Optimum contributions: The problem

Utilisation and conservation of farm animal genetic resources 161

Chapter 7. Genetic contributions and inbreeding

Box 7.3. Expected contributions in a population with one sex. he results on predicting gene low and ΔF are easiest to follow in the case of a single sexed population with random mating and mass selection, where a population of T individuals are created by N selected parents and the trait selected for has heritability h 2 with breeding value for individual i denoted A i . In mass selection the breeding value is a selective advantage that is inherited in part by the ofspring. Let r i be the long-term contribution of an ancestor i, then ignoring the case of seling to simplify: r i = ½ Σ j ofspring r j To calculate the expected gene low from an individual, it is necessary to calculate μ i = E[r i | A i ] and here a linear model is itted so μ i = α+βA i −A bari . To calculate this expectation it is simplest to consider irst E[r i | A i ,n i ] = ½ n i E[r j | A i ] where n i is the number selected from the progeny of i. he transfer of selective advantage across generations can be modelled by A j −A barj = πA i −A bari where A bar. is the mean breeding value of the whole selected group in a generation, so that E[r j | A i ] = α+βπA i −A bari ; this assumes a near-equilibrium over generations that will occur ater a small number of generations. Furthermore superior ancestors will have more ofspring selected, so although the average number selected per parent of a diploid species will be 2, it is modelled better by the linear approximation 21+λA i −A bari . herefore again assuming the near equilibrium α+βA i −A bari = ½.2.1+λA i −A bari α+βπ A i −A bari , which allows expression of β in terms of α by equating terms in A i −A bari i.e. β=1−π -1 λα. It is easily seen that on average the selected parents must have equal contributions so α=N -1 ; standard selection theory gives λ = iσ P -1 where i is the intensity of selection and σ P is the standard deviation; also π = ½1-kh 2 where k is the variance reduction coeicient; and consequently β = 2iN -1 1+kh 2 -1 . Note that the value of the selective advantage is reduced by more than half in selection since π ½ because of the increased competitiveness of the other parents. herefore μ i = N -1 [1+2i1+kh 2 -1 σ P -1 A i −A bari ] From this expression of μ i , Woolliams and Bijma 2000 show that ΔF = ½ E[μ i 2 ] assuming Poisson litter sizes. herefore for the single sex population in mass selection, ΔF = 2N -1 [1+4i 2 1+kh 2 -2 h 2 1-kh 2 ] ignoring terms of ON -2 , since ater selection VarA i −A bari = h 2 σ P 2 1-kh 2 . he power of this result is that it requires only the mean conditional on the selective advantages to be modelled, which can be done for a wide class of genetic structures using the methods of Woolliams et al. 1999. hese were developed into predictive formulae for mass selection by Bijma et al. 2000, and for truncation selection on BLUP by Bijma and Woolliams 2000. 162 Utilisation and conservation of farm animal genetic resources John Woolliams measured precisely by phenotype. he graph also shows how ΔF increases with family size, since larger family sizes permit greater intensity of selection: note that only mass selection is shown in Figure 7.4, but for the same family size ΔF with truncation on BLUP will remain much greater than for mass selection until h 2 approaches 1. Note also, a point not shown in Figure 7.4, that the relationship between ΔF and ΔG is non-linear in these truncation selection schemes: for a given h 2 , moving from random selection to mass selection achieves genetic gain but increases ΔF, but moving from mass selection to truncation on BLUP increases gain relatively little but ΔF dramatically.

6. Guidelines for best practice

Breeding schemes may be broadly classiied into two groups: a those with sophisticated extensive pedigree available and where BLUP is used for evaluations; and b schemes which are less sophisticated or are limited in their scope to accumulate full pedigrees on ofspring e.g. aquaculture of some ish species. he former schemes can have high ΔF if the estimates of breeding value are used naively, since their additional accuracy comes primarily through the use of information on relatives. As a consequence there is an increased chance of co-selection of relatives such as full-sibs, creating inappropriate variance in the long-term contributions of the parents. However, using selection algorithms such as the optimisation of contributions described in paragraph 4. allows ΔF to be managed explicitly at sustainable levels whilst retaining the primary beneit of BLUP, namely that it provides the best estimates of breeding value. Schemes suiciently sophisticated to use BLUP are suiciently sophisticated to utilise optimum contributions. his may not be an option in the less sophisticated schemes but note: optimum contributions can be utilised with breeding values estimated from phenotype alone, therefore guidance from predictions is most necessary in practice for those breeding schemes using mass selection. FAO 1998 present simple recommendations for numbers of parents to achieve Ne = 50, i.e. ΔF = 0.01, in mass selection or simpler breeding schemes with the aid of 3 scenarios: 1 selection strictly within families; 2 selection that is strictly random a dangerous assumption; and 3 mass selection with h 2 = 0.4. he value of 0.4 was chosen since it is at the let edge of the plateau in the relationship between h 2 and ΔF, and would be considered as a high heritability in practice. It is always safest to plan with case 3. Recommendations for these scenarios, based on Equation 6 of Bijma et al. 2000, are given Table 7.2 but in a slightly diferent format from FAO 1998. Table 7.2 shows the minimum number of sires required to achieve Ne = 50 in a generation, for a wide range of mating ratios and lifetime family sizes for a female. In the table it is assumed that the number of dams per male is always 1 or more, that mating is hierarchical a conservative assumption, and that generations are discrete. An