The genetic analysis of age-dependent traits: modeling the character process

The extension of classical quantitative genetics to deal with function-valued characters (also called infinite-dimensional characters) such as growth curves, mortality curves, and reaction norms, was begun by Kirkpatrick and co-workers. In this theory, the analogs of variance components for single traits are covariance functions for function-valued traits. In the approach presented here, we employ a variety of parametric models for covariance functions that have a number of desirable properties: the functions (1) are positive definite, (2) can be estimated using procedures like those currently used for single traits, (3) have a small number of parameters, and (4) allow simple hypotheses to be easily tested. The methods are illustrated using data from a large experiment that examined the effects of spontaneous mutations on age-specific mortality rates in Drosophila melanogaster. Our methods are shown to work better than a standard multivariate analysis, which assumes the character value at each age is a distinct character. Advantages over existing methods that model covariance functions as a series of orthogonal polynomials are discussed.     

Multivariate character process models for the analysis of two or more correlated function-valued traits

Various methods, including random regression, structured antedependence models, and character process models, have been proposed for the genetic analysis of longitudinal data and other function-valued traits. For univariate problems, the character process models have been shown to perform well in comparison to alternative methods. The aim of this article is to present an extension of these models to the simultaneous analysis of two or more correlated function-valued traits. Analytical forms for stationary and nonstationary cross-covariance functions are studied. Comparisons with the other approaches are presented in a simulation study and in an example of a bivariate analysis of genetic covariance in age-specific fecundity and mortality in Drosophila. As in the univariate case, bivariate character process models with an exponential correlation were found to be quite close to first-order structured antedependence models. The simulation study showed that the choice of the most appropriate methodology is highly dependent on the covariance structure of the data. The bivariate character process approach proved to be able to deal with quite complex nonstationary and nonsymmetric cross-correlation structures and was found to be the most appropriate for the real data example of the fruit fly Drosophila melanogaster.     

Hypothesis testing in comparative and experimental studies of function-valued traits

Many traits of evolutionary interest, when placed in their developmental, physiological, or environmental contexts, are function-valued. For instance, gene expression during development is typically a function of the age of an organism and physiological processes are often a function of environment. In comparative and experimental studies, a fundamental question is whether the function-valued trait of one group is different from another. To address this question, evolutionary biologists have several statistical methods available. These methods can be classified into one of two types: multivariate and functional. Multivariate methods, including univariate repeated-measures analysis of variance (ANOVA), treat each trait as a finite list of data. Functional methods, such as repeated-measures regression, view the data as a sample of points drawn from an underlying function. A key difference between multivariate and functional methods is that functional methods retain information about the ordering and spacing of a set of data values, information that is discarded by multivariate methods. In this study, we evaluated the importance of that discarded information in statistical analyses of function-valued traits. Our results indicate that functional methods tend to have substantially greater statistical power than multivariate approaches to detect differences in a function-valued trait between groups.     

Structured antedependence models for genetic analysis of repeated measures on multiple quantitative traits

Simultaneous analysis of correlated traits that change with time is an important issue in genetic analyses. Several methodologies have already been proposed for the genetic analysis of longitudinal data on single traits, in particular random regression and character process models. Although the latter proved, in most cases, to compare favourably to alternative approaches for analysis of single function-valued traits, they do not allow a straightforward extension to the multivariate case. In this paper, another methodology (structured antedependence models) is proposed, and methods are derived for the genetic analysis of two or more correlated function-valued traits. Multivariate analyses are presented of fertility and mortality in Drosophila and of milk, fat and protein yields in dairy cattle. These models offer a substantial flexibility for the correlation structure, even in the case of complex non-stationary patterns, and perform better than multivariate random regression models, with fewer parameters.     

Variation,selection and evolution of function-valued traits

We describe an emerging framework for understanding variation, selection and evolution of phenotypic traits that are mathematical functions. We use one specific empirical example – thermal performance curves (TPCs) for growth rates of caterpillars – to demonstrate how models for function-valued traits are natural extensions of more familiar, multivariate models for correlated, quantitative traits. We emphasize three main points. First, because function-valued traits are continuous functions, there are important constraints on their patterns of variation that are not captured by multivariate models. Phenotypic and genetic variation in function-valued traits can be quantified in terms of variance-covariance functions and their associated eigenfunctions: we illustrate how these are estimated as well as their biological interpretations for TPCs. Second, selection on a function-valued trait is itself a function, defined in terms of selection gradient functions. For TPCs, the selection gradient describes how the relationship between an organism's performance and its fitness varies as a function of its temperature. We show how the form of the selection gradient function for TPCs relates to the frequency distribution of environmental states (caterpillar temperatures) during selection. Third, we can predict evolutionary responses of function-valued traits in terms of the genetic variance-covariance and the selection gradient functions. We illustrate how non-linear evolutionary responses of TPCs may occur even when the mean phenotype and the selection gradient are themselves linear functions of temperature. Finally, we discuss some of the methodological and empirical challenges for future studies of the evolution of function-valued traits.     

Direct estimation of genetic principal components: simplified analysis of complex phenotypes

Estimating the genetic and environmental variances for multivariate and function-valued phenotypes poses problems for estimation and interpretation. Even when the phenotype of interest has a large number of dimensions, most variation is typically associated with a small number of principal components (eigen-vectors or eigenfunctions). We propose an approach that directly estimates these leading principal components; these then give estimates for the covariance matrices (or functions). Direct estimation of the principal components reduces the number of parameters to be estimated, uses the data efficiently, and provides the basis for new estimation algorithms. We develop these concepts for both multivariate and function-valued phenotypes and illustrate their application in the restricted maximum-likelihood framework.     

On the generalized poisson regression mixture model for mapping quantitative trait loci with count data

Statistical methods for mapping quantitative trait loci (QTL) have been extensively studied. While most existing methods assume normal distribution of the phenotype, the normality assumption could be easily violated when phenotypes are measured in counts. One natural choice to deal with count traits is to apply the classical Poisson regression model. However, conditional on covariates, the Poisson assumption of mean-variance equality may not be valid when data are potentially under- or overdispersed. In this article, we propose an interval-mapping approach for phenotypes measured in counts. We model the effects of QTL through a generalized Poisson regression model and develop efficient likelihood-based inference procedures. This approach, implemented with the EM algorithm, allows for a genomewide scan for the existence of QTL throughout the entire genome. The performance of the proposed method is evaluated through extensive simulation studies along with comparisons with existing approaches such as the Poisson regression and the generalized estimating equation approach. An application to a rice tiller number data set is given. Our approach provides a standard procedure for mapping QTL involved in the genetic control of complex traits measured in counts.     

Nonparametric mixed effects models for unequally sampled noisy curves

We propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients. The method is applicable when the individual curves are sampled at variable and irregularly spaced points. This produces a low-rank, low-frequency approximation to the covariance structure, which can be estimated naturally by the EM algorithm. Smooth curves for individual trajectories are constructed as best linear unbiased predictor (BLUP) estimates, combining data from that individual and the entire collection. This framework leads naturally to methods for examining the effects of covariates on the shapes of the curves. We use model selection techniques--Akaike information criterion (AIC), Bayesian information criterion (BIC), and cross-validation--to select the number of breakpoints for the spline approximation. We believe that the methodology we propose provides a simple, flexible, and computationally efficient means of functional data analysis.     

Variation in continuous reaction norms: quantifying directions of biological interest

Thermal performance curves are an example of continuous reaction norm curves of common shape. Three modes of variation in these curves--vertical shift, horizontal shift, and generalist-specialist trade-offs--are of special interest to evolutionary biologists. Since two of these modes are nonlinear, traditional methods such as principal components analysis fail to decompose the variation into biological modes and to quantify the variation associated with each mode. Here we present the results of a new method, template mode of variation (TMV), that decomposes the variation into predetermined modes of variation for a particular set of thermal performance curves. We illustrate the method using data on thermal sensitivity of growth rate in Pieris rapae caterpillars. The TMV model explains 67% of the variation in thermal performance curves among families; generalist-specialist trade-offs account for 38% of the total between-family variation. The TMV method implemented here is applicable to both differences in mean and patterns of variation, and it can be used with either phenotypic or quantitative genetic data for thermal performance curves or other continuous reaction norms that have a template shape with a single maximum. The TMV approach may also apply to growth trajectories, age-specific life-history traits, and other function-valued traits.     

A Bayesian Nonparametric Approach for Mapping Dynamic Quantitative Traits

In biology, many quantitative traits are dynamic in nature. They can often be described by some smooth functions or curves. A joint analysis of all the repeated measurements of the dynamic traits by functional quantitative trait loci (QTL) mapping methods has the benefits to (1) understand the genetic control of the whole dynamic process of the quantitative traits and (2) improve the statistical power to detect QTL. One crucial issue in functional QTL mapping is how to correctly describe the smoothness of trajectories of functional valued traits. We develop an efficient Bayesian nonparametric multiple-loci procedure for mapping dynamic traits. The method uses the Bayesian P-splines with (nonparametric) B-spline bases to specify the functional form of a QTL trajectory and a random walk prior to automatically determine its degree of smoothness. An efficient deterministic variational Bayes algorithm is used to implement both (1) the search of an optimal subset of QTL among large marker panels and (2) estimation of the genetic effects of the selected QTL changing over time. Our method can be fast even on some large-scale data sets. The advantages of our method are illustrated on both simulated and real data sets.     

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 Population Genetic Signal of Polygenic Adaptation

Adaptation in response to selection on polygenic phenotypes may occur via subtle allele frequencies shifts at many loci. Current population genomic techniques are not well posed to identify such signals. In the past decade, detailed knowledge about the specific loci underlying polygenic traits has begun to emerge from genome-wide association studies (GWAS). Here we combine this knowledge from GWAS with robust population genetic modeling to identify traits that may have been influenced by local adaptation. We exploit the fact that GWAS provide an estimate of the additive effect size of many loci to estimate the mean additive genetic value for a given phenotype across many populations as simple weighted sums of allele frequencies. We use a general model of neutral genetic value drift for an arbitrary number of populations with an arbitrary relatedness structure. Based on this model, we develop methods for detecting unusually strong correlations between genetic values and specific environmental variables, as well as a generalization of comparisons to test for over-dispersion of genetic values among populations. Finally we lay out a framework to identify the individual populations or groups of populations that contribute to the signal of overdispersion. These tests have considerably greater power than their single locus equivalents due to the fact that they look for positive covariance between like effect alleles, and also significantly outperform methods that do not account for population structure. We apply our tests to the Human Genome Diversity Panel (HGDP) dataset using GWAS data for height, skin pigmentation, type 2 diabetes, body mass index, and two inflammatory bowel disease datasets. This analysis uncovers a number of putative signals of local adaptation, and we discuss the biological interpretation and caveats of these results.     

Patterns of phenotypic covariation and correlation in modern humans as viewed from morphological integration

Proportionality of phenotypic and genetic distance is of crucial importance to adequately focus on population history and structure, and it depends on the proportionality of genetic and phenotypic covariance. Constancy of phenotypic covariances is unlikely without constancy of genetic covariation if the latter is a substantial component of the former. If phenotypic patterns are found to be relatively stable, the most probable explanation is that genetic covariance matrices are also stable. Factors like morphological integration account for such stability. Morphological integration can be studied by analyzing the relationships among morphological traits. We present here a comparison of phenotypic correlation and covariance structure among worldwide human populations. Correlation and covariance matrices between 47 cranial traits were obtained for 28 populations, and compared with design matrices representing functional and developmental constraints. Among-population differences in patterns of correlation and covariation were tested for association with matrices of genetic distances (obtained after an examination of 10 Alu-insertions) and with Mahalanobis distances (computed after craniometrical traits). All matrix correlations were estimated by means of Mantel tests. Results indicate that correlation and covariance structure in our species is stable, and that among-group correlation/covariance similarity is not related to genetic or phenotypic distance. Conversely, genetic and morphological distance matrices were highly correlated. Correlation and covariation patterns were largely associated with functional and developmental factors, which probably account for the stability of covariance patterns.     

A penalized spline approach to functional mixed effects model analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.     

Quantitative genetics of continuous reaction norms: thermal sensitivity of caterpillar growth rates

A continuous reaction norm or performance curve represents a phenotypic trait of an individual or genotype in which the trait value may vary with some continuous environmental variable. We explore patterns of genetic variation in thermal performance curves of short-term caterpillar growth rate in a population of Pieris rapae. We compare multivariate methods, which treat performance at each test temperature as a distinct trait, with function-valued methods that treat a performance curve as a continuous function. Mean growth rate increased with increasing temperatures from 8 to 35 degrees C, was highest at 35 degrees C, and declined at 40 degrees C. There was substantial and significant variation among full-sib families in their thermal performance curves. Estimates of broad-sense genetic variances and covariances showed that genetic variance in growth rate increased more than 30-fold from low (8-11 degrees C) to high (35-40 degrees C) temperatures, even after differences in mean growth rate across temperatures were removed. Growth rate at 35 and 40 degrees C was negatively correlated genetically, suggesting a genetic trade-off in growth rate at these temperatures; this trade-off may represent either a generalist-specialist trade-off and/or variation in the optimal temperature for growth. The estimated genetic variance-covariance function (G function), the function-valued analog of the variance-covariance matrix (G matrix), was quite bumpy compared with the estimated G matrix; and results of principal component analyses of the G function were difficult to interpret. The use of orthogonal polynomials as the basis functions in current function-valued estimation methods may generate artifacts when the true G function has prominent local features, such as strong negative covariances at nearby temperatures (e.g. at 35 and 40 degrees C); this may be a particular issue for thermal performance curves and other highly nonlinear reaction norms.     

Functional mapping of quantitative trait loci underlying the character process: a theoretical framework

Unlike a character measured at a finite set of landmark points, function-valued traits are those that change as a function of some independent and continuous variable. These traits, also called infinite-dimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termed functional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponential growth curve (e.g., logistic or S-shaped). A maximum-likelihood approach based on a logistic-mixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.     

A general model of functional constraints on phenotypic evolution

A general model of the functional constraints on the rate and direction of phenotypic evolution is developed using a decomposition of the Lande-Arnold model of multivariate phenotypic evolution. The important feature of the model is the F matrix of performance coefficients reflecting the causal relationship between morphophysiological (m-p) and functional performance traits. The structure of F, which reflects the functional architecture of the organism, constrains the shape of the adaptive landscape and thus the rate and direction of m-p trait evolution. The rate of m-p trait evolution is a function of the pattern of coefficients in a row of F. The sums and variances of these rows are related to current concepts of evolvability. The direction of m-p trait evolution through m-p trait space is a function of the functional covariances among m-p traits. The functional covariance between a pair of m-p traits is a measure of how much the traits function together and is computed as the covariance between rows of F. Finally, it is shown that genetic covariances between m-p traits and performance traits are a function of the F matrix, but a G matrix that includes these covariances cannot be used to model functional constraints effectively.     

Genetic analysis of growth curves using the SAEM algorithm

The analysis of nonlinear function-valued characters is very important in genetic studies, especially for growth traits of agricultural and laboratory species. Inference in nonlinear mixed effects models is, however, quite complex and is usually based on likelihood approximations or Bayesian methods. The aim of this paper was to present an efficient stochastic EM procedure, namely the SAEM algorithm, which is much faster to converge than the classical Monte Carlo EM algorithm and Bayesian estimation procedures, does not require specification of prior distributions and is quite robust to the choice of starting values. The key idea is to recycle the simulated values from one iteration to the next in the EM algorithm, which considerably accelerates the convergence. A simulation study is presented which confirms the advantages of this estimation procedure in the case of a genetic analysis. The SAEM algorithm was applied to real data sets on growth measurements in beef cattle and in chickens. The proposed estimation procedure, as the classical Monte Carlo EM algorithm, provides significance tests on the parameters and likelihood based model comparison criteria to compare the nonlinear models with other longitudinal methods.     

An algorithm for molecular dissection of tumor progression

The volumetric growth of tumor cells as a function of time is most often likely to be a complex trait, controlled by the combined influences of multiple genes and environmental influences. Genetic mapping has proven to be a powerful tool for detecting and identifying specific genes affecting complex traits, i.e., quantitative trait loci (QTL), based on polymorphic markers. In this article, we present a novel statistical model for genetic mapping of QTL governing tumor growth trajectories in humans. In principle, this model is a combination of functional mapping proposed to map function-valued traits and linkage disequilibrium mapping designed to provide high resolution mapping of QTL by making use of recombination events created at a historic time. We implement an EM-simplex hybrid algorithm for parameter estimation, in which a closed-form solution for the EM algorithm is derived to estimate the population genetic parameters of QTL including the allele frequencies and the coefficient of linkage disequilibrium, and the simplex algorithm incorporated to estimate the curve parameters describing the dynamic changes of cancer cells for different QTL genotypes. Extensive simulations are performed to investigate the statistical properties of our model. Through a number of hypothesis tests, our model allows for cutting-edge studies aimed to decipher the genetic mechanisms underlying cancer growth, development and differentiation. The implications of our model in gene therapy for cancer research are discussed.     

A mathematical framework for protein structure comparison

Comparison of protein structures is important for revealing the evolutionary relationship among proteins, predicting protein functions and predicting protein structures. Many methods have been developed in the past to align two or multiple protein structures. Despite the importance of this problem, rigorous mathematical or statistical frameworks have seldom been pursued for general protein structure comparison. One notable issue in this field is that with many different distances used to measure the similarity between protein structures, none of them are proper distances when protein structures of different sequences are compared. Statistical approaches based on those non-proper distances or similarity scores as random variables are thus not mathematically rigorous. In this work, we develop a mathematical framework for protein structure comparison by treating protein structures as three-dimensional curves. Using an elastic Riemannian metric on spaces of curves, geodesic distance, a proper distance on spaces of curves, can be computed for any two protein structures. In this framework, protein structures can be treated as random variables on the shape manifold, and means and covariance can be computed for populations of protein structures. Furthermore, these moments can be used to build Gaussian-type probability distributions of protein structures for use in hypothesis testing. The covariance of a population of protein structures can reveal the population-specific variations and be helpful in improving structure classification. With curves representing protein structures, the matching is performed using elastic shape analysis of curves, which can effectively model conformational changes and insertions/deletions. We show that our method performs comparably with commonly used methods in protein structure classification on a large manually annotated data set.