Functional mapping of genotype-environment interactions for soybean growth by a semiparametric approach
- Qin Li†1,
- Zhongwen Huang†2, 3,
- Meng Xu4,
- Chenguang Wang1,
- Junyi Gai3Email author,
- Youjun Huang5,
- Xiaoming Pang6, 7 and
- Rongling Wu6, 7Email author
© Li et al; licensee BioMed Central Ltd. 2010
Received: 15 January 2010
Accepted: 2 June 2010
Published: 2 June 2010
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.
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.
The model provides a basis for deciphering the genetic architecture of trait expression adjusted to different biotic and abiotic environments for any organism.
In nature, any biological trait of an organism is never isolated from other traits or variables, but rather all of them are integrated under the premise that natural selection has tended to optimize energy absorption and transport within fractal-like distribution networks [1, 2]. Such a tendency has led to several universal biological laws; for example, the change of metabolic rate or surface area scales as the 3/4-power of body mass [3, 4], organismic growth as a function of age follows a sigmoidal shape [5, 6] and metabolic rate increases exponentially with temperature . A number of explicit mathematical equations have been established to describe these biological laws that hold true for all manners of life forms under natural selection.
These mathematical functions have now played an important role in incorporating biological laws into a mapping framework to detect specific quantitative trait loci (QTLs) for growth trajectories and developmental events. R. Wu and group pioneered a series of statistical models, called functional mapping, to map growth and development QTLs through estimating genotype-specific mathematical parameters that define a biological process (reviewed in ). In statistics, functional mapping has proven powerful and stable by modeling the patterns of trait development and autocorrelations among different time points measured [9–12]. For some dynamic traits whose expression cannot be mathematically described, nonparametric modeling based on the Legendre polynomial orthogonal has been proposed, thus enhancing the flexibility of functional mapping [13, 14]. Functional mapping is genetically relevant, allowing the formulation and test of numerous biologically meaningful hypotheses about the genetic control of growth [15, 16]. Some of the most important hypotheses include the timing of a QTL to switch on or off, the duration of its genetic effect, and the pleiotropic effect of this QTL on different aspects of development.
Functional mapping has been further extended to model the genetic control of more complicated biological problems related to QTL-QTL interactions [15, 17] and QTL-environment interactions on growth trajectories [10, 11].  provided a general procedure within the functional mapping framework for testing the effects of individual genetic components, including the additive, dominance, additive-additive, additive-dominance, dominance-additive and dominance-dominance, on different stages of development. [10, 11] incorporated environmental regimes, such as climates or sexes, into functional mapping to explore the consequence of interactions between the QTL and environments in shaping developmental trajectories. Original functional mapping and its extensions were founded on simple interval mapping, whose utilization may be limited in a situation where there are more than one QTL located on a similar region of a chromosome. When such multiple QTL occur, interval mapping-based approaches may find some "ghost" QTLs which provide significant signals for their existence although they do not exist in reality. [18, 19] developed a so-called composite interval mapping to separate multiple linked QTLs by integrating the principle of interval mapping for the testing markers and partial regression analysis of all possible other markers as co-factors. Composite interval mapping has been instrumental for the identification of QTLs that are responsible for different traits .
The combination of functional mapping and composite interval mapping can be expected to improve the estimation of multiple QTLs for growth trajectories, but this computationally presents a high challenge in terms of the complexity of estimating growth parameters associated with genotypes at each marker as a co-factor. Fortunately, some of these issues have been solved by R. Wu and group . These authors integrate parametrical interval mapping (aimed to test a hypothesized QTL in a test interval) and Legendre polynomial-based nonparametric longitudinal regression analysis (aimed to control the genome background by choosing a proper set of markers as co-factors), facilitating computation, estimation and tests of functional mapping for multiple QTLs. In this article, we extend 's semiparametric idea to illustrate an analytical strategy for mapping QTLs that affect growth trajectories through their main effects or QTL-environment interaction effects. An instructive procedure is described to test the impacts of each of these components on the timing of development and stages of growth in a time course. We report on the detection of QTLs that affect plant height growth trajectories in soybeans by using the approach developed in this article.
Experimental Design and Regression Model
Consider a recombinant inbred line (RIL) population in which there are two homozygous genotypes for alternative alleles at each locus. Molecular markers are genotyped for each RIL progeny and analyzed to construct a genetic linkage map used to identify quantitative trait loci (QTLs) that affect growth trajectories. The RIL design allows the same genotype to be replicated in time and space. Assume that there are n RILs in a QTL mapping study which are grown in a randomized complete block design in L different environments with multiple replicates per environment. Each of the individuals studied is measured for a growth trait at a series of time points, say T. We will take means at each time point over replicates in each environment to describe growth trajectories for an RIL. Alternatively, we can use individual plants per RIL to conduct the functional mapping of growth trajectories by modeling the spatial structure of a covariance .
where and μ l (t) and α l (t) are the environment-specific population mean and additive genetic effect for the QTL at time t, respectively; is the indicator variables for individual i that specify the QTL genotypes, which is defined as 1 for genotype QQ and 0 for genotype qq; α kl (t) is the environment-dependent additive effect at time t associated with marker k (except for the interval constructed by the two markers); x ik is the indicator variables that specify the additive effect of marker k for individual i, respectively, which is defined similarly as ; and e il (t) is the environment-dependent residual error at time t, normally distributed as . The covariance between the residual errors at different time points t1 and t2 within environment l is denoted as σ l (t1,t2). All the variances and covariances in environment l form a (T × T) covariance matrix ∑ l .
where Ω = (ωj|i, Θ jl , Ψ l ; j = 1, 2, l = 1, ..., L) is a vector of known parameters, that is, the genomic location of the QTL, QTL genotype-specific parameters, and parameters common to all genotypes. In statistics, the location parameter ωj|iis the proportion of different mixture normals in equation (2), reflecting the segregation of the QTL in the population, which can be inferred from known marker genotypes at the test interval. For a RIL mapping population, n progeny can be classified into four different groups of two-marker genotypes. In each group, the mixture proportion or frequency of a QTL genotype is RIL-specific and can be expressed as the conditional probability of QTL genotype j for RIL i given its marker genotype . QTL genotype- and environment-specific distribution density, f jl (y il ; Θ jl , Ψ l ), is assumed to be multivariate normal with expected mean vector u jl = (u jl (1), ⋯, u jl (T)) for QTL genotype j and covariance matrix ∑ l .
Modeling Time-Dependent Genetic Effects
Combining the QTL and marker effects in composite functional mapping, time-dependent expected means for different QTL genotypes in environment l are modelled by if a logistic curve (3) is considered. The model allows the fitting of any other growth curves.
Modeling the Covariance Matrix
The covariance structure of serial measurements can be modeled by a number of approaches. One of the commonly used approaches for structuring the covariance is the first order au-toregressive (AR(1)) model . Its advantage lies in the existence of a general expression for calculating the determinant and inverse of the matrix for any number of time points measured. In practice, the assumptions of variance stationarity and correlation stationarity, i.e., the residual variance at different time points is the same, expressed as σ2, and the correlation between two different time points t1 and t2 decreases exponentially in ρ with time lag, expressed as , may not hold.  used  transform-both-sides (TBS) model to meet the first assumption because heteroscedastic variances can be stable after the data are log-transformed. For the AR(1) model, we only need to use to model the structure of environment-specific covariance matrix ∑ l . Some other approaches for modeling the covariance structure include the structured antedepedence model [26, 27].
To estimate the MLEs of the parameters that model the time-dependent QTL effects and covariance matrix, we implement the simplex or Newton-Raphson algorithm in the estimation process with the EM algorithm [28, 29]. As shown in , the closed forms for estimating the determinant and inverse of the AR(1) covariance matrix can be derived, whose implementation will increase the computational efficiency of functional mapping.
where ℓ is the log-likelihood function and x and z are any parameters of interest in the current model. Taking the inverse of the negative values of the partial derivative matrix, the estimate of the asymptotic sample covariance matrix can be obtained.
where are the MLEs of parameters under the Legendre polynomial of order dimension, represents the number of independent parameters under order , and n is the total number of observation at a particular time point. The optimal model is one that displays the minimum BIC value. Note that the model selection procedure also incorporates the choice of the best number of markers (flanking a given test interval) as co-factors in composite functional mapping.
Existence of a QTL
and the covariance matrix with the AR(1) structure. The threshold for the tested interval is estimated as the 5% percentile of the LR values from 1000 simulation replicates. A genome-wide critical threshold is determined by scanning through the entire linkage map, although this process is computationally extensive.
Pleiotropic Effect of a QTL
for environment l2. If the null hypotheses above (8 and 9) are both rejected, this implies that the significant QTL detected affects pleiotropically growth trajectories in the two environments. Otherwise, this QTL is operational only in one environment, which causes genotype by environment interactions for growth trajectories. The thresholds for each of the hypotheses (8 and 9) corresponding to two different environments can be determined using the simulation approach as described above.
QTL by Environment Interaction
The rejection of the above null hypothesis means that a significant genotype by environment interaction exists due to allelic sensitivity to a varying environment. The log-likelihood ratio test statistics for hypothesis (10) can be determined from simulation studies.
The model described above was used to analyze a real example from a soybean genome project at Nanjing Agricultural University, China. Two original inbred lines of soybean, Kefeng No. 1 and Nannong 1138-2, as parents were crossed to generate an F1 population which was selfed for 7 generations to produce an RIL population composed of two groups of homozygous genotypes each containing two identical alleles from a different parental line. Let 1 and 2 denote the homozygotes derived from the Kefeng No. 1 alleles and Nannong 1138-2 alleles, respectively. A total of 184 RILs were genotyped for 488 molecular markers (restricted fragment length polymorphisms, simple sequence repeats and amplified fragment length polymorphsim) that construct a linkage map with 25 linkage groups covering 4,151.2 cM of the soybean genome .
The RILs were planted in a simple lattice design with multiple replicates randomly grown in a plot at the Jiangpu Agricultural Experiment Station of Nanjing Agricultural University, China. The plants were measured for their plant height growth for six to eight times with the first time at the 28th day after emergence and successive seven times every 10 days thereafter. The same study was repeated for year 2005 and 2006. Thus, different years are viewed as two different environments.
BIC values under different models of composite functional mapping by choosing markers within different window lengths (in cM) as co-factors. Different orders of the Legendre polynomial are considered under each model.
Order of Legendre Polynomials
Window Width (cM)
The MLEs of genotypic curve parameters at the two QTLs detected on chromosomes 3, 6, and 24 and sampling errors (in parentheses) of the estimates by composite functional mapping under the optimal order of the Legendre polynomial and the optimal number of co-factors (see Table 1) for plant height growth trajectories of soybeans in two different years (2005 and 2006).
By estimating and testing the inflection point of a logistic curve using ln b/r, we further investigate how the detected QTLs exert significant impacts on the timing of maximum growth rate (Fig. 3). The inflection point of growth curves in 2005 displays a difference of three days between the two genotypes (32 vs. 29) at the QTL on chromosome 3, whereas this difference in 2006 is as large as 21 days (45 vs. 24). For the QTL on chromosome 6, we detect a similar pattern for the difference in the inflection point, i.e., 4 days (31 vs. 27) in 2005 and 25 days (41 vs. 16) in 2006. Yet, unlike these two QTLs, there is no year-specific difference in the inflection point for the QTL on chromosome 24, which does not exhibit a significant effect on this developmental characteristic in both years. Although favorable alleles that increase plant growth are contributed by parent Kefeng No. 1 for QTLs detected on chromosomes 3 and 6, parent Nannong 1138-2 contributes such a favorable allele for the QTL on chromosome 24 (Fig. 3).
Genetic mapping has proven to be a powerful approach for map individual genes or quantitative trait loci (QTLs) that control a quantitatively inherited trait . However, this approach would not gain too much insight into the genetic control mechanisms for phenotypic variation if some statistical and biological issues related to the approach are not resolved. Zeng and others are among the first who systematically investigate the effects of multiple linked QTLs on the power of genetic mapping, and further proposed so-called composite interval mapping to separate linked QTLs through using the markers outside the test interval to make the background control of the genome [18, 19, 36]. Many other approaches that take into account a specific statistical, computational or genetic issue of genetic mapping have been developed [37–43]. All these approaches have been instrumental for the characterization of QTLs that control quantitative traits of interest to agriculture, biology and health sciences [44, 20, 45, 46].
More recently, genetic mapping has been integrated with some fundamental biological principles, aimed to generate biologically more meaningful discoveries related to trait formation and development. One of the most important products for this integration is the formulation of a series of statistical models, called functional mapping [9, 15, 16, 24, 8]. Functional mapping capitalizes on the mathematical aspects of biological processes to model the temporal pattern of genetic effects exerted by a QTL in time course. It offers tremendous advantages in the generation of testable biological hypotheses and synthesis of different disciplines for a more comprehensive understanding of biology. Functional mapping have now been extended to explore the roles gene-gene, gene-environment, gene-sex interactions play in directing growth trajectories of a complex trait [15, 10, 11]. Combined with composite interval mapping, functional mapping has been shown to have more power to separate linked QTLs on the same chromosome . This composite functional mapping uses a parametric approach to model the temporal effect of the QTL effect and a nonparametric approach based on the Lengendre polynomial [22, 47, 13] to model the temporal effects of different markers as co-factors. Parametric modeling preserves the biological relevance of the original functional mapping, whereas nonparametric modeling increases the flexibility of functional mapping and its computational efficiency.
In this report, we incorporate composite functional mapping to map QTLs that interact with environments to regulate the process of trait development. While traditional functional mapping detected significant signals for the existence of QTLs for plant height growth trajectories in a recombinant inbred line (RIL) population of soybeans, the new composite functional mapping shows great power to separate multiple linked QTLs on the same chromosome. In total, we detected three significant QTLs on chromosomes 3, 6 and 24 that affect height growth in different years. These QTLs were also found by an allometric model . By integrating environmental factors, the new model detected these QTLs to be operational in both years. However, the magnitudes of their effects vary between different years, showing significant QTL by year interactions. It seems that plant height growth is under strong genetic control. In the functional mapping of rice,  found several significant QTLs for plant heights in two contrasting climates although the expression of these QTLs is not environment-dependent.
Currently, there are three different hypotheses proposed to explain the mechanisms of genotype × environment interactions-heterozygosity, allelic sensitivity, and gene regulation [48, 49]. The new model provides a genome-wide search for QTLs that act in terms of the second hypothesis, i.e., the environment-dependent change of a trait is caused by differentiate expression of a QTL in different environments. However, it is possible to modify the model to test the other hypotheses. For example, the gene regulation hypothesis can be tested by assuming different but epistatically interacting QTLs for mean growth curves and individual growth curves across different environments [50, 51]. Our approach combines powerful statistics and molecular genetics with developmental and ecological mechanisms underlying biological features, relationships and processes to shed light on the genetic basis of complex traits. Such a mechanistic strategy will be powerful to address fundamental questions about plant development and plastic response to changing environments.
This manuscript is partially supported by grants from NSF (No. 0540745), the China National Key Basic Research Program (2006CB1017, 2009CB1184, 2010CB125906), the China National Hightech R & D Program (2006AA100104), the Natural Science Foundation of China (30671266), the China MOE 111 Project (B08025), and the Changjiang Scholars Award.
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