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.     

Wavelet-based parametric functional mapping of developmental trajectories with high-dimensional data

The biological and statistical advantages of functional mapping result from joint modeling of the mean-covariance structures for developmental trajectories of a complex trait measured at a series of time points. While an increased number of time points can better describe the dynamic pattern of trait development, significant difficulties in performing functional mapping arise from prohibitive computational times required as well as from modeling the structure of a high-dimensional covariance matrix. In this article, we develop a statistical model for functional mapping of quantitative trait loci (QTL) that govern the developmental process of a quantitative trait on the basis of wavelet dimension reduction. By breaking an original signal down into a spectrum by taking its averages (smooth coefficients) and differences (detail coefficients), we used the discrete Haar wavelet shrinkage technique to transform an inherently high-dimensional biological problem into its tractable low-dimensional representation within the framework of functional mapping constructed by a Gaussian mixture model. Unlike conventional nonparametric modeling of wavelet shrinkage, we incorporate mathematical aspects of developmental trajectories into the smooth coefficients used for QTL mapping, thus preserving the biological relevance of functional mapping in formulating a number of hypothesis tests at the interplay between gene actions/interactions and developmental patterns for complex phenotypes. This wavelet-based parametric functional mapping has been statistically examined and compared with full-dimensional functional mapping through simulation studies. It holds great promise as a powerful statistical tool to unravel the genetic machinery of developmental trajectories with large-scale high-dimensional data.     

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.     

A likelihood approach for mapping growth trajectories using dominant markers in a phase-unknown full-sib family

Dominant markers have been commonly used in mapping quantitative trait loci (QTLs) in outcrossing species, in which not much prior genome information is available. But the dominant nature of these markers may lead to reduced QTL mapping precision and power. A new statistical method is proposed to incorporate growth laws into a QTL mapping framework, under which the use of the efficiency of dominant markers can be increased. This new method can be used to identify specific QTLs affecting differentiation in growth trajectories, and further estimate the timing of a QTL to turn on, or turn off, affecting growth during the entire ontogeny of a species. Using this method based on dominant markers we have successfully mapped a QTL for stem height growth trajectories to a linkage group in a forest tree. The implications of this method for the understanding of the genetic architecture of growth using dominant markers are discussed.Communicated by F. Salamini     

A general framework for analyzing the genetic architecture of developmental characteristics

The genetic architecture of growth traits plays a central role in shaping the growth, development, and evolution of organisms. While a limited number of models have been devised to estimate genetic effects on complex phenotypes, no model has been available to examine how gene actions and interactions alter the ontogenetic development of an organism and transform the altered ontogeny into descendants. In this article, we present a novel statistical model for mapping quantitative trait loci (QTL) determining the developmental process of complex traits. Our model is constructed within the traditional maximum-likelihood framework implemented with the EM algorithm. We employ biologically meaningful growth curve equations to model time-specific expected genetic values and the AR(1) model to structure the residual variance-covariance matrix among different time points. Because of a reduced number of parameters being estimated and the incorporation of biological principles, the new model displays increased statistical power to detect QTL exerting an effect on the shape of ontogenetic growth and development. The model allows for the tests of a number of biological hypotheses regarding the role of epistasis in determining biological growth, form, and shape and for the resolution of developmental problems at the interface with evolution. Using our newly developed model, we have successfully detected significant additive x additive epistatic effects on stem height growth trajectories in a forest tree.     

Quantitative trait loci for growth trajectories in Populus

Growth trajectories are a biological process important to plant and animal breeding, and to evolutionary genetic studies. In this article, we report the detection of quantitative trait loci (QTLs) responsible for growth trajectories in poplars that are used as a model system for the study of forest biology. These QTLs were localized on a genetic linkage map of polymorphic markers using a statistical mapping method incorporating growth-curve models. The effects of the QTLs on growth are described as a function of age, so that age-specific changes in QTL effects can be readily projected throughout the entire growth process. The QTLs identified display increased effects on growth when trees age, yet the timing of QTL activation is earlier for stem height than diameter, which is consistent with the ecological viewpoint of canopy competition. The implications of the results for breeding and silviculture are discussed.     

A unified statistical model for functional mapping of environment-dependent genetic expression and genotype x environment interactions for ontogenetic development

The effects of quantitative trait loci (QTL) on phenotypic development may depend on the environment (QTL x environment interaction), other QTL (genetic epistasis), or both. In this article, we present a new statistical model for characterizing specific QTL that display environment-dependent genetic expressions and genotype x environment interactions for developmental trajectories. Our model was derived within the maximum-likelihood-based mixture model framework, incorporated by biologically meaningful growth equations and environment-dependent genetic effects of QTL, and implemented with the EM algorithm. With this model, we can characterize the dynamic patterns of genetic effects of QTL governing growth curves and estimate the global effect of the underlying QTL during the course of growth and development. In a real example with rice, our model has successfully detected several QTL that produce differences in their genetic expression between two contrasting environments. These detected QTL cause significant genotype x environment interactions for some fundamental aspects of growth trajectories. The model provides the basis for deciphering the genetic architecture of trait expression adjusted to different biotic and abiotic environments and genetic relationships for growth rates and the timing of life-history events for any organism.     

A statistical model for the genetic origin of allometric scaling laws in biology

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling (power-law) relationships between size and rate. Although such allometric relationships may be under genetic determination, their precise genetic mechanisms have not been clearly understood due to a lack of a statistical analytical method. In this paper, we present a basic statistical framework for mapping quantitative genes (or quantitative trait loci, QTL) responsible for universal quarter-power scaling laws of organic structure and function with the entire body size. Our model framework allows the testing of whether a single QTL affects the allometric relationship of two traits or whether more than one linked QTL is segregating. Like traditional multi-trait mapping, this new model can increase the power to detect the underlying QTL and the precision of its localization on the genome. Beyond the traditional method, this model is integrated with pervasive scaling laws to take advantage of the mechanistic relationships of biological structures and processes. Simulation studies indicate that the estimation precision of the QTL position and effect can be improved when the scaling relationship of the two traits is considered. The application of our model in a real example from forest trees leads to successful detection of a QTL governing the allometric relationship of third-year stem height with third-year stem biomass. The model proposed here has implications for genetic, evolutionary, biomedicinal and breeding research.     

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.     

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.     

Semiparametric functional mapping of quantitative trait loci governing long-term HIV dynamics

MOTIVATION: Functional mapping has proven to be powerful for characterizing quantitative trait loci (QTL) that control complex dynamic traits. More recently, functional mapping has been extended to identify the host QTL responsible for HIV dynamics by incorporating a parametric bi-exponential function for earlier stages of viral load trajectories. However, existing functional mapping cannot be used to map long-term HIV dynamics because no mathematical functions are available for later stages of HIV dynamic changes. RESULTS: We derived a statistical model for functional mapping of dynamic QTL through characterizing HIV load trajectories during a long-term period semiparametrically. The new model was constructed within the maximum likelihood framework and implemented with the EM-simplex algorithm. It allows for the test of differences in the genetic control of short- and long-term HIV dynamics and the characterization of the effects of viral-host genome interaction. Extensive simulation studies have been performed to test the statistical behavior of this model. The new model will provide an important tool for genetic and genomic studies of human complex diseases like HIV/AIDS and their pathological progression. AVAILABILITY: Available on request from the corresponding author.     

A general statistical framework for mapping quantitative trait loci in nonmodel systems: issue for characterizing linkage phases

Because of uncertainty about linkage phases of founders, linkage mapping in nonmodel, outcrossing systems using molecular markers presents one of the major statistical challenges in genetic research. In this article, we devise a statistical method for mapping QTL affecting a complex trait by incorporating all possible QTL-marker linkage phases within a mapping framework. The advantage of this model is the simultaneous estimation of linkage phases and QTL location and effect parameters. These estimates are obtained through maximum-likelihood methods implemented with the EM algorithm. Extensive simulation studies are performed to investigate the statistical properties of our model. In a case study from a forest tree, this model has successfully identified a significant QTL affecting wood density. Also, the probability of the linkage phase between this QTL and its flanking markers is estimated. The implications of our model and its extension to more general circumstances are discussed.     

A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure

A mixture model for determining quantitative trait loci (QTL) affecting growth trajectories has been proposed in the literature. In this article, we extend this model to a more general situation in which longitudinal traits for each subject are measured at unequally spaced time intervals, different subjects have different measurement patterns, and the residual correlation within subjects is nonstationary. We derive an EM-simplex hybrid algorithm to estimate the allele frequencies, Hardy-Weinberg disequilibrium, and linkage disequilibrium between QTL in the original population and parameters contained in the growth equation and in the covariance structure. A worked example of head circumference growth in 145 children is used to validate our extended model. A simulation study is performed to examine the statistical properties of the parameter estimation obtained from this example. Finally, we discuss the implications and extensions of our model for detecting QTL that affect growth trajectories.     

Functional mapping of dynamic traits with robust t-distribution

Functional mapping has been a powerful tool in mapping quantitative trait loci (QTL) underlying dynamic traits of agricultural or biomedical interest. In functional mapping, multivariate normality is often assumed for the underlying data distribution, partially due to the ease of parameter estimation. The normality assumption however could be easily violated in real applications due to various reasons such as heavy tails or extreme observations. Departure from normality has negative effect on testing power and inference for QTL identification. In this work, we relax the normality assumption and propose a robust multivariate t-distribution mapping framework for QTL identification in functional mapping. Simulation studies show increased mapping power and precision with the t distribution than that of a normal distribution. The utility of the method is demonstrated through a real data analysis.     

Functional mapping of growth and development

Understanding how an organism develops into a fully functioning adult from a mass of undifferentiated cells may reveal different strategies that allow the organism to survive under limiting conditions. Here, we review an analytical model for characterizing quantitative trait loci (QTLs) that underlie variation in growth trajectories and developmental timing. This model, called functional mapping, incorporates fundamental principles behind biological processes or networks that are bridged with mathematical functions into a statistical mapping framework. Functional mapping estimates parameters that determine the shape and function of a particular biological process, thus providing a flexible platform to test biologically meaningful hypotheses regarding the complex relationships between gene action and development.     

QTL linkage analysis of connected populations using ancestral marker and pedigree information

The common assumption in quantitative trait locus (QTL) linkage mapping studies that parents of multiple connected populations are unrelated is unrealistic for many plant breeding programs. We remove this assumption and propose a Bayesian approach that clusters the alleles of the parents of the current mapping populations from locus-specific identity by descent (IBD) matrices that capture ancestral marker and pedigree information. Moreover, we demonstrate how the parental IBD data can be incorporated into a QTL linkage analysis framework by using two approaches: a Threshold IBD model (TIBD) and a Latent Ancestral Allele Model (LAAM). The TIBD and LAAM models are empirically tested via numerical simulation based on the structure of a commercial maize breeding program. The simulations included a pilot dataset with closely linked QTL on a single linkage group and 100 replicated datasets with five linkage groups harboring four unlinked QTL. The simulation results show that including parental IBD data (similarly for TIBD and LAAM) significantly improves the power and particularly accuracy of QTL mapping, e.g., position, effect size and individuals’ genotype probability without significantly increasing computational demand.     

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.     

A semiparametric approach for composite functional mapping of dynamic quantitative traits

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.     

Mapping temporally varying quantitative trait loci in time-to-failure experiments

Existing methods for mapping quantitative trait loci (QTL) in time-to-failure experiments assume that the QTL effect is constant over the course of the study. This assumption may be violated when the gene(s) underlying the QTL are up- or downregulated on a biologically meaningful timescale. In such situations, models that assume a constant effect can fail to detect QTL in a whole-genome scan. To investigate this possibility, we utilize an extension of the Cox model (EC model) within an interval-mapping framework. In its simplest form, this model assumes that the QTL effect changes at some time point t0 and follows a linear function before and after this change point. The approximate time point at which this change occurs is estimated. Using simulated and real data, we compare the mapping performance of the EC model to the Cox proportional hazards (CPH) model, which explicitly assumes a constant effect. The results show that the EC model detects time-dependent QTL, which the CPH model fails to detect. At the same time, the EC model recovers all of the QTL the CPH model detects. We conclude that potentially important QTL may be missed if their time-dependent effects are not accounted for.     

Functional mapping of quantitative trait loci that interact with the hg mutation to regulate growth trajectories in mice

The high growth (hg) mutation increases body size in mice by 30-50%. Given the complexity of the genetic regulation of animal growth, it is likely that the effect of this major locus is mediated by other quantitative trait loci (QTL) with smaller effects within a web of gene interactions. In this article, we extend our functional mapping model to characterize modifier QTL that interact with the hg locus during ontogenetic growth. Our model is derived within the maximum-likelihood context, incorporated by mathematical aspects of growth laws and implemented with the EM algorithm. In an F2 population founded by a congenic high growth (HG) line and non-HG line, a highly additive effect due to the hg gene was detected on growth trajectories. Three QTL located on chromosomes 2 and X were identified to trigger significant additive and/or dominant effects on the process of growth. The most significant finding made from our model is that these QTL interact with the hg locus to affect the shapes of the growth process. Our model provides a powerful means for understanding the genetic architecture and regulation of growth rate and body size in mammals.