Functional Mapping of Quantitative Trait Loci Underlying Growth Trajectories Using a Transform-Both-Sides Logistic Model

Genome-wide association study of multiyear dynamic growth traits in hybrid Liriodendron identifies robust genetic loci associated with growth trajectories

The genetic factors underlying growth traits differ over time points or stages. However, most current studies of phenotypes at single time points do not capture all loci or explain the genetic differences underlying growth trajectories. Hybrid Liriodendron exhibits obvious heterosis and is widely cultivated, although its complex genetic mechanism underlying growth traits remains unknown. A genome-wide association study (GWAS) is an effective method for elucidating the genetic architecture by identifying genetic loci underlying complex quantitative traits. In the present study, using a GWAS, we identified robust loci associated with growth trajectories in hybrid Liriodendron populations. We selected 233 hybrid progenies derived from 25 crosses for resequencing, and measured their tree height (H) and diameter at breast height (DBH) for 11 consecutive years; 192 972 high-quality single nucleotide polymorphisms (SNPs) were obtained. The dynamics of the multiyear single-trait GWAS showed that year-specific SNPs predominated, and only five robust SNPs for DBH were identified in at least three different years. Multitrait GWAS analysis with model parameters as latent variables also revealed 62 SNPs for H and 52 for DBH associated with the growth trajectory, displaying different biomass accumulation patterns, among which four SNPs exerted pleiotropic effects. All identified SNPs also exhibited temporal variations in effect sizes and inheritance patterns potentially related to different growth and developmental stages. The haplotypes resulting from these significant SNPs might pyramid favorable loci, benefitting the selection of superior genotypes. The present study provides insights into the genetic architecture of dynamic growth traits and lays a basis for future molecular-assisted breeding.

Dynamic analysis of QTLs on plant height with single segment substitution lines in rice

Dynamic regulation of QTLs remains mysterious. Single segment substitution lines (SSSLs) and conditional QTL mapping and functional QTL mappings are ideal materials and methods to explore dynamics of QTLs for complex traits. This paper analyzed the dynamics of QTLs on plant height with SSSLs in rice. Five SSSLs were verified with plant height QTLs first. All five QTLs had significant positive effects at one or more developmental stages except QTL₁. They interacted each other, with negative effects before 49 d after transplanting and positive effects since then. The five QTLs selectively expressed in specific periods, mainly in the periods from 35 to 42 d and from 49 to 56 d after transplanting. Expressions of epistasis were dispersedly in various periods, negative effects appearing mainly before 35 d. The five QTLs brought the inflexion point ahead of schedule, accelerated growth and degradation, and changed the peak plant height, while their interactions had the opposite effects. The information will be helpful to understand the genetic mechanism for developmental traits.

Functional mapping of tillering QTLs using the Wang-Lan-Ding model and a SSSL population

Understanding dynamic changes in the genetic architecture of quantitative traits is crucial in developmental genetics. Functional mapping is an appropriate method that passes a mathematical equation to describe a biological developmental process with the genetic mapping framework. Appropriate genetic model and applicable mapping population are indispensable condition for functional mapping of important agronomic traits in plants. Based on the Wang-Lan-Ding model, we ever applied a DH population to carry out functional mapping QTLs underlying growth trajectory on tiller number. However, inconsistent genetic background among the DH lines might disturb the mapping results. With the advent of innovative research materials, single segment substitution lines, allows us to do more precise genetic analyses. Thus functional mapping was again conducted on tiller number using the Wang-Lan-Ding model and a single segment substitution line population with the genetic background of Huajingxian 74 so as to explore QTL dynamic mechanism to regulate developmental traits. We detected that all five single segment substitution lines harbored tillering QTLs with additives and/or dominances to influence the four functional parameters, the optimum tillering time (t₀), the maximum tiller number (K), the tillering increased rate (r) and the tillering decreased rate (c), which were estimated from the Wang-Lan-Ding model and with some biological meaning. They mainly brought the inflexion point (t₀) delay, the peak increase (K) and the degradation (c) acceleration, while the growth (r) slow down. Moreover, epistatic interactions among these QTLs were confirmed to be prevalent. A total of 39 significant epistatic effects were detected to associate with the four parameters, occupying 34.8% of 112 pairs of epistatic interactions investigated. Contrary to the QTL effects, these epistatic effects largely decreased t₀, K and c, while increased r. Our results indicated that the five QTL effects and their epistatic effects significantly changed the shape and trajectory of tiller number via influence of the four functional parameters. Rational use of these QTLs is expected to improve tillering number of rice by molecular design breeding.

Integrating High-Throughput Phenotyping and Statistical Genomic Methods to Genetically Improve Longitudinal Traits in Crops

The rapid development of remote sensing in agronomic research allows the dynamic nature of longitudinal traits to be adequately described, which may enhance the genetic improvement of crop efficiency. For traits such as light interception, biomass accumulation, and responses to stressors, the data generated by the various high-throughput phenotyping (HTP) methods requires adequate statistical techniques to evaluate phenotypic records throughout time. As a consequence, information about plant functioning and activation of genes, as well as the interaction of gene networks at different stages of plant development and in response to environmental stimulus can be exploited. In this review, we outline the current analytical approaches in quantitative genetics that are applied to longitudinal traits in crops throughout development, describe the advantages and pitfalls of each approach, and indicate future research directions and opportunities.

np2 QTL: networking phenotypic plasticity quantitative trait loci across heterogeneous environments

Despite its critical importance to our understanding of plant growth and adaptation, the question of how environment-induced plastic response is affected genetically remains elusive. Previous studies have shown that the reaction norm of an organism across environmental index obeys the allometrical scaling law of part-whole relationships. The implementation of this phenomenon into functional mapping can characterize how quantitative trait loci (QTLs) modulate the phenotypic plasticity of complex traits to heterogeneous environments. Here, we assemble functional mapping and allometry theory through Lokta-Volterra ordinary differential equations (LVODE) into an R-based computing platform, np² QTL, aimed to map and visualize phenotypic plasticity QTLs. Based on LVODE parameters, np² QTL constructs a bidirectional, signed and weighted network of QTL-QTL epistasis, whose emergent properties reflect the ecological mechanisms for genotype-environment interactions over any range of environmental change. The utility of np² QTL was validated by comprehending the genetic architecture of phenotypic plasticity via the reanalysis of published plant height data involving 3502 recombinant inbred lines of maize planted in multiple discrete environments. np² QTL also provides a tool for constructing a predictive model of phenotypic responses in extreme environments relative to the median environment.

Evo-Devo-EpiR: a genome-wide search platform for epistatic control on the evolution of development

Evo-devo is a theory proposed to study how phenotypes evolve by comparing the developmental processes of different organisms or the same organism experiencing changing environments. It has been recognized that nonallelic interactions at different genes or quantitative trait loci, known as epistasis, may play a pivotal role in the evolution of development, but it has proven difficult to quantify and elucidate this role into a coherent picture. We implement a high-dimensional genome-wide association study model into the evo-devo paradigm and pack it into the R-based Evo-Devo-EpiR, aimed at facilitating the genome-wide landscaping of epistasis for the diversification of phenotypic development. By analyzing a high-throughput assay of DNA markers and their pairs simultaneously, Evo-Devo-EpiR is equipped with a capacity to systematically characterize various epistatic interactions that impact on the pattern and timing of development and its evolution. Enabling a global search for all possible genetic interactions for developmental processes throughout the whole genome, Evo-Devo-EpiR provides a computational tool to illustrate a precise genotype-phenotype map at interface between epistasis, development and evolution.

E-Index for Differentiating Complex Dynamic Traits

While it is a daunting challenge in current biology to understand how the underlying network of genes regulates complex dynamic traits, functional mapping, a tool for mapping quantitative trait loci (QTLs) and single nucleotide polymorphisms (SNPs), has been applied in a variety of cases to tackle this challenge. Though useful and powerful, functional mapping performs well only when one or more model parameters are clearly responsible for the developmental trajectory, typically being a logistic curve. Moreover, it does not work when the curves are more complex than that, especially when they are not monotonic. To overcome this inadaptability, we therefore propose a mathematical-biological concept and measurement, E-index (earliness-index), which cumulatively measures the earliness degree to which a variable (or a dynamic trait) increases or decreases its value. Theoretical proofs and simulation studies show that E-index is more general than functional mapping and can be applied to any complex dynamic traits, including those with logistic curves and those with nonmonotonic curves. Meanwhile, E-index vector is proposed as well to capture more subtle differences of developmental patterns.

BAYESIAN GROUP LASSO FOR NONPARAMETRIC VARYING-COEFFICIENT MODELS WITH APPLICATION TO FUNCTIONAL GENOME-WIDE ASSOCIATION STUDIES

Although genome-wide association studies (GWAS) have proven powerful for comprehending the genetic architecture of complex traits, they are challenged by a high dimension of single-nucleotide polymorphisms (SNPs) as predictors, the presence of complex environmental factors, and longitudinal or functional natures of many complex traits or diseases. To address these challenges, we propose a high-dimensional varying-coefficient model for incorporating functional aspects of phenotypic traits into GWAS to formulate a so-called functional GWAS or fGWAS. Bayesian group lasso and the associated MCMC algorithms are developed to identify significant SNPs and estimate how they affect longitudinal traits through time-varying genetic actions. The model is generalized to analyze the genetic control of complex traits using subject-specific sparse longitudinal data. The statistical properties of the new model are investigated through simulation studies. We use the new model to analyze a real GWAS data set from the Framingham Heart Study, leading to the identification of several significant SNPs associated with age-specific changes of body mass index. The fGWAS model, equipped with Bayesian group lassso, will provide a useful tool for genetic and developmental analysis of complex traits or diseases.

An allometric model for mapping seed development in plants

Despite a tremendous effort to map quantitative trait loci (QTLs) responsible for agriculturally and biologically important traits in plants, our understanding of how a QTL governs the developmental process of plant seeds remains elusive. In this article, we address this issue by describing a model for functional mapping of seed development through the incorporation of the relationship between vegetative and reproductive growth. The time difference of reproductive from vegetative growth is described by Reeve and Huxley’s allometric equation. Thus, the implementation of this equation into the framework of functional mapping allows dynamic QTLs for seed development to be identified more precisely. By estimating and testing mathematical parameters that define Reeve and Huxley’s allometric equations of seed growth, the dynamic pattern of the genetic effects of the QTLs identified can be analyzed. We used the model to analyze a soybean data, leading to the detection of QTLs that control the growth of seed dry weight. Three dynamic QTLs, located in two different linkage groups, were detected to affect growth curves of seed dry weight. The QTLs detected may be used to improve seed yield with marker-assisted selection by altering the pattern of seed development in a hope to achieve a maximum size of seeds at a harvest time.

Dynamic quantitative trait locus analysis of seed vigor at three maturity stages in rice

Seed vigor is an important characteristic of seed quality. In this study, one rice population of recombinant inbred lines (RILs) was used to determine the genetic characteristics of seed vigor, including the germination potential, germination rate, germination index and time for 50% of germination, at 4 (early), 5 (middle) and 6 weeks (late) after heading in two years. A total of 24 additive and 9 epistatic quantitative trait loci (QTL) for seed vigor were identified using QTL Cartographer and QTLNetwork program respectively in 2012; while 32 simple sequence repeat (SSR) markers associated with seed vigor were detected using bulked segregant analysis (BSA) in 2013. The additive, epistatic and QTL × development interaction effects regulated the dry maturity developmental process to improve seed vigor in rice. The phenotypic variation explained by each additive, epistatic QTL and QTL × development interaction ranged from 5.86 to 40.67%, 4.64 to 11.28% and 0.01 to 1.17%, respectively. The QTLs were rarely co-localized among the different maturity stages; more QTLs were expressed at the early maturity stage followed by the late and middle stages. Twenty additive QTLs were stably expressed in two years which might play important roles in establishment of seed vigor in different environments. By comparing chromosomal positions of these stably expressed additive QTLs with those previously identified, the regions of QTL for seed vigor are likely to coincide with QTL for grain size, low temperature germinability and seed dormancy; while 5 additive QTL might represent novel genes. Using four selected RILs, three cross combinations of seed vigor for the development of RIL populations were predicted; 19 elite alleles could be pyramided by each combination.

Statistical resolution of missing longitudinal data in clinical pharmacogenomics

Clinical pharmacogenomics, integrating genomic information with clinical practices to facilitate the prediction of drug response, has recently emerged as a vital area of public health. In clinical trials, phenotypic data on drug response are often longitudinal, with some patients dropping out early due to physiological or other unpredictable reasons. The genetic analysis of such missing longitudinal data presents a significant challenge in clinical pharmacogenomics. We develop a statistical algorithm for detecting haplotypes that control longitudinal responses subject to non-ignorable dropout. The model was derived by incorporating the selection model into a dynamic model - functional mapping, aimed to discover genetic variants that contribute to phenotypic variation in longitudinal traits. The selection models is a statistical approach for analyzing missing longitudinal data by assuming that dropout depends on the outcome of drug response. The model derived can jointly characterize the genetic control of longitudinal responses and dropout events. Simulation studies were performed to investigate the statistical properties of the model and validate its practical usefulness. The model will find its implications for clinical pharmacogenomics toward personalized medicine.

A novel method for analyzing genetic association with longitudinal phenotypes

Knowledge of genes influencing longitudinal patterns may offer information about predicting disease progression. We developed a systematic procedure for testing association between SNP genotypes and longitudinal phenotypes. We evaluated false positive rates and statistical power to localize genes for disease progression. We used genome-wide SNP data from the Framingham Heart Study. With longitudinal data from two real studies unrelated to Framingham, we estimated three trajectory curves from each study. We performed simulations by randomly selecting 500 individuals. In each simulation replicate, we assigned each individual to one of the three trajectory groups based on the underlying hypothesis (null or alternative), and generated corresponding longitudinal data. Individual Bayesian posterior probabilities (BPPs) for belonging to a specific trajectory curve were estimated. These BPPs were treated as a quantitative trait and tested (using the Wald test) for genome-wide association. Empirical false positive rates and power were calculated. Our method maintained the expected false positive rate for all simulation models. Also, our method achieved high empirical power for most simulations. Our work presents a method for disease progression gene mapping. This method is potentially clinically significant as it may allow doctors to predict disease progression based on genotype and determine treatment accordingly.

Bayesian B-spline mapping for dynamic quantitative traits

Owing to their ability and flexibility to describe individual gene expression at different time points, random regression (RR) analyses have become a popular procedure for the genetic analysis of dynamic traits whose phenotypes are collected over time. Specifically, when modelling the dynamic patterns of gene expressions in the RR framework, B-splines have been proved successful as an alternative to orthogonal polynomials. In the so-called Bayesian B-spline quantitative trait locus (QTL) mapping, B-splines are used to characterize the patterns of QTL effects and individual-specific time-dependent environmental errors over time, and the Bayesian shrinkage estimation method is employed to estimate model parameters. Extensive simulations demonstrate that (1) in terms of statistical power, Bayesian B-spline mapping outperforms the interval mapping based on the maximum likelihood; (2) for the simulated dataset with complicated growth curve simulated by B-splines, Legendre polynomial-based Bayesian mapping is not capable of identifying the designed QTLs accurately, even when higher-order Legendre polynomials are considered and (3) for the simulated dataset using Legendre polynomials, the Bayesian B-spline mapping can find the same QTLs as those identified by Legendre polynomial analysis. All simulation results support the necessity and flexibility of B-spline in Bayesian mapping of dynamic traits. The proposed method is also applied to a real dataset, where QTLs controlling the growth trajectory of stem diameters in Populus are located.

Dynamic semiparametric Bayesian models for genetic mapping of complex trait with irregular longitudinal data

Many phenomena of fundamental importance to biology and biomedicine arise as a dynamic curve, such as organ growth and HIV dynamics. The genetic mapping of these traits is challenged by longitudinal variables measured at irregular and possibly subject-specific time points, in which case nonnegative definiteness of the estimated covariance matrix needs to be guaranteed. We present a semiparametric approach for genetic mapping within the mixture-model setting by jointly modeling mean and covariance structures for irregular longitudinal data. Penalized spline is used to model the mean functions of individual quantitative trait locus (QTL) genotypes as latent variables, whereas an extended generalized linear model is used to approximate the covariance matrix. The parameters for modeling the mean-covariances are estimated by MCMC, using the Gibbs sampler and the Metropolis-Hastings algorithm. We derive the full conditional distributions for the mean and covariance parameters and compute Bayes factors to test the hypothesis about the existence of significant QTLs. We used the model to screen the existence of specific QTLs for age-specific change of body mass index with a sparse longitudinal data set. The new model provides powerful means for broadening the application of genetic mapping to reveal the genetic control of dynamic traits.

A Bayesian Framework for Functional Mapping through Joint Modeling of Longitudinal and Time-to-Event Data

The most powerful and comprehensive approach of study in modern biology is to understand the whole process of development and all events of importance to development which occur in the process. As a consequence, joint modeling of developmental processes and events has become one of the most demanding tasks in statistical research. Here, we propose a joint modeling framework for functional mapping of specific quantitative trait loci (QTLs) which controls developmental processes and the timing of development and their causal correlation over time. The joint model contains two submodels, one for a developmental process, known as a longitudinal trait, and the other for a developmental event, known as the time to event, which are connected through a QTL mapping framework. A nonparametric approach is used to model the mean and covariance function of the longitudinal trait while the traditional Cox proportional hazard (PH) model is used to model the event time. The joint model is applied to map QTLs that control whole-plant vegetative biomass growth and time to first flower in soybeans. Results show that this model should be broadly useful for detecting genes controlling physiological and pathological processes and other events of interest in biomedicine.

Gene-environment interactions on growth trajectories

It has been suggested that children with larger brains tend to perform better on IQ tests or cognitive function tests. Prenatal head growth and head growth in infancy are two crucial periods for subsequent intelligence. Studies have shown that environmental exposure to air pollutants during pregnancy is associated with fetal growth reduction, developmental delay, and reduced IQ. Meanwhile, genetic polymorphisms may modify the effect of environment on head growth. However, studies on gene-environment or gene-gene interactions on growth trajectories have been quite limited partly due to the difficulty to quantitatively measure interactions on growth trajectories. Moreover, it is known that assessing the significance of gene-environment or gene-gene interactions on cross-sectional outcomes empirically using the permutation procedures may bring substantial errors in the tests. We proposed a score that quantitatively measures interactions on growth trajectories and developed an algorithm with a parametric bootstrap procedure to empirically assess the significance of the interactions on growth trajectories under the likelihood framework. We also derived a Wald statistic to test for interactions on growth trajectories and compared it to the proposed parametric bootstrap procedure. Through extensive simulation studies, we demonstrated the feasibility and power of the proposed testing procedures. We applied our method to a real dataset with head circumference measures from birth to age 7 on a cohort currently being conducted by the Columbia Center for Children's Environmental Health (CCCEH) in Krakow, Poland, and identified several significant gene-environment interactions on head circumference growth trajectories.

A mathematical framework for functional mapping of complex phenotypes using delay differential equations

All biological phenomena occurring at different levels of organization from cells to organisms can be modeled as a dynamic system, in which the underlying components interact dynamically to comprehend its biological function. Such a systems modeling approach facilitates the use of biochemically and biophysically detailed mathematical models to describe and quantify "living cells," leading to an in-depth and precise understanding of the behavior, development and function of a biological system. Here, we illustrate how this approach can be used to map genes or quantitative trait loci (QTLs) that control a complex trait using the example of the circadian rhythm system which has been at the forefront of analytical mathematical modeling for many years. We integrate a system of biologically meaningful delay differential equations (DDEs) into functional mapping, a statistical model designed to map dynamic QTLs involved in biological processes. The DDEs model the ability of circadian rhythm to generate autonomously sustained oscillations with a period close to 24h, in terms of time-varying mRNA and protein abundances. By incorporating the Runge-Kutta fourth order algorithm within the likelihood-based context of functional mapping, we estimated the genetic parameters that define the periodic pattern of QTL effects on time-varying mRNA and protein abundances and their dynamic association as well as the linkage disequilibrium of the QTL and a marker. We prove theorems about how to choose appropriate parameters to guarantee periodic oscillations. We further used simulation studies to investigate how a QTL influences the period and the amplitude of circadian oscillations through changing model parameters. The model provides a quantitative framework for assessing the interplay between genetic effects of QTLs and rhythmic responses.

Modeling the genetic etiology of pharmacokinetic-pharmacodynamic links with the ARMA process

Substantial variability exists among different patients in response to drugs. The identification of genetic factors that contribute to the interpersonal differentiation has been an important task for pharmacogenetic research and drug discovery. In this article, we have derived a high-dimensional statistical model for unveiling the genetic machinery for drug response by integrating two different but biologically related processes--pharmacokinetics (PK) and pharmacodynamics (PD)--into a genetic mapping framework. Using an integrated model of PK and PD, we can identify specific DNA sequence variants and test how they relate to the differential effect of the body to the drug (PK) and the effect of the drug on the body (PD). To effectively model a two-stage hierarchic structure of the covariance matrix at the PD and PK level, we have for the first time introduced an autoregressive moving-average (ARMA) process to the mixture-based likelihood function for sequence mapping. Closed-form estimates of the determinant and inverse of the ARMA-based covariance matrix are incorporated into the estimation step, which significantly increases the computational efficiency. Simulation studies have been performed to test the statistical behavior of our model. Potential applications of this model to pharmacogenetic research are discussed.

Functional mapping of genotype-environment interactions for soybean growth by a semiparametric approach

BACKGROUND

Functional mapping is a powerful approach for mapping quantitative trait loci (QTLs) that control biological processes. Functional mapping incorporates mathematical aspects of growth and development into a general QTL mapping framework and has been recently integrated with composite interval mapping to build up a so-called composite functional mapping model, aimed to separate multiple linked QTLs on the same chromosomal region.

RESULTS

This article reports the principle of using composite functional mapping to estimate the effects of QTL-environment interactions on growth trajectories by parametrically modeling the tested QTL in a marker interval and nonparametrically modeling the markers outside the interval as co-factors. With this new model, we can characterize the dynamic patterns of the genetic effects of QTLs governing growth trajectories, estimate the global effects of the underlying QTLs during the course of growth and development, and test the differentiation in the shapes of QTL genotype-specific growth curves between different environments. By analyzing a real example from a soybean genome project, our model detects several QTLs that cause significant genotype-environment interactions for plant height growth processes.

CONCLUSIONS

The model provides a basis for deciphering the genetic architecture of trait expression adjusted to different biotic and abiotic environments for any organism.

Functional mapping of drug response with pharmacodynamic-pharmacokinetic principles

Recent research in pharmacogenomics has inspired our hope to predict drug response by linking it with DNA information extracted from the human genome. However, many genetic models of drug response do not incorporate biochemical principles of host-drug interactions, limiting the effectiveness of the predictive models. We argue that functional mapping, a computational tool aimed at identifying genes and genetic networks that control dynamic traits, can help explain the detailed genetic architecture of drug response by incorporating pharmacokinetic and pharmacodynamic processes. Functional mapping is particularly powerful in determining the genetic commonality and differences of drug efficacy vs. drug toxicity and drug sensitivity vs. drug resistance. We pinpoint several future directions in which functional mapping can be coupled with systems biology to unravel the genetic and metabolic machinery of drug response.

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.

An optimal strategy for functional mapping of dynamic trait loci

As an emerging powerful approach for mapping quantitative trait loci (QTLs) responsible for dynamic traits, functional mapping models the time-dependent mean vector with biologically meaningful equations and are likely to generate biologically relevant and interpretable results. Given the autocorrelation nature of a dynamic trait, functional mapping needs the implementation of the models for the structure of the covariance matrix. In this article, we have provided a comprehensive set of approaches for modelling the covariance structure and incorporated each of these approaches into the framework of functional mapping. The Bayesian information criterion (BIC) values are used as a model selection criterion to choose the optimal combination of the submodels for the mean vector and covariance structure. In an example for leaf age growth from a rice molecular genetic project, the best submodel combination was found between the Gaussian model for the correlation structure, power equation of order 1 for the variance and the power curve for the mean vector. Under this combination, several significant QTLs for leaf age growth trajectories were detected on different chromosomes. Our model can be well used to study the genetic architecture of dynamic traits of agricultural values.

Nonparametric modeling of longitudinal covariance structure in functional mapping of quantitative trait loci

Estimation of the covariance structure of longitudinal processes is a fundamental prerequisite for the practical deployment of functional mapping designed to study the genetic regulation and network of quantitative variation in dynamic complex traits. We present a nonparametric approach for estimating the covariance structure of a quantitative trait measured repeatedly at a series of time points. Specifically, we adopt Huang et al.'s (2006, Biometrika 93, 85-98) approach of invoking the modified Cholesky decomposition and converting the problem into modeling a sequence of regressions of responses. A regularized covariance estimator is obtained using a normal penalized likelihood with an L(2) penalty. This approach, embedded within a mixture likelihood framework, leads to enhanced accuracy, precision, and flexibility of functional mapping while preserving its biological relevance. Simulation studies are performed to reveal the statistical properties and advantages of the proposed method. A real example from a mouse genome project is analyzed to illustrate the utilization of the methodology. The new method will provide a useful tool for genome-wide scanning for the existence and distribution of quantitative trait loci underlying a dynamic trait important to agriculture, biology, and health sciences.

A statistical model for genetic mapping of viral infection by integrating epidemiological behavior

Large-scale studies of genetic variation may be helpful for understanding the genetic control mechanisms of viral infection and, ultimately, predicting and eliminating infectious disease outbreaks. We propose a new statistical model for detecting specific DNA sequence variants that are responsible for viral infection. This model considers additive, dominance and epistatic effects of haplotypes from three different genomes, recipient, transmitter and virus, through an epidemiological process. The model is constructed within the maximum likelihood framework and implemented with the EM algorithm. A number of hypothesis tests about population genetic structure and diversity and the pattern of genetic control are formulated. A series of closed forms for the EM algorithm to estimate haplotype frequencies and haplotype effects in a network of genetic interactions among three genomes are derived. Simulation studies were performed to test the statistical properties of the model, recommending necessary sample sizes for obtaining reasonably good accuracy and precision of parameter estimation. By integrating, for the first time, the epidemiological principle of viral infection into genetic mapping, the new model shall find an immediate application to studying the genetic architecture of viral infection.

Functional mapping of human growth trajectories

Human height is an important trait from biological and social perspectives. Genes have been widely recognized to be involved in human body growth, but their detailed controlling mechanisms are poorly understood. Here, we present a computational model for functional mapping of quantitative trait loci (QTLs) that control trajectories of human height growth through an interactive network. The model integrates mathematical equations of human growth curves into the mixture model-based functional mapping framework, allowing the identification and mapping of individual QTLs responsible for the developmental pattern of human growth. The model was derived on a random sample of subjects from a natural population, for each of which molecular markers within candidate genes or throughout the entire genome are typed and height data from childhood to adulthood are collected. A series of testable hypotheses are formulated about the genetic control of developmental timing and duration at different stages. The model was used to characterize epistatic QTLs for height growth hidden in 548 Japanese girls which is a semi-real data set with simulated the marker genotypes. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the genetic and developmental mechanisms of human body growth and its associated diseases.

A computational approach to the functional clustering of periodic gene-expression profiles

DNA microarray analysis has emerged as a leading technology to enhance our understanding of gene regulation and function in cellular mechanism controls on a genomic scale. This technology has advanced to unravel the genetic machinery of biological rhythms by collecting massive gene-expression data in a time course. Here, we present a statistical model for clustering periodic patterns of gene expression in terms of different transcriptional profiles. The model incorporates biologically meaningful Fourier series approximations of gene periodic expression into a mixture-model-based likelihood function, thus producing results that are likely to be closer to biological relevance, as compared to those from existing models. Also because the structures of the time-dependent means and covariance matrix are modeled, the new approach displays increased statistical power and precision of parameter estimation. The approach was used to reanalyze a real example with 800 periodically expressed transcriptional genes in yeast, leading to the identification of 13 distinct patterns of gene-expression cycles. The model proposed can be useful for characterizing the complex biological effects of gene expression and generate testable hypotheses about the workings of developmental systems in a more precise quantitative way.

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.

Nonparametric functional mapping of quantitative trait loci

Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.

A statistical model for the identification of genes governing the incidence of cancer with age

The cancer incidence increases with age. This epidemiological pattern of cancer incidence can be attributed to molecular and cellular processes of individual subjects. Also, the incidence of cancer with ages can be controlled by genes. Here we present a dynamic statistical model for explaining the epidemiological pattern of cancer incidence based on individual genes that regulate cancer formation and progression. We incorporate the mathematical equations of age-specific cancer incidence into a framework for functional mapping aimed at identifying quantitative trait loci (QTLs) for dynamic changes of a complex trait. The mathematical parameters that specify differences in the curve of cancer incidence among QTL genotypes are estimated within the context of maximum likelihood. The model provides testable quantitative hypotheses about the initiation and duration of genetic expression for QTLs involved in cancer progression. Computer simulation was used to examine the statistical behavior of the model. The model can be used as a tool for explaining the epidemiological pattern of cancer incidence.

Functional mapping imprinted quantitative trait loci underlying developmental characteristics

Background

Genomic imprinting, a phenomenon referring to nonequivalent expression of alleles depending on their parental origins, has been widely observed in nature. It has been shown recently that the epigenetic modification of an imprinted gene can be detected through a genetic mapping approach. Such an approach is developed based on traditional quantitative trait loci (QTL) mapping focusing on single trait analysis. Recent studies have shown that most imprinted genes in mammals play an important role in controlling embryonic growth and post-natal development. For a developmental character such as growth, current approach is less efficient in dissecting the dynamic genetic effect of imprinted genes during individual ontology.

Results

Functional mapping has been emerging as a powerful framework for mapping quantitative trait loci underlying complex traits showing developmental characteristics. To understand the genetic architecture of dynamic imprinted traits, we propose a mapping strategy by integrating the functional mapping approach with genomic imprinting. We demonstrate the approach through mapping imprinted QTL controlling growth trajectories in an inbred F₂ population. The statistical behavior of the approach is shown through simulation studies, in which the parameters can be estimated with reasonable precision under different simulation scenarios. The utility of the approach is illustrated through real data analysis in an F₂ family derived from LG/J and SM/J mouse stains. Three maternally imprinted QTLs are identified as regulating the growth trajectory of mouse body weight.

Conclusion

The functional iQTL mapping approach developed here provides a quantitative and testable framework for assessing the interplay between imprinted genes and a developmental process, and will have important implications for elucidating the genetic architecture of imprinted traits.

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.

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 computational approach for functional mapping of quantitative trait loci that regulate thermal performance curves

Whether and how thermal reaction norm is under genetic control is fundamental to understand the mechanistic basis of adaptation to novel thermal environments. However, the genetic study of thermal reaction norm is difficult because it is often expressed as a continuous function or curve. Here we derive a statistical model for dissecting thermal performance curves into individual quantitative trait loci (QTL) with the aid of a genetic linkage map. The model is constructed within the maximum likelihood context and implemented with the EM algorithm. It integrates the biological principle of responses to temperature into a framework for genetic mapping through rigorous mathematical functions established to describe the pattern and shape of thermal reaction norms. The biological advantages of the model lie in the decomposition of the genetic causes for thermal reaction norm into its biologically interpretable modes, such as hotter-colder, faster-slower and generalist-specialist, as well as the formulation of a series of hypotheses at the interface between genetic actions/interactions and temperature-dependent sensitivity. The model is also meritorious in statistics because the precision of parameter estimation and power of QTLdetection can be increased by modeling the mean-covariance structure with a small set of parameters. The results from simulation studies suggest that the model displays favorable statistical properties and can be robust in practical genetic applications. The model provides a conceptual platform for testing many ecologically relevant hypotheses regarding organismic adaptation within the Eco-Devo paradigm.

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.

Modeling sequence-sequence interactions for drug response

MOTIVATION

Genetic interactions or epistasis may play an important role in the genetic etiology of drug response. With the availability of large-scale, high-density single nucleotide polymorphism markers, a great challenge is how to associate haplotype structures and complex drug response through its underlying pharmacodynamic mechanisms.

RESULTS

We have derived a general statistical model for detecting an interactive network of DNA sequence variants that encode pharmacodynamic processes based on the haplotype map constructed by single nucleotide polymorphisms. The model was validated by a pharmacogenetic study for two predominant beta-adrenergic receptor (betaAR) subtypes expressed in the heart, beta1AR and beta2AR. Haplotypes from these two receptors trigger significant interaction effects on the response of heart rate to different dose levels of dobutamine. This model will have implications for pharmacogenetic and pharmacogenomic research and drug discovery.

AVAILABILITY

A computer program written in Matlab can be downloaded from the webpage of statistical genetics group at the University of Florida.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Identification of quantitative trait nucleotides that regulate cancer growth: a simulation approach

A general growth model derived from basic cellular properties can be used to describe the dynamic process of cancer growth with mathematical equations. It has been recognized that cancer growth is under genetic control, with a multitude of interacting genes each segregating in a Mendelian fashion and displaying environmental sensitivity. In this article, we integrate the mathematical aspects of the pervasive growth model into a statistical framework for the identification of quantitative trait nucleotides that underlie cancer growth. This integrative framework is constructed with a single nucleotide polymorphism-based haplotype blocking analysis. Simulation studies have been performed to demonstrate the usefulness of the model. The proposed model provides a generic platform model for testing and detecting specific DNA sequence variants that regulates the timing of cancer emergence, growth and differentiation.

Functional mapping - how to map and study the genetic architecture of dynamic complex traits

The development of any organism is a complex dynamic process that is controlled by a network of genes as well as by environmental factors. Traditional mapping approaches for analysing phenotypic data measured at a single time point are too simple to reveal the genetic control of developmental processes. A general statistical mapping framework, called functional mapping, has been proposed to characterize, in a single step, the quantitative trait loci (QTLs) or nucleotides (QTNs) that underlie a complex dynamic trait. Functional mapping estimates mathematical parameters that describe the developmental mechanisms of trait formation and expression for each QTL or QTN. The approach provides a useful quantitative and testable framework for assessing the interplay between gene actions or interactions and developmental changes.

A joint model for nonparametric functional mapping of longitudinal trajectory and time-to-event

Background

The characterization of the relationship between a longitudinal response process and a time-to-event has been a pressing challenge in biostatistical research. This has emerged as an important issue in genetic studies when one attempts to detect the common genes or quantitative trait loci (QTL) that govern both a longitudinal trajectory and developmental event.

Results

We present a joint statistical model for functional mapping of dynamic traits in which the event times and longitudinal traits are taken to depend on a common set of genetic mechanisms. By fitting the Legendre polynomial of orthogonal properties for the time-dependent mean vector, our model does not rely on any curve, which is different from earlier parametric models of functional mapping. This newly developed nonparametric model is demonstrated and validated by an example for a forest tree in which stemwood growth and the time to first flower are jointly modelled.

Conclusion

Our model allows for the detection of specific QTL that govern both longitudinal traits and developmental processes through either pleiotropic effects or close linkage, or both. This model will have great implications for integrating longitudinal and event data to gain better insights into comprehensive biology and biomedicine.

Functional mapping for genetic control of programmed cell death

"Naturally occurring" or "programmed" cell death (PCD) in which the cell uses specialized cellular machinery to kill itself is a ubiquitous phenomenon that occurs early in organ development. Such a cell suicide mechanism that enables metazoans to control cell number and eliminate cells threatening the organism's survival has been thought to be under genetic control. In this report, we develop a novel statistical model for mapping specific genes or quantitative trait loci (QTL) that are responsible for the PCD process based on polymorphic molecular markers. This model incorporates the biological mechanisms of PCD that undergoes two different developmental stages, exponential growth and polynomial death. We derived a parametric approach to model the exponential growth and a nonparametric approach based on the Legendre function to model the polynomial death. A series of stationary and nonstationary models has been used to approximate the structure of the covariance matrix among cell numbers at a multitude of different times. The statistical behavior of our model is investigated through simulation studies and validated by a real example in rice.

A framework to monitor environment-induced major genes for developmental trajectories: implication for a prenatal cocaine exposure study

Whether there are specific genes involved in response to different environmental agents and how such genes regulate developmental trajectories during lifetime are of fundamental importance in health, clinical and pharmaceutical research. In this article, we present a novel statistical model for monitoring environment-induced genes of major effects on longitudinal outcomes of a trait. This model is derived within the maximum likelihood framework, incorporated by mathematical aspects of growth and developmental processes. A typical structural model is implemented to approximate time-dependent covariance matrices for the longitudinal trait. This model allows for a number of biologically meaningful hypothesis tests regarding the effects of major genes on overall growth trajectories or particular stages of development. It can be used to test whether and how major genetic effects are expressed differently under altered environmental agents. In a well-designed case-control study, our model has been employed to detect cocaine-dependent genes that affect growth trajectories for head circumference during childhood. The detected gene triggers significant effects on growth curves in both cocaine-exposed (case) and unexposed groups (control), but with different extents. Significant genotype-environment interactions due to this so-called environment-sensitive gene are promising for further studies toward its genomic mapping using polymorphic molecular markers.

Multilocus linkage disequilibrium mapping of epistatic quantitative trait loci that regulate HIV dynamics: a simulation approach

The time-dependent change of HIV particle load, i.e. HIV dynamics, is likely to be controlled by a multitude of quantitative trait loci (QTL) that interact with each other as well as with various developmental and environmental factors in a coordinated manner. In this article, we have derived a new statistical model for mapping the epistatic QTL responsible for HIV dynamics in a natural human population. This model, constructed on the integrated theme of functional mapping and linkage disequilibrium (LD) mapping, can make use of information from multiple markers genotyped from the human genome. It allows for the test and estimation of genetic actions and interactions involved in the control of HIV progression and provides a general platform to identify the detailed genetic architecture of resistance or susceptibility of humans to HIV on a dynamic scale. We have generalized this model to accommodate various complicated clincal designs for AIDS studies. Simulation studies with different scenarios are performed to examine the statistical behaviour of the model. The genetic and statistical extensions of this mapping model to HIV/AIDS genomic research are discussed.

A hyperspace model to decipher the genetic architecture of developmental processes: allometry meets ontogeny

To better utilize limited resources for their survival and reproduction, all organisms undergo developmental changes in both body size and shape during ontogeny. The genetic analysis of size change with increasing age, i.e., growth, has received considerable attention in quantitative developmental genetic studies, but the genetic architecture of ontogenetic changes in body shape and its associated allometry have been poorly understood partly due to the lack of analytical tools. In this article, we attempt to construct a multivariate statistical framework for studying the genetic regulation of ontogenetic growth and shape. We have integrated biologically meaningful mathematical functions of growth curves and developmental allometry into the estimation process of genetic mapping aimed at identifying individual quantitative trait loci (QTL) for phenotypic variation. This model defined with high dimensions can characterize the ontogenetic patterns of genetic effects of QTL over the lifetime of an organism and assess the interplay between genetic actions/interactions and phenotypic integration. The closed forms for the residual covariance matrix and its determinant and inverse were derived to overcome the computational complexity typical of our high-dimensional model. We used a worked example to validate the utility of this model. The implications of this model for genetic research of evo-devo are discussed.