Coefficients of inbreeding and homozygosity in recurrent selection: The case of m linked loci

Summary Methods of calculating the coefficients of inbreeding and homozygosity in a finite population undergoing recurrent selection (self-select-intercross in succeeding generations) are investigated for the case of m linked loci and effective directional selection. These coefficients are derived in terms of vectors whose components reflect the various possible patterns of genes being identical at a given stage of the recurrent selection breeding program.For the case of two linked loci the progress of the panmictic index and/or the index of total heterozygosity through twenty-five cycles of recurrent selection is traced by means of computer-simulated populations ranging in sizes from ten through one hundred, assuming varying recombination probabilities, and assuming both minimum and maximum inbreeding selection patterns.Results indicate that the coefficient of relationship in the source population is extremely important in tracing the progress of the degree of inbreeding and/or total homozygosity, that linkage plays a major role in promoting heterozygosity in a recurrent selection system, and that careful intercrossing rather than random mating in alternate generations of the recurrent selection cycle is important in promoting maximum heterozygosity in the selected population. In the simulated populations the effect of small population sizes is observed and, in general, indications are that unless more than five complete recurrent cycles are contemplated, increasing the population size results in only relatively minor increases in panmixia, especially when linked loci are involved in the selected trait and when care is taken to avoid a maximum inbreeding selection pattern.     

Genetic response from marker assisted selection in an outbred population for differing marker bracket sizes and with two identified quantitative trait loci.

Effect of flanking quantitative trait loci (QTL)-marker bracket size on genetic response to marker assisted selection in an outbred population was studied by simulation of a nucleus breeding scheme. In addition, genetic response with marker assisted selection (MAS) from two quantitative trait loci on the same and different chromosome(s) was investigated. QTL that explained either 5% or 10% of phenotypic variance were simulated. A polygenic component was simulated in addition to the quantitative trait loci. In total, 35% of the phenotypic variance was due to genetic factors. The trait was measured on females only. Having smaller marker brackets flanking the QTL increased the genetic response from MAS selection. This was due to the greater ability to trace the QTL transmission from one generation to the next with the smaller flanking QTL-marker bracket, which increased the accuracy of estimation of the QTL allelic effects. Greater negative covariance between effects at both QTL was observed when two QTL were located on the same chromosome compared to different chromosomes. Genetic response with MAS was greater when the QTL were on the same chromosome in the early generations and greater when they were on different chromosomes in the later generations of MAS.     

Predictions of patterns of response to artificial selection in lines derived from natural populations

The pattern of response to artificial selection on quantitative traits in laboratory populations can tell us something of the genetic architecture in the natural population from which they were derived. We modeled artificial selection in samples drawn from natural populations in which variation had been maintained by recurrent mutation, with genes having an effect on the trait, which was subject to real stabilizing selection, and a pleitropic effect on fitness (the joint-effect model). Natural selection leads to an inverse correlation between effects and frequencies of genes, such that the frequency distribution of genes increasing the trait has an extreme U-shape. In contrast to the classical infinitesimal model, an early accelerated response and a larger variance of response among replicates were predicted. However, these are reduced if the base population has been maintained in the laboratory for some generations by random sampling prior to artificial selection. When multiple loci and linkage are also taken into account, the gametic disequilibria generated by the Bulmer and Hill-Robertson effects are such that little or no increase in variance and acceleration of response in early generations of artificial selection are predicted; further, the patterns of predicted responses for the joint-effect model now become close to those of the infinitesimal model. Comparison with data from laboratory selection experiments shows that, overall, the analysis did not provide clear support for the joint-effect model or a clear case for rejection.     

Identifying signatures of sexual selection using genomewide selection components analysis

Sexual selection must affect the genome for it to have an evolutionary impact, yet signatures of selection remain elusive. Here we use an individual‐based model to investigate the utility of genome‐wide selection components analysis, which compares allele frequencies of individuals at different life history stages within a single population to detect selection without requiring a priori knowledge of traits under selection. We modeled a diploid, sexually reproducing population and introduced strong mate choice on a quantitative trait to simulate sexual selection. Genome‐wide allele frequencies in adults and offspring were compared using weighted F_ST values. The average number of outlier peaks (i.e., those with significantly large F_ST values) with a quantitative trait locus in close proximity (“real” peaks) represented correct diagnoses of loci under selection, whereas peaks above the F_ST significance threshold without a quantitative trait locus reflected spurious peaks. We found that, even with moderate sample sizes, signatures of strong sexual selection were detectable, but larger sample sizes improved detection rates. The model was better able to detect selection with more neutral markers, and when quantitative trait loci and neutral markers were distributed across multiple chromosomes. Although environmental variation decreased detection rates, the identification of real peaks nevertheless remained feasible. We also found that detection rates can be improved by sampling multiple populations experiencing similar selection regimes. In short, genome‐wide selection components analysis is a challenging but feasible approach for the identification of regions of the genome under selection.     

The relation between the genetic architecture of quantitative traits and long-term genetic response

The genetic architecture of a quantitative trait refers to the number of genetic variants, allele frequencies, and effect sizes of variants that affect a trait and their mode of gene action. This study was conducted to investigate the effect of four shapes of allelic frequency distributions (constant, uniform, L-shaped and U-shaped) and different number of trait-affecting loci (50, 100, 200, 500) on allelic frequency changes, long term genetic response, and maintaining genetic variance. To this end, a population of 440 individuals composed of 40 males and 400 females as well as a genome of 200 cM consisting of two chromosomes and with a mutation rate of 2.5?×?10^?5 per locus was simulated. Selection of superior animals was done using best linear unbiased prediction (BLUP) with assumption of infinitesimal model. Selection intensity was constant over 30 generations of selection. The highest genetic progress obtained when the allelic frequency had L-shaped distribution and number of trait-affecting loci was high (500). Although quantitative genetic theories predict the extinction of genetic variance due to artificial selection in long time, our results showed that under L- and U-shapped allelic frequency distributions, the additive genetic variance is persistent after 30 generations of selection. Further, presence or absence of selection limit can be an indication of low (<50) or high (>100) number of trait-affecting loci, respectively. It was concluded that the genetic architecture of complex traits is an important subject which should be considered in studies concerning long-term response to selection.     

Genomewide selection in oil palm: increasing selection gain per unit time and cost with small populations

Oil palm (Elaeis guineensis Jacq.) requires 19 years per cycle of phenotypic selection. The use of molecular markers may reduce the generation interval and the cost of oil-palm breeding. Our objectives were to compare, by simulation, the response to phenotypic selection, marker-assisted recurrent selection (MARS), and genomewide selection with small population sizes in oil palm, and assess the efficiency of each method in terms of years and cost per unit gain. Markers significantly associated with the trait were used to calculate the marker scores in MARS, whereas all markers were used (without significance tests) to calculate the marker scores in genomewide selection. Responses to phenotypic selection and genomewide selection were consistently greater than the response to MARS. With population sizes of N = 50 or 70, responses to genomewide selection were 4–25% larger than the corresponding responses to phenotypic selection, depending on the heritability and number of quantitative trait loci. Cost per unit gain was 26–57% lower with genomewide selection than with phenotypic selection when markers cost US $1.50 per data point, and 35–65% lower when markers cost $0.15 per data point. With population sizes of N = 50 or 70, time per unit gain was 11–23 years with genomewide selection and 14–25 years with phenotypic selection. We conclude that for a realistic yet relatively small population size of N = 50 in oil palm, genomewide selection is superior to MARS and phenotypic selection in terms of gain per unit cost and time. Our results should be generally applicable to other tree species that are characterized by long generation intervals, high costs of maintaining breeding plantations, and small population sizes in selection programs.     

APPLICABILITY OF THE HYPERGEOMETRIC PHENOTYPIC MODEL TO HAPLOID AND DIPLOID POPULATIONS

We show that the phenotypic hypergeometric model of a quantitative trait can exactly describe both haploid and diploid populations. The condition necessary for this is equiprobability of genotypes within each phenotype. This requires equal allele frequencies across the loci, which may be the case when the population is under disruptive selection.     

Moving from QTL experimental results to the utilization of QTL in breeding programmes

Results from quantitative trait loci studies cannot be readily implemented into breeding schemes through marker assisted selection because of uncertainty about whether the quantitative trait loci identified are real and whether the identified quantitative trait loci are segregating in the breeding population. The present paper outlines and discusses strategies to reduce uncertainty in the results from quantitative trait loci studies. One strategy to confirm results from quantitative trait loci studies is to combine P -values from many quantitative trait loci experiments, while another is to establish a confirmation study. The power of a confirmation study must be high to ensure that the postulated quantitative trait loci can be verified. In the calculation of the experimental power, there are many issues that have to be addressed: size of the quantitative trait loci to be detected, significance level required, experimental design and expected heterozygosity for the design. To ensure marker assisted selection can be quickly implemented once quantitative trait loci are confirmed, DNA samples should be retained from daughters, and the sires and dams of elite sires.     

The evolution of genetic architecture under frequency-dependent disruptive selection

We propose a model to analyze a quantitative trait under frequency-dependent disruptive selection. Selection on the trait is a combination of stabilizing selection and intraspecific competition, where competition is maximal between individuals with equal phenotypes. In addition, there is a density-dependent component induced by population regulation. The trait is determined additively by a number of biallelic loci, which can have different effects on the trait value. In contrast to most previous models, we assume that the allelic effects at the loci can evolve due to epistatic interactions with the genetic background. Using a modifier approach, we derive analytical results under the assumption of weak selection and constant population size, and we investigate the full model by numerical simulations. We find that frequency-dependent disruptive selection favors the evolution of a highly asymmetric genetic architecture, where most of the genetic variation is concentrated on a small number of loci. We show that the evolution of genetic architecture can be understood in terms of the ecological niches created by competition. The phenotypic distribution of a population with an adapted genetic architecture closely matches this niche structure. Thus, evolution of the genetic architecture seems to be a plausible way for populations to adapt to regimes of frequency-dependent disruptive selection. As such, it should be seen as a potential evolutionary pathway to discrete polymorphisms and as a potential alternative to other evolutionary responses, such as the evolution of sexual dimorphism or assortative mating.     

A population genetics model for multiple quantitative traits exhibiting pleiotropy and epistasis

We study a population genetics model of an organism with a genome of L(tot)loci that determine the values of T quantitative traits. Each trait is controlled by a subset of L loci assigned randomly from the genome. There is an optimum value for each trait, and stabilizing selection acts on the phenotype as a whole to maintain actual trait values close to their optima. The model contains pleiotropic effects (loci can affect more than one trait) and epistasis in fitness. We use adaptive walk simulations to find high-fitness genotypes and to study the way these genotypes are distributed in sequence space. We then simulate the evolution of haploid and diploid populations on these fitness landscapes and show that the genotypes of populations are able to drift through sequence space despite stabilizing selection on the phenotype. We study the way the rate of drift and the extent of the accessible region of sequence space is affected by mutation rate, selection strength, population size, recombination rate, and the parameters L and T that control the landscape shape. There are three regimes of the model. If LTL(tot), there are many small peaks that can be spread over a wide region of sequence space. Compensatory neutral mutations are important in the population dynamics in this case.     

THE ACTION OF STABILIZING SELECTION,MUTATION, AND DRIFT ON EPISTATIC QUANTITATIVE TRAITS

For a quantitative trait under stabilizing selection, the effect of epistasis on its genetic architecture and on the changes of genetic variance caused by bottlenecking were investigated using theory and simulation. Assuming empirical estimates of the rate and effects of mutations and the intensity of selection, we assessed the impact of two‐locus epistasis (synergistic/antagonistic) among linked or unlinked loci on the distribution of effects and frequencies of segregating loci in populations at the mutation‐selection‐drift balance. Strong pervasive epistasis did not modify substantially the genetic properties of the trait and, therefore, the most likely explanation for the low amount of variation usually accounted by the loci detected in genome‐wide association analyses is that many causal loci will pass undetected. We investigated the impact of epistasis on the changes in genetic variance components when large populations were subjected to successive bottlenecks of different sizes, considering the action of genetic drift, operating singly (D), or jointly with mutation (MD) and selection (MSD). An initial increase of the different components of the genetic variance, as well as a dramatic acceleration of the between‐line divergence, were always associated with synergistic epistasis but were strongly constrained by selection.     

Further insights of the variance component method for detecting QTL in livestock and aquacultural species: relaxing the assumption of additive effects

Complex traits may show some degree of dominance at the gene level that may influence the statistical power of simple models, i.e. assuming only additive effects to detect quantitative trait loci (QTL) using the variance component method. Little has been published on this topic even in species where relatively large family sizes can be obtained, such as poultry, pigs, and aquacultural species. This is important, when the idea is to select regions likely to be harbouring dominant QTL or in marker assisted selection. In this work, we investigated the empirical power and accuracy to both detect and localise dominant QTL with or without incorporating dominance effects explicitly in the model of analysis. For this purpose, populations with variable family sizes and constant population size and different values for dominance variance were simulated. The results show that when using only additive effects there was little loss in power to detect QTL and estimates of position, using or not using dominance, were empirically unbiased. Further, there was little gain in accuracy of positioning the QTL with most scenarios except when simulating an overdominant QTL.     

Selection mapping of loci for quantitative disease resistance in a diverse maize population

The selection response of a complex maize population improved primarily for quantitative disease resistance to northern leaf blight (NLB) and secondarily for common rust resistance and agronomic phenotypes was investigated at the molecular genetic level. A tiered marker analysis with 151 simple sequence repeat (SSR) markers in 90 individuals of the population indicated that on average six alleles per locus were available for selection. An improved test statistic for selection mapping was developed, in which quantitative trait loci (QTL) are identified through the analysis of allele-frequency shifts at mapped multiallelic loci over generations of selection. After correcting for the multiple tests performed, 25 SSR loci showed evidence of selection. Many of the putatively selected loci were unlinked and dispersed across the genome, which was consistent with the diffuse distribution of previously published QTL for NLB resistance. Compelling evidence for selection was found on maize chromosome 8, where several putatively selected loci colocalized with published NLB QTL and a race-specific resistance gene. Analysis of F(2) populations derived from the selection mapping population suggested that multiple linked loci in this chromosomal segment were, in part, responsible for the selection response for quantitative resistance to NLB.     

Evolution of Allometry in Antirrhinum

Correlated variation in shape and size (allometry) is a major component of natural diversity. We examined the evolutionary and genetic basis for allometry using leaves and flower petals of snapdragon species (Antirrhinum). A computational method was developed to capture shape and size variation in both types of organ within the Antirrhinum species group. The results show that the major component of variation between species involves positively correlated changes in leaf and petal size. The correlation was maintained in an F2 population derived from crossing two species with organs of different sizes, suggesting that developmental constraints were involved. Identification of the underlying genes as quantitative trait loci revealed that the larger species carried alleles that increased organ size at all loci. Although this was initially taken as evidence that directional selection has driven diversity in both leaf and petal size, simulations revealed that evolution without consistent directional selection, an undirected walk, could also account for the parental distribution of organ size alleles.     

Directional Selection and Variation in Finite Populations

Predictions are made of the equilibrium genetic variances and responses in a metric trait under the joint effects of directional selection, mutation and linkage in a finite population. The "infinitesimal model" is analyzed as the limiting case of many mutants of very small effect, otherwise Monte Carlo simulation is used. If the effects of mutant genes on the trait are symmetrically distributed and they are unlinked, the variance of mutant effects is not an important parameter. If the distribution is skewed, unless effects or the population size is small, the proportion of mutants that have increasing effect is the critical parameter. With linkage the distribution of genotypic values in the population becomes skewed downward and the equilibrium genetic variance and response are smaller as disequilibrium becomes important. Linkage effects are greater when the mutational variance is contributed by many genes of small effect than few of large effect, and are greater when the majority of mutants increase rather than decrease the trait because genes that are of large effect or are deleterious do not segregate for long. The most likely conditions for "Muller's ratchet" are investigated.     

Strategies for efficient implementation of molecular markers in wheat breeding

Although molecular markers allow more accurate selection in early generations than conventional screens, large numbers can make selection impracticable while screening in later generations may provide little or no advantage over conventional selection techniques. Investigation of different crossing strategies and consideration of when to screen, what proportion to retain and the impacts of dominant vs. codominant marker expression revealed important choices in the design of marker-assisted selection programs that can produce large efficiency gains. Using F₂ enrichment increased the frequency of selected alleles allowing large reductions in minimum population size for recovery of target genotypes (commonly around 90%) and/or selection at a greater number of loci. Increasing homozygosity by inbreeding from F₂ to F_2:3 also reduced population size by around 90% in some crosses with smaller incremental reductions in subsequent generations. Backcrossing was found to be a useful strategy to reduce population size compared with a biparental population where one parent contributed more target alleles than the other and was complementary to F₂ enrichment and increasing homozygosity. Codominant markers removed the need for progeny testing reducing the number of individuals that had to be screened to identify a target genotype. However, although codominant markers allow target alleles to be fixed in early generations, minimum population sizes are often so large in F₂ that it is not efficient to do so at this stage. Formulae and tables for calculating genotypic frequencies and minimum population sizes are provided to allow extension to different breeding systems, numbers of target loci, and probabilities of failure. Principles outlined are applicable to implementation of markers for both quantitative trait loci (QTL) and major genes.     

The Effects of Selection on Linkage Analysis for Quantitative Traits

The effects of within-sample selection on the outcome of analyses detecting linkage between genetic markers and quantitative traits were studied. It was found that selection by truncation for the trait of interest significantly reduces the differences between marker genotype means thus reducing the power to detect linked quantitative trait loci (QTL). The size of this reduction is a function of proportion selected, the magnitude of the QTL effect, recombination rate between the marker locus and the QTL, and the allele frequency of the QTL. Proportion selected was the most influential of these factors on bias, e.g., for an allele substitution effect of one standard deviation unit, selecting the top 80%, 50% or 20% of the population required 2, 6 or 24 times the number of progeny, respectively, to offset the loss of power caused by this selection. The effect on power was approximately linear with respect to the size of gene effect, almost invariant to recombination rate, and a complex function of QTL allele frequency. It was concluded that experimental samples from animal populations which have been subjected to even minor amounts of selection will be inefficient in yielding information on linkage between markers and loci influencing the quantitative trait under selection.     

The genetics of adaptive shape shift in stickleback: pleiotropy and effect size

The distribution of effect sizes of genes underlying adaptation is unknown ( Orr 2005 ). Are suites of traits that diverged under natural selection controlled by a few pleiotropic genes of large effect (major genes model), by many independently acting genes of small effect (infinitesimal model), or by a combination, with frequency inversely related to effect size (geometric model)? To address this we carried out a quantitative trait loci (QTL) study of a suite of 54 position traits describing body shapes of two threespine stickleback species: an ancestral Pacific marine form and a highly derived benthic species inhabiting a geologically young lake. About half of the 26 detected QTL affected just one coordinate and had small net effects, but several genomic regions affected multiple aspects of shape and had large net effects. The distribution of effect sizes followed the gamma distribution, as predicted by the geometric model of adaptation when detection limits are taken into account. The sex‐determining chromosome region had the largest effect of any QTL. Ancestral sexual dimorphism was similar to the direction of divergence, and was largely eliminated during freshwater adaptation, suggesting that sex differences may provide variation upon which selection can act. Several shape QTL are linked to Eda, a major gene responsible for reduction of lateral body armor in freshwater. Our results are consistent with predictions of the geometric model of adaptation. Shape evolution in stickleback results from a few genes with large and possibly widespread effects and multiple genes of smaller effect.     

Genetic architecture of Arabidopsis thaliana response to infection by Pseudomonas syringae

Plant pathogens can severely reduce host yield and fitness. Thus, investigating the genetic basis of plant response to pathogens is important to further understand plant-pathogen coevolution and to improve crop production. The interaction between Arabidopsis thaliana and Pseudomonas syringae is an important model for studying the genetic basis of plant-pathogen interactions. Studies in this model have led to the discovery of many genes that differentiate a resistant from a susceptible plant. However, little is known about the genetic basis of quantitative variation in response to P. syringae. In this study, we investigate the genetic basis of three aspects of A. thaliana's response to P. syringae: symptom severity, bacterial population size and fruit production using a quantitative trait loci (QTL) analysis. We found two QTL for symptom severity and two for fruit production (possible candidate genes for observed QTL are discussed). We also found significant two-locus epistatic effect on symptom severity and fruit production. Although bacterial population size and symptom severity were strongly phenotypically correlated, we did not detect any QTL for bacterial population size. Despite the detected genetic variation observed for susceptibility, we found only a weak overall relationship between susceptibility traits and fitness, suggesting that these traits may not respond to selection.     

The accuracy of marker-assisted selection for quantitative traits within populations in linkage equilibrium.

Using the concept of conditional coancestry, given observed markers, an explicit expression of the accuracy of marker-based selection is derived in situations of linkage equilibrium between markers and quantitative trait loci (QTL), for the general case of full-sib families nested within half-sib families. Such a selection scheme is rather inaccurate for moderate values of family sizes and QTL variance, and the accuracies predicted for linkage disequilibrium can never be reached. The result is used to predict the accuracy of marker-assisted combined selection (MACS) and is shown to agree with previous MACS results obtained by simulation of a best linear unbiased prediction animal model. Low gains in accuracy are generally to be expected compared to standard combined selection. The maximum gain, assuming infinite family size and all QTLs marked, is about 50%.