Prediction of genetic contributions and generation intervals in populations with overlapping generations under selection

A method to predict long-term genetic contributions of ancestors to future generations is studied in detail for a population with overlapping generations under mass or sib index selection. An existing method provides insight into the mechanisms determining the flow of genes through selected populations, and takes account of selection by modeling the long-term genetic contribution as a linear regression on breeding value. Total genetic contributions of age classes are modeled using a modified gene flow approach and long-term predictions are obtained assuming equilibrium genetic parameters. Generation interval was defined as the time in which genetic contributions sum to unity, which is equal to the turnover time of genes. Accurate predictions of long-term genetic contributions of individual animals, as well as total contributions of age classes were obtained. Due to selection, offspring of young parents had an above-average breeding value. Long-term genetic contributions of youngest age classes were therefore higher than expected from the age class distribution of parents, and generation interval was shorter than the average age of parents at birth of their offspring. Due to an increased selective advantage of offspring of young parents, generation interval decreased with increasing heritability and selection intensity. The method was compared to conventional gene flow and showed more accurate predictions of long-term genetic contributions.     

Minimization of rate of inbreeding for small populations with overlapping generations

We propose a method that minimizes the rate of inbreeding (delta F) for small unselected populations with overlapping generations and several reproductive age classes. It minimizes the increase in coancestry of parents and optimizes the contribution of each selection candidate. The carrying capacity of the population is limited to a fixed number of animals per year. When survival rate equalled 100%, only animals from the oldest age class were selected, which maximized the number of parents per generation, slowed down the turnover of generations and minimized the increase of coancestry across sublines. However, the population became split into sublines separated by age classes, which substantially increased inbreeding within sublines. Sublines were prevented by a restriction of selecting at least one sire and one dam from the second-oldest age class, which resulted in an L times lower delta F, where L equals the average generation interval of sires and dams. Minimum coancestry mating resulted in lower levels of inbreeding than random mating, but delta F was approximately the same. For schemes where the oldest animals were selected, delta F increased by 18-52% compared with the proposed method.     

Optimal mass selection policies for schemes with overlapping generations and restricted inbreeding

Optimum breeding schemes for maximising the rate of genetic progress with a restriction on the rate of inbreeding (per year or per generation) are investigated for populations with overlapping generations undergoing mass selection. The optimisation is for the numbers of males and females to be selected and for their distribution over age classes. Expected rates of genetic progress (ΔG) are combined with expected rates of inbreeding (ΔF) in a linear objective function (Φ = ΔG - λΔF) which is maximised. A simulated annealing algorithm is used to obtain the solutions. The restriction on inbreeding is achieved by increasing the number of parents and, in small schemes with severe restrictions, by increasing the generation interval. In the latter case the optimum strategy for obtaining the maximum genetic gain is far from truncation selection across age classes. In most situations, the optimum mating ratio is one but the differences in genetic gain obtained with different mating ratios are small. Optimisation of schemes when restricting the rate of inbreeding per generation leads to shorter generation intervals than optimisation when restricting the rate of inbreeding per year.     

Prediction of rates of inbreeding in populations selected on best linear unbiased prediction of breeding value

Predictions for the rate of inbreeding (DeltaF) in populations with discrete generations undergoing selection on best linear unbiased prediction (BLUP) of breeding value were developed. Predictions were based on the concept of long-term genetic contributions using a recently established relationship between expected contributions and rates of inbreeding and a known procedure for predicting expected contributions. Expected contributions of individuals were predicted using a linear model, u(i)(()(x)()) = alpha + betas(i), where s(i) denotes the selective advantage as a deviation from the contemporaries, which was the sum of the breeding values of the individual and the breeding values of its mates. The accuracy of predictions was evaluated for a wide range of population and genetic parameters. Accurate predictions were obtained for populations of 5-20 sires. For 20-80 sires, systematic underprediction of on average 11% was found, which was shown to be related to the goodness of fit of the linear model. Using simulation, it was shown that a quadratic model would give accurate predictions for those schemes. Furthermore, it was shown that, contrary to random selection, DeltaF less than halved when the number of parents was doubled and that in specific cases DeltaF may increase with the number of dams.     

Non-random mating for selection with restricted rates of inbreeding and overlapping generations

Minimum coancestry mating with a maximum of one offspring per mating pair (MC1) is compared with random mating schemes for populations with overlapping generations. Optimum contribution selection is used, whereby ΔF is restricted. For schemes with ΔF restricted to 0.25% per year, 256 animals born per year and heritability of 0.25, genetic gain increased with 18% compared with random mating. The effect of MC1 on genetic gain decreased for larger schemes and schemes with a less stringent restriction on inbreeding. Breeding schemes hardly changed when omitting the iteration on the generation interval to find an optimum distribution of parents over age-classes, which saves computer time, but inbreeding and genetic merit fluctuated more before the schemes had reached a steady-state. When bulls were progeny tested, these progeny tested bulls were selected instead of the young bulls, which led to increased generation intervals, increased selection intensity of bulls and increased genetic gain (35% compared to a scheme without progeny testing for random mating). The effect of MC1 decreased for schemes with progeny testing. MC1 mating increased genetic gain from 11–18% for overlapping and 1–4% for discrete generations, when comparing schemes with similar genetic gain and size.     

Predictions of response to selection caused by angling in a wild population of Atlantic salmon (Salmo salar)

1. Recreational angling activities in wild populations of Atlantic salmon may induce a selection pressure towards a reduction in body size and length if the angling season coincides with the return of the largest sea age fish class. 2. Using estimates of heritability for growth traits and estimates of the selection pressure from angling operating on growth, we predicted the response to selection expected to occur in a wild population of Atlantic salmon. 3. The dataset used here comprised individuals from two consecutive generations (parents and offspring) from the River Bidasoa (NW Spain). Offspring were assigned to parents using six highly polymorphic microsatellite loci. Use of restricted maximum likelihood methodology and the animal model allowed us to estimate the heritability for body length and body weight as well as their genetic correlation. 4. Estimated heritabilities (0.32 ± 0.12 for length and 0.32 ± 0.11 for weight) and selection pressure caused by angling were used to obtain predictions of response to selection because of angling. Our results suggested a decline of 1.9 mm in body length and 103.3 g in body weight per generation because of angling pressure. 5. The results derived from this study suggest that the angling season should be annually delayed in order to avoid selective angling of the multi‐year class and further reductions in body weight and length.     

Optimizing Selection on Multiple Identified Quantitative Trait Loci in Population with Overlapping Generations

A method was developed to model and optimize selection on multiple identified quantitative trait loci (QTLs) and polygenic estimated breeding value, in order to maximize a weighted sum of cumulative response to selection over multiple years in a population with overlapping generations. The model allows for a population with multiple sex-age classes, different number of age class between sires and dams, and varied genetic contribution of the age class. The optimization problem was formulated as a multiple-stage optimal control problem and solved by a forward and backward iteration loop. The practical utility of this method was illustrated in an example of pig breeding population with overlapping generations. The selection response of this method was compared with standard QTL selection and conventional best linear unbiased prediction (BLUP) selection. Simulation results show that optimal selection achieved greater selection response than either standard QTL or conventional BLUP selections. The influence of population structure on optimal selection was significant. Optimal QTL selection and standard QTL selection were more favorable in a population with overlapping generations than discrete generations, and obtained more benefits relative to conventional BLUP selection in a population with overlapping generations. Optimal QTL selection relative to conventional BLUP selection is also more favorable following increase of genetic contribution of two-year-old boars and sows in a population with overlapping generations.     

Predicting rates of inbreeding in populations undergoing selection

Tractable forms of predicting rates of inbreeding (DeltaF) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. DeltaF was shown to be approximately (1)/(4)(1 - omega) times the expected sum of squared lifetime contributions, where omega is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express DeltaF in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing (1)/(4) (since omega = 0) was increased to (1)/(2). Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting DeltaF with sib indices in discrete generations since previously published solutions had proved complex.     

Minimizing inbreeding by managing genetic contributions across generations

Here we present the strategy that achieves the lowest possible rate of inbreeding (DeltaF) for a population with unequal numbers of sires and dams with random mating. This new strategy results in a DeltaF as much as 10% lower than previously achieved. A simple and efficient approach to reducing inbreeding in small populations with sexes of unequal census number is to impose a breeding structure where parental success is controlled in each generation. This approach led to the development of strategies for selecting replacements each generation that were based upon parentage, e.g., a son replacing its sire. This study extends these strategies to a multigeneration round robin scheme where genetic contributions of ancestors to descendants are managed to remove all uncertainties about breeding roles over generations; i.e., male descendants are distributed as equally as possible among dams. In doing so, the sampling variance of genetic contributions within each breeding category is eliminated and consequently DeltaF is minimized. Using the concept of long-term genetic contributions, the asymptotic DeltaF of the new strategy for random mating, M sires and d dams per sire, is phi/(12M), where phi = [1 + 2((1)/(4))(d)]. Predictions were validated using Monte Carlo simulations. The scheme was shown to achieve the lowest possible DeltaF using pedigree alone and showed that further reductions in DeltaF below that obtained from random mating arise from preferential mating of relatives and not from their avoidance.     

Estimation of additive genetic variance in commercial layer poultry and simulated populations under selection

Changes in genetic parameters over generations for a selected commercial population and simulated populations of poultry with different sizes were studied. The traits analyzed from the commercial population were rate of lay, age at first egg, egg weight, deformation, and body weight. In the simulated population, a trait measured on both sexes and a sex-limited trait, measured only on one sex, each with a heritability of 0.1 and 0.5, were analyzed. In the commercial and simulated populations, males and females were selected on the basis of family selection indexes and data was available only after many generations of selection. Parameters for each generation were estimated by fitting an animal model using derivative free maximum likelihood (DFREML) with different data structures. In structure 1, data included the given (base) generation for which the parameters were to be estimated, and all subsequent generations. In structure 2, only data on birds in the given generation and their progeny were included. In both structures, parents of base-generation birds were assumed unrelated and pedigrees traced back to these parents. With commercial data using structure 1, estimates of _a ² and h² decreased by 14 to 37% across five generations. With structure 2, no trends were observed, though estimates were lower than for structure 1. For simulated data, with a heritability of 0.1, both structures yielded apparently unbiased estimates of the observed additive genetic variances in the (selected) base generation, no matter how many generations of data were utilized, for both sex-limited and normal traits. However, with a heritability of 0.5 the estimated additive genetic variance for both types of trait decreased with a decrease in the number of generations used in the estimation. Estimates based on the first two generations underestimated, while estimates based on five generations of data overestimated, the observed genetic variances in the defined base. The combinations of conditions that lead to varying degrees of bias remain undefined.     

Genetic correlation and response to selection in simulated populations

Summary Effects of truncation selection of a primary trait upon genetic correlation with a secondary trait were examined over 30 generations in genetic populations simulated by computer. Populations were 24 males and 24 females mated randomly with replacement; number of offspring was determined by intensity of selection. Each trait was controlled by 48 loci segregating independently, effects were equal at every locus, and gene frequency was arbitrarily set at 0.5 at each locus in the initial generation. All combinations of three genetic correlations, three intensities of selection, and three environmental variances were simulated. Gene action was additive. Genetic correlation was set by number of loci which affected both traits and was measured each generation as the product-moment correlation of genotypic values and estimated by two methods of combining phenotypic covariances between parent and offspring.Genetic correlations in each offspring generation remained consistently near initial correlations for all environmental variances when fraction of offspring saved as parents was as large as one-half. When the fraction of offspring saved was as small as one-fifth, genetic correlations decreased but most rapidly with heritability high and after the 15th generation of selection. Truncation selection caused genetic correlation to decrease in those offspring selected to become parents of the next generation. Amount of reduction depended on heritability of the selected trait rather than on degree of truncation selection. Estimates of genetic correlation from phenotypic covariances between parent and offspring fluctuated markedly from real correlations in the small populations simulated.Michigan Agricultural Experiment Station Journal Article 4836. Part of North Central Regional Project NC-2.     

Effects of Population Size and Selection Intensity on Short-Term Response to Selection for Postweaning Gain in Mice

The effects of population size and selection intensity on the mean response was examined after 14 generations of within full-sib family selection for postweaning gain in mice. Population sizes of 1, 2, 4, 8 and 16 pair matings were each evaluated at selection intensities of 100% (control), 50% and 25% in a replicated experiment. Selection response per generation increased as selection intensity increased. Selection response and realized heritability tended to increase with increasing population size. Replicate variability in realized heritability was large at population sizes of 1, 2 and 4 pairs. Genetic drift was implicated as the primary factor causing the reduced response and lowered repeatability at the smaller population sizes. Lines with intended effective population sizes of 62 yielded larger selection responses per unit selection differential than lines with effective population sizes of 30 or less.     

Polygenic Mutation in Drosophila Melanogaster: Non-Linear Divergence among Unselected Strains

A highly inbred strain of Drosophila melanogaster was subdivided into 20 replicate sublines that were maintained independently with 10 pairs of randomly sampled parents per generation for 180 generations. The variance between lines in abdominal and sternopleural bristle number increased little after 100 generations, in contrast to the neutral expectation of a linear increase; and the covariances of line means between different generations declined with increasing number of generations apart, in contrast to the neutral expectation of constant covariance. Thus, under a neutral model, the estimates of mutational variance were lower than for previous estimates from the first 100 generations of subline divergence. An autoregressive model was fitted to the variance of line means that indicated strong natural selection. There is no single unequivocal explanation for the results. Possible and nonexclusive alternatives include stabilizing selection on bristle number and deleterious effects on fitness of bristle mutations. The inferred strengths of selection on both traits are too high for stabilizing selection alone, and the between-line variance did not continue to increase sufficiently for pleiotropy alone to account for the observations. A third potential explanation that does not invoke selection is duplicate epistasis between mutations affecting bristle number.     

世代重叠群体多个QTL最优化选择

一种扩展的方法能够在一个世代重叠的群体内对多个数量性状位点选择进行最优化,目的是为了在整个计划期内获得最大的累积反应加权和。该模型允许群体有多个性别年龄组、公母畜间有不同的年龄组数、各年龄组有不同的遗传贡献。整个最优化问题被描述成一个多阶段系统优化控制问题,通过一个向前和向后的迭代循环解决。用一个世代重叠的实际育种猪群的参数来评价该方法的选择效果,并和标准QTL选择和常规BLUP选择进行比较。模拟结果表明,优化选择要优于标准QTL选择和常规BLUP选择。群体结构对优化选择的影响比较明显。优化QTL选择和标准QTL选择在世代重叠的群体内比在世代离散的群体内的选择优势更明显,相对于常规BLUP选择,能够获得更大的选择优势。在世代重叠群体内随着2岁公畜遗传贡献的增大,优化选择相对于常规BLUP选择的优势越明显。     

Inferring the trajectory of genetic variance in the course of artificial selection

A method is proposed to infer genetic parameters within a cohort, using data from all individuals in an experiment. An application is the study of changes in additive genetic variance over generations, employing data from all generations. Inferences about the genetic variance in a given generation are based on its marginal posterior distribution, estimated via Markov chain Monte Carlo methods. As defined, the additive genetic variance within the group is directly related to the amount of selection response to be expected if parents are chosen within the group. Results from a simulated selection experiment are used to illustrate properties of the method. Four sets of data are analysed: directional selection with and without environmental trend, and random selection, with and without environmental trend. In all cases, posterior credibility intervals of size 95% assign relatively high density to values of the additive genetic variance and heritability in the neighbourhood of the true values. Properties and generalizations of the method are discussed.     

Optimum proportions selected with unequal sex numbers

Summary Optimum proportions selected for each sex, giving the maximum selection response up to some specified time, have been investigated. Selection was carried out from a constant number of individuals scored per generation. It has been shown that the maximum response after t generations of selection is obtained when the number of individuals scored and the proportion selected is the same for males and females every generation. Specific situations, where the sex-ratio among scored or breeding individuals or the selected proportion of males or females is taken as given throughout the whole selection process, have been studied, optimum strategies being described in each case. The procedures leading to maximum advance at the selection limit have also been considered in all these situations.     

A note on effective population size with overlapping generations

A simple derivation is given for a formula obtained previously for the effective size of random-mating populations with overlapping generations. The effective population size is the same as that for a population with discrete generations having the same variance of lifetime family size and the same number of individuals entering the population per generation.     

Survival of the currently fittest: genetics of rainbow trout survival across time and space

As a fitness trait, survival is assumed to exhibit low heritability due to strong selection eroding genetic variation and/or spatio-temporal variation in mortality agents reducing genetic and increasing residual variation. The latter phenomenon in particular may contribute to low heritability in multigeneration data, even if certain cohorts exhibit significant genetic variation. Analysis of survival data from 10 year classes of rainbow trout reared at three test stations showed that treating survival as a single trait across all generations resulted in low heritability (h(2) = 0.08-0.17). However, when heritabilities were estimated from homogeneous generation and test station-specific cohorts, a wide range of heritability values was revealed (h(2) = 0.04-0.71). Of 64 genetic correlations between different cohorts, 20 were positive, but 16 were significantly negative, confirming that genetic architecture of survival is not stable across generations and environments. These results reveal the existence of hidden genetic variation for survival and demonstrate that treating survival as one trait over several generations may not reveal its true genetic architecture. Negative genetic correlations between cohorts indicate that overall survival has limited potential to predict general resistance, and care should be taken when using it as selection criterion.     

Prediction of rates of inbreeding in selected populations

A method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into 'contribution' matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, Ne approximately 2N/(mu 2r + sigma 2r), where N is the number of individuals per generation and mu r and sigma 2r are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing 'competitiveness' of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.     

Efficiency of including first-generation information in second-generation ranking and selection: results of computer simulation

Using computer simulation, we evaluated the impact of using first-generation information to increase selection efficiency in a second-generation breeding program. Selection efficiency was compared in terms of increase in rank correlation between estimated and true breeding values (i.e., ranking accuracy), reduction in coefficient of variation of correlation coefficients (i.e., ranking reliability), and increase in realized gain, with best linear unbiased prediction (BLUP). The test populations were generated with varying parameters: selection strategy (forward vs backward selection of parents); number of parents (24∼96); number of crosses per parent (1∼8); heritability (0.05∼0.35); ratio of dominance to additive variance (0∼3); ratio of additive-by-site to additive variance (0∼3); and ratio of dominance-by-site to additive variance (0∼3). The two selection strategies gave distinct results. When parents of the second-generation crosses had been selected via backward selection, adding first-generation information markedly increased selection efficiency. Conversely, when parents had been selected via forward selection, first-generation information provided little increase in efficiency. The amount of increase depended more on heritabilities in both generations and less on dominance and genotype–by–environment effects. Including first-generation information helped more when there were many parents and few crosses per parent in the second generation. Only in the case of extremely low first-generation heritabilities was there no benefit to adding first-generation information in terms of improved ranking reliability and accuracy.