Models to estimate maternally controlled genetic variation in quantitative seed characters

Summary Estimating quantitative contributions to specific traits can be accomplished from a variety of genetic models (Mather 1949; Mather and Jinks 1971; Falconer 1981). Residual genetic effects, those beyond main and interaction effects of the embryo genotype, are often pooled under a single classification, termed maternal effects. Maternal contributions to seed-related traits can originate from various maternal sources (e.g., endosperm, testa and cytoplasm). Quantitative contributions of a maternal nature are not predictable from parental performance and effects are largely non-persistent over generations (Jinks et al. 1972). The methods used to determine maternal effects in quantitative traits often do not measure quantitative genetic parameters, while those that do are either complex or partially resolve potential contributions of individual sources of maternal effects. We present simple genetic models for estimating quantitative genetic parameters which take into account maternal effects expressed in the major seed tissues of higher plants.     

Runaway sexual selection without genetic correlations: social environments and flexible mate choice initiate and enhance the fisher process

Female mating preferences are often flexible, reflecting the social environment in which they are expressed. Associated indirect genetic effects (IGEs) can affect the rate and direction of evolutionary change, but sexual selection models do not capture these dynamics. We incorporate IGEs into quantitative genetic models to explore how variation in social environments and mate choice flexibility influence Fisherian sexual selection. The importance of IGEs is that runaway sexual selection can occur in the absence of a genetic correlation between male traits and female preferences. Social influences can facilitate the initiation of the runaway process and increase the rate of trait elaboration. Incorporating costs to choice do not alter the main findings. Our model provides testable predictions: (1) genetic covariances between male traits and female preferences may not exist, (2) social flexibility in female choice will be common in populations experiencing strong sexual selection, (3) variation in social environments should be associated with rapid sexual trait divergence, and (4) secondary sexual traits will be more elaborate than previously predicted. Allowing feedback from the social environment resolves discrepancies between theoretical predictions and empirical data, such as why indirect selection on female preferences, theoretically weak, might be sufficient for preferences to become elaborated.     

Developmental quantitative genetic models of evolutionary change

Discussions about evolutionary change in developmental processes or morphological structures are predicated on specific quantitative genetic models whose parameters predict whether evolutionary change can occur, its relative rate and direction, and if correlated change will occur in other related and unrelated structures. The appropriate genetic model should reflect the relevant genetical and developmental biology of the organisms, yet be simple enough in its parameters so that deductions can be made and hypotheses tested. As a consequence, the choice of the most appropriate genetic model for polygenically controlled traits is a complex tissue and the eventual choice of model is often a compromise between completeness of the model and computational expediency. Herein, we discuss several developmental quantitative genetic models for the evolution of development and morphology. The models range from the classical direct effects model to complex epigenetic models. Further, we demonstrate the algebraic equivalency of the Cowley and Atchley epigenetic model and Wagner's developmental mapping model. Finally, we propose a new multivariate model for continuous growth trajectories. The relative efficacy of these various models for understanding evolutionary change in developmental and morphological traits is discussed. © 1994 Wiley-Liss, Inc.     

The role of maternal and paternal effects in the evolution of parental quality by sexual selection

Genetic models of maternal effects and models of mate choice have focused on the evolutionary effects of variation in parental quality. There have been, however, few attempts to combine these into a single model for the evolution of sexually selected traits. We present a quantitative genetic model that considers how male and female parental quality (together or separately) affect the expression of a sexually selected offspring trait. We allow female choice of males based on this parentally affected trait and examine the evolution of mate choice, parental quality and the indicator trait. Our model reveals a number of consequences of maternal and paternal effects. (1) The force of sexual selection owing to adaptive mate choice can displace parental quality from its natural selection optimum. (2) The force of sexual selection can displace female parental quality from its natural selection optimum even when nonadaptive mate choice occurs (e.g. runaway sexual selection), because females of higher parental quality produce more attractive sons and these sons counterbalance the loss in fitness owing to over-investment in each offspring. (3) Maternal and paternal effects can provide a source of genetic variation for offspring traits, allowing evolution by sexual selection even when those traits do not show direct genetic variation (i.e. are not heritable). (4) The correlation between paternal investment and the offspring trait influenced by the parental effects can result in adaptive mate choice and lead to the elaboration of both female preference and the male sexually selected trait. When parental effects exist, sexual selection can drive the evolution of parental quality when investment increases the attractiveness of offspring, leading to the elaboration of indicator traits and higher than expected levels of parental investment.     

Analysis of cytoplasmic and maternal effects. II. Genetic models for triploid endosperms

Genetic models for quantitative traits of triploid endosperms are proposed for the analysis of direct gene effects, cytoplasmic effects, and maternal gene effects. The maternal effect is partitioned into maternal additive and dominance components. In the full genetic model, the direct effect is partitioned into direct additive and dominance components and high-order dominance component, which are the cumulative effects of three-allele interactions. If the high-order dominance effects are of no importance, a reduced genetic model can be used. Monte Carlo simulations were conducted in this study for demonstrating unbiasedness of estimated variance and covariance components from the MINQUE (0/1) procedure, which is a minimum norm quadratic unbiased estimation (MINQUE) method setting 0 for all the prior covariances and 1 for all the prior variances. Robustness of estimating variance and covariance components for the genetic models was tested by simulations. Both full and reduced genetic models are shown to be robust for estimating variance and covariance components under several situations of no specific effects. Efficiency of predicting random genetic effects for the genetic models by the MINQUE (0/1) procedure was compared with the best linear unbiased prediction (BLUP). A worked example is given to illustrate the use of the reduced genetic model for kernel growth characteristics in corn (Zea mays L.).     

SASQuant: a SAS software program to estimate genetic effects and heritabilities of quantitative traits in populations consisting of 6 related generations

Plant breeders are interested in the analysis of phenotypic data to measure genetic effects and heritability of quantitative traits and predict gain from selection. Measurement of phenotypic values of 6 related generations (parents, F(1), F(2), and backcrosses) allows for the simultaneous analysis of both Mendelian and quantitative traits. In 1997, Liu et al. released a SAS software based program (SASGENE) for the analysis of inheritance and linkage of qualitative traits. We have developed a new program (SASQuant) that estimates gene effects (Hayman's model), genetic variances, heritability, predicted gain from selection (Wright's and Warner's models), and number of effective factors (Wright's, Mather's, and Lande's models). SASQuant makes use of traditional genetic models and allows for their easy application to complex data sets. SASQuant is freely available and is intended for scientists studying quantitative traits in plant populations.     

VARIATION IN SEED CHARACTERS IN NEMOPHILA MENZIESII: EVIDENCE OF A GENETIC BASIS FOR MATERNAL EFFECT

A growing body of evidence indicates that phenotypic selection on juvenile traits of both plants and animals may be considerable. Because juvenile traits are typically subject to maternal effects and often have low heritabilities, adaptive responses to natural selection on these traits may seem unlikely. To determine the potential for evolutionary response to selection on juvenile traits of Nemophila menziesii (Hydrophyllaceae), we conducted two quantitative genetic studies. A reciprocal factorial cross, involving 16 parents and 1960 progeny, demonstrated a significant maternal component of variance in seed mass and additive genetic component of variance in germination time. This experiment also suggested that interaction between parents, though small, provides highly significant contributions to the variance of both traits. Such a parental interaction could arise by diverse mechanisms, including dependence of nuclear gene expression on cytoplasmic genotype, but the design of this experiment could not distinguish this from other possible causes, such as effects on progeny phenotype of interaction between the environmental conditions of both parents. The second experiment, spanning three generations with over 11,000 observations, was designed for investigation of the additive genetic variance in maternal effect, assessment of paternal effects, as well as further partitioning of the parental interaction identified in the reciprocal factorial experiment. It yielded no consistent evidence of paternal effects on seed mass, nor of parental interactions. Our inference of such interaction effects from the first experiment was evidently an artifact of failing to account for the substantial variance among fruits within crosses. The maternal effect was found to have a large additive genetic component, accounting for at least 20% of the variation in individual seed mass. This result suggests that there is appreciable potential for response to selection on seed mass through evolution of the maternal effect. We discuss aspects that may nevertheless limit response to individual selection on seed mass, including trade-offs between the size of individual seeds and germination time and between the number of seeds a maternal plant can mature and their mean size.     

Gene-environment interaction and the mapping of complex traits: some statistical models and their implications

The manifestation of many complex diseases or traits is very likely the result of an inextricable interplay of the biological and the environmental. Yet the role of environmental effect has traditionally been played down, for various reasons. In this paper, some simple statistical models that incorporate gene-environment interaction (GEI) have been proposed and their behavior and implications investigated. These implications concern the conditional independence assumption in likelihood calculation of pedigree data, the fine-tuning of the sib pair method for mapping quantitative traits, apportioning of disease or trait variation due to specific causes. In addition, they concern properties of gene mapping methods that do not take GEI into account, and they bring into question the utility of commonly used measures of genetic effects such as recurrence risk ratio for relative pairs, twin concordance rates, and heritability coefficients. In the presence of GEI, all these measures are functions not only of genetic effects and gene frequency, but also of environmental effects, the distribution of environmental factors in the population, and of GEI. Above all, these measures are all measures of familial aggregation, since they can be significant even in the absence of any genetic component of the disease. Thus their use as indicators of the genetic basis of complex diseases is cast into doubt.     

Analysis of genetic effects of nuclear–cytoplasmic interaction on quantitative traits: Genetic models for seed traits of plants

Two Genetic models (an embryo model and an endosperm model) were proposed for analyzing genetic effects of nuclear genes, cytoplasmic genes, maternal genes, and nuclear–cytoplasmic interaction (NCI) as well as their genotype by environment interaction for quantitative traits of plant seed. In these models, the NCI effects were partitioned into direct additive and dominance NCI components. Mixed linear model approaches were employed for statistical analysis. For both balanced and unbalanced diallel cross designs, Monte Carlo simulations were conducted to evaluate unbiasedness and precision of estimated variance components of these models. The results showed that the proposed methods work well. Random genetic effects were predicted with an adjusted unbiased prediction method. Seed traits (protein content and oil content) of Upland cotton (Gossypium hirsutum L.) were analyzed as worked examples to demonstrate the use of the models.     

Analysis of genetic effects on nutrient quality traits in indica rice

Nine cytoplasmic male-sterile lines and five restorer lines were used in an incomplete diallel cross to analyze seed effects, cytoplasmic effects, and maternal gene effects on nutrient quality traits of indica rice (Oryza sauva L.). The results indicated that nutrient quality traits were controlled by cytoplasmic and maternal effects as well as by seed direct effects. Maternal effects for lysine content (LC), lysine index (LI), and the ratio of lysine content to protein content (RLP) were more important than seed direct effects, while protein content (PC) and protein index (PI) were mainly affected by seed direct effects. Cytoplasmic effects accounted for 2.41–20.80% of the total genetic variation and were significant for all nutrient quality traits. Additive genetic effects were much more important than dominance effects for all of the traits studied, so that selection could be applied for these traits in early generations.     

EVOLUTIONARY POTENTIAL OF A LARGE MARINE VERTEBRATE: QUANTITATIVE GENETIC PARAMETERS IN A WILD POPULATION

Estimating quantitative genetic parameters ideally takes place in natural populations, but relatively few studies have overcome the inherent logistical difficulties. For this reason, no estimates currently exist for the genetic basis of life-history traits in natural populations of large marine vertebrates. And yet such estimates are likely to be important given the exposure of this taxon to changing selection pressures, and the relevance of life-history traits to population productivity. We report such estimates from a long-term (1995–2007) study of lemon sharks ( Negaprion brevirostris ) conducted at Bimini, Bahamas. We obtained these estimates by genetically reconstructing a population pedigree (117 dams, 487 sires, and 1351 offspring) and then using an "animal model" approach to estimate quantitative genetic parameters. We find significant additive genetic (co)variance, and hence moderate heritability, for juvenile length and mass. We also find substantial maternal effects for these traits at age-0, but not age-1, confirming that genotype–phenotype interactions between mother and offspring are strongest at birth; although these effects could not be parsed into their genetic and nongenetic components. Our results suggest that human-imposed selection pressures (e.g., size-selective harvesting) might impose noteworthy evolutionary change even in large marine vertebrates. We therefore use our findings to explain how maternal effects may sometimes promote maladaptive juvenile traits, and how lemon sharks at different nursery sites may show "constrained local adaptation." We also show how single-generation pedigrees, and even simple marker-based regression methods, can provide accurate estimates of quantitative genetic parameters in at least some natural systems.     

When three traits make a line: evolution of phenotypic plasticity and genetic assimilation through linear reaction norms in stochastic environments

Genetic assimilation emerges from selection on phenotypic plasticity. Yet, commonly used quantitative genetics models of linear reaction norms considering intercept and slope as traits do not mimic the full process of genetic assimilation. We argue that intercept–slope reaction norm models are insufficient representations of genetic effects on linear reaction norms and that considering reaction norm intercept as a trait is unfortunate because the definition of this trait relates to a specific environmental value (zero) and confounds genetic effects on reaction norm elevation with genetic effects on environmental perception. Instead, we suggest a model with three traits representing genetic effects that, respectively, (i) are independent of the environment, (ii) alter the sensitivity of the phenotype to the environment and (iii) determine how the organism perceives the environment. The model predicts that, given sufficient additive genetic variation in environmental perception, the environmental value at which reaction norms tend to cross will respond rapidly to selection after an abrupt environmental change, and eventually becomes equal to the new mean environment. This readjustment of the zone of canalization becomes completed without changes in genetic correlations, genetic drift or imposing any fitness costs of maintaining plasticity. The asymptotic evolutionary outcome of this three‐trait linear reaction norm generally entails a lower degree of phenotypic plasticity than the two‐trait model, and maximum expected fitness does not occur at the mean trait values in the population.     

Fixed‐effect variance and the estimation of repeatabilities and heritabilities: issues and solutions

Linear mixed‐effects models are frequently used for estimating quantitative genetic parameters, including the heritability, as well as the repeatability, of traits. Heritability acts as a filter that determines how efficiently phenotypic selection translates into evolutionary change, whereas repeatability informs us about the individual consistency of phenotypic traits. As quantities of biological interest, it is important that the denominator, the phenotypic variance in both cases, reflects the amount of phenotypic variance in the relevant ecological setting. The current practice of quantifying heritabilities and repeatabilities from mixed‐effects models frequently deprives their denominator of variance explained by fixed effects (often leading to upward bias of heritabilities and repeatabilities), and it has been suggested to omit fixed effects when estimating heritabilities in particular. We advocate an alternative option of fitting models incorporating all relevant effects, while including the variance explained by fixed effects into the estimation of the phenotypic variance. The approach is easily implemented and allows optimizing the estimation of phenotypic variance, for example by the exclusion of variance arising from experimental design effects while still including all biologically relevant sources of variation. We address the estimation and interpretation of heritabilities in situations in which potential covariates are themselves heritable traits of the organism. Furthermore, we discuss complications that arise in generalized and nonlinear mixed models with fixed effects. In these cases, the variance parameters on the data scale depend on the location of the intercept and hence on the scaling of the fixed effects. Integration over the biologically relevant range of fixed effects offers a preferred solution in those situations.     

Allelic penetrance approach as a tool to model two-locus interaction in complex binary traits

Many binary phenotypes do not follow a classical Mendelian inheritance pattern. Interaction between genetic and environmental factors is thought to contribute to the incomplete penetrance phenomena often observed in these complex binary traits. Several two-locus models for penetrance have been proposed to aid the genetic dissection of binary traits. Such models assume linear genetic effects of both loci in different mathematical scales of penetrance, resembling the analytical framework of quantitative traits. However, changes in phenotypic scale are difficult to envisage in binary traits and limited genetic interpretation is extractable from current modeling of penetrance. To overcome this limitation, we derived an allelic penetrance approach that attributes incomplete penetrance to the stochastic expression of the alleles controlling the phenotype, the genetic background and environmental factors. We applied this approach to formulate dominance and recessiveness in a single diallelic locus and to model different genetic mechanisms for the joint action of two diallelic loci. We fit the models to data on the genetic susceptibility of mice following infections with Listeria monocytogenes and Plasmodium berghei. These models gain in genetic interpretation, because they specify the alleles that are responsible for the genetic (inter)action and their genetic nature (dominant or recessive), and predict genotypic combinations determining the phenotype. Further, we show via computer simulations that the proposed models produce penetrance patterns not captured by traditional two-locus models. This approach provides a new analysis framework for dissecting mechanisms of interlocus joint action in binary traits using genetic crosses.     

Mapping of epistatic quantitative trait loci in four-way crosses

Four-way crosses (4WC) involving four different inbred lines often appear in plant and animal commercial breeding programs. Direct mapping of quantitative trait loci (QTL) in these commercial populations is both economical and practical. However, the existing statistical methods for mapping QTL in a 4WC population are built on the single-QTL genetic model. This simple genetic model fails to take into account QTL interactions, which play an important role in the genetic architecture of complex traits. In this paper, therefore, we attempted to develop a statistical method to detect epistatic QTL in 4WC population. Conditional probabilities of QTL genotypes, computed by the multi-point single locus method, were used to sample the genotypes of all putative QTL in the entire genome. The sampled genotypes were used to construct the design matrix for QTL effects. All QTL effects, including main and epistatic effects, were simultaneously estimated by the penalized maximum likelihood method. The proposed method was confirmed by a series of Monte Carlo simulation studies and real data analysis of cotton. The new method will provide novel tools for the genetic dissection of complex traits, construction of QTL networks, and analysis of heterosis.     

It takes two: Heritable male effects on reproductive timing but not clutch size in a wild bird population*

Within-population variation in the traits underpinning reproductive output has long been of central interest to biologists. Since they are strongly linked to lifetime reproductive success, these traits are expected to be subject to strong selection and, if heritable, to evolve. Despite the formation of durable pair bonds in many animal taxa, reproductive traits are often regarded as female-specific, and estimates of quantitative genetic variation seldom consider a potential role for heritable male effects. Yet reliable estimates of such social genetic effects are important since they influence the amount of heritable variation available to selection. Based on a 52-year study of a nestbox-breeding great tit (Parus major) population, we apply “extended” bivariate animal models in which the heritable effects of both sexes are modeled to assess the extent to which males contribute to heritable variation in seasonal reproductive timing (egg laying date) and clutch size, while accommodating the covariance between the two traits. Our analyses show that reproductive timing is a jointly expressed trait in this species, with (positively covarying) heritable variation for laydate being expressed in both members of a breeding pair, such that the total heritable variance is 50% larger than estimated by traditional models. This result was robust to explicit consideration of a potential male-biased environmental confound arising through sexually dimorphic dispersal. In contrast to laydate, males’ contribution to heritable variation in clutch size was limited. Our study thus highlights the contrasting extent of social determination for two major components of annual reproductive success, and emphasizes the need to consider the social context of what are often considered individual-level traits.     

How to separate genetic and environmental causes of similarity between relatives

Related individuals often have similar phenotypes, but this similarity may be due to the effects of shared environments as much as to the effects of shared genes. We consider here alternative approaches to separating the relative contributions of these two sources to phenotypic covariances, comparing experimental approaches such as cross-fostering, traditional statistical techniques and more complex statistical models, specifically the 'animal model'. Using both simulation studies and empirical data from wild populations, we demonstrate the ability of the animal model to reduce bias due to shared environment effects such as maternal or brood effects, especially where pedigrees contain multiple generations and immigration rates are low. However, where common environment effects are strong, a combination of both cross-fostering and an animal model provides the best way to avoid bias. We illustrate ways of partitioning phenotypic variance into components of additive genetic, maternal genetic, maternal environment, common environment, permanent environment and temporal effects, but also show how substantial confounding between these different effects may occur. Whilst the flexibility of the mixed model approach is extremely useful for incorporating the spatial, temporal and social heterogeneity typical of natural populations, the advantages will inevitably be restricted by the quality of pedigree information and care needs to be taken in specifying models that are appropriate to the data.     

The use of quantitative traits in the study of human population structure

Studies of human population structure and history have tended to use demographic and/or serological data for analysis. This paper reviews the methods and studies that incorporate quantitative traits (usually polygenic traits) in such analyses. Methods of assessing the degree and pattern of among-group variation are discussed, and are characterized as being model-free or model-bound. Model-free methods deal with the measure of overall populational differentiation and with comparative methods for describing the pattern of differentiation. Model-bound methods are used for direct incorporation into theoretical models of population structure in order to estimate genetic parameters, such as those in admixture and isolation by distance models. To date, studies have indicated that quantitative traits may often be used successfully in studies of human population structure, and show effects of microevolutionary forces on quantitative variation among populations.     

Marker-assisted prediction of non-additive genetic values

It has become increasingly clear from systems biology arguments that interaction and non-linearity play an important role in genetic regulation of phenotypic variation for complex traits. Marker-assisted prediction of genetic values assuming additive gene action has been widely investigated because of its relevance in artificial selection. On the other hand, it has been less well-studied when non-additive effects hold. Here, we explored a nonparametric model, radial basis function (RBF) regression, for predicting quantitative traits under different gene action modes (additivity, dominance and epistasis). Using simulation, it was found that RBF had better ability (higher predictive correlations and lower predictive mean square errors) of predicting merit of individuals in future generations in the presence of non-additive effects than a linear additive model, the Bayesian Lasso. This was true for populations undergoing either directional or random selection over several generations. Under additive gene action, RBF was slightly worse than the Bayesian Lasso. While prediction of genetic values under additive gene action is well handled by a variety of parametric models, nonparametric RBF regression is a useful counterpart for dealing with situations where non-additive gene action is suspected, and it is robust irrespective of mode of gene action.