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
Utilisation and conservation of farm animal genetic resources 159
Chapter 7. Genetic contributions and inbreeding
Box 7.2. he relationship between ΔF and contributions. he following will assume for simplicity a single population of diploids with random mating.
Consider a base population of N individuals at time t=0, with 2N alleles considered by convention to be neutral and distinct. Let Q be one such allele, then the allele frequency at time 0 is f
Q
= 2N
-1
. Deine the additive trait for individual i by p
Q
i = 0 when i has no Q alleles, ½ when heterozygous for Q, and 1 when homozygous for Q. In the base population, one individual i has
p
Q
i = ½, whilst the remainder are all zero. In all subsequent generations the frequency of Q can be decomposed into the sum:
f
Q
t = Σ
base individuals j
r
j
0,tA
Q
j + Σ
generation u = 1…t
Σ
individuals j
r
j
u,ta
Q
j where A
Q
j is the breeding value for j for trait p
Q
½ or 0 and a
Q
j is the Mendelian sampling term for j for trait p
Q
. Because this is a unique allele in the base its contribution to inbreeding at time t for random mating, F
Q
t = E[f
Q
t
2
]. F
Q
t has a very simple form, since all the cross product terms have an expectation of 0, so: F
Q
t = E [ Σ
base individuals j
r
j
0,t
2
A
Q
j
2
] + E[ Σ
generation u = 1…t
Σ
individuals j
r
j
u,t
2
a
Q
j
2
] Since the allele is neutral there is no covariance between the r
j 2
and the A
Q
j
2
or a
Q
j
2
, and E[A
Q
j
2
] = 4N
-1
. Further note that since the base is arbitrary in a scheme of constant structure subject to constant selection pressures, then E[Σ
generation
r
j
u,t
2
] is a constant, say E[Σr
j 2
] for all generation from 0 up to the last few where long-term contributions have yet to occur: as
will be seen later this will not afect the proof and so this lack of convergence will be ignored to simplify terms. he calculation of Mendelian sampling variances will not be described, but
at time t, E[a
Q
j
2
] ≈ 8N
-1
1-ΔF
t
with the approximation made here ignoring second order terms. herefore:
F
Q
t = E[Σr
j 2
]4N
-1
+ E[Σr
j 2
] 8N
-1
Σ
generation 1to t
1-ΔF
u
Now note that the inbreeding coeicient at time t is Ft = Σ
base alleles
F
.
t. Since Q was an arbitrary choice Ft is 2NF
Q
t = ¼ E[Σr
j 2
] 1 + Σ
0 to t
1-ΔF
u
. Let t go to ininity then Ft=1, and the sum Σ
0 to t
1-ΔF
u
is ΔF
-1
, so ater arranging terms ΔF = ¼ E[Σr
j 2
] 1+ΔF ≈ ¼ E[Σr
j 2
]. So ΔF is a direct function of ¼ E[Σr
j 2
], and the error in ΔF = ¼ E[Σr
j 2
] is OΔF which is small for all practical purposes. his proof can be extended to two sexes with the same
result Woolliams and Bijma, 2000 and to non-random mating where the factor becomes ¼1-α, where further α relates to the departure from Hardy-Weinberg equilibrium. Note the ¼ arises from
the terms describing Mendelian sampling variance, since it describes the increment in inbreeding due to the contributions uniquely attributable to each individual.
160 Utilisation and conservation of farm animal genetic resources
John Woolliams
generations, and a number of inheritance models, e.g. imprinting, and selection indices. Ronnegard and Woolliams 2003 developed models with maternal efects. Equation
7.3 is then used to develop predictions of ΔF for index selection Woolliams and Bijma, 2000, overlapping generations and mass selection Bijma
et al., 2000 and truncation selection with BLUP Bijma and Woolliams, 2000. hese predictions can be obtained
from sotware such as SelAction Rutten et al., 2002.
Some typical results are shown in Figure 7.4, where comparisons are made of the same breeding scheme structure either using random selection hence no gain, mass
selection or truncation selection with BLUP estimated breeding values. It is clear that mass selection increases ΔF, but to a much lesser degree than truncation selection
on BLUP. However there is a clear diference in the relationship to heritability: for mass selection, ΔF increases as h
2
for the trait selected increases from 0, reaching an approximate plateau when h
2
lies between 0.4 and 0.7 before reducing again; whilst for truncation on BLUP, ΔF decreases steadily, converging with mass selection as h
2
tends to 1 since additional family information is of no value when the breeding value is
0.000 0.005
0.010 0.015
0.020 0.025
0.030 0.035
0.040
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1
Heritability ∆
F
Figure 7.4. Relation of predicted lines and simulated symbols rates of inbreeding ΔF with heritability h
2
for populations with discrete generations, with 20 sires and 20 dams and diferent numbers of ofspring per dam n
o
, assumed ixed within a population, ½n
o
of each sex: −−−, random selection, n
o
=8; solid squares, mass selection, n
o
=8; solid circles, truncation on BLUP, n
o
= 8; open squares, mass selection, n
o
= 32. Based upon Bijma et al. 2000 and Bijma and Woolliams 2000.
