MEASURING SELECTION AND CONSTRAINT IN THE EVOLUTION OF GROWTH

We present a quantitative genetic model for the evolution of growth trajectories that makes no assumptions about the shapes of growth trajectories that are possible. Evolution of a population's mean growth trajectory is governed by the selection gradient function and the additive genetic covariance function. The selection gradient function is determined by the impact of changes in size on the birth and death rates at different ages, and can be estimated for natural populations. The additive genetic covariance function can also be estimated empirically, as we demonstrate with four vertebrate populations. Using the genetic data from mice, a computer simulation shows that evolution of a growth trajectory can be constrained by the absence of genetic variation for certain changes in the trajectory's shape. These constraints can be visualized with an analysis of the covariance function. Results from four vertebrate populations show that while each has substantial genetic variation for some evolutionary changes in its growth trajectory, most types of changes have little or no variation available. This suggests that constraints may often play an important role in the evolution of growth.     

Constraints on the evolution of function‐valued traits: a study of growth in Tribolium castaneum

Growth trajectories often impact individual fitness. They are continuous by nature and so are amenable to analysis using a function‐valued (FV) trait framework to reveal their underlying genetic architecture. Previous studies have found high levels of standing additive genetic (co)variance for growth trajectories despite the expectation that growth should be responding to frequent strong directional selection. In this study, the FV framework is used to estimate the additive genetic covariance function for growth trajectories in larval Tribolium castaneum to address questions about standing additive genetic (co)variance and possible evolutionary constraints on growth and to predict responses to four plausible selection regimes. Results show that additive genetic (co)variance is high at the early ages, but decreases towards later ages in the larval period. A selection gradient function of the same size and in the same direction of the first eigenfunction of the G‐function should give the maximal response. However, evolutionary constraints may be acting to keep this maximal response from being realized, through either conflicting effects on survivability and fecundity of larger body size, few evolutionary directions having sufficient additive variance for a response, genetic trade‐offs with other traits or physiological regulatory mechanisms. More light may be shed on these constraints through the development of more sophisticated statistical approaches and implementation of additional empirical studies to explicitly test for specific types of constraints.     

Genetic covariance structure of growth in the salamander Ambystoma macrodactylum

The size of an organism at any point during ontogeny often has fitness consequences through either direct selection on size or through selection on size-related morphological, performance, or life history traits. However, the evolutionary response to selection on size across ontogeny (a growth trajectory) may be limited by genetic correlations across ages. Here we characterize the phenotypic and genetic covariance structure of length and mass growth trajectories in a natural population of larval Ambystoma macrodactylum using function-valued quantitative genetic analyses and principal component decomposition. Most of the phenotypic and genetic variation in both growth trajectories appears to be confined to a single principal component describing a pattern of positive covariation among sizes across all ages. Higher order principal components with no significant associated genetic variation were identified for both trajectories, suggesting that evolution towards certain patterns of negative covariation between sizes across ages is constrained. The well-characterized positive relationship between size at metamorphosis and fitness in pond-breeding amphibians predicts that the across-age covariance structure will strongly limit evolution only if there is negative selection on size prior to metamorphosis. The pattern of genetic covariation observed in this study is similar to that observed in other vertebrate taxa, indicating that size may often be highly genetically and phenotypically integrated across ontogeny. Additionally, we find that phenotypic and genetic analyses of growth trajectories can yield qualitatively similar patterns of covariance structure.     

A quantitative genetic model for growth,shape, reaction norms,and other infinite-dimensional characters

Infinite-dimensional characters are those in which the phenotype of an individual is described by a function, rather than by a finite set of measurements. Examples include growth trajectories, morphological shapes, and norms of reaction. Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters. A quantitative-genetic model is developed, and the recursion equation for the evolution of the population mean phenotype of an infinite-dimensional character is derived. The infinite-dimensional method offers three advantages over conventional finite-dimensional methods when applied to this kind of trait: (1) it describes the trait at all points rather than at a finite number of landmarks, (2) it eliminates errors in predicting the evolutionary response to selection made by conventional methods because they neglect the effects of selection on some parts of the trait, and (3) it estimates parameters of interest more efficiently.     

Analysis of cytoplasmic and maternal effects I. A genetic model for diploid plant seeds and animals

A genetic model for modified diallel crosses is proposed for estimating variance and covariance components of cytoplasmic, maternal additive and dominance effects, as well as direct additive and dominance effects. Monte Carlo simulations were conducted to compare the efficiencies of minimum norm quadratic unbiased estimation (MINQUE) methods. For both balanced and unbalanced mating designs, MINQUE (0/1), which has 0 for all the prior covariances and 1 for all the prior variances, has similar efficiency to MINQUE(), which has parameter values for the prior values. Unbiased estimates of variance and covariance components and their sampling variances could be obtained with MINQUE(0/1) and jackknifing. A t-test following jackknifing is applicable to test hypotheses for zero variance and covariance components. The genetic model is robust for estimating variance and covariance components under several situations of no specific effects. A MINQUE(0/1) procedure is suggested for unbiased estimation of covariance components between two traits with equal design matrices. Methods of unbiased prediction for random genetic effects are discussed. A linear unbiased prediction (LUP) method is shown to be efficient for the genetic model. An example is given for a demonstration of estimating variance and covariance components and predicting genetic effects.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.     

Age-specific genetic and maternal effects in fecundity of preindustrial Finnish women

A population's potential for evolutionary change depends on the amount of genetic variability expressed in traits under selection. Studies attempting to measure this variability typically do so over the life span of individuals, but theory suggests that the amount of additive genetic variance can change during the course of individuals' lives. Here we use pedigree data from historical Finns and a quantitative genetic framework to investigate how female fecundity, throughout an individual's reproductive life, is influenced by "maternal" versus additive genetic effects. We show that although maternal effects explain variation in female fecundity early in life, these effects wane with female age. Moreover, this decline in maternal effects is associated with a concomitant increase in additive genetic variance with age. Our results thus highlight that single over-lifetime estimates of trait heritability may give a misleading view of a trait's potential to respond to changing selection pressures.     

A non-stationary model for functional mapping of complex traits

SUMMARY: Understanding the genetic control of growth is fundamental to agricultural, evolutionary and biomedical genetic research. In this article, we present a statistical model for mapping quantitative trait loci (QTL) that are responsible for genetic differences in growth trajectories during ontogenetic development. This model is derived within the maximum likelihood context, implemented with the expectation-maximization algorithm. We incorporate mathematical aspects of growth processes to model the mean vector and structured antedependence models to approximate time-dependent covariance matrices for longitudinal traits. Our model has been employed to map QTL that affect body mass growth trajectories in both male and female mice of an F2 population derived from the Large and Small mouse strains. The results from this model are compared with those from the autoregressive-based functional mapping approach. Based on results from computer simulation studies, we suggest that these two models are alternative to one another and should be used simultaneously for the same dataset.     

Natural selection, fitness entropy, and the dynamics of coevolution

The coevolutionary dynamics of interacting populations were studied by combining continuous time Lotka-Volterra models of population growth with single-locus genetic models of weak selection. The effects of natural selection on population growth were evaluated using Ginzburg's fitness entropy function as a measure of the deviation of a population's initial allele frequencies from their polymorphic equilibrium values. This entropy measure was used to relate the dynamics of a community composed of evolving populations to the dynamics of a "reference community" whose populations are initially in genetic equilibrium. Specifically, a quantity called the "selective difference area" was defined as the total difference between the population size trajectories of a reference and evolving population. The selective difference area represents the amount of extra life a species would realize if the entire community were at genetic equilibrium. It was shown that this selective difference area is a simple linear function of the initial fitness entropies of each species. This prediction is independent of the strength of selection and holds for any arbitrary set of initial population densities. Numerical examples were presented to illustrate the results. Under the assumption of weak selection, a generalization for arbitrary population growth models was outlined.     

Evolutionary quantitative genetics of nonlinear developmental systems

In quantitative genetics, the effects of developmental relationships among traits on microevolution are generally represented by the contribution of pleiotropy to additive genetic covariances. Pleiotropic additive genetic covariances arise only from the average effects of alleles on multiple traits, and therefore the evolutionary importance of nonlinearities in development is generally neglected in quantitative genetic views on evolution. However, nonlinearities in relationships among traits at the level of whole organisms are undeniably important to biology in general, and therefore critical to understanding evolution. I outline a system for characterizing key quantitative parameters in nonlinear developmental systems, which yields expressions for quantities such as trait means and phenotypic and genetic covariance matrices. I then develop a system for quantitative prediction of evolution in nonlinear developmental systems. I apply the system to generating a new hypothesis for why direct stabilizing selection is rarely observed. Other uses will include separation of purely correlative from direct and indirect causal effects in studying mechanisms of selection, generation of predictions of medium‐term evolutionary trajectories rather than immediate predictions of evolutionary change over single generation time‐steps, and the development of efficient and biologically motivated models for separating additive from epistatic genetic variances and covariances.     

THE DETERMINATION OF GENETIC COVARIANCES AND PREDICTION OF EVOLUTIONARY TRAJECTORIES BASED ON A GENETIC CORRELATION MATRIX

There is much interest in measuring selection, quantifying evolutionary constraints, and predicting evolutionary trajectories in natural populations. For these studies, genetic (co)variances among fitness traits play a central role. We explore the conditions that determine the sign of genetic covariances and demonstrate a critical role of selection in shaping genetic covariances. In addition, we show that genetic covariance matrices rather than genetic correlation matrices should be characterized and studied in order to infer genetic basis of population differentiation and/or to predict evolutionary trajectories.     

Evolution in a changing environment: a case study with great tit fledging mass

Heritable phenotypic traits under significant and consistent directional selection often fail to show the expected evolutionary response. A potential explanation for this contradiction is that because environmental conditions change constantly, environmental change can mask an evolutionary response to selection. We combined an "animal model" analysis with 36 years of data from a long-term study of great tits (Parus major) to explore selection on and evolution of a morphological trait: body mass at fledging. We found significant heritability of this trait, but despite consistent positive directional selection on both the phenotypic and the additive genetic component of body mass, the population mean phenotypic value declined rather than increased over time. However, the mean breeding value for body mass at fledging increased over time, presumably in response to selection. We show that the divergence between the response to selection observed at the levels of genotype and phenotype can be explained by a change in environmental conditions over time, that is, related both to increased spring temperature before breeding and elevated population density. Our results support the suggestion that measuring phenotypes may not always give a reliable impression of evolutionary trajectories and that understanding patterns of phenotypic evolution in nature requires an understanding of how the environment has itself changed.     

Genetic covariance between components of male reproductive success: within‐pair vs. extra‐pair paternity in song sparrows

The evolutionary trajectories of reproductive systems, including both male and female multiple mating and hence polygyny and polyandry, are expected to depend on the additive genetic variances and covariances in and among components of male reproductive success achieved through different reproductive tactics. However, genetic covariances among key components of male reproductive success have not been estimated in wild populations. We used comprehensive paternity data from socially monogamous but genetically polygynandrous song sparrows (Melospiza melodia) to estimate additive genetic variance and covariance in the total number of offspring a male sired per year outside his social pairings (i.e. his total extra‐pair reproductive success achieved through multiple mating) and his liability to sire offspring produced by his socially paired female (i.e. his success in defending within‐pair paternity). Both components of male fitness showed nonzero additive genetic variance, and the estimated genetic covariance was positive, implying that males with high additive genetic value for extra‐pair reproduction also have high additive genetic propensity to sire their socially paired female's offspring. There was consequently no evidence of a genetic or phenotypic trade‐off between male within‐pair paternity success and extra‐pair reproductive success. Such positive genetic covariance might be expected to facilitate ongoing evolution of polygyny and could also shape the ongoing evolution of polyandry through indirect selection.     

Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate

We present the application of a nonparametric method to performing functional principal component analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal component scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing of healthy adults with a small tracer dose of 14C-folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken. The proposed methodology, incorporating shrinkage and data-adaptive features, is seen to be well suited for describing population kinetics of 14C-folate-specific activity and random effects, and can also be applied to other functional data analysis problems.     

A mechanistic model for genetic machinery of ontogenetic growth

Two different genetic mechanisms can be proposed to explain variation in growth trajectories. The allelic sensitivity hypothesis states that growth trajectory is controlled by the time-dependent expression of alleles at the deterministic quantitative trait loci (dQTL) formed during embryogenesis. The gene regulation hypothesis states that the differentiation in growth process is due to the opportunistic quantitative trait loci (oQTL) through their mediation with new developmental signals. These two hypotheses of genetic control have been elucidated in the literature. Here, we propose a new statistical model for discerning these two mechanisms in the context of growth trajectories by integrating growth laws within a QTL-mapping framework. This model is developed within the maximum-likelihood context, implemented with a grid approach for estimating the genomic positions of the deterministic and opportunistic QTL and the simplex algorithm for estimating the growth curve parameters of the genotypes at these QTL and the parameters modeling the residual (co)variance matrix. Our model allows for extensive hypothesis tests for the genetic control of growth processes and developmental events by these two types of QTL. The application of this new model to an F(2) progeny in mice leads to the detection of deterministic and opportunistic QTL on chromosome 1 for mouse body mass growth. The estimates of QTL positions and effects from our model are broadly in agreement with those by traditional interval-mapping approaches. The implications of this model for biological and biomedical research are discussed.     

Ontogeny of additive and maternal genetic effects: lessons from domestic mammals

Evolution of size and growth depends on heritable variation arising from additive and maternal genetic effects. Levels of heritable (and nonheritable) variation might change over ontogeny, increasing through "variance compounding" or decreasing through "compensatory growth." We test for these processes using a meta-analysis of age-specific weight traits in domestic ungulates. Generally, mean standardized variance components decrease with age, consistent with compensatory growth. Phenotypic convergence among adult sheep occurs through decreasing environmental and maternal genetic variation. Maternal variation similarly declines in cattle. Maternal genetic effects are thus reduced with age (both in absolute and relative terms). Significant trends in heritability (decreasing in cattle, increasing in sheep) result from declining maternal and environmental components rather than from changing additive variation. There was no evidence for increasing standardized variance components. Any compounding must therefore be masked by more important compensatory processes. While extrapolation of these patterns to processes in natural population is difficult, our results highlight the inadequacy of assuming constancy in genetic parameters over ontogeny. Negative covariance between direct and maternal genetic effects was common. Negative correlations with additive and maternal genetic variances indicate that antagonistic pleiotropy (between additive and maternal genetic effects) may maintain genetic variance and limit responses to selection.     

Genetic Correlations and Maternal Effect Coefficients Obtained from Offspring-Parent Regression

Additive genetic variances and covariances of quantitative characters are necessary to predict the evolutionary response of the mean phenotype vector in a population to natural or artificial selection. Standard formulas for estimating these parameters, from the resemblance between relatives in one or two characters at a time, are biased by natural selection on the parents and by maternal effects. We show how these biases can be removed using a multivariate analysis of offspring-parent regressions. A dynamic model of maternal effects demonstrates that, in addition to the phenotypic variance-covariance matrix of the characters, sufficient parameters for predicting the response of the mean phenotype vector to weak selection are the additive genetic variance-covariance matrix and a set of causal coefficients for maternal effects. These can be simultaneously estimated from offspring-parent regressions alone, in some cases just from the daughter-mother regressions, if all of the important selected and maternal characters have been measured and included in the analysis.     

EMPIRICAL COMPARISON OF G MATRIX TEST STATISTICS: FINDING BIOLOGICALLY RELEVANT CHANGE

A central assumption of quantitative genetic theory is that the breeder's equation ( R = GP ⁻¹ S ) accurately predicts the evolutionary response to selection. Recent studies highlight the fact that the additive genetic variance–covariance matrix ( G ) may change over time, rendering the breeder's equation incapable of predicting evolutionary change over more than a few generations. Although some consensus on whether G changes over time has been reached, multiple, often-incompatible methods for comparing G matrices are currently used. A major challenge of G matrix comparison is determining the biological relevance of observed change. Here, we develop a "selection skewers" G matrix comparison statistic that uses the breeder's equation to compare the response to selection given two G matrices while holding selection intensity constant. We present a bootstrap algorithm that determines the significance of G matrix differences using the selection skewers method, random skewers, Mantel's and Bartlett's tests, and eigenanalysis. We then compare these methods by applying the bootstrap to a dataset of laboratory populations of Tribolium castaneum . We find that the results of matrix comparison statistics are inconsistent based on differing a priori goals of each test, and that the selection skewers method is useful for identifying biologically relevant G matrix differences.     

COMPARING STRENGTHS OF DIRECTIONAL SELECTION: HOW STRONG IS STRONG?

The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the variance of the selected trait yields the response in units of standard deviation as the product of the heritability and the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and biases in publication or choice of traits to study.