Design
We used stochastic simulation to estimate rates of genetic gain \((\Delta \text{G})\) realised for PWS at 1% rate of pedigree inbreeding \((\Delta F)\) in optimum-contribution selection (OCS) when we genotyped both live and dead animals and when we genotyped only live animals. We did this by simulating three genotyping strategies:
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i)
Increasing the number of animals genotyped by increasing the number of dead animals genotyped. All (100%) live animals and 0, 20, 40, 60, 80, or 100% dead animals were genotyped.
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ii)
Fixed number of animals genotyped by varying the proportion of live and dead animals genotyped. All live animals were genotyped excluding the equivalent number of dead animals genotyped. That is, 0, 20, 40, 60, 80, or 100% dead animals were randomly chosen to be genotyped and an equivalent number of live animals were randomly chosen not to be genotyped.
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iii)
Increasing number of live animals genotyped; no dead animals genotyped: 0, 20, 40, 60, 80, or 100% of live animals were genotyped. Live animals were chosen randomly.
These genotyping strategies were carried out in breeding schemes with three initial PWS (50, 90, and 95%), two litter sizes (six and 10), and two EBV prediction models (threshold and linear models with genomic and pedigree information). Pedigree selection was simulated using pedigree-based EBV as a reference for two prediction methods. PWS had a heritability of 0.02 on the observed scale. The trait was controlled by 7702 biallelic QTL (quantitative trait loci). The QTL were randomly distributed across a 30 M genome that consisted of 18 167-cM chromosome pairs. The genome contained 54,218 biallelic markers that were used to calculate genomic EBV. The number of chromosomes and LD between alleles at the markers were simulated to resemble those in three commercial breeds of Danish pigs [10]. Breeding schemes were run for 10 discrete generations ( t = 1 … 10). Animals in the base populations were randomly selected in generation t = 1. In generations t = 2 … 10, selection candidates were allocated matings by OCS. All animals were genotyped before selection in generations t = 2 … 10. Each combination of genotyping strategy, initial PWS, litter size, and prediction model was replicated 50 times.
For each replicate, variance components were estimated on the simulated data with both the threshold and linear models with no selection in each replicate. The breeding scenarios varying in litter size and an initial PWS for estimating variance components were run for six discrete generations where live animals were randomly selected from generations \(\text{t}\) = 1 to 6, without use of OCS to control inbreeding. Pedigree information was used to estimate variance components because variance component estimates can be biased if genomic information differs between selective genotyping strategies [9, 11]. Estimated variance components were used for calculating EBV with both the threshold and linear models.
Breeding schemes
In each breeding scheme, three hundred matings were allocated to either 1800 selection candidates—900 males and 900 females or 3000 selection candidates—1500 males and 1500 females—by OCS in generations t = 2 to 10. Three hundred females were allocated a single mating and males were allocated 0, 1, 2…, or 40 matings. No restriction was imposed on the number of males used as parents. The 300 dams selected by OCS were paired randomly to the selected sires. Each pair (dam) produced either six or 10 offspring, resulting in 20 full sib families and 1800 offspring or 3000 offspring. Offspring were assigned as males or females with a probability of 0.5. The PWS rate in generation t = l was set to 90, 95, or 50% and litter size was set to six or 10, resulting in breeding schemes that will be referred to as LS6S90, LS6S95, LS650 and LS10S90.
Simulation procedure
Generations − 1000 to − 1: founder population
The simulation of the pig genome in the founder population is described in [12]. Linkage disequilibrium (LD) between the 54,218 markers and 7702 QTL was established in a founder population of 25 males and 25 females using a Fisher-Wright inheritance model [13, 14].
The 54,218 markers and 7702 QTL in our simulated breeding scenarios were all segregating in generation \(t\) = − 1 of the founder population. The additive-genetic effects of the bi-allelic alleles at the 7702 segregating QTL were standardised so that the total additive-genetic variance on the underlying scale for the trait under selection was equal to 0.03, 0.06 and 0.09 for PWS of 50, 90, 95% in founder population. No new mutations were generated after the founder population was simulated. Chromosomes from the 25 males and 25 females in generation \(t\) = − 1 of the founder population were pooled: 18 pools of 100 chromosomes. Each pool consisted of 50 chromosome pairs of the \(i\)th chromosome (\(i\) = 1 … 18) from 50 founder animals. The breeding scenarios were initiated by sampling base populations from these chromosome pools.
Generation 0: base populations
Each replicate combination of genotyping strategy, litter size and prediction method was initiated by sampling a unique base population. Twenty males and 300 females were sampled in the simulated breeding scheme. The genotype of each base animal was sampled from the 18 pools of chromosomes in generation \(t\) = − 1 of the founder population. For chromosome \(i\) (\(i\) = 1 … 18), two chromosomes were randomly sampled without replacement from the \(i\) th pool of 100 chromosomes. The sampled chromosomes were replaced before the next base animal was sampled. Base animals were assumed to be unrelated and non-inbred based on pedigree and IBD alleles. They were genotyped, but not phenotyped for the trait under selection.
Generation 1: Random selection in base populations
In the simulated breeding scheme, 20 sires and 300 dams were selected. Each sire was mated with 15 dam. Each dam produced six or 10 offspring.
Generations 2 to 10: optimum-contribution selection
Animals were selected based on best linear unbiased prediction (BLUP) of EBVs using pedigree or genomic information in generations \(t\) = 2 … 10. Residual maximum lilkelihood (REML) estimates of variance components on the simulated data were used in each BLUP run across the generations. Animals were allocated matings by OCS in generations \(t\) = 2 … 10.
PWS
PWS was assessed as a binary trait and assumed to follow a threshold-liability model [15]. PWS of the ith pig was PWSi = 0 (died) if Ui ≤ t and PWSi = 1 (survived) if Ui > t, where Ui is the pig’s unobserved underlying liability and t is a fixed threshold set to attain the predefined average PWS (50, 90, or 95%). The unobserved underlying liability of pig i was:
$${\text{U}}_{\text{i}}= {\text{a}}_{\text{i}}+ {\text{c}}_{\text{i}}+ {\text{e}}_{\text{i}},$$
(1)
where \({\text{a}}_{\text{i}}\) is the TBV of the animals, calculated as the sum of additive genetic effects for the 7702 QTL, \({\text{c}}_{\text{i}}\) is a litter effect sampled from \({\text{c}}_{\text{i}}\sim N(0,{\upsigma }_{\text{c}}^{2})\), and \({\text{e}}_{\text{i}}\) is the residual value sampled from \({\text{e}}_{\text{i}}\sim N(0, {\upsigma }_{\text{e}}^{2})\). For PWS equal to 50, 90, and 95%, \({\upsigma }_{\text{c}}^{2}\) was set to 0.06, 0.15, and 0.22, and \({\upsigma }_{\text{e}}^{2}\) was set to 0.91, 0.79, and 0.69 to obtain an heritability on the observed scale of \({\text{h}}_{\text{o}}^{2}\) of 0.02.
The heritability on the observed-scaled (\({\text{h}}_{\text{o}}^{2}\)) of 0.02 can be transformed to heritability on the underlying scale using the formula of Dempster and Lerner [16]:
$${\text{h}}_{\text{l}}^{2}=\frac{{\text{h}}_{\text{o}}^{2}\text{K}(1-\text{K})}{{\text{z}}^{2}},$$
(2)
where \(\text{K}\) is the percentage of PWS which, in this study, was assumed to be 50%, 90% and 95%, \(\text{z}\) is the height of the normal distribution curve at the threshold, \({\text{h}}_{\text{l}}^{2}\) is the heritability on the liability scale and \({\text{h}}_{\text{o}}^{2}\) is the observed-scale heritability. For PWS equal to 50, 90, and 95%, the heritability of PWS on the unobserved liability scale was \({\text{h}}_{\text{l}}^{2}\) = 0.03, 0.06, and 0.09, respectively.
Estimating variance components for PWS
The REML estimates of variance components for PWS were calculated using both animal and sire models with the threshold and linear methods on the simulated data. The linear model assumes normality of the binary trait, while the threshold model assumes that the observed binary responses are the result of underlying normally distributed latent variables [17]. Preliminary results showed that variance components estimated with the sire model were the most consistent with the true heritability (0.02, on the observed scale). Therefore, in this study, variance components derived from the sire model were used to predict EBV under the animal model in each replicate for PWS.
PWS assessed as a binary trait was assumed to follow a threshold liability model [15], which postulates that there is an unobserved underlying variable (i.e., liability) for the \({i}^{th}\) animal, \({U}_{i}\), where \({Y}_{i}=0\) (died) if \({U}_{i}\le t\), \({Y}_{i}=1\) (survived) if \({U}_{i}>t\), and \(t\) is a fixed threshold set to zero.
$$\mathbf{U}={\mathbf{X}\mathbf{b}+\mathbf{Z}}_\mathbf{1}\mathbf{s}+ {\mathbf{Z}}_\mathbf{2}\mathbf{c}+\mathbf{e},$$
(3)
\(f(y=0)\) if \({U}_{i}\le t\),
\(f(y=1)\) if \({U}_{i}>t\), where \(\mathbf{U}\) is the vector of underlying liabilities to survive and the estimated variance components from this model were on the underlying scale, \(\mathbf{b}\) is the vector of fixed generation effects, \(\mathbf{s}\) is the vector of the sire genetic effect, \(\mathbf{c}\) is the random litter effect and \(\mathbf{e}\) is the vector of the residual effect. The distribution of the random effects was as follows:
$$\left(\begin{array}{c}\mathbf{s}\\ \mathbf{c}\\ \mathbf{e}\end{array}\right) \sim \text{ N }\left(\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right], \left[\begin{array}{ccc}{{\mathbf{A}}_{\mathbf{s}}\upsigma }_{\text{s}}^{2}& 0& 0\\ 0& {\mathbf{I}\upsigma }_{{\text{c}}_{\text{s}}}^{2}& 0\\ 0& 0& {\mathbf{I}\upsigma }_{{\text{e}}_{\text{s}}}^{2}\end{array}\right]\right),$$
where, \({\mathbf{A}}_{\mathbf{s}}\) is the pedigree relationship among sires, \({\upsigma }_{\text{s}}^{2}\) is the sire genetic variance, and \({\upsigma }_{{\text{c}}_{\text{s}}}^{2}\text{ and}\) \({\upsigma }_{{\text{e}}_{\text{s}}}^{2}\) are the litter variance and residual variance in the threshold sire model. A linear sire model was also used to estimate the variance components in the simulated data where the observations of PWS were treated as a normally distributed trait.
Estimating breeding values for PWS
Breeding values for PWS were estimated using both linear and threshold animal models similar to Eq. (3) and the pedigree relationships among sires in Eq. (3) were replaced with pedigree relationships among all animals. The same variance components from the sire linear and threshold models were used to predict EBV for all generations within a replicate (Table 1). The genetic variance for calculating EBV was set to four times the estimated sire variance, the litter variance was adjusted by subtracting the estimated sire variance.The single-step genomic breeding value was estimated using the \(\mathbf{H}\) matrix instead of \(\mathbf{A}\) [18].
For the threshold model, dstimated breeding values were transformed to the probability scale using the equation from Hidalgo et al. [19]:
$${\text{P}}_{\text{i}}=1-\text{ \O }\left(\frac{{\upmu }_{\text{i}}-{\upmu }_{\text{u}}}{{\upsigma }_{\text{u}}}\right),$$
where \({\text{P}}_{\text{i}}\) is the probability of survival for animal i, Ø is the standard cumulative distribution function, \({\upmu }_{\text{i}}\) is the estimated breeding value of animal \(\text{i}\), \({\upmu }_{\text{u}}\) is the mean estimated breeding value of all animals, \({\upsigma }_{\text{u}}\) is the standard deviation of estimated breeding values of all animals.
Criteria used for to assess scenarios
The rate of true genetic gain (\({\Delta \text{G}}_{\text{true}}\)) on the underlying scale was calculated for each scenario and accuracy and bias of EBV of live animals were used to investigate the mechanisms that underlie the differences between scenarios.
\({\Delta \text{G}}_{\text{true}}\) was calculated for each replicate as a linear regression of \({\text{G}}_{\text{t}}\) on time \(\text{t}\), where \({\text{G}}_{\text{t}}\) is the average TBV of the animals born in generations \(\text{t}\) = 5 to 10 on the underlying scale. The accuracy and bias of the EBV of live animals were calculated in each replicate as the correlation and regression of TBV with and on EBV of live animals, with EBV expressed on the probability scale. Accuracy and bias of EBV were compared between the genotyping strategies for both the linear and the threshold models. All results were presented as the mean of 50 replicates.
Software
Breeding programs were simulated using the ADAM program [20], the OCS was run using the EVA program [21]. Variance componenets and EBVs were calculated using the DMU software [22].
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