Finite Locus Model Research Articles

Response to Selection in Finite Locus Models with Nonadditive Effects.

Under the finite-locus model in the absence of mutation, the additive genetic variation is expected to decrease when directional selection is acting on a population, according to quantitative-genetic theory. However, some theoretical studies of selection suggest that the level of additive variance can be sustained or even increased when nonadditive genetic effects are present. We tested the hypothesis that finite-locus models with both additive and nonadditive genetic effects maintain more additive genetic variance (VA) and realize larger medium- to long-term genetic gains than models with only additive effects when the trait under selection is subject to truncation selection. Four genetic models that included additive, dominance, and additive-by-additive epistatic effects were simulated. The simulated genome for individuals consisted of 25 chromosomes, each with a length of 1 M. One hundred bi-allelic QTL, 4 on each chromosome, were considered. In each generation, 100 sires and 100 dams were mated, producing 5 progeny per mating. The population was selected for a single trait (h2 = 0.1) for 100 discrete generations with selection on phenotype or BLUP-EBV. VA decreased with directional truncation selection even in presence of nonadditive genetic effects. Nonadditive effects influenced long-term response to selection and among genetic models additive gene action had highest response to selection. In addition, in all genetic models, BLUP-EBV resulted in a greater fixation of favorable and unfavorable alleles and higher response than phenotypic selection. In conclusion, for the schemes we simulated, the presence of nonadditive genetic effects had little effect in changes of additive variance and VA decreased by directional selection.

Single versus subdivided population strategies in breeding against an adverse genetic correlation

In advanced conifer breeding programmes, the simultaneous genetic improvement of adversely correlated traits constitutes a major challenge. Population subdivision strategies have been proposed to deal with breeding objective uncertainty, to reduce inbreeding depression in production populations and to reduce genetic correlation adversity. We used Monte Carlo simulations based on a finite locus model to study the effect of a two-breeding-population strategy applying selection for each trait in each breeding population on the genetic correlation and on genetic gains in breeding populations (BP) and the production population (PP) within a time frame of ten generations. A single-BP and a two-subline strategy both applying multitrait index selection with equal trait weights were used as references. Two BP strategy simulations indicated that simultaneous genetic gain for the two traits could be achieved in the PP despite adverse pleiotropy. The adversity of the genetic correlations decreased in BPs of the two-BP strategy, in contrast to single-BP and subline strategies, but the adversity reduction came at the cost of a lower rate of aggregated (summed) genetic gain in the PP for the two-BP strategy compared to the single-BP or subline strategies. The subline strategy exhibited increased genetic gain in the PP at equal levels of inbreeding as intended. Two BP strategies could be useful to develop breeds specialised on different traits and to simultaneously reduce adverse genetic correlations. However, if the aggregated genetic gain should be maximised, the single-BP strategy appears a better choice.

Distribution and location of genetic effects for dairy traits

Genetic effects for many dairy traits and for total economic merit are evenly distributed across all chromosomes. A high-density scan using 38,416 single nucleotide polymorphism markers for 5,285 bulls confirmed 2 previously known major genes on Bos taurus autosomes (BTA) 6 and 14 but revealed few other large effects. Markers on BTA18 had the largest effects on calving ease, several conformation traits, longevity, and total merit. Prediction accuracy was highest using a heavy-tailed prior assuming that each marker had an effect on each trait, rather than assuming a normal distribution of effects as in a linear model, or that only some loci have nonzero effects. A prior model combining heavy tails with finite alleles produced results that were intermediate compared with the individual models. Differences between models were small (1 to 2%) for traits with no major genes and larger for heavy tails with traits having known quantitative trait loci (QTL; 6 to 8%). Analysis of bull recessive codes suggested that marker effects from genomic selection may be used to identify regions of chromosomes to search in detail for candidate genes, but individual single nucleotide polymorphisms were not tracking causative mutations with the exception of diacylglycerol O-acyltransferase 1. Additive genetic merits were constructed for each chromosome, and the distribution of BTA14-specific estimated breeding value (EBV) showed that selection primarily for milk yield has not changed the distribution of EBV for fat percentage even in the presence of a known QTL. Such chromosomal EBV also may be useful for identifying complementary mates in breeding programs. The QTL affecting dystocia, conformation, and economic merit on BTA18 appear to be related to calf size or birth weight and may be the result of longer gestation lengths. Results validate quantitative genetic assumptions that most traits are due to the contributions of a large number of genes of small additive effect, rather than support the finite locus model.

Assessment of the dynamics of microparasite infections in genetically homogeneous and heterogeneous populations using a stochastic epidemic model1

The aim of this paper was to explore the effect of genetic heterogeneity in host resistance to infection on the population-level risks and outcomes of epidemics. This was done using a stochastic epidemiological model in which the model parameters were assumed to be genetically controlled traits of the host. A finite locus model was explored, with a gene controlling the transmission coefficient (i.e., host susceptibility to infection) and a gene controlling the recovery period. Both genes were simulated to have 2 alleles with underlying additive or dominance inheritance and an independent assortment of alleles. The model was parameterized for a viral pig disease (transmissible gastroenteritis), and complete homogeneous mixing among genotypes was assumed. Mean population genotype dramatically affected epidemic outcomes, and subtle effects of heterogeneity on epidemic properties were also observed. Genetic variation in the transmission coefficient led to probabilities of epidemics occurring that were slightly greater than expected, but genetic variation in the recovery rate had no such effect. Epidemics were generally less severe in genetically heterogeneous populations than expected from the constituent subpopulations. Furthermore, the genotype of the initial infected animal had a marked effect on epidemic probabilities, particularly when genetic variation was for recovery rate. The results of this model provide useful information to determine the optimum population structures and to exploit genetic variation in resistance to infection. Applications of the proposed model in genetically heterogeneous populations for identifying practical disease management strategies are also discussed.

Heritability of refractive value and ocular biometrics

The aim of this work was to analyse genetic influences on ocular refractive value and axial length using the hypothesis of a polygenic control. The genealogical records of 55 families were used in the analyses. The cohort included 723 individuals and clinical data were collected for 445 individuals with a mean age of 37.86years. Ocular refraction was determined by standard autorefractometry. Axial length was evaluated by scan ultrasonography. Gender, age and ethnic origin were included as covariates in the statistical analyses. Using variance component analysis via a Markov Chain Monte Carlo (MCMC) method, we estimated the heritability of refractive value and axial length in the pedigree. We then performed a segregation analysis, using Loki, a (MCMC) linkage analysis program for multilocus inheritance models, examining different inheritance models with polygenic components. Polygenic control was modelled under an additive infinitesimal model (which assumes infinite loci with small effects, with additive actions) and under a finite locus model (i.e. several causal loci). The estimates of heritability were 0.20 (95% confidence interval (CI) 0.04–0.36) for refractive value and 0.20 (95% CI 0.03–0.43) for axial length. Segregation analyses suggested that ocular refraction and axial length are under a polygenic control. A finite number of genes were identified with or without a polygenic, infinitesimal component. Ocular refraction is mildly-moderately heritable in the studied population.

A study on the minimum number of loci required for genetic evaluation using a finite locus model

For a finite locus model, Markov chain Monte Carlo (MCMC) methods can be used to estimate the conditional mean of genotypic values given phenotypes, which is also known as the best predictor (BP). When computationally feasible, this type of genetic prediction provides an elegant solution to the problem of genetic evaluation under non-additive inheritance, especially for crossbred data. Successful application of MCMC methods for genetic evaluation using finite locus models depends, among other factors, on the number of loci assumed in the model. The effect of the assumed number of loci on evaluations obtained by BP was investigated using data simulated with about 100 loci. For several small pedigrees, genetic evaluations obtained by best linear prediction (BLP) were compared to genetic evaluations obtained by BP. For BLP evaluation, used here as the standard of comparison, only the first and second moments of the joint distribution of the genotypic and phenotypic values must be known. These moments were calculated from the gene frequencies and genotypic effects used in the simulation model. BP evaluation requires the complete distribution to be known. For each model used for BP evaluation, the gene frequencies and genotypic effects, which completely specify the required distribution, were derived such that the genotypic mean, the additive variance, and the dominance variance were the same as in the simulation model. For lowly heritable traits, evaluations obtained by BP under models with up to three loci closely matched the evaluations obtained by BLP for both purebred and crossbred data. For highly heritable traits, models with up to six loci were needed to match the evaluations obtained by BLP.

A comparison of�alternative methods to�compute conditional genotype probabilities for�genetic evaluation with�finite locus models

An increased availability of genotypes at marker loci has prompted the development of models that include the effect of individual genes. Selection based on these models is known as marker-assisted selection (MAS). MAS is known to be efficient especially for traits that have low heritability and non-additive gene action. BLUP methodology under non-additive gene action is not feasible for large inbred or crossbred pedigrees. It is easy to incorporate non-additive gene action in a finite locus model. Under such a model, the unobservable genotypic values can be predicted using the conditional mean of the genotypic values given the data. To compute this conditional mean, conditional genotype probabilities must be computed. In this study these probabilities were computed using iterative peeling, and three Markov chain Monte Carlo (MCMC) methods – scalar Gibbs, blocking Gibbs, and a sampler that combines the Elston Stewart algorithm with iterative peeling (ESIP). The performance of these four methods was assessed using simulated data. For pedigrees with loops, iterative peeling fails to provide accurate genotype probability estimates for some pedigree members. Also, computing time is exponentially related to the number of loci in the model. For MCMC methods, a linear relationship can be maintained by sampling genotypes one locus at a time. Out of the three MCMC methods considered, ESIP, performed the best while scalar Gibbs performed the worst.

The contribution of genetic diversity to the spread of infectious diseases in livestock populations.

This article uses stochastic simulations with a compartmental epidemic model to quantify the impact of genetic diversity within animal populations on the transmission of infectious disease. Genetic diversity is defined by the number of distinct genotypes in the population conferring resistance to microparasitic (e.g., viral or bacterial) infections. Scenarios include homogeneous populations and populations composed of few (finite-locus model) or many (infinitesimal model) genotypes. Genetic heterogeneity has no impact upon the expected value of the basic reproductive ratio (the primary description of the transmission of infection) but affects the variability of this parameter. Consequently, increasing genetic heterogeneity is associated with an increased probability of minor epidemics and decreased probabilities of both major (catastrophic) epidemics and no epidemics. Additionally, heterogeneity per se is associated with a breakdown in the expected relationship between the basic reproductive ratio and epidemic severity, which has been developed for homogeneous populations, with increasing heterogeneity generally resulting in fewer infected animals than expected. Furthermore, increased heterogeneity is associated with decreased disease-dependent mortality in major epidemics and a complex trend toward decreased duration of these epidemics. In summary, more heterogeneous populations are not expected to suffer fewer epidemics on average, but are less likely to suffer catastrophic epidemics.

Strategies for Continual Application of Marker-Assisted Selection in an Open Nucleus Population

The objectives of this study were to develop and simulate the implementation of several strategies for repeated application of quantitative trait loci (QTL) detection and marker-assisted selection (MAS) and to compare the short-term and continual genetic responses. A finite locus model was simulated with 20 QTL randomly distributed across 30 chromosome. Three hundred markers were evenly spaced across the genome. Allelic effects were sampled from a double exponential distribution. A daughter design was used every generation to determine the marker alleles favorably associated to QTL alleles. The MAS was applied within family to young bulls, before progeny testing, as part of an open nucleus. Young bulls were selected using strategies based on 1) the single marker with greatest contrast (BEST1), 2) the sum of n greatest contrasts (BESTn), 3) the best n contrasts, limited to one per chromosome (LIMn), 4) the sum of all contrasts exceeding a given threshold n (THRESn), and 5) the sum of contrasts exceeding a threshold, but limited to one per chromosome (LIMT). The maximum progress was achieved by strategies that selected upon several markers flanking multiple QTL in each generation. When THRES was applied, the mean true breeding value (TBV) of selected bulls was increased by 11.98% (over conventional selection) versus 6.73% for BEST1 in the first generation. Applying a full genome scan in each generation allowed selection for different QTL across time. By selecting for multiple QTL over time, MAS maintained superiority over conventional selection for many generations.

Use of marker haplotypes to refine covariances among relatives for breeding value estimation

Markers flanking DNA regions, where quantitative trait loci (QTL) have been previously spotted, can be used to trace the common inheritance of major genes for a better definition of covariances among animals. A practical approach to the use of marker data to refine the additive relationship matrix used in the traditional best linear unbiased prediction (BLUP) methodology is presented. The technique allows the number of the mixed model equations to be kept to an animal level, blending polygenic pedigree data with marker haplotype information. The advantage of this marker‐assisted selection (MAS) approach over BLUP selection has been assessed through a stochastic simulation. A finite locus model with 32 independent biallelic loci was generated with normally distributed allelic effects. The heritability of the trait, measured on both sexes and on females only, was set to 0.2 and 0.5. Five‐allelic markers 2, 10 and 20 cM apart, bracketed the QTL with the largest effect on the trait, accounting for 17% of the genetic variance. The bracketed QTL had two or eight alleles and its position was undefined within the bracket. Results show a moderate 2% advantage of MAS over BLUP in terms of higher genetic response when trait was recorded on both sexes and heritability was 0.2. The benefit is in the short term, but it lasts longer with polyallelic QTL. When the trait was recorded on females only, MAS produced only a small and insignificant genetic gain, but reduced the overall inbreeding in the population. MAS was also inefficient when heritability was 0.5.

Estimation of additive, dominance and epistatic variance components using finite locus models implemented with a single-site Gibbs and a descent graph sampler.

In a previous contribution, we implemented a finite locus model (FLM) for estimating additive and dominance genetic variances via a Bayesian method and a single-site Gibbs sampler. We observed a dependency of dominance variance estimates on locus number in the analysis FLM. Here, we extended the FLM to include two-locus epistasis, and implemented the analysis with two genotype samplers (Gibbs and descent graph) and three different priors for genetic effects (uniform and variable across loci, uniform and constant across loci, and normal). Phenotypic data were simulated for two pedigrees with 6300 and 12,300 individuals in closed populations, using several different, non-additive genetic models. Replications of these data were analysed with FLMs differing in the number of loci. Simulation results indicate that the dependency of non-additive genetic variance estimates on locus number persisted in all implementation strategies we investigated. However, this dependency was considerably diminished with normal priors for genetic effects as compared with uniform priors (constant or variable across loci). Descent graph sampling of genotypes modestly improved variance components estimation compared with Gibbs sampling. Moreover, a larger pedigree produced considerably better variance components estimation, suggesting this dependency might originate from data insufficiency. As the FLM represents an appealing alternative to the infinitesimal model for genetic parameter estimation and for inclusion of polygenic background variation in QTL mapping analyses, further improvements are warranted and might be achieved via improvement of the sampler or treatment of the number of loci as an unknown.

The interaction of selection intensity, inbreeding depression, and random genetic drift on short- and long-term response to selection: Results using finite locus and finite population size models incorporating directional dominance

The interaction of selection intensity, inbreeding depression, and random genetic drift on short- and long-term response to selection: Results using finite locus and finite population size models incorporating directional dominance

Behaviour of the additive finite locus model

A finite locus model to estimate additive variance and the breeding values was implemented using Gibbs sampling. Four different distributions for the size of the gene effects across the loci were considered: i) uniform with loci of different effects, ii) uniform with all loci having equal effects, iii) exponential, and iv) normal. Stochastic simulation was used to study the influence of the number of loci and the distribution of their effect assumed in the model analysis. The assumption of loci with different and uniformly distributed effects resulted in an increase in the estimate of the additive variance according to the number of loci assumed in the model of analysis, causing biases in the estimated breeding values. When the gene effects were assumed to be exponentially distributed, the estimate of the additive variance was still dependent on the number of loci assumed in the model of analysis, but this influence was much less. When assuming that all the loci have the same gene effects or when they were normally distributed, the additive variance estimate was the same regardless of the number of loci assumed in the model of analysis. The estimates were not significantly different from either the true simulated values or from those obtained when using the standard mixed model approach where an infinitesimal model is assumed. The results indicate that if the number of loci has to be assumed a priori, the most useful finite locus models are those assuming loci with equal effects or normally distributed effects. © Inra/Elsevier, Paris '

Behaviour of the additive finite locus model

Behaviour of the additive finite locus model