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 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.     

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.     

远交群体动态性状基因定位的似然分析Ⅰ.理论方法

受动物遗传育种中用来估计动态性状育种值的随机回归测定日模型思想的启发 ,将关于时间 (测定日期 )的Legendre多项式镶嵌在遗传模型的每个遗传效应中 ,以刻画QTL对动态性状变化过程的作用 ,从而建立起动态性状基因定位的数学模型。利用远交设计群体 ,阐述了动态性状基因定位的似然分析原理 ,推导了定位参数似然估计的EM法两步求解过程。结合动态性状遗传分析的特点和普通数量性状基因定位研究进展 ,还提出了有关动态性状基因定位进一步研究的设想     

Functional mapping of quantitative trait loci underlying growth trajectories using a transform-both-sides logistic model

The incorporation of developmental control mechanisms of growth has proven to be a powerful tool in mapping quantitative trait loci (QTL) underlying growth trajectories. A theoretical framework for implementing a QTL mapping strategy with growth laws has been established. This framework can be generalized to an arbitrary number of time points, where growth is measured, and becomes computationally more tractable, when the assumption of variance stationarity is made. In practice, however, this assumption is likely to be violated for age-specific growth traits due to a scale effect. In this article, we present a new statistical model for mapping growth QTL, which also addresses the problem of variance stationarity, by using a transform-both-sides (TBS) model advocated by Carroll and Ruppert (1984, Journal of the American Statistical Association 79, 321-328). The TBS-based model for mapping growth QTL cannot only maintain the original biological properties of a growth model, but also can increase the accuracy and precision of parameter estimation and the power to detect a QTL responsible for growth differentiation. Using the TBS-based model, we successfully map a QTL governing growth trajectories to a linkage group in an example of forest trees. The statistical and biological properties of the estimates of this growth QTL position and effect are investigated using Monte Carlo simulation studies. The implications of our model for understanding the genetic architecture of growth are discussed.     

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.     

Mapping epigenetic quantitative trait loci (QTL) altering a developmental trajectory.

Genetic variation in a quantitative trait that changes with age is important to both evolutionary biologists and breeders. A traditional analysis of the dynamics of genetic variation is based on the genetic variance-covariance matrix among different ages estimated from a quantitative genetic model. Such an analysis, however, cannot reveal the mechanistic basis of the genetic variation for a growth trait during ontogeny. Age-specific genetic variance at time t conditional on the causal genetic effect at time t - 1 implies the generation of episodes of new genetic variation arising during the interval t - 1 to t. In the present paper, the conditional genetic variance estimated from Zhu's (1995) conditional model was partitioned into its underlying individual quantitative trait loci (QTL) using molecular markers in an F2 progeny of poplars (Populus trichocarpa and Populus deltoides). These QTL, defined as epigenetic QTL, govern the alterations of growth trajectory in a population. Three epigenetic QTL were detected to contribute significantly to variation in growth trajectory during the period from the establishment year to the subsequent year in the field. It is suggested that the activation and expression of epigenetic QTL are influenced by the developmental status of trees and the environment in which they are grown.     

Genetic mapping of developmental instability: design, model and algorithm

Developmental instability or noise, defined as the phenotypic imprecision of an organism in the face of internal or external stochastic disturbances, has been thought to play an important role in shaping evolutionary processes and patterns. The genetic studies of developmental instability have been based on fluctuating asymmetry (FA) that measures random differences between the left and the right sides of bilateral traits. In this article, we frame an experimental design characterized by a spatial autocorrelation structure for determining the genetic control of developmental instability for those traits that cannot be bilaterally measured. This design allows the residual environmental variance of a quantitative trait to be dissolved into two components due to permanent and random environmental factors. The degree of developmental instability is quantified by the relative proportion of the random residual variance to the total residual variance. We formulate a mixture model to estimate and test the genetic effects of quantitative trait loci (QTL) on the developmental instability of the trait. The genetic parameters including the QTL position, the QTL effects, and spatial autocorrelations are estimated by implementing the EM algorithm within the mixture model framework. Simulation studies were performed to investigate the statistical behavior of the model. A live example for poplar trees was used to map the QTL that control root length growth and its developmental instability from cuttings in water culture.     

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.     

Multiple interval mapping for quantitative trait loci.

A new statistical method for mapping quantitative trait loci (QTL), called multiple interval mapping (MIM), is presented. It uses multiple marker intervals simultaneously to fit multiple putative QTL directly in the model for mapping QTL. The MIM model is based on Cockerham's model for interpreting genetic parameters and the method of maximum likelihood for estimating genetic parameters. With the MIM approach, the precision and power of QTL mapping could be improved. Also, epistasis between QTL, genotypic values of individuals, and heritabilities of quantitative traits can be readily estimated and analyzed. Using the MIM model, a stepwise selection procedure with likelihood ratio test statistic as a criterion is proposed to identify QTL. This MIM method was applied to a mapping data set of radiata pine on three traits: brown cone number, tree diameter, and branch quality scores. Based on the MIM result, seven, six, and five QTL were detected for the three traits, respectively. The detected QTL individually contributed from approximately 1 to 27% of the total genetic variation. Significant epistasis between four pairs of QTL in two traits was detected, and the four pairs of QTL contributed approximately 10.38 and 14.14% of the total genetic variation. The asymptotic variances of QTL positions and effects were also provided to construct the confidence intervals. The estimated heritabilities were 0.5606, 0.5226, and 0. 3630 for the three traits, respectively. With the estimated QTL effects and positions, the best strategy of marker-assisted selection for trait improvement for a specific purpose and requirement can be explored. The MIM FORTRAN program is available on the worldwide web (http://www.stat.sinica.edu.tw/chkao/).     

Genotype-environment interaction for quantitative trait loci affecting life span in Drosophila melanogaster

The nature of genetic variation for Drosophila longevity in a population of recombinant inbred lines was investigated by estimating quantitative genetic parameters and mapping quantitative trait loci (QTL) for adult life span in five environments: standard culture conditions, high and low temperature, and heat-shock and starvation stress. There was highly significant genetic variation for life span within each sex and environment. In the analysis of variance of life span pooled over sexes and environments, however, the significant genetic variation appeared in the genotype x sex and genotype x environment interaction terms. The genetic correlation of longevity across the sexes and environments was not significantly different from zero in these lines. We estimated map positions and effects of QTL affecting life span by linkage to highly polymorphic roo transposable element markers, using a multiple-trait composite interval mapping procedure. A minimum of 17 QTL were detected; all were sex and/or environment-specific. Ten of the QTL had sexually antagonistic or antagonistic pleiotropic effects in different environments. These data provide support for the pleiotropy theory of senescence and the hypothesis that variation for longevity might be maintained by opposing selection pressures in males and females and variable environments. Further work is necessary to assess the generality of these results, using different strains, to determine heterozygous effects and to map the life span QTL to the level of genetic loci.     

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     

Bayesian point estimation of quantitative trait loci

In this article, we consider the problem of the estimation of quantitative trait loci (QTL), those chromosomal regions at which genetic information affecting some quantitative trait is encoded. Generally the number of such encoding sites is unknown, and associations between neutral molecular marker genotypes and observed trait phenotypes are sought to locate them. We consider a Bayesian model for simple experimental designs, and discuss the existing approaches to inference for this problem. In particular, we focus on locating positions of the best candidate markers segregating for the trait, a situation which is of primary interest in comparative mapping. We introduce a loss function for estimating both the number of QTL and their location, and we illustrate its application via simulated and real data.     

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.     

Quantitative trait loci affecting components of wing shape in Drosophila melanogaster

Two composite multiple regression-interval mapping analyses were performed to identify candidate quantitative trait loci (QTL) affecting components of wing shape in Drosophila melanogaster defined by eight relative warp-based measures. A recombinant inbred line design was used to map QTL for the shape of two intervein regions in the anterior compartment of the wing, using a high resolution map of retrotransposon insertion sites between Oregon-R and Russian 2b. A total of 35 QTL representing up to 23 different loci were identified, many of which are located near components of the epidermal growth factor-Ras signal transduction pathway that regulates vein vs. intervein decision making and vein placement. Over one-half of the loci were detected in both sexes, and just under one-half were detected at two different growth temperatures. Different loci were found to affect aspects of shape in each intervein region, confirming that the shape of the whole wing should be regarded as a compound trait composed of several developmental units. In addition, a reciprocal backcross design was used to map QTL affecting shape in the posterior compartment of the wings of 831 flies, using a molecular map of 16 allele-specific oligohybridization single nucleotide polymorphism (SNP) markers between two divergent inbred lines. A total of 13 QTL were detected and shown to have generally additive effects on separable components of shape, in both sexes. By contrast, 8 QTL that affected wing size in these backcrosses were nearly dominant in their effects. The results confirm at the genetic level that wing shape is regulated independent of wing size and set up the hypothesis that wing shape is regulated in part through the regulation of the length and positioning of wing veins, involving quantitative regulation of the activity of secreted growth factors.     

Mapping-Linked Quantitative Trait Loci Using Bayesian Analysis and Markov Chain Monte Carlo Algorithms

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL and no additional or an unlinked QTL.     

Mapping quantitative trait Loci interactions from the maternal and offspring genomes

The expression of most developmental or behavioral traits involves complex interactions between quantitative trait loci (QTL) from the maternal and offspring genomes. The maternal-offspring interactions play a pivotal role in shaping the direction and rate of evolution in terms of their substantial contribution to quantitative genetic (co)variation. To study the genetics and evolution of maternal-offspring interactions, a unifying statistical framework that embraces both the direct and indirect genetic effects of maternal and offspring QTL on any complex trait is developed. This model is derived for a simple backcross design within the maximum-likelihood context, implemented with the EM algorithm. Results from extensive simulations suggest that this model can provide reasonable estimation of additive and dominant effects of the QTL at different generations and their interaction effects derived from the maternal and offspring genomes. Although our model is framed to characterize the actions and interactions of maternal and offspring QTL affecting offspring traits, the idea can be readily extended to decipher the genetic machinery of maternal traits, such as maternal care. Our model provides a powerful means for studying the evolutionary significance of indirect genetic effects in any sexually reproductive organisms.     

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.     

An ecophysiologically based mapping model identifies a major pleiotropic QTL for leaf growth trajectories of Phaseolus vulgaris

Crop modeling, a widely used tool to predict plant growth and development in heterogeneous environments, has been increasingly integrated with genetic information to improve its predictability. This integration can also shed light on the mechanistic path that connects the genotype to a particular phenotype under specific environments. We implemented a bivariate statistical procedure to map and identify quantitative trait loci (QTLs) that can predict the form of plant growth by estimating cultivar‐specific growth parameters and incorporating these parameters into a mapping framework. The procedure enables the characterization of how QTLs act differently in response to developmental and environmental cues. We used this procedure to map growth parameters of leaf area and mass in a mapping population of the common bean (Phaseolus vulgaris L.). Different sets of QTLs are responsible for various aspects of growth, including the initiation time of growth, growth rate, inflection point and asymptotic growth. A major QTL of a large effect was identified to pleiotropically affect trait expression in distinct environments and different traits expressed on the same organism. The integration of crop models and QTL mapping through our statistical procedure provides a powerful means of building a more precise predictive model of genotype‐phenotype relationships for crops.     

The genetic architecture of embryonic developmental rate and genetic covariation with age at maturation in rainbow trout Oncorhynchus mykiss

The genetic architecture underlying variation in embryonic developmental rate (DR) and genetic covariation with age of maturation (MAT) was investigated in rainbow trout Oncorhynchus mykiss. Highly significant additive parental effects and more limited evidence of epistatic effects on progeny hatching time were detected in three diallel sets of families. Genome scans with an average of 142 microsatellite loci from all 29 linkage groups in two families detected significant quantitative trait loci (QTL) for developmental rate on RT-8 and RT-30 with genome-wide and chromosome-wide effects, respectively. The QTL on linkage group RT-8 explained 23·7% of the phenotypic variation and supports results from previous studies. The co-localization of QTL for both DR and MAT to several linkage groups and the observation that alleles associated with faster developmental rate were found significantly more often in early maturing rather than typical and later maturing male ancestors supports the hypothesis of genetic covariation between DR and MAT. The maturation background and schedule of additional sires, however, did not have a consistent association with their progeny hatching times, suggesting that other genetic, environmental and physiological effects contribute to variation in these life-history traits.