Management of diversity and inbreeding when importing new stock into an inbred population

Abstract This article relates to breeding programs that seek to manage genetic diversity. The method maximizes a multicomponent objective function, applicable across breeding scenarios. However, this paper focuses on breeding decisions following immigration of 10 unrelated individuals into a highly inbred simulated population (F ≈ 0.34). We use Optimal Contribution Selection to maximize retention of genetic diversity. However, some treatments add Coancestry Assortative Mating (CAM). This helps to avoid early dilution of immigrant genetic material, maximizing its ability to contribute to genetic diversity in the longer term. After 20 generations, this resulted in considerably increased genetic diversity, with mean coancestries 59% of what random pairing gave. To manage progeny inbreeding, common practice is to reject matings above an upper limit. As a suboptimal rules-based approach, this resulted in 26% decreased genetic diversity and 8% increased inbreeding in the long term, compared with random pairing. In contrast, including mean progeny inbreeding as a continuous variable in the overall objective function decreased final inbreeding by 37% compared with random pairing. Adding some emphasis on selection for a single trait resulted in a similar pattern of effects on coancestry and inbreeding, with 12% higher trait response under CAM. Results indicate the properties of alternative methods, but we encourage users to do their own investigations of particular scenarios, such as including inbreeding depression. Practical implementation of these methods is discussed: they have been widely adopted in domestic animal breeding and are highly flexible to accommodate a wide range of technical and logistical objectives and constraints.

Figures as low as 33% in the top six rows of Table S1 show that OCS without CAM can result in a threefold increase in loss of diversity compared to the wild population. The 59% result for initial ̅ =0.34 is more realistic, but still substantial. An aberrant figure in Table 1 is 132% for mean inbreeding in the top row. This is seen in figure 4 and explained in the main text.
The impact of trait selection to give this result under ̅ =0.34 is damped at ̅ = 0.49 and especially at 0.95, probably because of the more dominant effect of diversity issues, with high variance of coancestries in these populations. The bottom three rows in Table S1 show that placing a limit on progeny inbreeding resulted in mean coancestry and inbreeding between 26% and 102% higher than where an appropriate weighting against mean progeny inbreeding is used. However, the highest figures are influenced by an altered balance between diversity and trait response, as discussed in the main text, such that the 102% increase in inbreeding is associated with at 7% increase in response. Figure S4 shows a slow start to selection response in the trait. With a mean inbreeding coefficient of 0.95 there was little genetic variation at the time of importing 10 new individuals, and it would take some generations of segregation for the new variation that they bring to be available for selection across many candidates. As a less important effect, given unknown pedigree for the immigrants, accuracy of EBVs would take a few generations to build up. Figure S1. Results for treatments targeting genetic diversity alone, and for simulation of initial inbred population Mean inbreeding coefficient Figure S2. Results over years for treatments that place equal emphasis on genetic diversity and genetic gain, and for simulation of initial inbred population to ̅ =0.49.  Example where the Ranked MK Selection algorithm gives an incorrect result.
Consider that a single pair mating is to be made from all the animals in the pedigree shown in Figure S5. Males are odd-numbered and females are even-numbered. Relationships are dictated by the pedigree, with the exception that animals 3 to 10 inclusive are from a single population and are lowly related to each other, with kinships of value k < 2.27% between each pair. Animals 1 and 2 were unrelated immigrants in the previous generation. The optimal solution is to mate male 1 with female 2, as they are the only totally unrelated animals.
However, under Ranked MK selection, animals 1 and 2 are given the lowest priority (also under Static and Dynamic MK selection): they are the first two animals to be placed in the sex-specific lists described in Table 2, because they have some close relatives and they have the highest MK values (8.33%, compared to 4.79% for animals 3 to 10, and 7.94% for animals 11 to 18, at k = 1%. See FigS5Results.xlsx for the worked example). The method used in this paper selects animals 1 and 2 as the single mating pair. Figure S5 has 4 matings per immigrant whereupon MK = ̅ in the main population has to be below 1/44 = 2.27% for animals 1 and 2 to be the first rejected. As shown in FigS5Results.xlsx, this figure increases asymptotically to 1/12 = 8.33% as the number of matings per immigrant increases. This is well below ̅ = 34% generated in the simulations in this paper.
Where more than one mating pair is to be selected, animals 1 and 2 remain the lowest priority

Scaling of objective function components
The impact of weightings applied under the treatments described depends on scaling of the objective function components involved, and this is described here for researchers using a … where C is parental coancestry, with constraint to exclude solutions above TD. In this case, function value ranges 0 at Cmax to 1 at the minimum parental coancestry at the prevailing TD.
This means that for the 90-degree treatments used in this paper, all solutions are <= 90 degrees and so progeny index does not play a role, and low parental coancestry is favoured without constraint.
Progeny inbreeding and progeny coancestry were not scaled, as these are conveniently on a scale of 0 to 1. This means that for treatments involving Fy (weighting y on mean progeny inbreeding) the result for mean progeny inbreeding is simply multiplied by y to give that component of the overall objective function. Figure S6 shows partial pedigree diagrams for the first replicate simulations of treatments