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John M. Hickey - One of the best experts on this subject based on the ideXlab platform.
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optimal cross selection for long term Genetic Gain in two part programs with rapid recurrent genomic selection
Theoretical and Applied Genetics, 2018Co-Authors: Gregor Gorjanc, Chris R Gaynor, John M. HickeyAbstract:Abstract Key message Optimal cross selection increases long-term Genetic Gain of two-part programs with rapid recurrent genomic selection. It achieves this by optimising efficiency of converting Genetic diversity into Genetic Gain through reducing the loss of Genetic diversity and reducing the drop of genomic prediction accuracy with rapid cycling.
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optimal cross selection for long term Genetic Gain in two part programs with rapid recurrent genomic selection
bioRxiv, 2017Co-Authors: Gregor Gorjanc, Robert Gaynor, John M. HickeyAbstract:This study evaluates optimal cross selection for balancing selection and maintenance of Genetic diversity in two-part plant breeding programs with rapid recurrent genomic selection. The two-part program reorganizes a conventional breeding program into population improvement component with recurrent genomic selection to increase the mean of germplasm and product development component with standard methods to develop new lines. Rapid recurrent genomic selection has a large potential, but is challenging due to genotyping costs or Genetic drift. Here we simulate a wheat breeding program for 20 years and compare optimal cross selection aGainst truncation selection in the population improvement with one to six cycles per year. With truncation selection we crossed a small or a large number of parents. With optimal cross selection we jointly optimised selection, maintenance of Genetic diversity, and cross allocation with AlphaMate program. The results show that the two-part program with optimal cross selection delivered the largest Genetic Gain that increased with the increasing number of cycles. With four cycles per year optimal cross selection had 78% (15%) higher long-term Genetic Gain than truncation selection with a small (large) number of parents. Higher Genetic Gain was achieved through higher efficiency of converting Genetic diversity into Genetic Gain; optimal cross selection quadrupled (doubled) efficiency of truncation selection with a small (large) number of parents. Optimal cross selection also reduced the drop of genomic selection accuracy due to the drift between training and prediction populations. In conclusion, optimal cross-selection enables optimal management and exploitation of population improvement germplasm in two-part programs.
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The potential of shifting recombination hotspots to increase Genetic Gain in livestock breeding
Genetics Selection Evolution, 2017Co-Authors: Serap Gonen, Mara Battagin, Susan E. Johnston, Gregor Gorjanc, John M. HickeyAbstract:This study uses simulation to explore and quantify the potential effect of shifting recombination hotspots on Genetic Gain in livestock breeding programs. We simulated three scenarios that differed in the locations of quantitative trait nucleotides (QTN) and recombination hotspots in the genome. In scenario 1, QTN were randomly distributed along the chromosomes and recombination was restricted to occur within specific genomic regions (i.e. recombination hotspots). In the other two scenarios, both QTN and recombination hotspots were located in specific regions, but differed in whether the QTN occurred outside of (scenario 2) or inside (scenario 3) recombination hotspots. We split each chromosome into 250, 500 or 1000 regions per chromosome of which 10% were recombination hotspots and/or contained QTN. The breeding program was run for 21 generations of selection, after which recombination hotspot regions were kept the same or were shifted to adjacent regions for a further 80 generations of selection. We evaluated the effect of shifting recombination hotspots on Genetic Gain, Genetic variance and genic variance. Our results show that shifting recombination hotspots reduced the decline of Genetic and genic variance by releasing standing allelic variation in the form of new allele combinations. This in turn resulted in larger increases in Genetic Gain. However, the benefit of shifting recombination hotspots for increased Genetic Gain was only observed when QTN were initially outside recombination hotspots. If QTN were initially inside recombination hotspots then shifting them decreased Genetic Gain. Shifting recombination hotspots to regions of the genome where recombination had not occurred for 21 generations of selection (i.e. recombination deserts) released more of the standing allelic variation available in each generation and thus increased Genetic Gain. However, whether and how much increase in Genetic Gain was achieved by shifting recombination hotspots depended on the distribution of QTN in the genome, the number of recombination hotspots and whether QTN were initially inside or outside recombination hotspots. Our findings show future scope for targeted modification of recombination hotspots e.g. through changes in zinc-finger motifs of the PRDM9 protein to increase Genetic Gain in production species.
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potential of gene drives with genome editing to increase Genetic Gain in livestock breeding programs
Genetics Selection Evolution, 2017Co-Authors: Serap Gonen, Gregor Gorjanc, Janez Jenko, Alan J Mileham, Bruce C A Whitelaw, John M. HickeyAbstract:Background This paper uses simulation to explore how gene drives can increase Genetic Gain in livestock breeding programs. Gene drives are naturally occurring phenomena that cause a mutation on one chromosome to copy itself onto its homologous chromosome.
Gregor Gorjanc - One of the best experts on this subject based on the ideXlab platform.
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optimal cross selection for long term Genetic Gain in two part programs with rapid recurrent genomic selection
Theoretical and Applied Genetics, 2018Co-Authors: Gregor Gorjanc, Chris R Gaynor, John M. HickeyAbstract:Abstract Key message Optimal cross selection increases long-term Genetic Gain of two-part programs with rapid recurrent genomic selection. It achieves this by optimising efficiency of converting Genetic diversity into Genetic Gain through reducing the loss of Genetic diversity and reducing the drop of genomic prediction accuracy with rapid cycling.
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optimal cross selection for long term Genetic Gain in two part programs with rapid recurrent genomic selection
bioRxiv, 2017Co-Authors: Gregor Gorjanc, Robert Gaynor, John M. HickeyAbstract:This study evaluates optimal cross selection for balancing selection and maintenance of Genetic diversity in two-part plant breeding programs with rapid recurrent genomic selection. The two-part program reorganizes a conventional breeding program into population improvement component with recurrent genomic selection to increase the mean of germplasm and product development component with standard methods to develop new lines. Rapid recurrent genomic selection has a large potential, but is challenging due to genotyping costs or Genetic drift. Here we simulate a wheat breeding program for 20 years and compare optimal cross selection aGainst truncation selection in the population improvement with one to six cycles per year. With truncation selection we crossed a small or a large number of parents. With optimal cross selection we jointly optimised selection, maintenance of Genetic diversity, and cross allocation with AlphaMate program. The results show that the two-part program with optimal cross selection delivered the largest Genetic Gain that increased with the increasing number of cycles. With four cycles per year optimal cross selection had 78% (15%) higher long-term Genetic Gain than truncation selection with a small (large) number of parents. Higher Genetic Gain was achieved through higher efficiency of converting Genetic diversity into Genetic Gain; optimal cross selection quadrupled (doubled) efficiency of truncation selection with a small (large) number of parents. Optimal cross selection also reduced the drop of genomic selection accuracy due to the drift between training and prediction populations. In conclusion, optimal cross-selection enables optimal management and exploitation of population improvement germplasm in two-part programs.
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The potential of shifting recombination hotspots to increase Genetic Gain in livestock breeding
Genetics Selection Evolution, 2017Co-Authors: Serap Gonen, Mara Battagin, Susan E. Johnston, Gregor Gorjanc, John M. HickeyAbstract:This study uses simulation to explore and quantify the potential effect of shifting recombination hotspots on Genetic Gain in livestock breeding programs. We simulated three scenarios that differed in the locations of quantitative trait nucleotides (QTN) and recombination hotspots in the genome. In scenario 1, QTN were randomly distributed along the chromosomes and recombination was restricted to occur within specific genomic regions (i.e. recombination hotspots). In the other two scenarios, both QTN and recombination hotspots were located in specific regions, but differed in whether the QTN occurred outside of (scenario 2) or inside (scenario 3) recombination hotspots. We split each chromosome into 250, 500 or 1000 regions per chromosome of which 10% were recombination hotspots and/or contained QTN. The breeding program was run for 21 generations of selection, after which recombination hotspot regions were kept the same or were shifted to adjacent regions for a further 80 generations of selection. We evaluated the effect of shifting recombination hotspots on Genetic Gain, Genetic variance and genic variance. Our results show that shifting recombination hotspots reduced the decline of Genetic and genic variance by releasing standing allelic variation in the form of new allele combinations. This in turn resulted in larger increases in Genetic Gain. However, the benefit of shifting recombination hotspots for increased Genetic Gain was only observed when QTN were initially outside recombination hotspots. If QTN were initially inside recombination hotspots then shifting them decreased Genetic Gain. Shifting recombination hotspots to regions of the genome where recombination had not occurred for 21 generations of selection (i.e. recombination deserts) released more of the standing allelic variation available in each generation and thus increased Genetic Gain. However, whether and how much increase in Genetic Gain was achieved by shifting recombination hotspots depended on the distribution of QTN in the genome, the number of recombination hotspots and whether QTN were initially inside or outside recombination hotspots. Our findings show future scope for targeted modification of recombination hotspots e.g. through changes in zinc-finger motifs of the PRDM9 protein to increase Genetic Gain in production species.
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potential of gene drives with genome editing to increase Genetic Gain in livestock breeding programs
Genetics Selection Evolution, 2017Co-Authors: Serap Gonen, Gregor Gorjanc, Janez Jenko, Alan J Mileham, Bruce C A Whitelaw, John M. HickeyAbstract:Background This paper uses simulation to explore how gene drives can increase Genetic Gain in livestock breeding programs. Gene drives are naturally occurring phenomena that cause a mutation on one chromosome to copy itself onto its homologous chromosome.
Ben J. Hayes - One of the best experts on this subject based on the ideXlab platform.
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Boosting Genetic Gain in allogamous crops via speed breeding and genomic selection
Frontiers in Plant Science, 2019Co-Authors: Abdulqader Jighly, Ben J. Hayes, Luke W. Pembleton, Noel O. I. Cogan, German Spangenberg, Hans D. DaetwylerAbstract:Breeding schemes that utilize modern breeding methods like genomic selection (GS) and speed breeding (SB) have the potential to accelerate Genetic Gain for different crops. We investigated through stochastic computer simulation the advantages and disadvantages of adopting both GS and SB (SpeedGS) into commercial breeding programs for allogamous crops. In addition, we studied the effect of omitting one or two selection stages from the conventional phenotypic scheme on GS accuracy, Genetic Gain, and inbreeding. As an example, we simulated GS and SB for five traits (heading date, forage yield, seed yield, persistency, and quality) with different Genetic architectures and heritabilities (0.7, 0.3, 0.4, 0.1, and 0.3; respectively) for a tall fescue breeding program. We developed a new method to simulate correlated traits with complex architectures of which effects can be sampled from multiple distributions, e.g. to simulate the presence of both minor and major genes. The phenotypic selection scheme required 11 years, while the proposed SpeedGS schemes required four to nine years per cycle. Generally, SpeedGS schemes resulted in higher Genetic Gain per year for all traits especially for traits with low heritability such as persistency. Our results showed that running more SB rounds resulted in higher Genetic Gain per cycle when compared to phenotypic or GS only schemes and this increase was more pronounced per year when cycle time was shortened by omitting cycle stages. While GS accuracy declined with additional SB rounds, the decline was less in round three than in round two, and it stabilized after the fourth SB round. However, more SB rounds resulted in higher inbreeding rate, which could limit long-term Genetic Gain. The inbreeding rate was reduced by approximately 30% when generating the initial population for each cycle through random crosses instead of generating half-sib families. Our study demonstrated a large potential for additional Genetic Gain from combining GS and SB. Nevertheless, methods to mitigate inbreeding should be considered for optimal utilization of these highly accelerated breeding programs.
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“SpeedGS” to Accelerate Genetic Gain in Spring Wheat
Applications of Genetic and Genomic Research in Cereals, 2019Co-Authors: Kai P. Voss-fels, Eva Herzog, Susanne Dreisigacker, Sivakumar Sukumaran, Amy Watson, Matthias Frisch, Ben J. Hayes, Lee T HickeyAbstract:Abstract The Genetic improvement of modern wheat varieties has been very successful throughout the history of breeding; yet, future wheat production remains challenging. While annual yield increases need to be doubled over the next few decades, the global production trends in all major wheat growing regions indicate a yield plateau. To overcome this, innovative strategies that efficiently integrate modern technologies in breeding programs are required. Using simulations based on real wheat data sets, we exemplify how genomic selection and “speed breeding,” a novel rapid generation advancement technology, can be combined to substantially reduce the length of the breeding cycle and maximize Genetic Gain per unit time. We outline the opportunities and challenges associated with the fusion of these breeding tools and reinforce the importance of integrating novel Genetic diversity in breeding programs to achieve sustainable long-term Genetic Gain.
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environmental characterization facilitate g e interaction to highlight the role of stay green traits for Genetic Gain
2019Co-Authors: Asad Amin, Lee T Hickey, Ben J. Hayes, Jack Christopher, Behnam Ababaei, Mark E Cooper, Kai P Vossfels, Karine ChenuAbstract:Environmental characterization (EC) one influential approach for understanding the performance of genotypes in different environments. Sometimes interactions between environment and genotype limit the Genetic Gain for complex traits in breeding programs, especially drought. Stay green lines are able to retain green leaf area longer than standard lines leading to superior adaptation under water-limitation. Modelling framework has been used analytically in breeding to dissect complex traits, such as yield under water limitation, into critical trait components (e.g. stay-green, flowering time, root architecture). Characterization can help to select more heritable genotypes that can be subjected to high throughout phenotyping, and make more sensible targets for genomic selection by the following search aims: (1) Characterise stressed environments (accounting for climate and soil characteristics, management practices, and crop development) to characterise the timing and severity of the stress and non-stress, (2) Identify the potentially adaptive cultivars and traits in each specific environment, and (3) Determine the correlation between stay-green traits and yield in the different environment types.
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mitigation of inbreeding while preserving Genetic Gain in genomic breeding programs for outbred plants
Theoretical and Applied Genetics, 2017Co-Authors: Ben J. Hayes, Hans D. DaetwylerAbstract:Key message Heuristic genomic inbreeding controls reduce inbreeding in genomic breeding schemes without reducing Genetic Gain.
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selection on optimal haploid value increases Genetic Gain and preserves more Genetic diversity relative to genomic selection
Genetics, 2015Co-Authors: Hans D. Daetwyler, German Spangenberg, Matthew J Hayden, Ben J. HayesAbstract:Doubled haploids are routinely created and phenotypically selected in plant breeding programs to accelerate the breeding cycle. Genomic selection, which makes use of both phenotypes and genotypes, has been shown to further improve Genetic Gain through prediction of performance before or without phenotypic characterization of novel germplasm. Additional opportunities exist to combine genomic prediction methods with the creation of doubled haploids. Here we propose an extension to genomic selection, optimal haploid value (OHV) selection, which predicts the best doubled haploid that can be produced from a segregating plant. This method focuses selection on the haplotype and optimizes the breeding program toward its end goal of generating an elite fixed line. We rigorously tested OHV selection breeding programs, using computer simulation, and show that it results in up to 0.6 standard deviations more Genetic Gain than genomic selection. At the same time, OHV selection preserved a substantially greater amount of Genetic diversity in the population than genomic selection, which is important to achieve long-term Genetic Gain in breeding populations.
Rex Bernardo - One of the best experts on this subject based on the ideXlab platform.
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Targeted recombination to increase Genetic Gain in self-pollinated species
Theoretical and Applied Genetics, 2019Co-Authors: Sushan Ru, Rex BernardoAbstract:Key message If we can induce or select for recombination at targeted marker intervals, Genetic Gains for quantitative traits in self-pollinated species may be doubled. Abstract Targeted recombination refers to inducing or selecting for a recombination event at genomic positions that maximize Genetic Gain in a cross. A previous study indicated that targeted recombination could double the rate of Genetic Gains in maize ( Zea mays L.), a cross-pollinated crop for which historical Genetic Gains have been large. Our objectives were to determine whether targeted recombination can sufficiently increase predicted Gains in self-pollinated species, and whether prospective Gains from targeted recombination vary across crops, populations, traits, and chromosomes. Genomewide marker effects were estimated from previously published marker and phenotypic data on 21 biparental populations of soybean [ Glycine max (L.) Merr.], wheat ( Triticum aestivum L.), barley ( Hordeum vulgare L.), and pea ( Pisum sativum L.). With the predicted Gain from nontargeted recombination as the baseline, the relative Gains from creating a doubled haploid with up to one targeted recombination [ RG _( x ≤ 1)] and two targeted recombinations [ RG _( x ≤ 2)] per chromosome or linkage group were calculated. Targeted recombination significantly ( P = 0.05) increased the predicted Genetic Gain compared to nontargeted recombination for all traits and all populations, except for plant height in barley. The mean RG _( x ≤ 1) was 211%, whereas the mean RG _( x ≤ 2) was 243%. The predicted Gain varied among traits and populations. For most traits and populations, having targeted recombination on less than a third of all the chromosomes led to the same or higher predicted Gain than nontargeted recombination. Together with previous findings in maize, our results suggested that targeted recombination could double the Genetic Gains in both self- and cross-pollinated crops.
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targeted recombination to increase Genetic Gain in self pollinated species
Theoretical and Applied Genetics, 2019Co-Authors: Sushan Ru, Rex BernardoAbstract:If we can induce or select for recombination at targeted marker intervals, Genetic Gains for quantitative traits in self-pollinated species may be doubled. Targeted recombination refers to inducing or selecting for a recombination event at genomic positions that maximize Genetic Gain in a cross. A previous study indicated that targeted recombination could double the rate of Genetic Gains in maize (Zea mays L.), a cross-pollinated crop for which historical Genetic Gains have been large. Our objectives were to determine whether targeted recombination can sufficiently increase predicted Gains in self-pollinated species, and whether prospective Gains from targeted recombination vary across crops, populations, traits, and chromosomes. Genomewide marker effects were estimated from previously published marker and phenotypic data on 21 biparental populations of soybean [Glycine max (L.) Merr.], wheat (Triticum aestivum L.), barley (Hordeum vulgare L.), and pea (Pisum sativum L.). With the predicted Gain from nontargeted recombination as the baseline, the relative Gains from creating a doubled haploid with up to one targeted recombination [RG(x ≤ 1)] and two targeted recombinations [RG(x ≤ 2)] per chromosome or linkage group were calculated. Targeted recombination significantly (P = 0.05) increased the predicted Genetic Gain compared to nontargeted recombination for all traits and all populations, except for plant height in barley. The mean RG(x ≤ 1) was 211%, whereas the mean RG(x ≤ 2) was 243%. The predicted Gain varied among traits and populations. For most traits and populations, having targeted recombination on less than a third of all the chromosomes led to the same or higher predicted Gain than nontargeted recombination. Together with previous findings in maize, our results suggested that targeted recombination could double the Genetic Gains in both self- and cross-pollinated crops.
B. Villanueva - One of the best experts on this subject based on the ideXlab platform.
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Effect of assortative mating on Genetic Gain and inbreeding in aquaculture selective breeding programs
Aquaculture, 2017Co-Authors: María Saura, B. Villanueva, Jesús Fernández, Miguel A. ToroAbstract:Abstract In this simulation study, the effect of the mating scheme on Genetic Gain and inbreeding has been explored for aquaculture selection programs where tank effects and large family sizes are common. Different selection methods were investigated (individual, family, sib, combined and within-family selection). Our results suggest that under family and sib selection, Genetic Gain was increased with assortative mating in comparison to random mating. The advantage of assortative mating increased when common environmental effects were present. Contrarily, a decrease in Genetic Gain was observed with disassortative mating, except for the case of within-family selection. The advantage of assortative mating over random mating was due to the increase in the between-family component of the additive Genetic variance that was exacerbated with the presence of common environmental effects. Under family and sib selection, the join effect of assortative mating and common environmental effects produced an increase in Genetic Gain of around 80 and 40% at early generations, and around 10 and 60% at later generations, respectively. Inbreeding was low under family selection for all mating schemes but much higher under sib selection when assortative mating was performed. In fact, the inbreeding coefficient after 10 generations of selection was 300% higher when assortative matings were performed under sib selection, compared to random matings. This was due to the fact that under sib selection, matings were based on family means, leading to an increased frequency of within-family matings. To our knowledge, this is the first study that investigates the effect of the mating scheme on Genetic Gain and inbreeding in an aquaculture context where family sizes are large and tank effects are present, and shows that assortative mating can substantially enhance the response to selection, particularly when family selection methods are applied. Statement of relevance Our article complies with the Policy Statement for submission of manuscripts to the Genetics Section, as it provides insight into the issue of breeding programs. Here, we have connected previous work in the field to address new questions, focusing on how the mating scheme may affect both Genetic Gain and inbreeding in aquaculture selection programs, where family sizes are typically large and tank effects are usually present. In fish species, it is possible to consider different mating schemes because fecundity is high and because in vitro fertilization is often possible. A particular problem in aquaculture breeding programs is the impossibility of tagging physically newborn individuals. Given this, a common practice in aquaculture is to rear families in separate tanks until the fish are large enough to be individually tagged. This introduces an environmental effect common to the members of the same family (tank effect) which can lead to a reduction of the response to selection that needs to be considered. We studied here the efficiency of different selection methods in terms of Genetic Gain and inbreeding and investigated the effect of the mating scheme to optimize breeding programs in aquaculture when tank effects are present. We have shown that assortative mating can substantially enhance the response to selection, particularly when family selection methods are applied and tank effects are present. To our knowledge, the effect of the mating scheme in an aquaculture context has never been addressed before. Our results suggest that assortative mating in the presence of common environmental variance may be considered in selection programs in aquaculture. Our conclusions will help breeders make optimal mating choices.
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Effect of assortative mating on Genetic Gain and inbreeding in aqueaculture selective breeding programs
2016Co-Authors: María Saura, B. Villanueva, Jesús Fernández, Miguel Angel Toro IbañezAbstract:In this simulation study, the effect of the mating scheme on Genetic Gain and inbreeding has been explored for aquaculture selection programs where tank effects and large family sizes are common. Different selection methods were investigated (individual, family, sib, combined and within-family selection). Our results suggest that under family and sib selection, Genetic Gain was increased with assortative mating in comparison to random mating. The advantage of assortative mating increased when common environmental effects were present. Contrarily, a decrease in Genetic Gain was observed with disassortative mating, except for the case of within-family selection. The advantage of assortative mating over random mating was due to the increase in the between-family component of the additive Genetic variance that was exacerbated with the presence of common environmental effects. Under family and sib selection, the join effect of assortative mating and common environmental effects produced an increase in Genetic Gain of around 80 and 40% at early generations, and around 10 and 60% at later generations, respectively. Inbreeding was low under family selection for all mating schemes but much higher under sib selection when assortative mating was performed. In fact, the inbreeding coefficient after 10 generations of selection was 300% higher when assortative matings were performed under sib selection, compared to random matings. This was due to the fact that under sib selection, matings were based on family means, leading to an increased frequency of within-family matings. To our knowledge, this is the first study that investigates the effect of the mating scheme on Genetic Gain and inbreeding in an aquaculture context where family sizes are large and tank effects are present, and shows that assortative mating can substantially enhance the response to selection, particularly when family selection methods are applied.
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Prediction of Genetic Gain from quadratic optimisation with constrained rates of inbreeding
Genetics Selection Evolution, 2006Co-Authors: B. Villanueva, Santiago Avendaño, John A WoolliamsAbstract:There are selection methods available that allow the optimisation of Genetic contributions of selection candidates for maximising the rate of Genetic Gain while restricting the rate of inbreeding. These methods imply selection on quadratic indices as the selection merit of a particular individual is a quadratic function of its estimated breeding value. This study provides deterministic predictions of Genetic Gain from selection on quadratic indices for a given set of resources (the number of candidates), heritability, and target rate of inbreeding. The rate of Gain was obtained as a function of the accuracy of the Mendelian sampling term at the time of convergence of long-term contributions of selected candidates and the theoretical ideal rate of Gain for a given rate of inbreeding after an exact allocation of long-term contributions to Mendelian sampling terms. The expected benefits from quadratic indices over traditional linear indices ( i.e. truncation selection), both using BLUP breeding values, were quantified. The results clearly indicate higher Gains from quadratic optimisation than from truncation selection. With constant rate of inbreeding and number of candidates, the benefits were generally largest for intermediate heritabilities but evident over the entire range. The advantage of quadratic indices was not highly sensitive to the rate of inbreeding for the constraints considered.
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expected Genetic contributions and their impact on gene flow and Genetic Gain
Genetics, 1999Co-Authors: John Woolliams, P Bijma, B. VillanuevaAbstract:Long-term Genetic contributions ( r i ) measure lasting gene flow from an individual i. By accounting for linkage disequilibrium generated by selection both within and between breeding groups (categories), assuming the infinitesimal model, a general formula was derived for the expected contribution of ancestor i in category q (μ i ( q ) ), given its selective advantages ( s i ( q ) ). Results were applied to overlapping generations and to a variety of modes of inheritance and selection indices. Genetic Gain was related to the covariance between r i and the Mendelian sampling deviation ( a i ), thereby linking Gain to pedigree development. When s i ( q ) includes a i , Gain was related to E [μ i ( q ) a i ], decomposing it into components attributable to within and between families, within each category, for each element of s i ( q ) . The formula for μ i ( q ) was consistent with previous index theory for predicting Gain in discrete generations. For overlapping generations, accurate predictions of gene flow were obtained among and within categories in contrast to previous theory that gave qualitative errors among categories and no predictions within. The generation interval was defined as the period for which μ i ( q ) , summed over all ancestors born in that period, equaled 1. Predictive accuracy was supported by simulation results for Gain and contributions with sib-indices, BLUP selection, and selection with imprinted variation.