The phenotype of an individual is controlled not only by its genes, but also by the environment in which it grows.
A growing body of evidence shows that the extent to which phenotypic changes are driven by the environment, known as phenotypic plasticity, is also under genetic control, but an overall picture of genetic variation for phenotypic plasticity remains elusive.
Here, we develop a model for mapping quantitative trait loci (QTLs) that regulate environment-induced plastic response. This model enables geneticists to test whether there exist actual QTLs that determine phenotypic plasticity and, if there are, further test how plasticity QTLs control the costs of plastic response by dissecting the genetic correlation of phenotypic plasticity and trait value.
The model was used to analyze real data for grain yield of winter wheat (Triticum aestivum), leading to the detection of pleiotropic QTLs and epistatic QTLs that affect phenotypic plasticity and its cost in this crop.
Phenotypic plasticity is the intrinsic capacity of an organism to alter its phenotype when the environment it grows in changes (Schlichting 1986; Sultan, 2000). In response to elevated CO2 and warmer temperature following industrialization, for example, plants are expected to decrease their leaf lifespan and seed longevity and shift their phenology in response to earlier springs (Nicotra et al., 2010). As a ubiquitous phenomenon, phenotypic plasticity has been extensively studied in terms of its pattern of occurrence and significance for evolution and breeding (Baldwin, 1896; Waddington, 1942; Scheiner, 1993; Agraval, 2001; West-Eberhard, 2005; Fusco & Minelli, 2010; Nicotra et al., 2010; Pfennig et al., 2010). Much research on phenotypic plasticity focuses on its role in the phenotypic innovation and diversity created by the organism in response to environmental changes (Sultan, 2010) as well as on whether such environment-induced plasticity is adaptive (Scheiner, 1993; Nicotra et al., 2010). Increasing effort has been devoted to studying the evolution and regulatory mechanisms of phenotypic plasticity through the identification of genes involved in the plastic response (Wu, 1998; Li et al., 2006; Ma et al., 2008; Beldade et al., 2011; Sommer & Ogawa, 2011). Zhou et al. (2012) provided a comprehensive survey of the differentiation of genome-wide gene expression in Drosophila under different physiological, social, nutritional, chemical and physical environments.
Until now, however, we have not been able to systematically illustrate the genetic architecture of quantitative variation in the plastic response among different individuals. It is possible that a gene(s) pleiotropically controls the same phenotype expressed in different environments. Yet, if the size of the pleiotropic effect exerted by the gene is environment-dependent, such a gene, called the plasticity gene, will create genetic variation in phenotypic plasticity through allelic sensitivity, giving rise to a genotype–environment interaction (Scheiner, 1993; Via et al., 1995). It has been recognized that possessing alternative phenotypes in different environments may be costly or limited because more energy is needed to maintain, produce and acquire a plastic response (DeWitt et al., 1998; Relyea, 2002; Auld et al., 2009; Nicotra et al., 2010). The cost of being phenotypically plastic causes the organism to compromise between the magnitude of phenotypic plasticity and the benefit of producing a particular phenotype. This tradeoff may be mediated by plasticity genes that interact with other genes, such as those for trait expression in a particular environment, via linkage, pleiotropy, or epistasis (Relyea, 2002). Thus, it is essential to identify those plasticity genes, their environment-dependent pleiotropic effects and their interactions with trait genes, in order to better obtain a complete picture of the genetic basis of phenotypic plasticity.
To test the genetic mechanisms by which a quantitative trait is phenotypically plastic, genetic mapping has been developed and used as a powerful tool (Lander & Botstein, 1989; Wu, 1998; Lacaze et al., 2009; Li et al., 2010; Ward et al., 2012). In mapping inflorescence development in Arabidopsis, Ungerer et al. (2003) reported quantitative trait loci (QTLs) that pleiotropically affect trait values expressed in different environments to varying extents. Ward et al. (2012) detected a QTL responsible for the response of flowering time to elevated CO2 in Arabidopsis, based on the difference in trait value between two treatments of 380 and 700 ppm CO2. Other studies have identified specific genes that are involved in the regulation of the plastic response (Li et al., 2006; Zhou et al., 2012). Although a number of plasticity QTLs have been identified, no study has been undertaken to obtain an overall picture of genetic variation for phenotypic plasticity and its biological function and limit.
The purpose of this study was to provide a general synthesis for studying the genetic architecture of phenotypic plasticity and plasticity costs in terms of a web of pleiotropic and epistatic control. The underlying model is based on genetic mapping that dissects phenotypic variation into individual QTLs and their interactions. The model considers multiple QTLs, located at different chromosomal positions, which affect the performance and phenotypic change of a trait over environments. A model for testing epistatic interactions for complex traits is not new (Cheverud et al., 1996; Routman & Cheverud, 1997; Cheverud, 2000), but the idea to jointly test whether epistatic QTLs pleiotropically affect the phenotypic values of a trait in different environments, but to a varying extent, and whether the QTLs for trait values epistatically interact with those for between-environment differences (Schlichting & Smith, 2002) has not previously been reported. The characterization of the genetic basis underlying the tradeoff between phenotypic plasticity and trait performance is of special interest to evolutionary biologists (DeWitt et al., 1998; Relyea, 2002; Auld et al., 2009; Nicotra et al., 2010), but a statistical model that allows this genetic basis to be investigated and quantified has not been available to date. Our model is constructed by defining the cost of plasticity as a negative correlation of phenotypic plasticity with trait performance (Relyea, 2002; Auld et al., 2009). By re-analyzing a published data set for genetic mapping of winter wheat (Triticum aestivum), we have demonstrated and validated the power of the model. The statistical properties of the model are examined through computer simulation.
We assume a mapping population composed of n recombinant inbred lines (RILs) in which there are two different homozygotes at each locus. A genetic linkage map is constructed with molecular markers for this population. RILs allow replicates of the same genotype in different environments, providing excellent material for studying phenotypic plasticity. Suppose these RILs are planted in a randomized block design under two contrasting treatments. The phenotypic plasticity of an RIL is defined as the difference or absolute difference in this genotype between the two treatments. The first definition measures both the direction and the size of phenotypic plasticity, whereas the second one only measures the size of phenotypic plasticity. The performance of an RIL over different treatments is simply described by its across-treatment mean value.
Let y1i and y2i denote the values of a trait for a given RIL i in treatments 1 and 2, respectively. The between-treatment difference and mean of this RIL are calculated as
( reflects the phenotypic plasticity of RIL i; describes the mean performance of this RIL.) We will test whether there are specific QTLs that control phenotypic plasticity and, if any, how they affect phenotypic plasticity through allelic sensitivity. We will further identify QTLs that control the costs of phenotypic plasticity by testing the genetic correlation of trait performance with phenotypic plasticity.
Quantitative genetic effects
Suppose that there are two epistatic QTLs that are located in different genomic locations. Specifically, each QTL is assumed to reside between a different pair of flanking markers. Two genotypes at QTL λ (λ = 1, 2) are symbolized by jλ, with jλ = 1 for QλQλ and 2 for qλqλ. The two QTLs form four joint QTL genotypes expressed as j1j2. The conditional probability of a given genotype at a single QTL, conditional upon the genotype of the flanking markers, for an RIL population is given in Wu et al. (2007). We assume that the two QTLs are located in different marker intervals so that they are independent of each other under the assumption of no interference. Thus, the conditional probability of a joint genotype at the two QTLs, given an interval-marker genotype, is expressed as the product of the conditional probabilities of the corresponding single-QTL genotypes.
The values of phenotypic plasticity () and mean () calculated are the sum of genotypic values at the two QTLs and residual errors, expressed as
where xi is the dummy variable that is designated as 1 if RIL i carries genotype j1j2 and 0 otherwise, and are the genotypic values of j1j2 for phenotypic plasticity and mean, respectively, expressed as
and and are the residual errors that follow a bivariate normal distribution with zero mean vector and covariance matrix
are the variances for the between-treatment difference and across-treatment mean, respectively, and
is the covariance between the difference and mean. Note that and are the genotypic values of j1j2 for the traits expressed in treatment 1s and 2, respectively, and are the variances of the trait expressed in treatments 1 and 2, respectively, and R is the across-treatment correlation.
Plasticity gene detection
According to quantitative genetic theory, the value of a joint QTL genotype (j1j2) is partitioned into the overall mean, additive effects at two different QTLs, and their additive × additive epistasis. Thus, Eqn 2 is re-expressed as
(μ1 and μ2, the overall mean of the trait expressed in treatments 1 and 2, respectively; a11 and a12, the additive effects of QTLs 1 and 2 on the trait in treatment 1, respectively; I1, the additive × additive epistatic effect between the two QTLs for the trait in treatment 1; a21 and a22, the additive effects of QTLs 1 and 2 on the trait in treatment 2, respectively; I2, the additive × additive epistatic effect between the two QTLs for the trait in treatment 2.) Plasticity genes are suggested to exist if at least one of the differences a21 – a11, a22 – a12, and I2 – I1 is different from zero.
Genetic basis of the tradeoff
To show the joint effects of QTLs on phenotypic plasticity and mean performance, we partition the genotypic values (Eqn 2) into their underlying components in the following way:
(μ− and μ+, the overall means of phenotypic plasticity and mean performance, respectively; and , the additive effects of QTLs 1 and 2 on phenotypic plasticity, respectively; and , the additive effects of QTLs 1 and 2 on mean performance, respectively; I− and I+, the additive × additive epistatic effects between the two QTLs on phenotypic plasticity and mean performance, respectively.) The plastic response may be expressed at a cost of producing a specific phenotype, leading to the tradeoff between phenotypic plasticity and trait performance. The genetic control of the tradeoff can be tested by comparing the genetic effects of QTLs on these two characteristics. Also, we can use the genetic correlation to describe the genetic control of the tradeoff, expressed as
(, the genetic variance for phenotypic plasticity; , the genetic variance for mean performance.) The overall genetic correlation includes the genetic components attributable to QTL 1 (), QTL 2 () and their interactions (rI = I−I+/(σPPσMP)).
With the phenotypic values and marker information measured for n RILs, we construct the likelihood of QTLs based on a mixture model,
where and are the conditional probability of a genotype jl and j2 at QTLs l and 2, respectively, given the marker genotype of RIL i, and is a bivariate normal distribution of RIL i, expressed as
is the correlation between the between-treatment difference and across-treatment mean.
The EM algorithm is implemented to obtain the maximum likelihood estimates (MLEs) of unknown parameters (Supporting Information Notes S1). With the MLEs of , we obtain the MLEs of the overall means, and additive and epistatic effects, by solving Eqn 5, as
The residual variances of the trait in treatments 1 () and 2 (), and the across-treatment correlation (R) can be estimated from a group of Eqns 3, 4, and 9. The estimates of these parameters can be used to estimate the proportions of total phenotypic variances explained by specific QTLs in each treatment.
After the genetic parameters have been obtained, we need to make hypothesis tests for the existence of QTLs for phenotypic plasticity and mean performance and the action mechanisms of these QTLs. The existence of a QTL can be tested by formulating the following hypotheses:
H1: at least one equality in the H0 does not hold,
where the null hypothesis H0 states that there is no QTL involved in phenotypic plasticity and mean performance, whereas the alternative hypothesis H1 suggests that there are different QTL genotypes, showing the existence of QTLs. The test statistic is the log-likelihood ratio (LR) of the full (H1) over reduced (H0) model. The critical threshold for declaring the existence of QTLs can be determined from permutation tests (Churchill & Doerge, 1994).
If significant QTLs are found, we will need to test how the QTLs detected trigger their genetic effects on phenotypic plasticity. The mode of QTL actions includes the additive effects at individual QTLs and the additive × additive epistatic effects between the two QTLs. Each of these effects may operate individually for phenotypic plasticity. This can be tested by formulating a null hypothesis in which an effect considered is set to be zero, expressed as
For a given null hypothesis, parameter estimates can still be obtained using the EM algorithm described above, with a constraint that the genotypic value of one QTL genotype is expressed as a function of those of the rest, derived from Eqns 16-21, under the null hypothesis.
If the null hypotheses Eqns 11 and 12 are both rejected, this suggests that QTL 1 pleiotropically affects the trait values expressed in treatments 1 and 2. The rejection of both hypotheses Eqns 13 and 14 implies a significant pleiotropic effect of QTL 2. Similarly, QTLs 1 and 2 interact with each other to pleiotropically affect the trait values expressed in treatments 1 and 2 if hypotheses Eqns 15 and 16 are both rejected.
Now, we can test through which mode a QTL affects phenotypic plasticity. The allelic sensitivity mechanism can be tested through the null hypothesis:
The rejection of the above null hypotheses suggests that phenotypic plasticity is caused by the additive effect of QTL 1, additive effect of QTL 2, and epistatic effect of QTLs 1 and 2, respectively.
Because of its costs, phenotypic plasticity exhibits a tradeoff with the benefit of producing a particular phenotype in an averaged environment. The genetic basis of this tradeoff involves plasticity genes and genes for trait mean which act via linkage, pleiotropy, or epistasis (Relyea, 2002). We can test whether phenotypic plasticity and mean performance have a shared genetic basis by formulating two null hypotheses
If both null hypotheses above are rejected, this implies that the significant QTLs detected affect pleiotropically these two biological characteristics. Otherwise, the QTLs only affect one of the characteristics.
If phenotypic plasticity and mean performance share a similar genetic basis, we need to test the mode of genetic actions by individual QTLs. How each genetic effect jointly affects phenotypic plasticity and mean performance can be tested by formulating the corresponding pairs of null hypotheses; for example,
for testing the pleiotropic effect of QTL 1,
for testing the pleiotropic effect of QTL 2, and
for testing the pleiotropic effect of the QTL–QTL interaction. The rejection of both null hypotheses in each pair implies that this mode of action is pleiotropic for phenotypic plasticity and mean performance. The directions of each effect on these two characteristics determine whether their relationship is synergistic or antagonistic.
We used a mapping population of 222 doubled haploid (DH) lines derived from two inbred cultivars of winter wheat (T. aestivum), Arche and Recital, which differ in the response to N deficiency (Le Gouis et al., 2000). A linkage map was constructed for this mapping population using 190 markers. Assuming Haldane's mapping function, the map is 2614 cM long, composed of markers distributed in 30 linkage groups. More details on the genetic map are given in Laperche et al. (2006). The DH population was planted with replicates in three locations during two consecutive years under two nitrogen levels: a high N supply (1) and a low N supply (2). In each study, grain yield (g m−2) was measured for each plant and the mean of each DH was then calculated. The detailed experimental design of this multi-location and multi-year study is described in Zheng et al. (2010). As an example used to demonstrate our new model, we selected a two-treatment study from one location and one year, in which the mean and difference of grain yield between the two treatments were calculated for each line.
Fig. 1 plots the trait values of grain yield for all mapping DH lines over the two treatments. Overall, grain yield does not display dramatic differences between the two levels of N supply, but the direction and extent of differences vary considerably among the lines. Such variation in phenotypic plasticity to N supply suggests the possible occurrence of genotype–environment interactions. By scanning the entire linkage map for the existence of two interacting QTLs, the model derived from likelihood (Eqn 8) can map and estimate QTLs that affect phenotypic plasticity and mean performance of grain yield. We drew a landscape of LR values across the map position calculated from the hypothesis of Eqn 10 to test the distribution of epistatic QTLs (Fig. 2). A total of four pairs of QTLs were detected to jointly affect phenotypic plasticity and mean performance, which are (A) a pair from chromosome 6, (B) a pair each from chromosomes 11 and 12, (C) a pair from chromosome 12, and (D) a pair each from chromosomes 12 and 22.
Table 1 shows the estimates of the main additive genetic effects for each QTL and additive × additive epistatic genetic effects between a pair of QTLs on yield grain for two levels of N supply. Hypothesis tests using Eqns 17-19 show that the QTLs detected are all sensitive to varying N supplies and that the values of epistatic interactions between all pairs of QTLs vary between the treatments. In particular, the QTL detected between markers rht_b1 and rht1 on chromosome 12 altered grain yield by as much as 3.25 m g−2 under high N supply but only by 0.28 m g−2 under low N supply. The additive × additive epistatic effect triggered by the QTL located between markers cfa2086 and gpw2046 on chromosome 6 and the QTL located between markers gwm526 and gwm382b on chromosome 6 changes from high to low N supply. All these treatment-dependent effects contribute to the phenotypic plasticity of yield grain through allelic sensitivity.
Table 1. Test for allelic sensitivity in winter wheat: additive genetic effects estimated for individual quantitative trait loci (QTLs) and epistatic interactions between a pair of QTLs at two levels of nitrogen (N) supply
In a scatter plot of phenotypic plasticity (z−) and mean performance (z+), it is found that less plastic DH lines tend to have higher yields over the two treatments (Fig. 3). This negative correlation can be explained as being attributable to the cost of being phenotypically plastic, leading to the tradeoff between phenotypic plasticity and trait mean. We further determined the genetic basis of the tradeoff. Table 2 provides the MLEs of the additive and epistatic effects between a pair of QTLs on phenotypic plasticity and trait mean. The result from the hypothesis test of Eqn 20 suggests that these two characteristics share a common genetic basis, because both null hypotheses are rejected. Three pairs of null hypotheses (Eqns 21-23) were used to test whether the shared genetic basis is attributable to the operation of pleiotropic QTLs (Table 2). It was found that, in most cases, QTLs and their interactions pleiotropically affect phenotypic plasticity and mean performance in different directions, thus explaining the genetic basis of their tradeoff. The genetic correlations between phenotypic plasticity and mean performance attributable to different genetic components explain the extent to which these two characteristics share the same genetic basis (Table 2).
Table 2. Test for the genetic basis of the tradeoff between phenotypic plasticity and mean performance in winter wheat: additive genetic effects of individual quantitative trait loci (QTLs) and their epistatic interaction effects on these two characteristics across two levels of nitrogen (N) supply
The genetic correlations between phenotypic plasticity and mean performance attributable to QTL 1, QTL 2 and their epistasis, calculated using Eqn 7, are given in parentheses.
We performed simulation studies to investigate the statistical properties of the models for plasticity mapping. By mimicking the real example above, we simulated 220 DH lines which are genotyped for 20 markers. The first 10 form a linkage group, whereas the second 10 are located in a different group. All markers are evenly spaced with a marker interval of 20 cM. The simulated population is grown in two environments and phenotyped for a trait expressed in the two environments. The simulation was designed to detect plasticity genes and to determine the genetic basis of plasticity costs.
To test the impact of allelic sensitivity on phenotypic plasticity, we assume two QTLs that are each located at a particular position in a different linkage group (in cM from the first marker on the left). These QTLs are expressed differently in the two environments (Table 3). With these genetic effects, we simulated the phenotypic values of each line expressed in the two environments by assuming bivariate normal errors that are used to adjust the heritability of the traits. The model was used to map the phenotypic plasticity and mean performance of the trait in the two environments. The QTLs that display allelic sensitivity can be estimated accurately, including their additive and epistatic effects, when the trait has a modest heritability (0.1) (Table 3). As expected, the precision of parameter estimation can be substantially improved when the heritability increases from 0.1 to 0.2.
Table 3. Simulation study for testing the allelic sensitivity model: maximum likelihood estimates (MLEs) of additive and epistatic genetic effects and other model parameters as well as the standard deviations (SDs) of the MLEs in each treatment from the simulated data obtained by mimicking the data structure of the wheat mapping population
H2 = 0.1
H2 = 0.2
MLE ± SD
MLE ± SD
Different heritabilities are considered.
54.92 ± 2.72
54.78 ± 1.51
105.47 ± 2.97
105.17 ± 1.73
42.27 ± 0.22
42.30 ± 0.15
41.20 ± 0.17
41.20 ± 0.12
−0.52 ± 0.24
−0.52 ± 0.16
0.15 ± 0.24
0.15 ± 0.16
0.13 ± 0.19
0.11 ± 0.12
−0.02 ± 0.18
−0.01 ± 0.12
−0.45 ± 0.25
−0.41 ± 0.17
−1.09 ± 0.17
−1.04 ± 0.13
6.73 ± 0.64
3.39 ± 0.32
9.70 ± 0.90
4.85 ± 0.45
−0.31 ± 0.06
−0.31 ± 0.06
The power of detecting additive and epistatic effects on phenotypic plasticity was estimated through simulation studies. In general, there is good power for the detection of plasticity QTLs, even when the sample size used is modest (250) (Table 4). If the heritability is low, a large sample size (say 500) is needed for plasticity QTL detection. We have also calculated the power of detecting the genetic control of plasticity costs by testing the genetic basis shared between phenotypic plasticity and mean performance based on the hypothesis of Eqn 20 (Table 4). The genetic control of plasticity costs is regarded as being significant only when two null hypotheses in Eqn (Eqn 20) are rejected. When the heritability is low, a relatively large sample size is required to determine the genetic basis of costs of plasticity. It was observed that the models for detecting plasticity QTLs and cost QTLs each have very small false positive rates (< 0.05).
Table 4. Power of detecting genetic effects on phenotypic plasticity and its costs under different heritabilities and sample sizes
Costs of plasticity
Phenotypic plasticity, that is, environment-induced change in the phenotype produced by a genotype for a particular trait, may have played a pivotal role in an organism's adaptation to varying environments (Schlichting, 1986; Scheiner, 1993; Sultan, 2000; Agraval, 2001; Pigliucci, 2005; West-Eberhard, 2005; Pfennig et al., 2010). An increasing body of evidence shows that specific genes may exist to determine the magnitude and direction of phenotypic plasticity (Wu, 1998; Ungerer et al., 2003; Ma et al., 2008; Lacaze et al., 2009; Li et al., 2010). Thus, revealing the genetic basis of phenotypic plasticity has emerged as an important goal in ecological, developmental and evolutionary genetic studies (Schlichting & Smith, 2002; Li et al., 2006). Genetic mapping using molecular markers has proved to be a powerful tool for mapping QTLs and genetic interactions that control complex traits. In this study, we developed and assessed a genetic mapping model for detecting the genetic architecture of phenotypic plasticity and its costs.
The allelic sensitivity of genes has been regarded as a genetic mechanism for phenotypic plasticity (Via et al., 1995; Schlichting & Smith, 2002; Ma et al., 2008), by which the organism responds rapidly to environmental perturbations without the time lag required for the response to natural selection on allelic variants (Zhou et al., 2012). By quantitatively measuring phenotypic plasticity as the difference between trait values expressed in different environments, our model provides a precise characterization of allelic sensitivity. The quantitative tests of various hypotheses about the pleiotropic and epistatic control of phenotypic plasticity allow us to better obtain a complete picture of the genetic architecture of this phenomenon. Although phenotypic plasticity is beneficial for an organism's adaptation, it can be costly or limited as a consequence of the extra energy required to maintain and produce it (DeWitt et al., 1998; Relyea, 2002; Auld et al., 2009; Nicotra et al., 2010). The idea of identifying the genetic basis of plasticity costs is not new (Callahan et al., 2005), but our model has for the first time presented a framework for mapping cost QTLs. More importantly, this model has synthesized a procedure for jointly identifying specific QTLs for phenotypic plasticity and its tradeoff with an overall phenotype produced, allowing tests of various modes of actions and interactions.
The model was used to re-analyze a published data set for mapping grain yield of winter wheat under two different treatments, high and low N supplies (Laperche et al., 2006; Zheng et al., 2010). Several QTLs have been detected that affect phenotypic plasticity through allelic sensitivity and the costs of producing a specific phenotype under two levels of N supply. Results from simulation studies by mimicking the example of the winter wheat mapping experiment suggest that the model possesses desirable statistical properties and, therefore, the identification of plasticity QTLs and plasticity-cost QTLs in winter wheat is reasonably precise. Results from simulation studies under a range of parameters have suggested that the model should well be applied in practical mapping settings.
The mapping model has focused on the characterization of QTLs that affect phenotypic plasticity and its costs. To illustrate the regulatory network of phenotypic plasticity for complex traits, ‘omics’ data should be collected and integrated with our mapping model to detect cis- or trans-acting QTLs. This integration will provide a foundation for testing mechanisms that link environmental inputs to alterations in gene and protein expression and for characterizing how the expression of environmentally sensitive genes is associated with the plasticity of organismal phenotypes (Zhou et al., 2012). In addition, as opposed to phenotypic plasticity, there is another important phenomenon, called environmental canalization, by which a genotype produces the same phenotype in different environments (Waddington, 1942). While phenotypic plasticity allows an organism to respond to environmental changes, environmental canalization buffers the organism's phenotypes against environmental perturbations. As a crucial prerequiste for optimal fitness, the organism should be equipped with the capacity to balance these two phenomena. The genetic underpinnings of the maintenance of this balance can be studied by identifying environmentally sensitive transcripts and environmentally robust transcripts (Zhou et al., 2012).
To comprehend the phenotypic plasticity of complex traits, our model should be extended in at least three areas. First, the model should allow more than two environments to be involved in genetic mapping, in which a reaction norm curve is used to describe the plastic response. Wang et al. (2013) provided a mathematical model for characterizing the pattern of reaction norm trajectories and integrating it into genetic mapping for the detection of environment-specific QTLs. By incorporating Wang et al.'s model, the idea presented in this article should be able to characterize the genetic basis of phenotypic plasticity and its costs over a range of environments. The consideration of a single environmental factor obviously is not adequate to draw a complete picture of the genetic architecture of how an organism responds to the environment in which it grows, given that the environment is composed of multiple signals. For example, the growth of a plant depends not only on sunlight, but also on water, nutrients and CO2. A more sophisticated model that describes the phenotypic plasticity of an organism to multiple environmental factors (Yap et al., 2011) should be developed. In addition, phenotype formation of a trait involves a series of developmental stages, at each of which genes may play a different role in the phenotypic plasticity of the trait (Wu & Lin, 2006; Nicoglou, 2011). Thus, the model should be integrated with developmental biology, with the aim of unraveling the genetic complexity and organization that underlie phenotypic alterations of an organism's response to environmental and developmental signals (Schlichting & Smith, 2002).
This work was partially supported by NSF/IOS-0923975, Changjiang Scholars Award, and ‘Thousand-person Plan’ Award.