Prediction of genetic gain from quadratic optimisation with constrained rates of inbreeding

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.     

Mating strategies with genomic information reduce rates of inbreeding in animal breeding schemes without compromising genetic gain

We tested the hypothesis that mating strategies with genomic information realise lower rates of inbreeding (∆F) than with pedigree information without compromising rates of genetic gain (∆G). We used stochastic simulation to compare ∆F and ∆G realised by two mating strategies with pedigree and genomic information in five breeding schemes. The two mating strategies were minimum-coancestry mating (MC) and minimising the covariance between ancestral genetic contributions (MCAC). We also simulated random mating (RAND) as a reference point. Generations were discrete. Animals were truncation-selected for a single trait that was controlled by 2000 quantitative trait loci, and the trait was observed for all selection candidates before selection. The criterion for selection was genomic-breeding values predicted by a ridge-regression model. Our results showed that MC and MCAC with genomic information realised 6% to 22% less ∆F than MC and MCAC with pedigree information without compromising ∆G across breeding schemes. MC and MCAC realised similar ∆F and ∆G. In turn, MC and MCAC with genomic information realised 28% to 44% less ∆F and up to 14% higher ∆G than RAND. These results indicated that MC and MCAC with genomic information are more effective than with pedigree information in controlling rates of inbreeding. This implies that genomic information should be applied to more than just prediction of breeding values in breeding schemes with truncation selection.     

Exploring the assumptions underlying genetic variation in host nematode resistance (Open Access publication)

The wide range of genetic parameter estimates for production traits and nematode resistance in sheep obtained from field studies gives rise to much speculation. Using a mathematical model describing host – parasite interactions in a genetically heterogeneous lamb population, we investigated the consequence of: (i) genetic relationships between underlying growth and immunological traits on estimated genetic parameters for performance and nematode resistance, and (ii) alterations in resource allocation on these parameter estimates. Altering genetic correlations between underlying growth and immunological traits had large impacts on estimated genetic parameters for production and resistance traits. Extreme parameter values observed from field studies could only be reproduced by assuming genetic relationships between the underlying input traits. Altering preferences in the resource allocation had less pronounced effects on the genetic parameters for the same traits. Effects were stronger when allocation shifted towards growth, in which case worm burden and faecal egg counts increased and genetic correlations between these resistance traits and body weight became stronger. Our study has implications for the biological interpretation of field data, and for the prediction of selection response from breeding for nematode resistance. It demonstrates the profound impact that moderate levels of pleiotropy and linkage may have on observed genetic parameters, and hence on outcomes of selection for nematode resistance.     

Non-additive genetic effects for fertility traits in Canadian Holstein cattle (Open Access publication )

The effects of additive, dominance, additive by dominance, additive by additive and dominance by dominance genetic effects on age at first service, non-return rates and interval from calving to first service were estimated. Practical considerations of computing additive and dominance relationships using the genomic relationship matrix are discussed. The final strategy utilized several groups of 1000 animals (heifers or cows) in which all animals had a non-zero dominance relationship with at least one other animal in the group. Direct inversion of relationship matrices was possible within the 1000 animal subsets. Estimates of variances were obtained using Bayesian methodology via Gibbs sampling. Estimated non-additive genetic variances were generally as large as or larger than the additive genetic variance in most cases, except for non-return rates and interval from calving to first service for cows. Non-additive genetic effects appear to be of sizeable magnitude for fertility traits and should be included in models intended for estimating additive genetic merit. However, computing additive and dominance relationships for all possible pairs of individuals is very time consuming in populations of more than 200 000 animals.     

A test of quantitative genetic theory using Drosophila- effects of inbreeding and rate of inbreeding on heritabilities and variance components

Inbreeding is expected to decrease the heritability within populations. However, results from empirical studies are inconclusive. In this study, we investigated the effects of three breeding treatments (fast and slow rate of inbreeding - inbred to the same absolute level - and a control) on heritability, phenotypic, genetic and environmental variances of sternopleural bristle number in Drosophila melanogaster. Heritability, and phenotypic, genetic and environmental variances were estimated in 10 replicate lines within each of the three treatments. Standard least squares regression models and Bayesian methods were used to analyse the data. Heritability and additive genetic variance within lines were higher in the control compared with both inbreeding treatments. Heritabilities and additive genetic variances within lines were higher in slow compared with fast inbred lines, indicating that slow inbred lines retain more evolutionary potential despite the same expected absolute level of inbreeding. The between line variance was larger with inbreeding and more than twice as large in the fast than in the slow inbred lines. The different pattern of redistribution of genetic variance within and between lines in the two inbred treatments cannot be explained invoking the standard model based on selective neutrality and additive gene action. Environmental variances were higher with inbreeding, and more so with fast inbreeding, indicating that inbreeding and the rate of inbreeding affect environmental sensitivity. The phenotypic variance decreased with inbreeding, but was not affected by the rate of inbreeding. No inbreeding depression for mean sternopleural bristle number was observed in this study. Considerable variance between lines in additive genetic variance within lines was observed, illustrating between line variation in evolutionary potential.     

The analysis of disease biomarker data using a mixed hidden Markov model (Open Access publication)

A mixed hidden Markov model (HMM) was developed for predicting breeding values of a biomarker (here, somatic cell score) and the individual probabilities of health and disease (here, mastitis) based upon the measurements of the biomarker. At a first level, the unobserved disease process (Markov model) was introduced and at a second level, the measurement process was modeled, making the link between the unobserved disease states and the observed biomarker values. This hierarchical formulation allows joint estimation of the parameters of both processes. The flexibility of this approach is illustrated on the simulated data. Firstly, lactation curves for the biomarker were generated based upon published parameters (mean, variance, and probabilities of infection) for cows with known clinical conditions (health or mastitis due to Escherichia coli or Staphylococcus aureus). Next, estimation of the parameters was performed via Gibbs sampling, assuming the health status was unknown. Results from the simulations and mathematics show that the mixed HMM is appropriate to estimate the quantities of interest although the accuracy of the estimates is moderate when the prevalence of the disease is low. The paper ends with some indications for further developments of the methodology.     

An approximate multitrait model for genetic evaluation in dairy cattle with a robust estimation of genetic trends (Open Access publication)

In a stochastic simulation study of a dairy cattle population three multitrait models for estimation of genetic parameters and prediction of breeding values were compared. The first model was an approximate multitrait model using a two-step procedure. The first step was a single trait model for all traits. The solutions for fixed effects from these analyses were subtracted from the phenotypes. A multitrait model only containing an overall mean, an additive genetic and a residual term was applied on these preadjusted data. The second model was similar to the first model, but the multitrait model also contained a year effect. The third model was a full multitrait model. Genetic trends for total merit and for the individual traits in the breeding goal were compared for the three scenarios to rank the models. The full multitrait model gave the highest genetic response, but was not significantly better than the approximate multitrait model including a year effect. The inclusion of a year effect into the second step of the approximate multitrait model significantly improved the genetic trend for total merit. In this study, estimation of genetic parameters for breeding value estimation using models corresponding to the ones used for prediction of breeding values increased the accuracy on the breeding values and thereby the genetic progress.     

Multigenerational effects of inbreeding in Cucurbita pepo ssp. texana (Cucurbitaceae)

The shape of the fitness function relating the decline in fitness with coefficient of inbreeding (f) can provide evidence concerning the genetic basis of inbreeding depression, but few studies have examined inbreeding depression across a range of f using noncultivated species. Futhermore, studies have rarely examined the effects of inbreeding depression in the maternal parent on offspring fitness. To estimate the shape of the fitness function, we examined the relationship between f and fitness across a range off from 0.000 to 0.875 for components of both male and female fitness in Cucurbita pepo ssp. texana. Each measure of female fitness declined with f, including pistillate flower number, fruit number, seed number per fruit, seed mass per fruit, and percentage seed germination. Several aspects of male fitness also declined with f, including staminate flower number, pollen number per flower, and the number of days of flowering, although cumulative inbreeding depression was less severe for male (0.34) than for female function (0.39). Fitness tended to decline linearly with f between f = 0.00 and f = 0.75 for most traits and across cumulative lifetime fitness (mean = 0.66), suggesting that individual genes causing inbreeding depression are additive and the result of many alleles of small effect. However, most traits also showed a small reduction in inbreeding depression between f = 0.75 and f = 0.875, and evidence of purging or diminishing epistasis was found for in vitro pollen-tube growth rate. To examine inbreeding depression as a maternal effect, we performed outcross pollinations on f = 0.0 and f = 0.5 mothers and found that depression due to maternal inbreeding was 0.07, compared to 0.10 for offspring produced through one generation of selfing. In at least some families, maternal inbreeding reduced fruit number, seed number and mass, staminate flower number, pollen diameter, and pollen-tube growth rate. Collectively these results suggest that, while the fitness function appears to be largely linear for most traits, maternal effects may compound the effects of inbreeding depression in multigenerational studies, though this may be partially offset by purging or diminishing epistasis.     

Parameter expansion for estimation of reduced rank covariance matrices (Open Access publication)

Parameter expanded and standard expectation maximisation algorithms are described for reduced rank estimation of covariance matrices by restricted maximum likelihood, fitting the leading principal components only. Convergence behaviour of these algorithms is examined for several examples and contrasted to that of the average information algorithm, and implications for practical analyses are discussed. It is shown that expectation maximisation type algorithms are readily adapted to reduced rank estimation and converge reliably. However, as is well known for the full rank case, the convergence is linear and thus slow. Hence, these algorithms are most useful in combination with the quadratically convergent average information algorithm, in particular in the initial stages of an iterative solution scheme.     

Relatedness and inbreeding in a French population of the unicolonial antIndomyrmex humilis (Mayr)

Summary Worker and queen samples of the Argentine antIridomyrmex humilis were analysed using allozyme electrophoresis. Two loci,peptidase andglucosephosphate isomerase, were found to be stainable and polymorphic. Both revealed an estimated relatedness in nestmate females that was close to zero, which is a consequence of the strong polygyny in this species. The inbreeding coefficient was found to be close to zero, which can be explained, in spite of intra-nest mating taking place inI. humilis, by the high mobility of mated queens.     

Impact of strong selection for the PrP major gene on genetic variability of four French sheep breeds (Open Access publication)

Effective selection on the PrP gene has been implemented since October 2001 in all French sheep breeds. After four years, the ARR "resistant" allele frequency increased by about 35% in young males. The aim of this study was to evaluate the impact of this strong selection on genetic variability. It is focussed on four French sheep breeds and based on the comparison of two groups of 94 animals within each breed: the first group of animals was born before the selection began, and the second, 3–4 years later. Genetic variability was assessed using genealogical and molecular data (29 microsatellite markers). The expected loss of genetic variability on the PrP gene was confirmed. Moreover, among the five markers located in the PrP region, only the three closest ones were affected. The evolution of the number of alleles, heterozygote deficiency within population, expected heterozygosity and the Reynolds distances agreed with the criteria from pedigree and pointed out that neutral genetic variability was not much affected. This trend depended on breed, i.e. on their initial states (population size, PrP frequencies) and on the selection strategies for improving scrapie resistance while carrying out selection for production traits.     

Estimating the inbreeding level and genetic relatedness in an isolated population of critically endangered Sichuan taimen (Hucho Bleekeri) using genome‐wide SNP markers

Sichuan taimen (Hucho bleekeri) is critically endangered fish listed in The Red List of Threatened Species compiled by the International Union for Conservation of Nature (IUCN). Specific locus amplified fragment sequencing (SLAF‐seq)‐based genotyping was performed for Sichuan taimen with 43 yearling individuals from three locations in Taibai River (a tributary of Yangtze River) that has been sequestered from its access to the ocean for more than 30 years since late 1980s. Applying the inbreeding level and genetic relatedness estimation using 15,396 genome‐wide SNP markers, we found that the inbreeding level of this whole isolated population was at a low level (2.6 × 10^?3 ± 0.079), and the means of coancestry coefficients within and between the three sampling locations were all very low (close to 0), too. Genomic differentiation was negatively correlated with the geographical distances between the sampling locations (p < .001), and the 43 individuals could be considered as genetically independent two groups. The low levels of genomic inbreeding and relatedness indicated a relatively large number of sexually mature individuals were involved in reproduction in Taibai River. This study suggested a genomic‐relatedness‐guided breeding and conservation strategy for wild fish species without pedigree information records.     

Population genetic structure and levels of inbreeding depression in the Mediterranean island endemic Cyclamen creticum (Primulaceae)

The variation and evolution of reproductive traits in island plants have much attention from conservation and evolutionary biologists. However, plants on islands in the Mediterranean region have very little attention. In the present study, we examine the floral biology and mating system of Cyclamen creticum , a diploid perennial herb endemic to Crete and Karpathos. Our purpose is to quantify (1) variation and covariation of floral traits related to the mating system, (2) the ability of the species to self in the absence of pollinators and its relative performance on selfing and outcrossing and (3) generic diversity within and among populations. Pollen/ovule ratios were indicative of a xenogamous species. A controlled pollination experiment showed that the species is self-compatible but is unable to set seed, in the absence of pollinators, probably due to stigma-anther separation. A multiplicative estimate of inbreeding depression based on fruit maturation, seed number and percentage seed germination gave δ= 0.38 Population genetic diversity was high, 54.76% polymorphic loci, a mean of 1.78 alleles per locus and a mean observed heterozygosity of 0.053. F -statistics nevertheless indicated high inbreeding rates (mean F ^is= 0.748) in natural populations, and low levels of population differentiation (mean F^is= 0.168). C. creticum thus appears to have a mixed-mating system with high levels of (pollinator) mediated inbreeding (either by facilitated selfing, geitonogamy or biparental inbreeding) in natural populations.     

Effect of inbreeding and salinity on quantitative characters and asymmetry of Tisbe holothuriae (Humes)

Effects of differing salinities on morphological characters have been found in marine and brackish populations of Tisbe holothuriae. The results suggested an active role of salinity as a factor promoting divergence. In order to test if the observed differentiation could be due to the level of heterozygosity, a study was performed in strains with low and high homozygosity. Six quantitative traits, and their respective fluctuating asymmetry, have been measured in outbred (coefficient of consanguineity, F = 0) and inbred (F = 0.5) strains, raised at three salinities; 20, 25 and 35. The analysis of variance was applied to test if the morphological measures were the same, and if the variations induced by decreased salinity had the same trend, in the two strains. Wilcoxon test was applied to the asymmetry, when it was not possible to render the variances homogeneous. The results showed that the measures and the effects of salinity on morphometry differed in the strains with low and high homozygosity. No effect of salinity on asymmetry was found. Length measures showed higher asymmetry in the most homozygous strain.     

Comparative effects of inbreeding and outbreeding on the ontogeny of Heliconius erato phyllis (Lepidoptera: Nymphalidae)

Inbreeding depression has been reported in various groups of organisms, including insects. Estimates of inbreeding consequences were obtained by comparing 12 life‐history and morphological traits among nine inbred families (F = 0.25) and 16 outbred families (F = 0) of the Neotropical butterfly Heliconius erato phyllis. A Student's t‐test showed statistically significant differences for pupal weight and right forewing area, both in males and in females, between inbred and outbred families. Survival during development, from egg hatching to adulthood, also differed significantly between inbred and outbred families. The average number of haploid lethal equivalents was 0.17 for pupal weight, 0.15 for forewing area and 0.71 for survival from hatching to adulthood. The results of this study confirm that the consequences of inbreeding are more deleterious to life history traits than to morphological ones.     

Experimental dissection of inbreeding and its adaptive significance in a flowering plant, Aquilegia canadensis (Ranunculaceae)

Inbreeding is a major component of the mating system in populations of many plants and animals, particularly hermaphroditic species. In flowering plants, inbreeding can occur through self-pollination within flowers (autogamy), self-pollination between flowers on the same plant (geitonogamy), or cross-pollination between closely related individuals (biparental inbreeding). We performed a floral emasculation experiment in 10 populations of Aquilegia canadensis (Ranunculaceae) and used allozyme markers to estimate the relative contribution of each mode of inbreeding to the mating system. We also examined how these modes of inbreeding were influenced by aspects of population structure and floral morphology and display predicted to affect the mating system. All populations engaged in substantial inbreeding. On average, only 25% of seed was produced by outcrossing (range among populations = 9-37%), which correlated positively with both population size (r = +0.61) and density (r = +0.64). Inbreeding occurred through autogamy and biparental inbreeding, and the relative contribution of each was highly variable among populations. Estimates of geitonogamy were not significantly greater than zero in any population. We detected substantial biparental inbreeding (mean = 14% of seeds, range = 4-24%) by estimating apparent selfing in emasculated plants with no opportunity for true selfing. This mode of inbreeding correlated negatively with population size (r = -0.87) and positively with canopy cover (r = +0.90), suggesting that population characteristics that increase outcross pollen transfer reduce biparental inbreeding. Autogamy was the largest component of the mating system in all populations (mean = 58%, range = 37-84%) and, as expected, was lowest in populations with the most herkogamous flowers (r = -0.59). Although autogamy provides reproductive assurance in natural populations of A. canadensis, it discounts ovules from making superior outcrossed seed. Hence, high autogamy in these populations seems disadvantageous, and therefore it is difficult to explain the extensive variation in herkogamy observed both among and especially within populations.     

Analysis of the real EADGENE data set: Multivariate approaches and post analysis (Open Access publication)

The aim of this paper was to describe, and when possible compare, the multivariate methods used by the participants in the EADGENE WP1.4 workshop. The first approach was for class discovery and class prediction using evidence from the data at hand. Several teams used hierarchical clustering (HC) or principal component analysis (PCA) to identify groups of differentially expressed genes with a similar expression pattern over time points and infective agent (E. coli or S. aureus). The main result from these analyses was that HC and PCA were able to separate tissue samples taken at 24 h following E. coli infection from the other samples. The second approach identified groups of differentially co-expressed genes, by identifying clusters of genes highly correlated when animals were infected with E. coli but not correlated more than expected by chance when the infective pathogen was S. aureus. The third approach looked at differential expression of predefined gene sets. Gene sets were defined based on information retrieved from biological databases such as Gene Ontology. Based on these annotation sources the teams used either the GlobalTest or the Fisher exact test to identify differentially expressed gene sets. The main result from these analyses was that gene sets involved in immune defence responses were differentially expressed.     

The risk of inbreeding due to habitat loss in the lion (Panthera leo)

A major concern in conservation biology is toavoid inbreeding depression in smallpopulations. Inbreeding has been shown to causeloss of fecundity and reduced survival innatural populations. To predict futureinbreeding levels, given changes in ecologicalconditions, data from large outbred populationsare needed. I use recent population geneticmodels to show how genetic data from such apopulation, combined with ecological data canbe used to predict future changes in levels ofinbreeding. In particular, data from the lion(Panthera leo) is used to simulate theeffect of habitat loss on the future levels ofinbreeding. To do this I used a deterministicpopulation genetics models and anindividual-based stochastic model. Two factorsstand out as particularly important: totalnumber of prides and male dispersal rates. Ifprides are very few (= 10) inbreeding (F)will increase from 0.0 in the initial state to0.26–0.45 after 30 generations, while if thenumber of prides is 100 this level is onlyaround 0.05 assuming no migration into thepopulation. Habitat heterogeneity had onlyminor effects. A reduction in male dispersalfrom their natal territory from 100% to 95%increased the level of inbreedingsubstantially, and never reached below F 0.25 regardless of the number ofprides. Therefore, to sustain a large outbredpopulation of lions, a continuous population ofat least 50 prides, but preferably 100 prides,with no limits to dispersal is required.     

Effects of inbreeding on traits that influence dispersal and progeny density in Cakile edentula var. lacustris (Brassicaceae)

Inbreeding may influence the intensity of sibling competition by altering the number of offspring produced or by changing plant morphology in ways that influence seed dispersion patterns. To test this possibility, effects of inbreeding on seed production and on traits that influence progeny density were measured using experimental pollinations of flowers of Cakile edentula var. lacustris. Different flowers on a plant were either hand pollinated with self pollen (with and without emasculation) or foreign pollen, or they were allowed to be pollinated naturally. Selfed flowers matured significantly fewer viable seeds than outcrossed flowers (10.3% less seed maturation with inbreeding depression of 19.2%), due in large part to a greater percentage of proximal seed abortions and lower germination success. Plants grown from selfed seeds tended to have lower seed production (37 fewer seeds on average, with inbreeding depression of 16.2%), caused in part by an increase in the percentage of fruits with proximal seed abortions, although this effect was not significant. Inbreeding depression in total fitness was 29.0%, which corresponds to a difference of 46 seeds per pollinated ovule. Selfing rate estimates were usually intermediate to high, indicating that inbreeding effects observed in this study would be present in naturally pollinated progeny. Although the influence of inbreeding directly on dispersal was negligible, the predicted reduction in sibling competition caused by reduced seed production resulted in an estimate of inbreeding depression of 17.5%, which is 11.5% lower than that measured under uniform conditions. Consequently, inbreeding depression estimated under natural dispersion patterns may be lower than that estimated under uniform conditions since seeds from self- and cross-pollination may not experience the same competitive environment in the field. Inbreeding in the maternal generation, therefore, could influence progeny fitness not only by determining the genetic composition of progeny, but also by influencing the competitive environment in which progeny grow.     

Isolation and characterization of microsatellite loci from two inbreeding bark beetle species (Coccotrypes)

We developed 14 microsatellite markers in Coccotrypes carpophagus and 14 in C. dactyliperda. These loci will be used for studying genetic structure and the level of inbreeding in populations in the Canary Islands and Madeira. As a result of long‐term inbreeding, genetic variability is relatively low in these bark beetle species. We found one to five alleles per locus in 29 C. carpophagus and 41 C. dactyliperda from various localities. Eleven of the markers developed for C. carpophagus amplified in C. dactyliperda and seven of the markers developed for C. dactyliperda amplified in C. carpophagus.