A note on increasing the limit of selection through selection within families

Strategies to Assure Optimal Trade-Offs Among Competing Objectives for the Genetic Improvement of Soybean

Plant breeding is a decision-making discipline based on understanding project objectives. Genetic improvement projects can have two competing objectives: maximize the rate of genetic improvement and minimize the loss of useful genetic variance. For commercial plant breeders, competition in the marketplace forces greater emphasis on maximizing immediate genetic improvements. In contrast, public plant breeders have an opportunity, perhaps an obligation, to place greater emphasis on minimizing the loss of useful genetic variance while realizing genetic improvements. Considerable research indicates that short-term genetic gains from genomic selection are much greater than phenotypic selection, while phenotypic selection provides better long-term genetic gains because it retains useful genetic diversity during the early cycles of selection. With limited resources, must a soybean breeder choose between the two extreme responses provided by genomic selection or phenotypic selection? Or is it possible to develop novel breeding strategies that will provide a desirable compromise between the competing objectives? To address these questions, we decomposed breeding strategies into decisions about selection methods, mating designs, and whether the breeding population should be organized as family islands. For breeding populations organized into islands, decisions about possible migration rules among family islands were included. From among 60 possible strategies, genetic improvement is maximized for the first five to 10 cycles using genomic selection and a hub network mating design, where the hub parents with the largest selection metric make large parental contributions. It also requires that the breeding populations be organized as fully connected family islands, where every island is connected to every other island, and migration rules allow the exchange of two lines among islands every other cycle of selection. If the objectives are to maximize both short-term and long-term gains, then the best compromise strategy is similar except that the mating design could be hub network, chain rule, or a multi-objective optimization method-based mating design. Weighted genomic selection applied to centralized populations also resulted in the realization of the greatest proportion of the genetic potential of the founders but required more cycles than the best compromise strategy.

Short-term effects of controlled mating and selection on the genetic variance of honeybee populations

Directional selection in a population yields reduced genetic variance due to the Bulmer effect. While this effect has been thoroughly investigated in mammals, it is poorly studied in social insects with biological peculiarities such as haplo-diploidy or the collective expression of traits. In addition to the natural adaptation to climate change, parasites, and pesticides, honeybees increasingly experience artificial selection pressure through modern breeding programs. Besides selection, many honeybee breeding schemes introduce controlled mating. We investigated which individual effects selection and controlled mating have on genetic variance. We derived formulas to describe short-term changes of genetic variance in honeybee populations and conducted computer simulations to confirm them. Thereby, we found that the changes in genetic variance depend on whether the variance is measured between queens (inheritance criterion), worker groups (selection criterion), or both (performance criterion). All three criteria showed reduced genetic variance under selection. In the selection and performance criteria, our formulas and simulations showed an increased genetic variance through controlled mating. This newly described effect counterbalanced and occasionally outweighed the Bulmer effect. It could not be observed in the inheritance criterion. A good understanding of the different notions of genetic variance in honeybees, therefore, appears crucial to interpreting population parameters correctly.

Long-Term Evaluation of Breeding Scheme Alternatives for Endangered Honeybee Subspecies

Modern breeding structures are emerging for European honeybee populations. However, while genetic evaluations of honeybees are becoming increasingly well understood, little is known about how selection decisions shape the populations' genetic structures. We performed simulations evaluating 100 different selection schemes, defined by selection rates for dams and sires, in populations of 200, 500, or 1000 colonies per year and considering four different quantitative traits, reflecting different genetic parameters and numbers of influential loci. Focusing on sustainability, we evaluated genetic progress over 100 years and related it to inbreeding developments. While all populations allowed for sustainable breeding with generational inbreeding rates below 1% per generation, optimal selection rates differed and sustainable selection was harder to achieve in smaller populations and for stronger negative correlations of maternal and direct effects in the selection trait. In small populations, a third or a fourth of all candidate queens should be selected as dams, whereas this number declined to a sixth for larger population sizes. Furthermore, our simulations indicated that, particularly in small populations, as many sires as possible should be provided. We conclude that carefully applied breeding provides good prospects for currently endangered honeybee subspecies, since sustainable genetic progress improves their attractiveness to beekeepers.

Comparison of infinitesimal and finite locus models for long-term breeding simulations with direct and maternal effects at the example of honeybees

Stochastic simulation studies of animal breeding have mostly relied on either the infinitesimal genetic model or finite polygenic models. In this study, we investigated the long-term effects of the chosen model on honeybee breeding schemes. We implemented the infinitesimal model, as well as finite locus models, with 200 and 400 gene loci and simulated populations of 300 and 1000 colonies per year over the course of 100 years. The selection was of a directly and maternally influenced trait with maternal heritability of [Formula: see text], direct heritability of [Formula: see text], and a negative correlation between the effects of rmd = - 0.18. Another set of simulations was run with parameters [Formula: see text], [Formula: see text], and rmd = - 0.53. All models showed similar behavior for the first 20 years. Throughout the study, we observed a higher genetic gain in the direct than in the maternal effects and a smaller gain with a stronger negative covariance. In the long-term, however, only the infinitesimal model predicted sustainable linear genetic progress, while the finite locus models showed sublinear behavior and, after 100 years, only reached between 58% and 62% of the mean breeding values in the infinitesimal model. While the infinitesimal model suggested a reduction of genetic variance by 33% to 49% after 100 years, the finite locus models saw a more drastic loss of 76% to 92%. When designing sustainable breeding strategies, one should, therefore, not blindly trust the infinitesimal model as the predictions may be overly optimistic. Instead, the more conservative choice of the finite locus model should be favored.

A modelling framework for the analysis of artificial-selection time series

Artificial-selection experiments constitute an important source of empirical information for breeders, geneticists and evolutionary biologists. Selected characters can generally be shifted far from their initial state, sometimes beyond what is usually considered as typical inter-specific divergence. A careful analysis of the data collected during such experiments may thus reveal the dynamical properties of the genetic architecture that underlies the trait under selection. Here, we propose a statistical framework describing the dynamics of selection-response time series. We highlight how both phenomenological models (which do not make assumptions on the nature of genetic phenomena) and mechanistic models (explaining the temporal trends in terms of e.g. mutations, epistasis or canalization) can be used to understand and interpret artificial-selection data. The practical use of the models and their implementation in a software package are demonstrated through the analysis of a selection experiment on the shape of the wing in Drosophila melanogaster.

Asymptotic rates of response from index selection

The reduction in additive genetic variance due to selection is investigated when index selection using family records is practised. A population of infinite size with no accumulation of inbreeding, an infinitesimal model and discrete generations are assumed. After several generations of selection, the additive genetic variance and the rate of response to selection reach an asymptote. A prediction of the asymptotic rate of response is considered to be more appropriate for comparing response from alternative breeding programmes and for comparing predicted and realized response than the response following the first generation of selection that is classically used. Algorithms to calculate asymptotic response rate are presented for selection based on indices which include some or all of the records of an individual, its full- and half-sibs and its parental estimated breeding values. An index using all this information is used to predict response when selection is based on breeding values estimated by using a Best Linear Unbiased Prediction (BLUP) animal model, and predictions agree well with simulation results. The predictions are extended to multiple trait selection.

Asymptotic responses are compared with one-generation responses for a variety of alternative breeding schemes differing in population structure, selection intensity and heritability of the trait. Asymptotic responses can be up to one-quarter less than one-generation responses, the difference increasing with selection intensity and accuracy of the index. Between family variance is reduced considerably by selection, perhaps to less than half its original value, so selection indices which do not account for this tend to place too much emphasis on family information. Asymptotic rates of response to selection, using indices including family information for traits not measurable on the individuals available for selection, such as sex limited or post-slaughter traits, are found to be as much as two-fifths less than their expected one-generation responses. Despite this, the ranking of the breeding schemes is not greatly altered when compared by one-generation rather than asymptotic responses, so the one-generation prediction is usually likely to be adequate for determining optimum breeding structure.

Systems of mating to reduce inbreeding in selected populations

Stochastic simulation is used to compare different systems of mating to reduce rates of inbreeding in selection programmes with phenotypic or animal model best linear unbiased prediction (BLUP) evaluation. Compensatory mating (the mating between individuals from the largest selected families to individuals from the smallest) turns out to be proportionately about 0-30 more effective than minimum coancestry matings for situations with low rates of inbreeding, such as phenotypic selection or high population size, although the advantage is less apparent if common environmental effects are important. A modification of this system of mating is proposed which can be applied for overlapping generations, and this is shown to reduce rates of inbreeding proportionately by about 0-50 more than for discrete generations. Under high inbreeding, however, such as for BLUP selection and small population size, minimum coancestry matings, or even avoidance of sib matings are more effective. A procedure combining compensatory and minimum coancestry matings is also simulated and gives the largest reductions in the rate of inbreeding. The effects of these and other systems of mating on the rate of inbreeding are shown to occur through a reduction in the cumulative effect of selection and a deviation from Hardy-Weinberg proportions.

The number of loci affecting a quantitative trait in Drosophila melanogaster revealed by artificial selection

Individual and within-full-sib family selection for low sternopleural bristle number was carried out for 17 generations, with six replicate lines for each selection method. Our results can be summarized as follows: (1) the response to selection was exhausted very quickly, (2) the additive variance of the selected lines declined rapidly, (3) the variation in response to selection decreased as selection progressed, (4) genetic differences among replicates at the selection limit were small, (5) individual selection resulted in a higher initial response than within-family selection, but similar limits were achieved with both procedures. These observations are consistent with the hypothesis that the pattern of response to selection is due to the segregation in the base population of only a few loci with large effects, at intermediate frequencies.

Positive assortative mating with selection restrictions on group coancestry enhances gain while conserving genetic diversity in long-term forest tree breeding

Selection and mating principles in a closed breeding population (BP) were studied by computer simulation. The BP was advanced, either by random assortment of mates (RAM), or by positive assortative mating (PAM). Selection was done with high precision using clonal testing. Selection considered both genetic gain and gene diversity by "group-merit selection", i.e. selection for breeding value weighted by group coancestry of the selected individuals. A range of weights on group coancestry was applied during selection to vary parent contributions and thereby adjust the balance between gain and diversity. This resulted in a series of scenarios with low to high effective population sizes measured by status effective number. Production populations (PP) were selected only for gain, as a subset of the BP. PAM improved gain in the PP substantially, by increasing the additive variance (i.e. the gain potential) of the BP. This effect was more pronounced under restricted selection when parent contributions to the next generation were more balanced with within-family selection as the extreme, i.e. when a higher status effective number was maintained in the BP. In that case, the additional gain over the BP mean for the clone PP and seed PPs was 32 and 84% higher, respectively, for PAM than for RAM in generation 5. PAM did not reduce gene diversity of the BP but increased inbreeding, and in that way caused a departure from Hardy-Weinberg equilibrium. The effect of inbreeding was eliminated by recombination during the production of seed orchard progeny. Also, for a given level of inbreeding in the seed orchard progeny or in a mixture of genotypes selected for clonal deployment, gain was higher for PAM than for RAM. After including inbreeding depression in the simulation, inbreeding was counteracted by selection, and the enhancement of PAM on production population gain was slightly reduced. In the presence of inbreeding depression the greatest PP gain was achieved at still higher levels of status effective number, i.e. when more gene diversity was conserved in the BP. Thus, the combination of precise selection and PAM resulted in close to maximal short-term PP gain, while conserving maximal gene diversity in the BP.

Balanced vs. slightly unbalanced selection

Balanced selection, defined as that all parents have equal contributions to the next generation, is expected to minimize the loss of gene diversity and thereby to allow a steady increase of selection response even at very advanced generations. In this study, balanced selection was compared with two degrees of slightly unbalanced selection (equal contributions from most of the families) and with unrestricted selection (unequal contributions) in terms of the retrieved response and genetic diversity consumed (group coancestry) over 30 generations of selection. A simulation was used varying the heritability and considering two population sizes and two family sizes. Results showed that a slight unbalance can bring a favourable relation between retrieved response and genetic diversity when compared to balanced selection at low and moderate heritabilities, and when small families and population size were assumed. The implicit constraint of a constant number of contributing families imposed in balanced and slightly unbalanced selection reduced the likelihood of gene loss compared to unrestricted selection. I conclude that a slight unbalance often could be more favourable than complete balance regarding selection efficiency, and that gain retrieved per unit of group coancestry declined with further unbalance.

Selection response in finite populations

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (sigma AW02) and mutational (sigma M2) variance. The cumulative response to selection until generation t(CRt) can be approximated as [equation: see text] where Ne is the effective population size, sigma AW infinity 2 = Ne sigma M2 is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R0 is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. beta is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R0 is the major determinant of the short term selection response, but sigma Me2 Ne and beta are also important for the long term. A selection method of high accuracy using family information gives a small Ne and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.

Balancing selection response and rate of inbreeding by including genetic relationships in selection decisions

An iterative selection strategy, based on estimated breeding values (EBV) and average relationship among selected individuals, is proposed to optimise the balance between genetic response and inbreeding. Stochastic simulation was used to compare rates of inbreeding and genetic gain with those of other strategies. For a range of heritabilities, population sizes and mating ratios, the iterative strategy, denoted ADJEBV, outperforms other strategies, giving the greatest genetic gain at a given rate of inbreeding and the least breeding at a given genetic gain. Where selection is currently by truncation on the EBV, with a restriction on the number of full-sibs selected, it should be possible to maintain similar levels of genetic gain and inbreeding with a reduction in population size of 10-30%, by changing to the iterative strategy. If performance is measured by the reduction in cumulative inbreeding without losing more than a given amount of genetic gain relative to results obtained under truncation selection on the EBV, then with the EBV based on a family index, the performance of ADJEBV is greater at low heritability, and is generally greater than where EBV are based on individual records. When comparisons of genetic response and inbreeding are made for alternative breeding scheme designs, schemes which give higher genetic gain within acceptable inbreeding levels would usually be favoured. If comparisons are made on this basis, then the selection method used should be ADJEBV, which maximises the genetic gain for a given level of inbreeding. The results indicated that all selection strategies used to reduce inbreeding had very small effects on the variance of gain, and so differences in this respect are unlikely to affect choices among selection strategies. Selection criteria are recommended based on maximising a selection objective which specifies the desired balance between genetic gain and inbreeding.

Implications of using an average relationship matrix in genetic evaluation for a population using multiple-sire matings

SUMMARY

Ambiguous paternity can be incorporated into the mixed model equations (MME) by including the average numerator relatinship matrix (average A), which averages the true sire-offspring relationship over the putative sires. A previous study has shown that some overestimation of genetic trend results from this substitution. A population of 40 breeding females and 2 breeding males was simulated 1,000 times with either random mating or sequential selection continuing for 8 breeding cycles. In the selection case candidates were ranked on estimated breeding values (EBVs) calculated from the MME with an animal model and the average A. Variances of the EBVs and prediction errors were computed. The results showed the average A incorrectly perceives both the variance of family sizes among males and the variance loss due to selection to be smaller. This will lead to an overestimation of genetic trend. ZUSAMMENFASSUNG: Folgerungen aus der Anwendung einer durchschnittlichen Verwandtschaftsmatrix bei der Zuchtwertschätzung für eine Population mit mehreren Vätern in einer Paarungsgruppe In den Mischmodellgleichungen kann eine unklare väterliche Abstammung durch die Verwendung einer durchschnittlichen Verwandtschaftsmatrix, die die Abstammung zu gleichen Teilen über die möglichen Väter aufteilt, berücksichtigt werden. Eine frühere Arbeit hat gezeigt, daß diese Maßnahme zu einer gewissen Überschätzung des genetischen Fortschritts führt. Eine Population mit 40 weiblichen und 2 männlichen Tieren wurde 1000mal über 8 Paarungsperioden simuliert und zwar mit und ohne gerichteter Selektion. Im Falle der Selektion wurden die Tiere aufgrund der mit einem Tiermodell und der durchschnittlichen Verwandtschaftsmatrix geschätzten Zuchtwerte geordnet. Die Varianzen der Zuchtwerte und der Schätzfehler wurden berechnet. Die Ergebnisse zeigen, daß durch eine durch-schnittliche Verwandtschaftsmatrix die Varianz der Größe der Nachkommensgruppen der Väter und der Verlust an genetischer Varianz aufgrund der Selektion unterschätzt wird. Dies führt zu einer Überschätzung des genetischen Fortschritts.

Long-term effects of selection based on the animal model BLUP in a finite population

Monte Carlo simulation was used to assess the long-term effects of truncation selection within small populations using indices (I=ωf+m) combining mid-parent [f=(a i+a d)/2] and Mendelian-sampling (m=a-f) evaluations provided by an animal model BLUP (a=f+m). Phenotypic values of panmictic populations were generated for 30 discrete generations. Assuming a purely additive polygenic model, heritability (h 2) values were 0.10, 0.25 or 0.50. Two population sizes were considered: five males and 25 females selected out of 50 candidates of each sex (small populations, S) and 50 males and 250 females selected out of 500 candidates in each sex (large populations, L). Selection was carried out on the index defined above with ω = 1 (animal model BLUP), ω=1/2, or ω=0 (selection on within-family deviations). Mass selection was also considered. Selection based on the animal model BLUP (ω=1) maximized the cumulative genetic gain in L populations. In S populations, selection using ω = 1/2 and mass selection were more efficient than selection under an animal model (+ 3 to + 7% and + 1 to + 4% respectively, depending on h (2)). Selection on within-family deviations always led to the lowest gains. In most cases, the variance of response to selection between replicates did not depend on the selection method. The within-replicate genetic variance and the average coefficient of inbreeding (F) were highly affected by selection with ω=1 or 1/2, especially in populations of size S. As expected, selection based on within-family deviations was less detrimental in that respect. The number of copies of founder neutral genes at a separate locus, and the probability vector of origin of the genes in reference to the founder animals, were also observed in addition to F values. The conclusion was that selection procedures placing less emphasis on family information might be interesting alternatives to selection based on animal model BLUP, especially for small populations with long-term selection objectives.

Predicting cumulated response to directional selection in finite panmictic populations

Accurate prediction of the cumulated genetic gain requires predicting genetic variance over time under the joint effects of selection and limited population size. An algorithm is proposed to quantify at each generation the effects of these factors on average coefficient of inbreeding, genetic variance, and genetic mean, under a purely additive polygenic model, with no mutation, and under the assumption of absence of inbreeding depression on viability affecting selection differentials. This algorithm is relevant to populations where mating is at random and generations do not overlap. It was tested via Monte Carlo simulation on a population of 3 males and 25 females mass selected out of 50 candidates of each sex, over 30 generations. For two values of the initial heritability of the selected trait, 0.5 and 0.9 (to represent high accuracy in index selection), predicted values of the genetic variance are in agreement with observed results up to the 12th and 19th generations, respectively. Beyond these generations, the variance is overestimated, due to an underestimation of the effect of selection on the rate of inbreeding. Finally, the algorithm provides predictions of the cumulated responses close to the observed values in both selected populations. It is concluded that, as regards the hypotheses of the study, the proposed algorithm is satisfactory, and could be used to optimize selection methods with respect to the cumulated genetic gain in the mid- or long-term. Possible extensions of the algorithm to more realistic situations are discussed.

Additive genetic variance within populations derived by single-seed descent and pod-bulk descent

Breeders of self-pollinated legumes commonly use single-seed descent (SSD) or pod-bulk descent (PBD) to produce segregating populations of highly inbred individuals. We presented equations for the expected value of the additive genetic variance within populations derived by SSD (E(V A)SSD) and PBD (E(V A)PBD) in terms of the initial population size (N 0), the number of seed harvested per pod (M), the probability of survival of an individual (θ), and the generation at which the population is evaluated (S t). Differences between (E(V A)SSD) and (E(V A)PBD) are due to differences in the expected amount of random drift which occurs with the two methods after the S 0 generation. With both methods, random drift occurs when progeny are sampled from heterozygous parents. An additional component of random drift occurs when sampled progeny fail to survive during SSD, or when sampling occurs amoung families during PBD. For values of N 0, M, θ, and S t that are typical of soybean (Glycine max (L.) Merr.) breeding programs, (E(V A)SSD) will be greater than (E(V A)PBD). The ratio of (E(V A)SSD) to (E(V A)PBD) will: (1) increase as M and θ increase; (2) approach a value of 1.00 as N 0 increases; and (3) be a curvilinear function of S t. Plant breeders should compare SSD and PBD based upon values of (E(V A)SSD) and (E(V A)PBD) and the expected cost of carrying out the two methods.

Effect of mass selection on the within-family genetic variance in finite populations

The adequacy of an expression for the withinfamily genetic variance under pure random drift in an additive infinitesimal model was tested via simulation in populations undergoing mass selection. Two hundred or one thousand unlinked loci with two alleles at initial frequencies of 1/2 were considered. The size of the population was 100 (50 males and 50 females). Full-sib matings were carried out for 15 generations with only one male and one female chosen as parents each generation, either randomly or on an individual phenotypic value. In the unselected population, results obtained from 200 replicates were in agreement with predictions. With mass selection, within-family genetic variance was overpredicted by theory from the 12th and 4th generations for the 1,000 and 200 loci cases, respectively. Taking into account the observed change in gene frequencies in the algorithm led to a much better agreement with observed values. Results for the distribution of gene frequencies and the withinlocus genetic covariance are presented. It is concluded that the expression for the within-family genetic variance derived for pure random drift holds well for mass selection within the limits of an additive infinitesimal model.

Two-way within-family and mass selection for 8-week body weight in different mouse populations

Two mouse populations, randombred albino mice and a cross of four inbred strains, were divergently selected for high (H8) and low (L8) 8-week body weight over 18 generations using within-family and individual selection. The crossbreds showed asymmetry of selection response and realized heritabilities (H80·29 ± 0·01; L80·17 ± 0·01). In the randombred population realized heritabilities were symmetrical (H80·23 ± 0·01; L80·22 ± 0·02). Over the first nine generations individual selection was nearly 40 per cent better than within-family selection, as was expected from the full sib correlation in both populations. As selection progressed, within-family selection reached 82% and 61% of the responses obtained with individual selection in the crossbreds and randombred respectively. Correlated responses for 3-week (weaning) and 5-week body weights agreed with observations made on direct responses, but selection for L8did not reduce weaning weight. Selection for L8decreased and selection for H8increased first litter size at birth. However, mass-selected L8-pairs had a higher life-reproduction and life-span than H8-pairs.

Computer simulation of within family selection in finite populations

The relative efficiency, in terms of selection limits, between mass selection and within family selection was compared by computer simulation methods. A 20-locus additive model was used to simulate a quantitative trait under selection. It was assumed that 50-75 percent of the genetic variance in the base population was controlled by four major genes initially at low frequencies.In populations of size N=100 no loss of major genes was found when either method of selection was used. When N=50 within family selection was generally superior to mass selection but when N=10 the situation was reversed. For N=30 within family selection was more efficient only under high selection intensity or high heritability situations.