**1. Introduction**

Complex traits are genetic traits controlled by multiple genes. They are sensitive to environmental changes and easily affected by the environment [1]. The phenotypic expression of complex traits in individuals within a population displays a continuous variation and generally a normal distribution. Most important biochemical, medical, and agronomic traits

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and the majority of human diseases are complex traits that are controlled by interconnected genetic networks and environmental factors. In these gene networks, the effect of each gene is small [2]. The characteristics also change at different developmental stages and demonstrate the dynamic features of complex traits. The study of these complex traits is an important topic in modern biology.

a trait by statistical hypothesis testing and parameter estimation, rather than directly estimating the genetic effects of the gene at all-time points. Because of the combination of biological laws (defined by mathematical models) and the reduced number of genetic parameters that need to be estimated, functional mapping increased the potency of detecting significant QTLs [14]. Statistical merits of functional mapping will be exemplified in which case data are recorded irregularly, bringing on data sparsity. Such sparse longitudinal data cannot be well analyzed by traditional mapping for two reasons. First, because the problem of missing data existing at a given time point, traditional method is unable to use all of individuals, thus leading to a biased parameter estimation and reduced power of QTL detection. Second, individuals are measured at a few number of time points, limiting the fit of growth equation. Functional mapping is robust for handling longitudinal sparse data in which no single time point has the phenotypic data for all individuals, facilitating the QTL mapping to study the

Functional Mapping of Plant Growth in *Arabidopsis thaliana*

http://dx.doi.org/10.5772/intechopen.74424

77

A set of tree data is used to assess the statistical validity of functional mapping. Several QTLs affecting the developmental trajectories of poplar were detected with the QTL mapping method based on the logistic-mixture model [15], and these QTLs were located on a genetic linkage map constructed by polymorphic markers. Studies have shown that functional mapping is useful in establishing gene-phenotype relationships and predicting QTL phenotypes of individual organisms based on the control of a trait. Functional mapping combines the principles of Mendelian genetics with statistical and developmental mechanisms, and is superior to traditional QTL mapping methods that combine the principles of statistics and genetics. To date, functional mapping has been used for mapping dynamic QTLs in poplars [16], jujubes [17], soybeans [18], rice [19], maize [20], yeast [21], oysters [22], mice [23], humans [24], and drug responses [25, 26].

*Arabidopsis thaliana* is a small, annual or winter annual, rosette plant. *A. thaliana* is a central genetic model and universal reference organism in plant and crop science. The successful integration of different fields of research in the study of *A. thaliana* has made a large contribution to our molecular understanding of key concepts in biology. The *Arabidopsis* reference genome sequence was the first published nuclear genome of a flowering plant in 2000 (http://www.arabidopsis.org) [27, 28]. The ease and speed with which experiments can be conducted on *A. thaliana* has allowed enormous fundamental progress in our knowledge of the molecular principles of plant development, cell biology, metabolism, physiology, genetics, and epigenetics [29]. The uses of *Arabidopsis* as the universal reference plant continue to expand, particularly in the field of systems biology [30, 31]. Important work has been done to investigate the molecular networks that mediate environmentally controlled developmental switches in *A*. *thaliana*. Examples include the transition from vegetative to reproductive development, also termed flowering time control [32–34], seed dormancy and germination control [35, 36], and the light-regulated development of seedlings [37, 38]. *A. thaliana* has also served as a model research organism for exploring many areas of fundamental biology, including photobiology, the circadian clock, DNA methylation, DNA repair, RNA silencing, protein degradation, and G-protein signaling, many of which have direct application in human health [39–42]. Insights into the functions of a multitude of individual genes, as well as the elucidation of biosynthetic pathways and regulatory networks, in *A*. *thaliana* have proven invaluable for identifying the genetic basis of agronomically important traits in crops such as plant height and flowering time [43, 44]. Classical molecular genetics approaches have been used in

genetic architecture of hard-to-measure traits.

Genetic mapping aimed at mapping underlying genes to genomic locations is a powerful tool for dissecting the genetic architecture of complex traits. Lander and Botstein have proposed an approach for mapping quantitative trait loci (QTLs) based on a sparse-density linkage map of molecular markers. This so-called interval mapping method can overcome the confounding problem of marker-QTL recombination [3]. Composite interval mapping includes other markers as covariates to control the overall genetic background, which displays increased power in QTL detection [4]. Considering QTL-QTL epistatic interactions in a linkage map, Kao et al. proposed using multiple marker intervals to map QTLs [5]. Currently, statistical methodologies for QTL mapping include regression analysis, maximum likelihood, and the Bayesian approach. With the development of high-throughput single nucleotide polymorphism (SNP) genotyping techniques, genome-wide association studies (GWAS) have provided a powerful means of mapping a complete set of genes underlying complex traits [6–8]. The genetic structure of a trait is explained by GWAS, which identify the numbers and chromosomal locations of each gene, the size of each gene's unique and pleiotropic effects, and the relative contributions of additive, dominant, and epistatic genetic effects. This provides an unprecedented tool for preparing a genotype-phenotype map [9]. So far, GWAS have detected many genetic variants for a wide range of complex traits, including those pertaining to agriculture, forestry, and human disease [9, 10].

It should be noted, however, that traits such as height and weight vary with time or other independent environmental stimuli. The traditional QTL mapping method directly associates a single marker with a single phenotype at a time point, which ignores the dynamic characteristics of organisms at different developmental stages, and cannot exactly reflect the whole genetic architecture of complex traits. Although thousands of QTLs were detected in many individuals, only a small number of QTLs were cloned and separated [11]. The reason for this problem is that the QTLs that have undergone rigorous statistical testing are divorced from biological relevance, which limited the projections of the genetic structure of traits. Genetic analysis of dynamic traits presents a serious statistical challenge. To solve these problems, Ma et al. [13] proposed a QTL mapping method based on a logistic-mixture model [12]. The QTL effect on developmental traits during ontogeny is considered as a function of time, and a series of growth formulas can be derived from the logistic curve describing plant height, size, and weight [13], arriving at a model which is expected to be improved in parameter estimation and statistical inference over previous models. Ma et al. [13] developed a maximum likelihood statistical framework based on a logistic-mixture model for the characteristics of function-valued traits, which change as a function of a specific variable. This QTL mapping strategy is called functional mapping.

Functional mapping combines mathematical functions that describe biological processes and assembles mathematical formulas into the statistical framework of QTL mapping to study the interactions between genes and phenotypic traits of organisms during growth and development. We estimate the parameters of a specific genotype that determines the development of a trait by statistical hypothesis testing and parameter estimation, rather than directly estimating the genetic effects of the gene at all-time points. Because of the combination of biological laws (defined by mathematical models) and the reduced number of genetic parameters that need to be estimated, functional mapping increased the potency of detecting significant QTLs [14]. Statistical merits of functional mapping will be exemplified in which case data are recorded irregularly, bringing on data sparsity. Such sparse longitudinal data cannot be well analyzed by traditional mapping for two reasons. First, because the problem of missing data existing at a given time point, traditional method is unable to use all of individuals, thus leading to a biased parameter estimation and reduced power of QTL detection. Second, individuals are measured at a few number of time points, limiting the fit of growth equation. Functional mapping is robust for handling longitudinal sparse data in which no single time point has the phenotypic data for all individuals, facilitating the QTL mapping to study the genetic architecture of hard-to-measure traits.

and the majority of human diseases are complex traits that are controlled by interconnected genetic networks and environmental factors. In these gene networks, the effect of each gene is small [2]. The characteristics also change at different developmental stages and demonstrate the dynamic features of complex traits. The study of these complex traits is an important topic

Genetic mapping aimed at mapping underlying genes to genomic locations is a powerful tool for dissecting the genetic architecture of complex traits. Lander and Botstein have proposed an approach for mapping quantitative trait loci (QTLs) based on a sparse-density linkage map of molecular markers. This so-called interval mapping method can overcome the confounding problem of marker-QTL recombination [3]. Composite interval mapping includes other markers as covariates to control the overall genetic background, which displays increased power in QTL detection [4]. Considering QTL-QTL epistatic interactions in a linkage map, Kao et al. proposed using multiple marker intervals to map QTLs [5]. Currently, statistical methodologies for QTL mapping include regression analysis, maximum likelihood, and the Bayesian approach. With the development of high-throughput single nucleotide polymorphism (SNP) genotyping techniques, genome-wide association studies (GWAS) have provided a powerful means of mapping a complete set of genes underlying complex traits [6–8]. The genetic structure of a trait is explained by GWAS, which identify the numbers and chromosomal locations of each gene, the size of each gene's unique and pleiotropic effects, and the relative contributions of additive, dominant, and epistatic genetic effects. This provides an unprecedented tool for preparing a genotype-phenotype map [9]. So far, GWAS have detected many genetic variants for a wide range of complex traits, including those pertaining to agri-

It should be noted, however, that traits such as height and weight vary with time or other independent environmental stimuli. The traditional QTL mapping method directly associates a single marker with a single phenotype at a time point, which ignores the dynamic characteristics of organisms at different developmental stages, and cannot exactly reflect the whole genetic architecture of complex traits. Although thousands of QTLs were detected in many individuals, only a small number of QTLs were cloned and separated [11]. The reason for this problem is that the QTLs that have undergone rigorous statistical testing are divorced from biological relevance, which limited the projections of the genetic structure of traits. Genetic analysis of dynamic traits presents a serious statistical challenge. To solve these problems, Ma et al. [13] proposed a QTL mapping method based on a logistic-mixture model [12]. The QTL effect on developmental traits during ontogeny is considered as a function of time, and a series of growth formulas can be derived from the logistic curve describing plant height, size, and weight [13], arriving at a model which is expected to be improved in parameter estimation and statistical inference over previous models. Ma et al. [13] developed a maximum likelihood statistical framework based on a logistic-mixture model for the characteristics of function-valued traits, which change as a

function of a specific variable. This QTL mapping strategy is called functional mapping.

Functional mapping combines mathematical functions that describe biological processes and assembles mathematical formulas into the statistical framework of QTL mapping to study the interactions between genes and phenotypic traits of organisms during growth and development. We estimate the parameters of a specific genotype that determines the development of

in modern biology.

76 Next Generation Plant Breeding

culture, forestry, and human disease [9, 10].

A set of tree data is used to assess the statistical validity of functional mapping. Several QTLs affecting the developmental trajectories of poplar were detected with the QTL mapping method based on the logistic-mixture model [15], and these QTLs were located on a genetic linkage map constructed by polymorphic markers. Studies have shown that functional mapping is useful in establishing gene-phenotype relationships and predicting QTL phenotypes of individual organisms based on the control of a trait. Functional mapping combines the principles of Mendelian genetics with statistical and developmental mechanisms, and is superior to traditional QTL mapping methods that combine the principles of statistics and genetics. To date, functional mapping has been used for mapping dynamic QTLs in poplars [16], jujubes [17], soybeans [18], rice [19], maize [20], yeast [21], oysters [22], mice [23], humans [24], and drug responses [25, 26].

*Arabidopsis thaliana* is a small, annual or winter annual, rosette plant. *A. thaliana* is a central genetic model and universal reference organism in plant and crop science. The successful integration of different fields of research in the study of *A. thaliana* has made a large contribution to our molecular understanding of key concepts in biology. The *Arabidopsis* reference genome sequence was the first published nuclear genome of a flowering plant in 2000 (http://www.arabidopsis.org) [27, 28]. The ease and speed with which experiments can be conducted on *A. thaliana* has allowed enormous fundamental progress in our knowledge of the molecular principles of plant development, cell biology, metabolism, physiology, genetics, and epigenetics [29]. The uses of *Arabidopsis* as the universal reference plant continue to expand, particularly in the field of systems biology [30, 31]. Important work has been done to investigate the molecular networks that mediate environmentally controlled developmental switches in *A*. *thaliana*. Examples include the transition from vegetative to reproductive development, also termed flowering time control [32–34], seed dormancy and germination control [35, 36], and the light-regulated development of seedlings [37, 38]. *A. thaliana* has also served as a model research organism for exploring many areas of fundamental biology, including photobiology, the circadian clock, DNA methylation, DNA repair, RNA silencing, protein degradation, and G-protein signaling, many of which have direct application in human health [39–42]. Insights into the functions of a multitude of individual genes, as well as the elucidation of biosynthetic pathways and regulatory networks, in *A*. *thaliana* have proven invaluable for identifying the genetic basis of agronomically important traits in crops such as plant height and flowering time [43, 44]. Classical molecular genetics approaches have been used in *A*. *thaliana* to dissect the patterning and development of flowers [45], embryos [46], leaves, and roots [47]. In this study, we describe the implementation of functional mapping to identify and map QTLs for height trajectories in a population of *A. thaliana*. The mapping population is composed of 144 recombinant inbred lines (RILs) derived from a Landsberg erecta (LER) cultivar and a Shadara (SHA) cultivar. Functional mapping identified 48 QTLs that determine the height growth of *A. thaliana*. The identification of these QTLs will help us address fundamental questions about the genetic mechanisms of height growth.
