referenced as Meuwissen 1997, although similar approaches were previously published by other authors see Woolliams
et al. 2002 for a more detailed history. he approach solves the problem of managing diversity in the
course of selection by inding the solution to the following: maximise c
T
g , subject to
ive constraints: ½c
T
Ac ≤ F, c
T
s = ½, c
T
d = ½, h ≤ c and c ≤ m, where c is a vector
of candidate contributions to the next generation, s and d are indicator vectors for
Box 7.1. Mendelian sampling terms. For all autosomal DNA, half the genes come from the sire and half the genes come from the
dam, and moreover the half that passed from each parent to the ofspring are chosen at random. herefore the expected breeding value of the ofspring A
of
is the average of the breeding values of its sire A
sire
and dam A
dam
, i.e. E[A
of
] = ½ A
sire
+ ½ A
dam
, where E[ ] denotes an expectation. Expressing this as a linear regression gives A
of
= ½ A
sire
+ ½ A
dam
+ a, where a is the
deviation of the ofspring from the average of its parents, and is called the Mendelian sampling term, with E[
a] = 0. We can also calculate the variance of a by considering the variance of both sides of the formula for A
of
, with the result that Var[ a] = ½1− ασ
A 2
where α is the deviation from random mating and σ
A 2
is the genetic variance in the base population prior to selection. herefore, for random mating, the Mendelian sampling term makes up ½ the genetic variation in
the base. he Mendelian sampling term arises because the actual alleles passed by each parent will vary from ofspring to ofspring, due to sampling among the two alleles it carries at each locus. he
term is important because it makes each individual unique, not just the average of its parents, and is the source of genetic variance within families, making full-sibs diferent from each other.
Utilisation and conservation of farm animal genetic resources 157
Chapter 7. Genetic contributions and inbreeding
males and females respectively, m and h are upper and lower bounds to contributions respectively. h e constraint c ≤ m is eliminated if no candidate has restriction on
maximum contribution. If no candidate has restriction on minimum contribution h ≤ c becomes 0 ≤ c. F is determined from the group coancestry of the current generation
of parents and the desired ΔF.
4.2. Optimum contributions: The design
h ere is an implicit design underlying the use of optimum contributions see Grundy et
al., 1998; Avendaño et al., 2004. First, in the absence of restrictions and constraints to the contrary, optimum contributions will treat males and females similarly, with equal
expected numbers of parents with the same distribution of contributions in relation to estimated genetic merit. Second, the distribution of contributions has a form shown in
Figure 7.3: a threshold linear relationship with estimated Mendelian sampling term, with the variance about the regression, tightly controlled in contrast to truncation
selection where this variance increases with the square of the mean.
Given the i nding that the estimated Mendelian sampling term is the selective advantage Avendaño
et al., 2004 it is tempting but very mistaken to interpret this as a form of within-family selection. What is occurring is that in each generation, from the
earliest possible opportunity dif erential contributions are being made in relation to the best estimate of the Mendelian sampling term available at the time, so that at all
stages a minimum of selection intensity is wasted between families. For example if one individual is predicted as meriting a greater long-term contribution than another why
give them equal mating proportions in the initial round of selection? As the optimum contributions algorithm becomes more and more restricted by practical constraints,
Estimated sampling term -0.5
0.5 1
Long-term contribution
0.14 0.07
Estimated sampling term -0.5
0.5 1
Long-term contribution
0.14 0.07
Figure 7.3. h e distribution of long-term contribution when gain is maximised but diversity
is managed in relation to estimated Mendelian sampling terms.
158 Utilisation and conservation of farm animal genetic resources
John Woolliams
e.g. on reproductive capacity, this design will become more and more obscured in practice although the principles will remain. It is important to realise that using the
optimum contribution algorithm to guide selection will maximise the gain made given
the current rate of inbreeding that the scheme has i.e. breeding companies cannot lose by implementing the selection algorithm, and by deinition
if it is not implemented the scheme is sub-optimal for gain he implementation of this method is described in
chapter 8.
5. Predicting ΔF
An important part of designing breeding schemes is the ability to predict the impact of events on ΔF. he relationship between ΔF and long-term contributions shown in
Equation 7.1 is limited for prediction since it is a function of observed contributions
and so ofers no predictions for the future, for example what happens if we select more intensely. he problem of predicting ΔF when selection in each generation is
unrelated to pedigree, such as random selection, is straightforward and has long been solved. However when selection is on a trait subject to any form of inheritance the
problem is more complex because each generation of selection cannot be considered independent of the previous generations. his is because the selective advantages that
help an individual to be selected as a parent are passed in part to the ofspring; therefore a genetically better parent is likely to have more ofspring selected, and each selected
ofspring is more likely to produce selected grand-ofspring since the grand-ofspring inherit in part, but to a lesser degree, the advantage of the grandparent. hus the selective
advantage of a parent inluences its long-term contribution over subsequent generations but with diminishing efect.
his problem is overcome by predicting the expected long-term contribution of an individual conditional upon its selective advantage. When selection is based
on phenotype termed mass selection with simple inheritance, the only selective advantage for an individual i is its breeding value A
i
, and μ
i
= E[r
i
given A
i
] or E[r
i
| A
i
]. Woolliams and Bijma 2000 show that for random mating α=0, with Poisson litter
sizes, Equation 7.1 can be replaced by: ΔF = ½ Σ E[μ
i 2
] Eq. 7.3
i.e. the observed contributions can be replaced by expectations providing the coeicient ¼ is replaced ½. For mass selection accurate predictions for ΔF can be obtained by
assuming a linear model for μ
i,
and the derivations of the coeicients in this model are described in Box 7.3, although these are simpliied by assuming single-sexed
diploids. Woolliams et al. 1999 show how μ
i
can be derived for two sexes, overlapping
