Genome-wide association study for age at puberty and resumption of cyclicity in a crossbred dairy cattle population.

Fertility is of primary economic importance in dairy cattle and the most common reason for involuntary culling. However, standard fertility traits have very low heritability that renders genetic selection slow and difficult. In this study, we explored fertility from an endocrine standpoint. A total of 1,163 crossbred Holstein-Normande females in a 3-generation familial design were studied for progesterone level measured every 10 d to determine age at puberty (PUB) and commencement of postpartum luteal activity (CPLA). Genetic parameters were estimated using REML with WOMBAT software. The heritability estimates were 0.38 ± 0.10 and 0.16 ± 0.07 for PUB and CPLA, respectively. Moreover, the 2 traits were genetically correlated (0.45 ± 0.23), suggesting a partially common determinism. Because of the family structure, a linkage disequilibrium and linkage analysis approach was preferred over standard genome-wide association study to map genomic regions associated with these traits. Ten quantitative trait loci (QTL) were detected for PUB on chromosomes 1, 3, 11, 13, 14, 21, and 29, whereas 3 QTL were associated with CPLA on chromosomes 21 and 26. Only the QTL on chromosome 21 was common to both traits. Four functional candidate genes (NCOA2, GAS2, OVOL1, and FOSL1) were identified in the detected regions. These findings will contribute to a clearer understanding of fertility determinism and enhance the value of introducing endocrinological data in fertility studies.

Short communication: Methods to compute genomic inbreeding for ungenotyped individuals.

The genomic measure of inbreeding is closer to the actual inbreeding than the pedigree-based measure. However, it cannot be computed for ungenotyped animals. An estimate of genomic inbreeding comes from the diagonal of matrix H used in single-step methods. This matrix projects genomic relationships to all ungenotyped members of the pedigree. The diagonal element of H-1 gives an estimate of the genomic inbreeding coefficient. However, so far no computational methods are available to compute the diagonal of H. Here we propose 3 exact methods to compute this diagonal. The first uses an already-existing algorithm to compute, for each ungenotyped individual, products of the form Hx to obtain the corresponding diagonal element of H. The second method computes, for each ungenotyped individual, a term that can be written as a quadratic form involving pedigree and genomic relationships. For both methods, the computational burden is linear in the number of ungenotyped animals. The last method reorders the computations of the second method so that they become linear in the number of genotyped animals, which is usually much smaller. We tested the methods in 3 small data sets (with ~2,000 genotyped animals and 30,000-500,000 animals in pedigree) and in a large simulated population (with 1,220,000 animals in pedigree and 36,000 genotyped animals). Tests resulted in satisfactory computing times (<10 min in the largest example using 10 parallel threads). Computing times were much shorter for the third method, as expected. Using these methods, estimates of genomic inbreeding in ungenotyped animals can be obtained on a regular basis.

Optimizing selection with several constraints in poultry breeding.

Poultry breeding schemes permanently face the need to control the evolution of coancestry and some critical traits, while selecting for a main breeding objective. The main aims of this article are first to present an efficient selection algorithm adapted to this situation and then to measure how the severity of constraints impacted on the degree of loss for the main trait, compared to BLUP selection on the main trait, without any constraint. Broiler dam and sire line schemes were mimicked by simulation over 10 generations and selection was carried out on the main trait under constraints for coancestry and for another trait, antagonistic with the main trait. The selection algorithm was a special simulated annealing (adaptative simulated annealing (ASA)). It was found to be rapid and able to meet constraints very accurately. A constraint on the second trait was found to induce an impact similar to or even greater than the impact of the constraint on coancestry. The family structure of selected poultry populations made it easy to control the evolution of coancestry at a reasonable cost but was not as useful for reducing the cost of controlling evolution of the antagonistic traits. Multiple constraints impacted almost additively on the genetic gain for the main trait. Adding constraints for several traits would therefore be justified in real life breeding schemes, possibly after evaluating their impact through simulated annealing.

MIM: an indirect method to assess inbreeding and coancestry in large incomplete pedigrees of selected dairy cattle.

In real data, inbreeding is usually underestimated because of missing pedigree information. A method adapted to the dairy cattle situation is presented to approximate inbreeding when the stored population pedigree is incomplete. Missing parents in incomplete pedigrees were given a dummy identification and assigned to groups (up to nine for a given birth date of progeny). These groups were linked to contemporary reference groups with known parents. An explicit model considered that polygenic breeding values in a censored group were centred on a function of the average breeding value in the corresponding reference group and deviated independently. Inbreeding coefficients were obtained progressively over birth dates starting from founders. For each date considered, the parameters pertaining to its groups were computed using the parameters already obtained from groups belonging to the previous dates. The updating algorithms were given in detail. An indirect method was implemented to expedite mass computations of the relationship coefficients involved (MIM). MIM was compared to Van Raden's (VR) method using simulated populations with 20 overlapping generations and different rates of missing sires and dams. In the situation of random matings, the average inbreeding coefficients by date obtained by MIM were close to true values, whereas they were strongly underestimated by VR. In the situation of assortative matings, MIM gave average inbreeding coefficients moderately underestimated, whereas those of VR's method were still strongly underestimated. The main conclusion of this study adapted to the situation of dairy cattle with incomplete pedigrees was that corrections for inbreeding and coancestry coefficients are more efficient with an explicit appropriate genetic model than without.

Optimized management of genetic variability in selected pig populations.

Controlling the increase of coancestry and inbreeding coefficients in selected populations is made possible through calculation of the optimal contributions allowed to breeding animals, given the current situation with regard to genetic diversity, and further, through optimal design of matings. The potential of such an approach for pig breeding was tested by retrospective optimization on the French Landrace population in reference to the matings actually carried out during a 21-week test period. The major constraint was that the average overall estimated breeding value (EBV) should be the same as the observed one, for not decreasing short-term genetic gain. Optimizing breeding allocations to boars would have led one to decrease coancestry and inbreeding coefficients by approximately 20%. This decrease would have even increased to approximately 30%, would have replacements and disposals been optimized after accounting for genetic variability, keeping the same constraint of genetic level identical to the observed one. These results showed the potential value, in the future, of completing each periodical calculation of EBVs by optimizations considering genetic variability and of releasing corresponding information to breeders, in order to enhance maintenance of genetic variability.

Estimation of genetic parameters for quantitative trait loci for dairy traits in the French Holstein population.

A marker-assisted selection program (MAS) has been implemented in dairy cattle in France. The efficiency of such a selection program depends on the use of correct genetic parameters for the marked quantitative trait loci (QTL). Therefore, the objective of this study was to estimate the proportion of genetic variance explained by 4 QTL described in previous studies (these QTL are segregating on chromosomes 6, 14, 20, and 26). Genotypes for 11 markers were available for 3,974 bulls grouped within 54 sire families of the French Holstein population undergoing MAS. The parameters were estimated for 4 QTL and 5 dairy traits: milk, fat and protein yields, and fat and protein percentages. The proportion of genetic variance explained by the QTL ranged from as low as 0.03 to 0.36%. Both lack of marker informativity and poor monitoring of QTL transmission might limit the accuracy of estimation. The QTL explained a larger proportion of genetic variance for milk composition traits. The QTL on chromosome 14 and chromosomes 6 and 20 have their largest influence on fat and protein percentages, respectively. The overall proportions of genetic variance explained by the QTL were 27.0, 30.7, 24.1, 48.2, and 33.6% for milk, fat and protein yields, and fat and protein percentages, respectively. These results clearly indicated that a large part of the genetic variance is explained by a small number of QTL and that their use in MAS might be beneficial for dairy cattle breeding programs.

A fast algorithm for computing inbreeding coefficients in large populations.

Inbreeding coefficients of animals are required in many genetic analyses of livestock records. A modification of Colleau's indirect algorithm to compute inbreeding coefficients in large populations is presented. With overlapping generations, the modified algorithm evaluated all progeny of each sire simultaneously in one back and forth exploration of a reduced pedigree. Simulation for a relatively large number of generations, different number of sires, family sizes and mating designs showed that Colleau's algorithm was faster (from 1.2 to 143 times) than two other algorithms under comparison (Tier, modified Meuwissen and Luo), in all situations investigated. Modifying Colleau's algorithm considerably decreased computation time (from 50 to 89%), resulting in a very fast algorithm. The number of sires mostly affected computational efficiency of the modified algorithm, whereas family size and mating design had virtually no effect. In the updating situation, when only animals born in the last year were evaluated, given known inbreeding coefficients for the other, the modified algorithm was also fast compared with the other three algorithms. Memory requirements for the algorithms were also discussed.

Predicting observed and latent responses to BLUP selection involving logistic variates.

Multivariate BLUPs can be derived when data are a mixture of continuous traits and observed discrete traits controlled by logistic latent traits. Algorithms were developed for predicting discrete responses to BLUP selection, and latent responses when the selection process included additional culling on scores. These algorithms were Taylor expansions using well-known expressions such as the probabilities and the two first moments of the truncated multinormal distribution, after appropriate re-parametrizations. They were compared to very accurate quadrature integrations. The test examples were suggested by a situation found in chickens where selection can involve body weight and leg deformity described by two logistic latent variates. Quadratic Taylor expansions generally provided a good accuracy. Therefore, they could be recommended when quadrature methods are too demanding, e.g., for complex breeding schemes.

Estimation of the genetic correlations between twisted legs and growth or conformation traits in broiler chickens.

SUMMARY: Genetic correlations between two types of leg deformities, valgus and varus angulations, and some growth or conformation traits were estimated in two commercial broiler strains. 14 264 chickens of both sexes in line A were measured for leg defects at 6 weeks and body weight at 3 (BW3) or 6 (BW6) weeks. The same measures were taken in line B on 8 164 chickens, as well as breast angle (BRA) and breast meat yield (BRM) at 6 weeks on 70% of the male birds. The multinomial logit model previously developed for the genetic analysis of valgus and varus deformities was extended to deal with the joint analysis of one unordered categorical trait and one continuous variable. The model assumed a competition between latent susceptibility variates related to the various deformities and linearly dependent on the continuous performances. Location parameters for latent susceptibilities and continuous trait were estimated by the 'Maximum A Posteriori' approach and dispersion parameters by the 'Maximum Marginal Likelihood' using a tilda-hat approximation. The genetic model took into account the effects of the sire, maternal grandsire and dam within maternal grandsire. As described in a previous study, leg deformities showed moderate heritabilities. Mean heritability estimate for both lines, based on the sire/maternal-grandsire (S/MGS) component, was equal to 0.22 for valgus and varus; when based on the dam component, mean estimates were equal to 0.37 and 0.29 for the two deformities respectively. Except for BRA, heritability of growth and conformation traits appeared to be smaller when based on S/MGS component (from 0.18 to 0.47) than on dam component (from 0.41 to 0.63). Very low genetic correlations were found between susceptibilities to leg deformities and growth performances: average estimates for both lines of the genetic correlation with BW3 were -0.03 and -0.05 for valgus and varus respectively. Respective genetic correlations with BW6 were estimated to be +0.05 and +0.01. According to a simulation study these small estimates were unlikely to be due to the negative back effects of severe disorders on growth performances. According to these results, including leg defects in breeding schemes would not delay improvement on growth through unfavourable genetic correlations. Susceptibility to valgus deformity appeared to be genetically independent of conformation traits (genetic correlation was estimated to be -0.06 and -0.08 with BRA and BRM respectively), whereas moderate unfavourable genetic correlations were found for varus (+0.16 and +0.19 with BRA and BRM respectively). Care must be taken when considering the impact of the actual intensive selection for greater conformation on the incidence of varus deformity. RÉSUMÉ: Les corrélations génétiques entre deux types de déformations osseuses, le valgus et le varus, et des caractères de croissance et de conformation ont été estimées dans deux lignées commerciales de poulet de type chair. Dans la lignée "A", 14 264 poulets des deux sexes ont été mesurés pour les problèmes de pattes à 6 semaines, ainsi que pour le poids vif aux âges de 3 et 6 semaines. Ces mêmes mesures ont été faites dans la lignée "B"sur 8 164 poulets; on disposait en plus pour un échantillon des animaux de cette lignée de la mesure de l'angle de poitrine et du rendement en filet à 6 semaines. Des développements du modèle logistique multinomial déjà utilisé pour l'analyse génétique des valgus et varus ont été réalisés pour permettre l'analyse conjointe de plusieurs caractères discrets non ordonnés et d'une variable continue. Le modèle d'analyse fait l'hypothèse d'une compétition entre plusieurs variables sous-jacentes de sensibilité aux déformations, dépendant linéairement de la performance continue. Les paramètres de position pour les sensibilités sous-jacentes et le caractère continu ont été estimés par l'approche bayésienne du "Maximum A Posteriori"et les paramètres de dispersion par une approximation de type tilde-hat du "Maximum de Vraisemblance Marginale". Le modèle génétique d'analyse comprenait les effets des père, grand-père maternel et mère intra grand-père. Comme démontré dans une étude précédente, la sensibilité aux problèmes de patte présente une héritabilité modérée. En moyenne sur les deux lignées, l'estimation obtenue par la voie père/grand-père maternel (P-GPM) est de 0.22 pour les deux déformations et celle pour la voie mère de 0.37 pour le valgus et de 0.29 pour le varus. A l'exception de l'angle de poitrine, l'héritabilité des caractères de croissance et de conformation apparaît largement supérieure par la voie mère (de 0.41 à 0.63) que par la voie P-GPM (de 0.18 à 0.47). Les corrélations génétiques entre les sensibilités aux déformations osseuses et les performances de croissance apparaissent très faibles: la moyenne des estimations de la corrélation avec le poids vif à 3 semaines est de -0.03 pour le valgus et -0.05 pour le varus. Les corrélations avec le poids à 6 semaines sont du même ordre, estimées à respectivement +0.05 et +0.01 pour les valgus et varus. Une étude par simulation a permis de vérifier que ces faibles corrélations n'étaient pas dues à des biais éventuels liés aux effets secondaires négatifs des pathologies sévères sur les performances de croissance. D'après ces rèsultats, la prise en compte en sélection de la sensibilité aux problèmes de pattes n'introdurait pas, par une corrélation génétique défavorable, de réponse indirecte négative sur le poids. Si la sensibilité au valgus apparaît génétiquement indépendante des caractères de conformation (avec des corrélations génétiques avec l'angle de poitrine et le pourcentage de filet estimées à -0.06 et -0.8 respectivement), la liaison génétique apparaît plutôt défavorable pour le varus: +0.16 et +0.19 respectivement avec l'angle et le pourcentage de filet. Ces résultats doivent inciter à surveiller l'impact sur l'incidence des varus des fortes pressions de sélection appliquées actuellement sur les caractères de conformation. ZUSAMMENFASSUNG: Genetische Korrelationen zwischen verbogenen Füßen und Wachstums- und Formmerkmalen in Broilern Genetische Korrelationen zwischen 2 Arten von Beindeformationen, Valgus und Varus Angulationen, und einigen Wachtums- und Formmerkmalen wurden bei zwei kommerziellen Broiler Herkünften geschätzt, 14 264 Hühner beiderlei Geschlechter wurden in Linie A auf Beinfehler bei 6 Wochen Alter und Körpergewicht bei 3 (BW3) und 6 Wochen (BW6) untersucht, in Linie B 8 164 Tiere, wo aber auch Brustwinkel (BRA) und Brustfleisch (BRM) von ca. 70% der Hähne erhoben worden ist. Das für die genetische Analyse von Valgus und Varus Deformationen entwickelte multinomiale logit Modell wurde für die gemeinsame Analyse eines ungeordneten kategorischen Merkmals und einer kontinuierlichen Variablen erweitert. Dieses unterstellt Kompetition zwischen latenter Anfälligkeiten für verschiedene Deformationen und lineare Beziehung zu kontinuierlich verteilter Leistung. Lokationsparameter wurden mittels "Maximum A Posteriori" Ansatz und Dispersionsparameter mittels "Maximum Marginaler Likelihood" unter Verwendung von 'tilde-hat' Approximation geschätzt. Das genetische Modell berücksichtigte Vater-, maternale Großvater- und Muttertier-innerhalb der letzteren-Wirkungen. Beindeformationen zeigen mittlere Heritabilitätswerte, 0.22 für Valgus and Varus aus Vater/maternalem Großvater Komponenten, 0.37 bez. 0.29 aus der Muttertierkomponente. Mit Ausnahme von BRA waren Heritabilitätswerte für Wachstum- und Formmerkmale aus S/MGS-Komponenten (0.18-0.47) kleiner als die aus Muttertierkomponenten (0.41-0.63). Genetische Korrelationen zwischen letzeren und Anfällikeiten waren sehr niedrig: zwischen BW3 und Valgus und Varus -0.03 bzw. -0.05, BW6 +0.05 und 0.01. Simulation zeigte, daß die niedrigen Werte kaum auf negative Rückwirkung der Defekte auf Leistung zurückzuführen sind, sodaß deren Berücksichtigung in der Selektion den Zuchtfortschritt nicht beeinträchtigen sollte. Valgusdeformation scheint genetisch unabhängig von Formmerkmalen zu sein (r(G) -0.06, -0.08 mit BRA und BRM), während Varus mäßig ungünstige Korrelationen zeigt (+0.16, -0.19 mit BRA und BRM), sodaß Selektion auf Bemuskelung dies zu beachten hat.

Marker-assisted selection for a sex-limited character in a nucleus breeding population.

The benefits of marker-assisted selection were examined by simulation of an adult multiple ovulation and embryo transfer nucleus breeding scheme. Animals were either typed for two polymorphic marker loci, 20 centimorgans apart, flanking a single biallelic quantitative trait locus and were evaluated using a model accounting for marker information, or animals were not typed but were evaluated by a conventional BLUP animal model. Selection was for a single trait measured on females, and each dam had 4 sons and 4 daughters. Nucleus foundation animals were chosen from a base population in linkage equilibrium. With the favorable allele at an initial frequency of 0.5, marker-assisted selection substantially increased responses at the quantitative trait locus but reduced the polygenic responses. Cumulative genetic gain increased by up to 3, 9, 12, and 6% after one, two, three, and six generations of selection, respectively. If the favorable allele was initially rare (frequency of 0.1), the merits of marker-assisted selection were even more pronounced (genetic gains increased by up to 9, 19, 24 and 15%, respectively). The superiority of marker-assisted selection over conventional BLUP increased when a restriction was placed on selection of full brothers and decreased when variance of the quantitative trait locus used in the evaluation model was overestimated.

Genetic steady-state under BLUP selection for an infinite and homogeneous population with discrete generations.

A matrix derivation is proposed to analytically calculate the asymptotic genetic variance-covariance matrix under BLUP selection according to the initial genetic parameters in a large population with discrete generations. The asymptotic genetic evolution of a homogeneous population with discrete generations is calculated for a selection operating on an index including all information (pedigree and records) from a non-inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is independent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction error variance depends on selection. Furthermore, records of several generations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order not to overestimate the asymptotic genetic gains and not to underestimate the asymptotic genetic variances.

Marker assisted selection for genetic improvement of animal populations when a single QTL is marked.

A Monte Carlo simulation study to evaluate the benefits of marker assisted selection (MAS) in small populations with one marked bi-allelic quantitative trait locus (QTL) is described. In the base generation, linkage phase equilibrium between the markers, QTL and polygenes was assumed and frequencies of 0.5 for the two QTL alleles were used. Six discrete generations of selection for a single character measured on both sexes followed. An additive genetic model was used with the QTL positioned midway between two highly polymorphic markers. Schemes were simulated with a distance of 10 cM between the QTL and either of the two markers and with the QTL explaining 1/8 of the total genetic variance in the base generation. Values of 0.5, 0.25 or 0.1 were assumed for the heritability. Eight males and 16, 32 or 64 females were selected each generation with each dam producing four sons and four daughters as candidates for the next generation. Animals were evaluated with a conventional BLUP animal model or with a model using marker information. MAS resulted in substantially higher QTL responses (4-54%), especially with low heritabilities, than conventional BLUP but lower polygenic responses (up to 4%) so that the overall effect on the total genetic response, although in the majority of cases favourable, was relatively small. With QTLs of larger size (explaining 25% of the genetic variance) comparable results were found. When the distance between the QTL and the markers was reduced to 2 cM, genetic responses were increased very slightly with a heritability of 0.5 whereas with a heritability of 0.1 responses were increased by up to 10%, compared with conventional BLUP. Results emphasize that MAS should be most useful for lowly heritable traits and that once QTLs for such traits have been identified the search for closely linked polymorphic markers should be intensified.

A simulation study of selection methods to improve mastitis resistance of dairy cows.

Mastitis problems were assumed to decrease profitability of dairy cows through milk price, treatment cost, and involuntary culling cost. Milk price decreased through a stepwise function of SCC (actual French conditions). A continuous latent variate was supposed to trigger other costs through appropriate thresholds. The relative weights for one genetic standard deviation in the selection objective were 1, -.07, and -.14 for yield, SCC, and mastitis liability, respectively. Annual genetic gains were predicted for a conventional breeding scheme using statistical parameters from the literature. Selection on yield and log SCC, with or without mastitis culling rate, increased genetic gains for the overall profitability of .7 or .9%, respectively, compared with selection on yield only. Increases of log SCC and mastitis problems because of selection were substantially reduced (40 to 60%). Consequences from constraint of the genetic trend for mastitis liability to zero depended on the method used to assess mastitis problems. Use of log SCC had a significant and variable negative impact (-8.9 to -36% according to parameters) on overall efficiency compared with the relevant unconstrained selection. Simultaneous use of log SCC and culling rate had a moderate effect (-4.9 to -7.4%) on overall efficiency compared with that from unconstrained selection.

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.

Using embryo sexing within closed mixed multiple ovulation and embryo transfer schemes for selection on dairy cattle.

Two types of multiple ovulation and embryo transfer schemes that included bull progeny testing were compared. In the juvenile schemes, embryos were collected at 16 to 18 mo of age without sexing, whereas, in the adult schemes, donors were chosen based on their first lactation record, and their embryos were systematically sexed. With the latter schemes, natural calves obtained at the first two calvings could compete with embryo transfer calves to be replacements. The optimal structure of this scheme was derived algebraically for the same number of transferred embryos as in the juvenile schemes. Predicted asymptotic annual genetic gains, after stabilization of genetic parameters taking into account the Bulmer effect, were found to be slightly in favor of the adult schemes for a given set of parameters (overall number of transferred embryos, number of embryos per collection, and embryo survival rate). In the adult schemes, the nucleus sizes were much larger than in the juvenile schemes, which allowed a higher selection differential on male paths, thus compensating for the longer generation interval. Asymptotic rate of genetic gain for Monte Carlo simulations were about 10 and 7% lower for juvenile and adult schemes, respectively, but still higher (20%) than the predicted value for the corresponding conventional scheme. Consequently, adult schemes with embryo sexing can be an efficient alternative to juvenile schemes without embryo sexing.

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.

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.