Epistatic pleiotropy and the genetic architecture of covariation within early and late-developing skull trait complexes in mice

The role of epistasis as a source of trait variation is well established, but its role as a source of covariation among traits (i.e., as a source of "epistatic pleiotropy") is rarely considered. In this study we examine the relative importance of epistatic pleiotropy in producing covariation within early and late-developing skull trait complexes in a population of mice derived from an intercross of the Large and Small inbred strains. Significant epistasis was found for several pairwise combinations of the 21 quantitative trait loci (QTL) affecting early developing traits and among the 20 QTL affecting late-developing traits. The majority of the epistatic effects were restricted to single traits but epistatic pleiotropy still contributed significantly to covariances. Because of their proportionally larger effects on variances than on covariances, epistatic effects tended to reduce within-group correlations of traits and reduce their overall degree of integration. The expected contributions of single-locus and two-locus epistatic pleiotropic QTL effects to the genetic covariance between traits were analyzed using a two-locus population genetic model. The model demonstrates that, for single-locus or epistatic pleiotropy to contribute to trait covariances in the study population, both traits must show the same pattern of single-locus or epistatic effects. As a result, a large number of the cases where loci show pleiotropic effects do not contribute to the covariance between traits in this population because the loci show a different pattern of effect on the different traits. In general, covariance patterns produced by single-locus and epistatic pleiotropy predicted by the model agreed well with actual values calculated from the QTL analysis. Nearly all single-locus and epistatic pleiotropic effects contributed positive components to covariances between traits, suggesting that genetic integration in the skull is achieved by a complex combination of pleiotropic effects.     

A genealogical interpretation of linkage disequilibrium

The degree of association between alleles at different loci, or linkage disequilibrium, is widely used to infer details of evolutionary processes. Here I explore how associations between alleles relate to properties of the underlying genealogy of sequences. Under the neutral, infinite-sites assumption I show that there is a direct correspondence between the covariance in coalescence times at different parts of the genome and the degree of linkage disequilibrium. These covariances can be calculated exactly under the standard neutral model and by Monte Carlo simulation under different demographic models. I show that the effects of population growth, population bottlenecks, and population structure on linkage disequilibrium can be described through their effects on the covariance in coalescence times.     

The evolution of recombination in a heterogeneous environment

Most models describing the evolution of recombination have focused on the case of a single population, implicitly assuming that all individuals are equally likely to mate and that spatial heterogeneity in selection is absent. In these models, the evolution of recombination is driven by linkage disequilibria generated either by epistatic selection or drift. Models based on epistatic selection show that recombination can be favored if epistasis is negative and weak compared to directional selection and if the recombination modifier locus is tightly linked to the selected loci. In this article, we examine the joint effects of spatial heterogeneity in selection and epistasis on the evolution of recombination. In a model with two patches, each subject to different selection regimes, we consider the cases of mutation-selection and migration-selection balance as well as the spread of beneficial alleles. We find that including spatial heterogeneity extends the range of epistasis over which recombination can be favored. Indeed, recombination can be favored without epistasis, with negative and even with positive epistasis depending on environmental circumstances. The selection pressure acting on recombination-modifier loci is often much stronger with spatial heterogeneity, and even loosely linked modifiers and free linkage may evolve. In each case, predicting whether recombination is favored requires knowledge of both the type of environmental heterogeneity and epistasis, as none of these factors alone is sufficient to predict the outcome.     

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

Epistasis,inbreeding depression,and the evolution of self-fertilization

Inbreeding depression resulting from partially recessive deleterious alleles is thought to be the main genetic factor preventing self-fertilizing mutants from spreading in outcrossing hermaphroditic populations. However, deleterious alleles may also generate an advantage to selfers in terms of more efficient purging, while the effects of epistasis among those alleles on inbreeding depression and mating system evolution remain little explored. In this article, we use a general model of selection to disentangle the effects of different forms of epistasis (additive-by-additive, additive-by-dominance, and dominance-by-dominance) on inbreeding depression and on the strength of selection for selfing. Models with fixed epistasis across loci, and models of stabilizing selection acting on quantitative traits (generating distributions of epistasis) are considered as special cases. Besides its effects on inbreeding depression, epistasis may increase the purging advantage associated with selfing (when it is negative on average), while the variance in epistasis favors selfing through the generation of linkage disequilibria that increase mean fitness. Approximations for the strengths of these effects are derived, and compared with individual-based simulation results.     

Genomic prediction of hybrid crops allows disentangling dominance and epistasis

We revisited, in a genomic context, the theory of hybrid genetic evaluation models of hybrid crosses of pure lines, as the current practice is largely based on infinitesimal model assumptions. Expressions for covariances between hybrids due to additive substitution effects and dominance and epistatic deviations were analytically derived. Using dense markers in a GBLUP analysis, it is possible to split specific combining ability into dominance and across-groups epistatic deviations, and to split general combining ability (GCA) into within-line additive effects and within-line additive by additive (and higher order) epistatic deviations. We analyzed a publicly available maize data set of Dent × Flint hybrids using our new model (called GCA-model) up to additive by additive epistasis. To model higher order interactions within GCAs, we also fitted “residual genetic” line effects. Our new GCA-model was compared with another genomic model which assumes a uniquely defined effect of genes across origins. Most variation in hybrids is accounted by GCA. Variances due to dominance and epistasis have similar magnitudes. Models based on defining effects either differently or identically across heterotic groups resulted in similar predictive abilities for hybrids. The currently used model inflates the estimated additive genetic variance. This is not important for hybrid predictions but has consequences for the breeding scheme—e.g. overestimation of the genetic gain within heterotic group. Therefore, we recommend using GCA-model, which is appropriate for genomic prediction and variance component estimation in hybrid crops using genomic data, and whose results can be practically interpreted and used for breeding purposes.     

Fine mapping of quantitative trait loci using linkage disequilibria with closely linked marker loci

A multimarker linkage disequilibrium mapping method was developed for the fine mapping of quantitative trait loci (QTL) using a dense marker map. The method compares the expected covariances between haplotype effects given a postulated QTL position to the covariances that are found in the data. The expected covariances between the haplotype effects are proportional to the probability that the QTL position is identical by descent (IBD) given the marker haplotype information, which is calculated using the genedropping method. Simulation results showed that a QTL was correctly positioned within a region of 3, 1.5, or 0.75 cM in 70, 62, and 68%, respectively, of the replicates using markers spaced at intervals of 1, 0.5, and 0.25 cM, respectively. These results were rather insensitive to the number of generations since the QTL occurred and to the effective population size, except that 10 generations yielded rather poor estimates of the QTL position. The position estimates of this multimarker disequilibrium mapping method were more accurate than those from a single marker transmission disequilibrium test. A general approach for identifying QTL is suggested, where several stages of disequilibrium mapping are used with increasingly dense marker spacing.     

Identifying quantitative trait locus by genetic background interactions in association studies

Association studies are designed to identify main effects of alleles across a potentially wide range of genetic backgrounds. To control for spurious associations, effects of the genetic background itself are often incorporated into the linear model, either in the form of subpopulation effects in the case of structure or in the form of genetic relationship matrices in the case of complex pedigrees. In this context epistatic interactions between loci can be captured as an interaction effect between the associated locus and the genetic background. In this study I developed genetic and statistical models to tie the locus by genetic background interaction idea back to more standard concepts of epistasis when genetic background is modeled using an additive relationship matrix. I also simulated epistatic interactions in four-generation randomly mating pedigrees and evaluated the ability of the statistical models to identify when a biallelic associated locus was epistatic to other loci. Under additive-by-additive epistasis, when interaction effects of the associated locus were quite large (explaining 20% of the phenotypic variance), epistasis was detected in 79% of pedigrees containing 320 individuals. The epistatic model also predicted the genotypic value of progeny better than a standard additive model in 78% of simulations. When interaction effects were smaller (although still fairly large, explaining 5% of the phenotypic variance), epistasis was detected in only 9% of pedigrees containing 320 individuals and the epistatic and additive models were equally effective at predicting the genotypic values of progeny. Epistasis was detected with the same power whether the overall epistatic effect was the result of a single pairwise interaction or the sum of nine pairwise interactions, each generating one ninth of the epistatic variance. The power to detect epistasis was highest (94%) at low QTL minor allele frequency, fell to a minimum (60%) at minor allele frequency of about 0.2, and then plateaued at about 80% as alleles reached intermediate frequencies. The power to detect epistasis declined when the linkage disequilibrium between the DNA marker and the functional polymorphism was not complete.     

Modeling epistasis of quantitative trait loci using Cockerham's model

We use the orthogonal contrast scales proposed by Cockerham to construct a genetic model, called Cockerham's model, for studying epistasis between genes. The properties of Cockerham's model in modeling and mapping epistatic genes under linkage equilibrium and disequilibrium are investigated and discussed. Because of its orthogonal property, Cockerham's model has several advantages in partitioning genetic variance into components, interpreting and estimating gene effects, and application to quantitative trait loci (QTL) mapping when compared to other models, and thus it can facilitate the study of epistasis between genes and be readily used in QTL mapping. The issues of QTL mapping with epistasis are also addressed. Real and simulated examples are used to illustrate Cockerham's model, compare different models, and map for epistatic QTL. Finally, we extend Cockerham's model to multiple loci and discuss its applications to QTL mapping.     

Linkage strategies for genetically complex traits. I. Multilocus models

In order to investigate linkage detection strategies for genetically complex traits, multilocus models of inheritance need to be specified. Here, two types of multilocus model are described: (1) a multiplicative model, representing epistasis (interaction) among loci, and (2) an additive model, which is shown to closely approximate genetic heterogeneity, which is characterized by no interlocus interaction. A ratio lambda R of risk for type R relatives that is compared with population prevalence is defined. For a single-locus model, lambda R - 1 decreases by a factor of two with each degree of relationship. The same holds true for an additive multilocus model. For a multiplicative (epistasis) model, lambda R - 1 decreases more rapidly than by a factor of two with degree of relationship. Examination of lambda R values for various classes of relatives can potentially suggest the presence of multiple loci and epistasis. For example, data for schizophrenia suggest multiple loci in interaction. It is shown in the second paper of this series that lambda R is the critical parameter in determining power to detect linkage by using affected relative pairs.     

Prediction of effects of genetic drift on variance components under a general model of epistasis

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.     

Genotype-environment interaction effects on reproductive performance in Tribolium castaneum

Summary The genetical expectations of the means, variances and covariances of populations of doubled haploid lines derived from f₁, F₂, F₃ and intermated F₂ (S₃) generations are presented. These expectations are identical, regardless of genetical architecture, providing there is no linkage disequilibrium. In the presence of linkage disequilibrium differences will occur whose magnitude and direction will depend on the degree of disequilibrium, recombination frequency and the presence or absence of epistasis.Data from an experiment to detect linkage disequilibrium in a cross between two spring barley varieties are presented. This involved a comparison of means, variances and covariances of doubled haploid populations derived from the F₁ and F₂ generations using the H. bulbosum system. Linkage disequilibrium was detected for important agronomic characters and the effect of this disequilibrium on the choice of generation for doubled haploid production is discussed.     

Statistical modeling of interlocus interactions in a complex disease: rejection of the multiplicative model of epistasis in type 1 diabetes

In general, common diseases do not follow a Mendelian inheritance pattern. To identify disease mechanisms and etiology, their genetic dissection may be assisted by evaluation of linkage in mouse models of human disease. Statistical modeling of multiple-locus linkage data from the nonobese diabetic (NOD) mouse model of type 1 diabetes has previously provided evidence for epistasis between alleles of several Idd (insulin-dependent diabetes) loci. The construction of NOD congenic strains containing selected segments of the diabetes-resistant strain genome allows analysis of the joint effects of alleles of different loci in isolation, without the complication of other segregating Idd loci. In this article, we analyze data from congenic strains carrying two chromosome intervals (a double congenic strain) for two pairs of loci: Idd3 and Idd10 and Idd3 and Idd5. The joint action of both pairs is consistent with models of additivity on either the log odds of the penetrance, or the liability scale, rather than with the previously proposed multiplicative model of epistasis. For Idd3 and Idd5 we would also not reject a model of additivity on the penetrance scale, which might indicate a disease model mediated by more than one pathway leading to beta-cell destruction and development of diabetes. However, there has been confusion between different definitions of interaction or epistasis as used in the biological, statistical, epidemiological, and quantitative and human genetics fields. The degree to which statistical analyses can elucidate underlying biologic mechanisms may be limited and may require prior knowledge of the underlying etiology.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

Linkage mechanisms in the vertebrate skull: Structure and function of three‐dimensional,parallel transmission systems

Many musculoskeletal systems, including the skulls of birds, fishes, and some lizards consist of interconnected chains of mobile skeletal elements, analogous to linkage mechanisms used in engineering. Biomechanical studies have applied linkage models to a diversity of musculoskeletal systems, with previous applications primarily focusing on two‐dimensional linkage geometries, bilaterally symmetrical pairs of planar linkages, or single four‐bar linkages. Here, we present new, three‐dimensional (3D), parallel linkage models of the skulls of birds and fishes and use these models (available as free kinematic simulation software), to investigate structure–function relationships in these systems. This new computational framework provides an accessible and integrated workflow for exploring the evolution of structure and function in complex musculoskeletal systems. Linkage simulations show that kinematic transmission, although a suitable functional metric for linkages with single rotating input and output links, can give misleading results when applied to linkages with substantial translational components or multiple output links. To take into account both linear and rotational displacement we define force mechanical advantage for a linkage (analogous to lever mechanical advantage) and apply this metric to measure transmission efficiency in the bird cranial mechanism. For linkages with multiple, expanding output points we propose a new functional metric, expansion advantage, to measure expansion amplification and apply this metric to the buccal expansion mechanism in fishes. Using the bird cranial linkage model, we quantify the inaccuracies that result from simplifying a 3D geometry into two dimensions. We also show that by combining single‐chain linkages into parallel linkages, more links can be simulated while decreasing or maintaining the same number of input parameters. This generalized framework for linkage simulation and analysis can accommodate linkages of differing geometries and configurations, enabling novel interpretations of the mechanics of force transmission across a diversity of vertebrate feeding mechanisms and enhancing our understanding of musculoskeletal function and evolution. J. Morphol. 277:1570–1583, 2016. © 2016 Wiley Periodicals, Inc.     

Epistasis between deleterious mutations and the evolution of recombination

Epistasis and the evolution of recombination are closely intertwined: epistasis generates linkage disequilibria (i.e. statistical associations between alleles), whereas recombination breaks them up. The mutational deterministic hypothesis (MDH) states that high recombination rates are maintained because the breaking up of linkage disequilibria generated by negative epistasis enables more efficient purging of deleterious mutations. However, recent theoretical and experimental work challenges the MDH. Experimental evidence suggests that negative epistasis, required by the MDH, is relatively uncommon. On the theoretical side, population genetic models suggest that, compared with the combined effects of drift and selection, epistasis generates a negligible amount of linkage disequilibria. Here, we assess these criticisms and discuss to what extent they invalidate the MDH as an explanation for the evolution of recombination.     

A comparison of methods to calculate a total merit index using stochastic simulation

Background

Modern dairy cattle breeding goals include several production and more and more functional traits. Estimated breeding values (EBV) that are combined in the total merit index usually come from single-trait models or from multivariate models for groups of traits. In most cases, a multivariate animal model based on phenotypic data for all traits is not feasible and approximate methods based on selection index theory are applied to derive the total merit index. Therefore, the objective of this study was to compare a full multitrait animal model with two approximate multitrait models and a selection index approach based on simulated data.

Methods

Three production and two functional traits were simulated to mimic the national Austrian Brown Swiss population. The reference method for derivation of the total merit index was a multitrait evaluation based on all phenotypic data. Two of the approximate methods were variations of an approximate multitrait model that used either yield deviations or de-regressed breeding values. The final method was an adaptation of the selection index method that is used in routine evaluations in Austria and Germany. Three scenarios with respect to residual covariances were set up: residual covariances were equal to zero, or half of or equal to the genetic covariances.

Results

Results of both approximate multitrait models were very close to those of the reference method, with rank correlations of 1. Both methods were nearly unbiased. Rank correlations for the selection index method showed good results when residual covariances were zero but correlations with the reference method decreased when residual covariances were large. Furthermore, EBV were biased when residual covariances were high.

Conclusions

We applied an approximate multitrait two-step procedure to yield deviations and de-regressed breeding values, which led to nearly unbiased results. De-regressed breeding values gave even slightly better results. Our results confirmed that ignoring residual covariances when a selection index approach is applied leads to remarkable bias. This could be relevant in terms of selection accuracy. Our findings suggest that the approximate multitrait approach applied to de-regressed breeding values can be used in routine genetic evaluation.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

Epistasis interaction of QTL effects as a genetic parameter influencing estimation of the genetic additive effect

Epistasis, an additive-by-additive interaction between quantitative trait loci, has been defined as a deviation from the sum of independent effects of individual genes. Epistasis between QTLs assayed in populations segregating for an entire genome has been found at a frequency close to that expected by chance alone. Recently, epistatic effects have been considered by many researchers as important for complex traits. In order to understand the genetic control of complex traits, it is necessary to clarify additive-by-additive interactions among genes. Herein we compare estimates of a parameter connected with the additive gene action calculated on the basis of two models: a model excluding epistasis and a model with additive-by-additive interaction effects. In this paper two data sets were analysed: 1) 150 barley doubled haploid lines derived from the Steptoe × Morex cross, and 2) 145 DH lines of barley obtained from the Harrington × TR306 cross. The results showed that in cases when the effect of epistasis was different from zero, the coefficient of determination was larger for the model with epistasis than for the one excluding epistasis. These results indicate that epistatic interaction plays an important role in controlling the expression of complex traits.