The effect of population structure on the relationship between heterosis and heterozygosity at marker loci

The relationship between heterozygosity at neutral marker loci and heterosis of F₁ hybrids is investigated using a theoretical model. Results emphasize that linkage disequilibrium between the markers and the loci implicated in heterosis [quantitative trait loci (QTLs) that exhibit dominance effects] is a necessary condition to finding a correlation ( _mh ) between heterozygosity at marker loci and the heterosis. The effect of population structure, in which the parental inbred lines of the hybrids belong to different heterotic groups, is considered. _mh is investigated for: (1) hybrids between lines that belong to the same heterotic group (within-group hybrids); (2) hybrids between lines that belong to different groups (between-group hybrids); and (3) all hybrids, both within and between-groups. Within a group, significant values of ( _mh ) may arise because of linkage disequilibrium generated by drift. At the between-group level, no correlation is expected since link-age disequilibrium should differ randomly from one group to the other, which is consistent with recent experimental results. Possible ways to achieve prediction of the heterosis in this situation are discussed. When all hybrids are considered simultaneously, divergence of allelic frequencies among groups for the markers and the QTLs produces a correlation between heterosis and heterozygosity at marker loci. This correlation increases with the number of markers that are considered.     

Genome-wide evaluation for quantitative trait loci under the variance component model

The identity-by-descent (IBD) based variance component analysis is an important method for mapping quantitative trait loci (QTL) in outbred populations. The interval-mapping approach and various modified versions of it may have limited use in evaluating the genetic variances of the entire genome because they require evaluation of multiple models and model selection. In this study, we developed a multiple variance component model for genome-wide evaluation using both the maximum likelihood (ML) method and the MCMC implemented Bayesian method. We placed one QTL in every few cM on the entire genome and estimated the QTL variances and positions simultaneously in a single model. Genomic regions that have no QTL usually showed no evidence of QTL while regions with large QTL always showed strong evidence of QTL. While the Bayesian method produced the optimal result, the ML method is computationally more efficient than the Bayesian method. Simulation experiments were conducted to demonstrate the efficacy of the new methods.     

Bayesian mapping of quantitative trait loci under the identity-by-descent-based variance component model

Variance component analysis of quantitative trait loci (QTL) is an important strategy of genetic mapping for complex traits in humans. The method is robust because it can handle an arbitrary number of alleles with arbitrary modes of gene actions. The variance component method is usually implemented using the proportion of alleles with identity-by-descent (IBD) shared by relatives. As a result, information about marker linkage phases in the parents is not required. The method has been studied extensively under either the maximum-likelihood framework or the sib-pair regression paradigm. However, virtually all investigations are limited to normally distributed traits under a single QTL model. In this study, we develop a Bayes method to map multiple QTL. We also extend the Bayesian mapping procedure to identify QTL responsible for the variation of complex binary diseases in humans under a threshold model. The method can also treat the number of QTL as a parameter and infer its posterior distribution. We use the reversible jump Markov chain Monte Carlo method to infer the posterior distributions of parameters of interest. The Bayesian mapping procedure ends with an estimation of the joint posterior distribution of the number of QTL and the locations and variances of the identified QTL. Utilities of the method are demonstrated using a simulated population consisting of multiple full-sib families.     

Relationship between heterosis and heterozygosity at marker loci: a theoretical computation

Summary In this paper we have studied the linear correlation between a genetic distance index between two parent lines (based on marker loci information) and the heterosis observed in the F₁ hybrid from the two lines, for a quantitative character (determined by several loci, or QTL). Theoretical computations of the correlation coefficient () between the distance index and the heterosis were made, assuming the biallelic model (defined by Fisher). When the alleles at both marker loci and QTL are equally distributed among the whole population of considered lines, the coefficient is a function of the squares of linkage disequilibria between alleles at marker loci and alleles at QTL. The QTL that are not marked by marker loci and marker loci that do not mark any QTL play symmetrical roles and can decrease greatly. We conclude that the prediction of F₁ hybrid heterosis based on marker loci would be more efficient if these markers were selected for their relationship to the alleles implicated in the heterotic traits considered.     

Linkage between quantitative and marker loci. V. Joint analysis of various marker and quantitative traits

New versions are suggested to analyse marker and quantitative characters combinations. Possible modes of application of the algorithms developed for recombination analysis are discussed, including: 1) the estimation of crossing-over frequency between markers with incomplete penetrance, 2) the quantitative character variability account to analyse genetic interference, 3) search for genetic factors affecting a set of quantitative characters, 4) the evaluation of differences between male and female meiosis at the crossingover level etc.     

Linkage relationship between retinoschisis and four marker loci

Summary The linkage relationship between the locus for juvenile retinoschisis (RS) and four X-chromosomal marker loci DXS9 (RC8), DXS16 (XUT23), DXS41 (99-6), and DXS43 (D2) has been studied in six families showing a history of this disease. Recombination with RS was found for all marker loci except DXS9. The maximum lod score is =2.66 for RS vs. SXS9 at a recombination fraction of =0.0. Multipoint linkage analysis was performed and the locus order best supported by our data is: RS-DXS9-DXS43-DXS16-DXS41.     

Sequential sampling in determining linkage between marker loci and quantitative trait loci

Summary As compared to classical, fixed sample size techniques, simulation studies showed that a proposed sequential sampling procedure can provide a substantial decrease (up to 50%, in some cases) in the mean sample size required for the detection of linkage between marker loci and quantitative trait loci. Sequential sampling with truncation set at the required sample size for the non-sequential test, produced a modest further decrease in mean sample size, accompanied by a modest increase in error probabilities. Sequential sampling with observations taken in groups produced a noticeable increase in mean sample size, with a considerable decrease in error probabilities, as compared to straightforward sequential sampling. It is concluded that sequential sampling has a particularly useful application to experiments aimed at investigating the genetics of differences between lines or strains that differ in some single outstanding trait.     

Linkage between loci of quantitative traits and marker loci: multi-trait analysis with a single marker

An efficient approach to increase the resolution power of linkage analysis between a quantitative trait locus (QTL) and a marker is described in this paper. It is based on a counting of the correlations between the QTs of interest. Such correlations may be caused by the segregation of other genes, environmental effects and physiological limitations. Let a QT locus A/a affect two correlated traits, x and y. Then, within the framework of mixture models, the accuracy of the parameter estimates may be seriously increased, if bivariate densities f _aa(x, y), f _Aa(x, y) and f _AA(x, y) rather than the marginals are considered as the basis for mixture decomposition. The efficiency of the proposed method was demonstrated employing Monte-Carlo simulations. Several types of progeny were considered, including backcross, F₂ and recombinant inbred lines. It was shown that provided the correlation between the traits involved was high enough, a good resolution to the problem is possible even if the QTL groups are strongly overlapping for their marginal densities.     

Comparison of genetic differentiation at marker loci and quantitative traits

The comparison of the degree of differentiation in neutral marker loci and genes coding quantitative traits with standardized and equivalent measures of genetic differentiation (F_ST and Q_ST, respectively) can provide insights into two important but seldom explored questions in evolutionary genetics: (i) what is the relative importance of random genetic drift and directional natural selection as causes of population differentiation in quantitative traits, and (ii) does the degree of divergence in neutral marker loci predict the degree of divergence in genes coding quantitative traits? Examination of data from 18 independent studies of plants and animals using both standard statistical and meta‐analytical methods revealed a number of interesting points. First, the degree of differentiation in quantitative traits (Q_ST) typically exceeds that observed in neutral marker genes (F_ST), suggesting a prominent role for natural selection in accounting for patterns of quantitative trait differentiation among contemporary populations. Second, the F_ST – Q_ST difference is more pronounced for allozyme markers and morphological traits, than for other kinds of molecular markers and life‐history traits. Third, very few studies reveal situations were Q_ST < F_ST, suggesting that selection pressures, and hence optimal phenotypes, in different populations of the same species are unlikely to be often similar. Fourth, there is a strong correlation between Q_ST and F_ST indices across the different studies for allozyme (r=0.81), microsatellite (r=0.87) and combined (r=0.75) marker data, suggesting that the degree of genetic differentiation in neutral marker loci is closely predictive of the degree of differentiation in loci coding quantitative traits. However, these interpretations are subject to a number of assumptions about the data and methods used to derive the estimates of population differentiation in the two sets of traits.     

On the mapping of quantitative trait loci at marker and non-marker locations

Previous studies have noted that the estimated positions of a large proportion of mapped quantitative trait loci (QTLs) coincide with marker locations and have suggested that this indicates a bias in the mapping methodology. In this study we predict the expected proportion of QTLs with positions estimated to be at the location of a marker and further examine the problem using simulated data. The results show that the higher proportion of putative QTLs estimated to be at marker positions compared with non-marker positions is an expected consequence of the estimation methods. The study initially focused on a single interval with no QTLs and was extended to include multiple intervals and QTLs of large effect. Further, the study demonstrated that the larger proportion of estimated QTL positions at the location of markers was not unique to linear regression mapping. Maximum likelihood produced similar results, although the accumulation of positional estimates at outermost markers was reduced when regions outside the linkage group were also considered. The bias towards marker positions is greatest under the null hypothesis of no QTLs or when QTL effects are small. This study discusses the impact the findings could have on the calculation of thresholds and confidence intervals produced by bootstrap methods.     

Detection of linkage between marker loci and loci affecting quantitative traits in crosses between segregating populations

Summary By making use of pedigree information and information on marker-genotypes of the parent and F-1 individuals crossed to form an F-2 population, it is possible to carry out a linkage analysis between marker loci and loci affecting quantitative traits in a cross between segregating parent populations that are at fixation for alternative alleles at the QTL, but share the same alleles at the marker loci. For two-allele systems, depending on marker allele frequencies in the parent populations, 2–4 times as many F-2 offspring will have to be raised and scored for markers and quantitative traits in order to provide power equivalent to that obtained in a cross between fully inbred lines. Major savings in number of F-2 offspring raised can be achieved by scoring each parent pair for a large number of markers in each chromosomal region and scoring F-1 and F-2 offspring only for those markers for which the parents were homozygous for alternative alleles. For multiple allele systems, particularly when dealing with hypervariable loci, only 10%–20% additional F-2 offspring will have to be raised and scored to provide power equivalent to that obtained in a cross between inbred lines. When a resource population contains novel favorable alleles at quantitative trait loci that are not present (or rare) in a commercial population, analyses of this sort will enable the loci of interest to be identified, mapped and manipulated effectively in breeding programs.Contribution no. 2124-E, 1987 series from The Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

Effects of partial selfing on the equilibrium genetic variance,mutation load,and inbreeding depression under stabilizing selection

The mating system of a species is expected to have important effects on its genetic diversity. In this article, we explore the effects of partial selfing on the equilibrium genetic variance V_g, mutation load L, and inbreeding depression δ under stabilizing selection acting on a arbitrary number n of quantitative traits coded by biallelic loci with additive effects. When the ratio is low (where U is the total haploid mutation rate on selected traits) and effective recombination rates are sufficiently high, genetic associations between loci are negligible and the genetic variance, mutation load, and inbreeding depression are well predicted by approximations based on single‐locus models. For higher values of and/or lower effective recombination, moderate genetic associations generated by epistasis tend to increase V_g, L, and δ, this regime being well predicted by approximations including the effects of pairwise associations between loci. For yet higher values of and/or lower effective recombination, a different regime is reached under which the maintenance of coadapted gene complexes reduces V_g, L, and δ. Simulations indicate that the values of V_g, L, and δ are little affected by assumptions regarding the number of possible alleles per locus.     

Optimum spacing of genetic markers for determining linkage between marker loci and quantitative trait loci

The cost of experiments aimed at determining linkage between marker loci and quantitative trait loci (QTL) was investigated as a function of marker spacing and number of individuals scored. It was found that for a variety of experimental designs, fairly wide marker spacings (ca. 50 cM) are optimum or close to optimum for initial studies of marker-QTL linkage, in the sense of minimizing overall cost of the experiment. Thus, even when large numbers of more or less evenly spaced markers are available, it will not always be cost effective to make full utilization of this capacity. This is particularly true when costs of rearing and trait evaluation per individual scored are low, as when marker data are obtained on individuals raised and evaluated for quantitative traits as part of existing programs. When costs of rearing and trait evaluation per individual scored are high, however, as in human family data collection carried out primarily for subsequent marker — QTL analyses, or when plants or animals are raised specifically for purposes of marker — QTL linkage experiments, optimum spacing may be rather narrow. It is noteworthy that when marginal costs of additional markers or individuals are constant, total resources allocated to a given experiment will determine total number of individuals sampled, but not the optimal marker spacing.     

Linkage between quantitative trait loci and marker loci: resolution power of three statistical approaches in single marker analysis

This paper presents a comparison of three methods of parameter estimation in analysis of linkage between a quantitative trait locus (QTL) and a marker locus: maximum likelihood, mean square for trait cumulative distribution function, and method of moments, employing simulated backcross data. The sensitivity of estimates to violation of assumptions of normality and equal variances were also studied. Some measures of discrepancy between the trait distributions in the QTL groups are considered to evaluate the potential dependence of the resolution capacity of the QTL substitution effect with respect to trait mean value and variance.     

Trait-based analyses for the detection of linkage between marker loci and quantitative trait loci in crosses between inbred lines

Summary Methods are presented for determining linkage between a marker locus and a nearby locus affecting a quantitative trait (quantitative trait locus=QTL), based on changes in the marker allele frequencies in selection lines derived from the F-2 of a cross between inbred lines, or in the high and low phenotypic classes of an F-2 or BC population. The power of such trait-based (TB) analyses was evaluated and compared with that of methods for determining linkage based on the mean quantitative trait value of marker genotypes in F-2 or BC populations [marker-based (MB) analyses]. TB analyses can be utilized for marker-QTL linkage determination in situations where the MB analysis is not applicable, including analysis of polygenic resistance traits where only a part of the population survives exposure to the Stressor and analysis of marker-allele frequency changes in selection lines. TB analyses may be a useful alternative to MB analyses when interest is centered on a single quantitative trait only and costs of scoring for markers are high compared with costs of raising and obtaining quantitative trait information on F-2 or BC individuals. In this case, a TB analysis will enable equivalent power to be obtained with fewer individuals scored for the marker, but more individuals scored for the quantitative trait. MB analyses remain the method of choice when more than one quantitative trait is to be analyzed in a given population.Contribution from the ARO, Bet Dagan, Israel. No. 1698-E, 1986 series     

On the power of experimental designs for the detection of linkage between marker loci and quantitative loci in crosses between inbred lines

Summary The power of experiments aimed at detecting linkage between a quantitative locus and a marker locus, both segregating in the backross or F₂ generation of a cross between two inbred lines, is examined. Given that the two lines are close to fixation for alternative alleles of both marker locus and quantitative locus, it is concluded that experiments involving a few thousand offspring should be able to detect close linkages involving quantitative loci (or groups of loci) having rather modest effects (i.e., that contribute, say, 1% of the total phenotypic variance in the F₂).     

On the mutual maintenance of a polymorphism in two nonepistatic loci with partial selfing

In a two-locus system of multiplicative fitnesses with a mixed selfing and random mating, the maintenance of a polymorphism in each locus is not independent of its maintenance in the other one.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

Genetic variance components and heritability of multiallelic heterozygosity under inbreeding

The maintenance of genetic diversity in fitness-related traits remains a central topic in evolutionary biology, for example, in the context of sexual selection for genetic benefits. Among the solutions that have been proposed is directional sexual selection for heterozygosity. The importance of such selection is highly debated. However, a critical evaluation requires knowledge of the heritability of heterozygosity, a quantity that is rarely estimated in this context, and often assumed to be zero. This is at least partly the result of the lack of a general framework that allows for its quantitative prediction in small and inbred populations, which are the focus of most empirical studies. Moreover, while current predictors are applicable only to biallelic loci, fitness-relevant loci are often multiallelic, as are the neutral markers typically used to estimate genome-wide heterozygosity. To this end, we first review previous, but little-known, work showing that under most circumstances, heterozygosity at biallelic loci and in the absence of inbreeding is heritable. We then derive the heritability of heterozygosity and the underlying variances for multiple alleles and any inbreeding level. We also show that heterozygosity at multiallelic loci can be highly heritable when allele frequencies are unequal, and that this heritability is reduced by inbreeding. Our quantitative genetic framework can provide new insights into the evolutionary dynamics of heterozygosity in inbred and outbred populations.