Genetic mapping of allometric scaling laws

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling relationships between rate and size (power laws). A statistical model for mapping specific quantitative trait loci (QTLs) that are responsible for allometric scaling laws has been developed. We present an improved model for allometric mapping of QTLs based on a more general allometry equation. This improved model includes two steps: (1) use model II regression analysis to estimate the parameters underlying universal allometric scaling laws, and (2) substitute the estimated allometric parameters in the mixture-based mapping model to obtain the estimation of QTL position and effects. This model has been validated by a real example for a mouse F2 progeny, in which two QTLs were detected on different chromosomes that determine the allometric relationship between growth rate and body weight.     

Exponential mapping of quantitative trait loci governing allometric relationships in organisms

Allometric scaling relationships or quarter-power rules, as a universal biological law, can be viewed as having some genetic component, and the particular genes (or quantitative trait loci, QTL) underlying these allometric relationships can be mapped using molecular markers. We develop a mathematical and statistical model for mapping allometric QTL on the basis of nonlinear power functions using Taylors approximation theory. Simulation studies indicate that the QTL position and effect can be estimated using our model, but the estimation precision can be improved from the higher- over lower-order approximation when the sample size used and gene effects are small. The application of our approach in a real example from forest trees leads to successful detection of a QTL governing the allometric relationship between 3rd-year stem height and 3rd-year stem biomass. It is expected that our model will have broad implications for genetic, evolutionary, biomedical and breeding research.     

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.     

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.     

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.     

Regression-based multi-trait QTL mapping using a structural equation model

Quantitative trait loci (QTL) mapping often results in data on a number of traits that have well-established causal relationships. Many multi-trait QTL mapping methods that account for the correlation among multiple traits have been developed to improve the statistical power and the precision of QTL parameter estimation. However, none of these methods are capable of incorporating the causal structure among the traits. Consequently, genetic functions of the QTL may not be fully understood. Structural equation modeling (SEM) allows researchers to explicitly characterize the causal structure among the variables and to decompose effects into direct, indirect, and total effects. In this paper, we developed a multi-trait SEM method of QTL mapping that takes into account the causal relationships among traits related to grain yield. Performance of the proposed method is evaluated by simulation study and applied to data from a wheat experiment. Compared with single trait analysis and the multi-trait least-squares analysis, our multi-trait SEM improves statistical power of QTL detection and provides important insight into how QTLs regulate traits by investigating the direct, indirect, and total QTL effects. The approach also helps build biological models that more realistically reflect the complex relationships among QTL and traits and is more precise and efficient in QTL mapping than single trait analysis.     

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     

作物数量性状研究进展

作物主要农艺性状和经济性状大多属于数量性状。传统数量遗传学采用数理统计方法,把控制数量性状的多基因系统作为一个整体进行研究。DNA分子标记技术的出现和发展,为数量性状研究提供了重要工具。自20世纪80年代以来,QTL定位的统计分析方法发展很快,先后提出单标记分析法、区间作图法及复合区间作图法等。目前,作物QTL研究取得了重要进展,一些重要作物、重要农艺性状的主效QTL基因已被相继克隆成功,作物数量性状的研究已经成为一个具有勃勃生机的热门领域。     

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.     

Advances in Bayesian multiple quantitative trait loci mapping in experimental crosses

Many complex human diseases and traits of biological and/or economic importance are determined by interacting networks of multiple quantitative trait loci (QTL) and environmental factors. Mapping QTL is critical for understanding the genetic basis of complex traits, and for ultimate identification of genes responsible. A variety of sophisticated statistical methods for QTL mapping have been developed. Among these developments, the evolution of Bayesian approaches for multiple QTL mapping over the past decade has been remarkable. Bayesian methods can jointly infer the number of QTL, their genomic positions and their genetic effects. Here, we review recently developed and still developing Bayesian methods and associated computer software for mapping multiple QTL in experimental crosses. We compare and contrast these methods to clearly describe the relationships among different Bayesian methods. We conclude this review by highlighting some areas of future research.     

植物数量性状基因定位研究概述

植物重要的性状多为数量性状。长期以来,人类一直寻求解释植物数量性状的遗传规律以便对其进行遗传操纵。现代分子生物技术的发展为植物数量性状基因的定位、分离等研究提供了条件。本文从数量性状基因座(QTL)作图群体类型及其特点,QTL定位方法,植物QTL研究现状,以及QTL精细定位、克隆、利用等方面进行了综述,并对今后植物QTL研究进行了展望。     

植物数量性状基因定位研究概述

植物重要的性状多为数量性状。长期以来,人类一直寻求解释植物数量性状的遗传规律以便对其进行遗传操纵。现代分子生物技术的发展为植物数量性状基因的定位、分离等研究提供了条件。本文从数量性状基因座(QTL)作图群体类型及其特点,QTL定位方法,植物QTL研究现状,以及QTL精细定位、克隆、利用等方面进行了综述,并对今后植物QTL研究进行了展望。     

Strategies for genetic mapping of categorical traits

The search for efficient and powerful statistical methods and optimal mapping strategies for categorical traits under various experimental designs continues to be one of the main tasks in genetic mapping studies. Methodologies for genetic mapping of categorical traits can generally be classified into two groups, linear and non-linear models. We develop a method based on a threshold model, termed mixture threshold model to handle ordinal (or binary) data from multiple families. Monte Carlo simulations are done to compare its statistical efficiencies and properties of the proposed non-linear model with a linear model for genetic mapping of categorical traits using multiple families. The mixture threshold model has notably higher statistical power than linear models. There may be an optimal sampling strategy (family size vs number of families) in which genetic mapping reaches its maximal power and minimal estimation errors. A single large-sibship family does not necessarily produce the maximal power for detection of quantitative trait loci (QTL) due to genetic sampling of QTL alleles. The QTL allelic model has a marked impact on efficiency of genetic mapping of categorical traits in terms of statistical power and QTL parameter estimation. Compared with a fixed number of QTL alleles (two or four), the model with an infinite number of QTL alleles and normally distributed allelic effects results in loss of statistical power. The results imply that inbred designs (e.g. F₂ or four-way crosses) with a few QTL alleles segregating or reducing number of QTL alleles (e.g. by selection) in outbred populations are desirable in genetic mapping of categorical traits using data from multiple families. This revised version was published online in July 2006 with corrections to the Cover Date.     

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/).     

Nonparametric functional mapping of quantitative trait loci

Summary .  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.     

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.     

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.     

Functional mapping of reaction norms to multiple environmental signals

Whether there are different genes involved in response to different environmental signals and how these genes interact to determine the final expression of the trait are of fundamental importance in agricultural and biological research. We present a statistical framework for mapping environment-induced genes (or quantitative trait loci, QTLs) of major effects on the expression of a trait that respond to changing environments. This framework is constructed with a maximum-likelihood-based mixture model, in which the mean and covariance structure of environment-induced responses is modelled. The means for responses to continuous environmental states, referred to as reaction norms, are approximated for different QTL genotypes by mathematical equations that were derived from fundamental biological principles or based on statistical goodness-of-fit to observational data. The residual covariance between different environmental states was modelled by autoregressive processes. Such an approach to studying the genetic control of reaction norms can be expected to be advantageous over traditional mapping approaches in which no biological principles and statistical structures are considered. We demonstrate the analytical procedure and power of this approach by modelling the photosynthetic rate process as a function of temperature and light irradiance. Our approach allows for testing how a QTL affects the reaction norm of photosynthetic rate to a specific environment and whether there exist different QTLs to mediate photosynthetic responses to temperature and light irradiance, respectively.     

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.