B RIEFINGS IN BIOINF ORMATICS . VOL 16. NO 1. 24 ^31 Advance Access published on 13 December 2013

doi:10.1093/bib/bbt089

A QTL model to map the common genetic basis for correlative phenotypic plasticity Tao Zhou*, Yafei Lyu*, Fang Xu,Wenhao Bo, Yi Zhai, Jian Zhang, Xiaoming Pang, Bingsong Zheng and Rongling Wu Submitted: 10th September 2013; Received (in revised form) : 13th November 2013

Abstract As an important mechanism for adaptation to heterogeneous environment, plastic responses of correlated traits to environmental alteration may also be genetically correlated, but less is known about the underlying genetic basis. We describe a statistical model for mapping specific quantitative trait loci (QTLs) that control the interrelationship of phenotypic plasticity between different traits. The model is constructed by a bivariate mixture setting, implemented with the EM algorithm to estimate the genetic effects of QTLs on correlative plastic response. We provide a series of procedure that test (1) how a QTL controls the phenotypic plasticity of a single trait; and (2) how the QTL determines the correlation of environment-induced changes of different traits. The model is readily extended to test how epistatic interactions among QTLs play a part in the correlations of different plastic traits. The model was validated through computer simulation and used to analyse multi-environment data of genetic mapping in winter wheat, showing its utilization in practice. Keywords: phenotypic plasticity; QTL; genetic correlation; EM algorithm

INTRODUCTION Phenotypic plasticity has been thought to be an important force for microevolution and species diversification [1–4]. It has been argued that selection on plasticity responses to environment pressures is not

independent among different traits [5, 6], because an organism always functions in an integrated manner [7]. An issue naturally arises about how genes that enable a functional response to the environment for one trait also affect adaptive changes of other traits.

Corresponding author. Rongling Wu, Center for Computational Biology, Beijing Forestry University, Beijing 100083, China. Tel: þ01186 10 6233 6283. Fax: þ01186 10 6233 6283. E-mail: [email protected]; Center for Statistical Genetics, Pennsylvania State University, Hershey, PA 17033, USA. Tel: þ001 717 531 2037. Fax: þ001 717 531 0480. E-mail: [email protected] *These authors contributed to this work equally. Tao Zhao is a PhD student in forest genetics and tree breeding in the Center for Computational Biology at Beijing Forestry University. He studies the genetic basis of salt tolerance in forest trees. Yafei Lyu is a PhD student in bioinformatics at The Pennsylvanis State University. His research focuses on the statistical genetics of quantitative traits. Fang Xu is a PhD student in forest genetics and tree breeding in the Center for Computational Biology at Beijing Forestry University. He studies the genetic architecture of quantitative traits using an integrative computational and experimental approach. Wenhao Bo is a post-doctoral researcher in the Center for Computational Biology at Beijing Forestry University. He studies the genetics of wood production and adaptive traits in forest trees. Yi Zhai is a research assistant in the Center for Statistical Genetics at The Pennsylvanis State University. Her research focuses on the statistical modeling of RNA-seq data. Jian Zhang is a PhD student in forest genetics and tree breeding in the Center for Computational Biology at Beijing Forestry University. His research interest lies in the identification of salt-resistance genes in woody plants. Xiaoming Pang is an Associate Professor of Tree Breeding at Beijing Forestry University. His research interest focuses on the utilization of molecular genetics and biotechnologies to study population genetic diversity and map quantitative trait loci in fruit trees. Bingsong Zheng is a Professor in forest genetics in the School of Forestry and Biotechnology at Zhejiang A&F University. His research focuses on the molecular genetics of complex traits in forest trees. Rongling Wu is a Changjiang Scholars Professor of Genetics and the Director of the Center for Computational Biology at Beijing Forestry University. His interest is to unravel the genetic roots for the outcome of a biological trait by dissecting the trait into its biochemical and developmental pathways. ß The Author 2013. Published by Oxford University Press. For Permissions, please email: [email protected]

Mapping the common genetic basis of phenotypic plasticity Genetic mapping offers unique power to identify genes or quantitative trait loci (QTLs) involved in phenotypic plasticity in response to environmental perturbations [8–10]. Several authors have detected specific QTLs that affect environment-induced response through allelic sensitivity [10–14]. Zhai et al. [15] developed a synthetic model that studies the genetic mechanisms of QTLs that contribute to phenotypic plasticity. This model allows several hypothesis tests about the environmental impact on QTL expression, its magnitude, direction and pleiotropy, providing an overall picture of genetic variation for phenotypic plasticity. In this article, we extended Zhai et al.’s model to map QTLs that leads to genetic correlations of environment-dependent alterations among two different traits. The resulting bivariate model can identify not only the allelic sensitivity of QTLs for single traits expressed in different environments, but also the degree and pattern of genetic covariance, triggered by QTLs, between plastic responses of different traits. The model is constructed within a mixture-model framework by which the genetic control of plasticity correlation can be tested and characterized. By re-analysing a published data set for QTL mapping of winter wheat planted in multiple environments, we have validated the utility and power of the model. Computer simulation was used to investigate statistical properties of the model.

MODEL Plasticity experiment Consider a mapping of n recombinant inbred lines (RILs), in which there are two genotypes for alternative alleles at each locus. A genetic linkage map is constructed with molecular markers for this population to identify QTLs that affect correlated phenotypic plasticity. All RILs are grown in two different environments. The phenotypic plasticity of an RIL is defined as the difference of this genotype between the two treatments. For a given RIL i, let y1i, y2i and z1i, z2i denote the values of two traits in environments 1 and 2, respectively. The between-environment differences of this RIL are calculated as y i ¼ y2i  y1i z i

¼ z2i  z1i

which reflect the phenotypic plasticity of traits y and z, respectively. Suppose there is a QTL which is

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located on the linkage map that affects the difference of two traits between environments. The genotypes at the QTL are symbolized by j, with j ¼ 1 for QQ and 2 for qq and the conditional probability of a QTL, conditional upon the genotype of the flanking markers, is given in Wu et al. [16]. Let myjk and mzjk denote the genotypic values of QTL genotype j expressed in environment k (k ¼ 1, 2). We partition the genotypic values of the phenotypic plasticity of each trait into the underlying components expressed as Trait y Environ 2  Environ 1

Trait z Environ 2  Environ 1

m m y1 ¼ my12  my11 z1 ¼ mz12  mz11 ¼ my2 þ ay2  my1  ay1 ¼ mz2 þ az2  mz1  az1   ¼ m þ a ¼m y y z þaz m m y2 ¼ my22  my21 z2 ¼ mz22  mz21 ¼ my2  ay2  my1 þay1 qq ¼ mz2  az2  mz1 þ az1   ¼ m  a ¼ m y y z  az

QQ

ð1Þ

where my1, my2 and mz1, mz2 are the overall means of traits y and z expressed in environments 1 and 2, respectively, ay1 and ay2 are the additive effects of QTL on trait y in environments 1 and 2, respectively, and az1 and az2 are the additive effects of QTL on trait z in environments 1 and 2, respectively. Plasticity genes that control the environment-dependent change of trait value are suggested to exist if  difference a y ¼ ay2  ay1 or az ¼ az2  az1 is different from zero. The genetic covariance between environment-dependent changes of traits y and z explained  by the QTL is calculated as a y az . Co-plasticity genes are defined as those that control the environmentdependent change of trait correlation. Co-plasticity  genes are thought to exist if a y az is different from zero.

Likelihood function  Let ðy i , zi Þ denote the vector of phenotypic plasticity of two traits for RIL i. Given phenotypic plasticity values and marker information, the likelihood function based on a mixture model can be constructed as

Lðy , z Þ ¼

n  Y

    o1ji f1 ðy i , zi Þþo2ji f2 ðyi , zi Þ ,

ð2Þ

i¼1

where ojji is the conditional probability of QTL genotype j, given the marker genotype of RIL i;

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Zhou et al.

 and fj ðy i , zi Þ is a bivariate normal distribution of RIL i, expressed as

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼  1  R2 s 2ps y z ( " !  2 y 1 i  myj exp  2ð1  R2 Þ s y !   2 #)     yi  myj zi  mzj zi  m zj 2R þ s s s y z z

     jji y z i  myj i  mzj i¼1 j¼1  , z m 2

n P 2  2 P y i myj i zj jji þ sz n sy 2

nR3  1þR sy sz R¼

 fj ðy i , zi Þ

 ojji fj ðy i , zi Þ

j0 ¼1

 oj0 ji fj0 ðy i , zi Þ

:

where ð3Þ r¼

ð4Þ

In the M step, the genotypic values of traits y and z and variance and correlation are estimated from the posterior probabilities using n P

m yj

¼

jji y i

i¼1 n P

ð5Þ jji

i¼1 n P

m zj ¼

jji z i

i¼1 n P

ð6Þ jji

i¼1

s2y

" n X 2  2 X 1   ¼   m y jji i yj nð1  R2 Þ i¼1 j¼1 #        R yi  myj zi  mzj r

" n X 2  2 X 1     m z s2z ¼ jji i zj nð1  R2 Þ i¼1 j¼1 #        0 R yi  myj zi  mzj r

ð9Þ

i¼1 j¼1

 with s y and sz are the standard deviations of the phenotypic plasticity of traits y and z, respectively, and R is the correlation between the two traits’ plasticity. The EM algorithm is implemented to estimate unknown parameters involved in the likelihood (2). The algorithm consists of two steps in each iteration. In the E step, the expected log-likelihood of unknown QTL genotypes given the observed phenotypic and maker data and current parameter estimates is estimated. In the M step, the expected log-likelihood is maximized. Specifically, in the E step, the posterior probability of a QTL genotype for RIL i is calculated by

jji ¼ P2

n P 2 P

ð7Þ

ð8Þ

sy sz and r0 ¼ : sz sy

Note that r and r0 on the right side of Equations (7) and (8) are based on the estimates of standard deviations in the previous iteration and R on the right side of Equation (9) is its estimate in the previous iteration. We can calculate the posterior probability in the E step (4) by given initial values for unknown parameters. Then, in the M steps (5)–(9), the estimates of the parameters are obtained. This process is repeated until when the estimates are stable to get the maximum-likelihood estimates (MLEs) of the parameters.

Hypothesis tests The existence of a QTL that affects phenotypic plasticity can be tested by formulating the following hypotheses:    H0 : m yj  my and mzj  mz

[ j ¼ 1, 2

ð10Þ

H1 : At least one equality in the H0 does not hold,

where the null hypothesis H0 states that there is no QTL involved in the phenotypic plasticity of two traits, whereas the alternative hypothesis H1 suggests the existence of such a QTL. The test statistic is the log-likelihood ratio (LR) of the full (H1) over reduced model (H0). Because the QTL position is not identifiable in the H0, the LR calculated may not obey a chi-square distribution with three degrees of freedom. Thus, we used an empirical approach from permutation tests to determine the critical threshold [17]. The null hypothesis in Equation (10) corresponds  to a y ¼ 0 and az ¼ 0, implying that allelic effects on the two traits studied are stable from one environment to next. To test whether the QTL pleiotropically affects the phenotypic plasticity of the two traits, we formulate the two null hypotheses as follows: H0 : a y ¼0

ð11Þ

H0 : a z ¼ 0:

ð12Þ

If both null hypotheses (11) and (12) are rejected, this means that the QTL exhibits a pleiotropic effect

Mapping the common genetic basis of phenotypic plasticity on plastic response of traits y and z, and this QTL is called the co-plasticity QTL. The direction of the genetic effect of the co-plasticity QTL on these two traits determines whether the relationship of their plastic responses is synergistic or antagonistic. The covariance of phenotypic plasticity between traits y and z, explained by the co-plasticity QTL, can be estimated as a^  ^ ^ ^ ya z . The ratio of a ya z to the total covariance of phenotypic plasticity between the two traits can be used as a parameter that measures the extent to which a co-plasticity QTL determines correlative environment-dependent response of different traits. For any given null hypothesis above, parameter estimates can still be obtained using the EM algorithm described, with a constraint that the genotypic value of one QTL genotype is expressed as a function of those of the rest, under the null hypothesis.

MODELVALIDATION Real example A mapping population of 222 doubled haploid (DH) lines was derived from two inbred cultivars of winter wheat (Triticum aestivum), Arche and Recital, which differ in reaction to N deficiency [18]. Using this mapping population, a linkage map of winter wheat was constructed from 190 markers. The markers covered 2614 cM of the genome distributed in 22 linkage groups. Laperche et al. [19] provided a detail on the genetic map. As described in Zheng et al. [20], 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, kernel number and thousand kernel weight were measured for each plant and the mean of each DH was then calculated. Here, a two-treatment study from one location and 1 year was selected to study phenotypic plasticity, in which the differences of kernel number and thousand kernel weight between the two treatments were calculated for each line. By scanning all linkage groups from 1 to 22 at every 2 cM, two peaks of the LR profile beyond the critical threshold were detected on linkage groups 12 and 14, suggesting possible existence of two QTLs (Figure 1). Table 1 gives the chromosomal positions of the QTLs detected and their genotypic values for kernel number and thousand kernel weight, respectively. Kernel number decreases for both genotypes at the QTL on linkage group 12

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from high N supply to low N supply, but the genotype for the Recital alleles decreases to a larger extent than that for the Arche alleles. Yet, both genotypes increase thousand kernel weight from high N supply to low N supply, though at different degrees. Overall, the genotype is more sensitive to N supply if it is derived from the Recital alleles than Arche alleles. The genetic effects on N-induced change of kernel number and thousand kernel  weight are calculated as a y ¼ 830 and az ¼ 0.64, i.e. 8% and 44% relative to the population mean, respectively. The hypothesis test separately for the two effects based on Equations (11) and (12) indicates that the QTL has a pleiotropic effect on the phenotypic plasticity of these two traits (P < 0.05). The test based on Equation (13) suggests that the  covariance (a y az ) between plastic responses of the two traits is significant (P < 0.05). A similar conclusion can be detected for the QTL on linkage group 14, but it seems that the direction of the genetic effect on the plastic response of the two traits is contrast to that of the QTL on linkage group 12. The genetic effects of the former are cal culated as a y ¼ 245 and az ¼ 0.64, i.e. 2% and 32% relative to the population mean, respectively. This QTL was detected to only affect kernel thousand weight, with non-significant effect on kernel number. It has no contribution to the interrelationship between the phenotypic plasticity of two traits based on the covariance test of Equation (13).

Computer simulation Simulation studies were performed to investigate statistical properties of the bivariate model for mapping co-plasticity genes. A DH population of 220 lines was simulated by mimicking the real example above. Two linkage groups each consisted of 10 markers spaced by 20 cM were simulated. Each linkage group was assumed to harbor a QTL for two traits. Grown in two distinct environments 1 and 2, the mapping population was simulated for two normally distributed traits y and z. Using given additive effect values of the QTLs in two environments, denoted as ay1, ay2; az1, az2 (1), the values of phenotypic plasticity of two traits were simulated, each with heritability of 0.05, 0.10 or 0.20. The bivariate model provides accurate estimates of the QTL positions and genotypic values of phenotypic plasticity (Tables 2 and 3), suggesting that the results detected in the example above from the model are reasonable and interpretable. With a

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Zhou et al.

Figure 1: The plot of the LRs across 22 linkage groups that test the existence of QTLs responsible for correlative phenotypic plasticity of kernel number and kernel thousand weight in a mapping population of winter wheat. The critical threshold for the 0.05 significance level, indicated by a horizontal line, was determined from 100 permutation tests. Two arrowed vertical lines indicate the positions of two QTLs detected.

Table 1: MLEs of the positions and genotypic values at the QTLs detected on linkage groups 12 and 14 for plastic response of kernel number (y) and kernel thousand weight (z) from low to high N Linkage Marker group interval

Position Genotypic value QQ m y1

1 12 2 14

gpw1108 -rht_b1 1019.4 gpw595-cfa2149 1252.7

qq m z1

m y2

m z2

9620 0.60 11280 1.89 10 881 2.11 10 392 0.83

modest sample size (200), all parameters can be better estimated even with a small heritability (0.05). As expected, when either sample or heritability, or both increases, the estimation precision of parameters increases dramatically. Additional simulation was performed to examine the power of QTL detection and false positive rates (FPRs) (Table 4). FPR was estimated by assuming that no QTL is involved in phenotypic plasticity (i.e. heritability is zero). In general, FPR is very low even with a modest sample size. The model was detected to display adequate power (0.73) for co-plasticity QTL identification when the sample size used is modest (200) and when the heritabilities of traits studied are small (0.05). The power increases dramatically with sample size and heritability.

DISCUSSION In nature, no biological trait can function in isolation; the plastic change of a specific trait to heterogeneous environments will always take place alongside other changes in phenotype [7, 21–23]. An understanding of how different traits are correlated in a particular environment, how trait correlation responds to environmental change, and how phenotypic plasticity of different traits is correlated is critical to advancing evolutionary, ecological and developmental studies. Although the first two questions have been heavily considered in the literature [21, 24], the last question has not received sufficient attention. In this article, we provide and assess a comprehensive framework to dissect how the phenotypic plasticity of one biological trait relates to the plastic response of another trait to heterogeneous environments through genetic control. We show that this framework offers a powerful tool for studying the genetic mechanisms of phenotypic plasticity. To examine the genetic basis of how two different traits are correlated in terms of their phenotypic plasticity, we need to replicate and grow a mapping population in two contrast environments in each of which two traits are measured for each mapping individual. A conventional approach for analysing this data is to develop a four-variate model that incorporates 2  2 ¼ 4 combinations of different traits and

Mapping the common genetic basis of phenotypic plasticity

29

Table 2: MLEs of QTL position and genotypic values for phenotypic plasticity of two traits from simulated data based on true parameters by mimicking the first QTL detected in the winter wheat example Sample size

H2

True 0.05 0.1 0.2 0.05 0.1 0.2 0.05 0.1 0.2

200

MLE

400

800

Position

QQ

qq

35

m y1 ¼ 9500

m z1 ¼ 2

m y2 ¼ 11 000

m z2 ¼ 0.6

34.94  4.66 34.91 3.68 34.95  2.58 35.02  3.24 35.04  2.47 35.04 1.84 35.01 2.30 35.05 1.64 34.96 1.25

9499  345 9498  242 9493 157 9501 246 9507 169 9503 114 9500 173 9502 117 9501 77

2.004  0.316 2.002  0.217 1.993  0.145 2.005  0.222 1.995  0.153 1.996  0.104 2.001 0.153 2.004  0.112 1.999  0.071

10 996  347 11001 229 11004 160 10 999  243 11002 168 10 998 115 10 991 179 10 997 121 11003  78

0.591 0.319 0.599  0.214 0.605  0.146 0.607  0.229 0.601 0.154 0.598  0.106 0.596  0.162 0.600  0.111 0.604  0.073

The standard errors of the estimates were calculated from1000 simulation replicates.

Table 3: MLEs of QTL position and genotypic values for phenotypic plasticity of two traits from simulated data based on true parameters by mimicking the second QTL detected in the winter wheat example Sample size

H2

True 0.05 0.1 0.2 0.05 0.1 0.2 0.05 0.1 0.2

200

MLE

400

800

Position

QQ

20

m y1

qq

20.50  9.29 20.65  6.68 20.33  3.80 20.63  6.43 20.30  4.00 20.19  2.61 20.48  3.86 20.26  2.52 20.03 1.44

10 806 117 10 804  76 10 801 53 10 805  80 10 800  56 10 801 36 10 803  55 10 800  38 10 801 26

¼ 10 800

m z1

¼ 2.2

m y2 ¼ 10 300

m z2 ¼ 0.8

2.217  0.324 2.222  0.217 2.202  0.143 2.216  0.225 2.203  0.157 2.202  0.109 2.203  0.152 2.199  0.111 2.198  0.073

10 289 118 10 299  80 10 296  51 12 931 79 10 297  55 10 299  35 10 299  56 10 301 38 10 300  25

0.775  0.327 0.791 0.224 0.792  0.138 0.785  0.224 0.793  0.159 0.799  0.103 0.794  0.159 0.806  0.107 0.797  0.074

The standard errors of the estimates were calculated from1000 simulation replicates.

Table 4: Power detection and FPRs under different sample size and heritability with 1000 simulation replicates Heritability

0 0.05 0.10 0.20

Sample size 200

400

800

0.013 73.1 98.3 100.0

0.006 98.4 100.0 100.0

0.005 99.8 100.0 100.0

different environments, allowing all possible traitwise and environment-wise comparisons. Despite being powerful, this approach requires the joint modeling and solution of four-dimensional variables,

which needs a considerably large sample size and heavy computational demand which may not be met in practice. By defining across-environment difference of a trait as its phenotypic plasticity [25], the framework described here reduces statistical modeling from four dimensions to two dimensions, making it feasible to study the genetic correlation of phenotypic plasticity. An additional merit this framework has is that it can characterize and map-specific QTLs that directly contribute to the co-response of different traits to environmental change. This merit is attributed to genetic mapping of phenotypic plasticity as a phenotypic trait, allowing any possible QTLs affecting phenotypic plasticity to be mapped. A bivariate model makes it possible to map the common genetic basis for phenotypic plasticity of different traits.

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Zhou et al.

By reanalysing a published data set for a mapping population of winter wheat grown under two nutritional levels [19, 20], the model has been validated for its practical usefulness. The model identified two QTLs that determine correlative plastic response of kernel number and thousand kernel weight to nutritional levels. The model displays particular power to detect the pattern of action of the two QTLs. In order to warrant the discovery, a simulation study was carried out by mimicking the two winter wheat QTLs detected in the segregating population. In studying phenotypic plasticity of two related traits, the model does not consider the pattern of biological interrelationship occurring between the traits. It has been recognized that traits covary with each other in a particular law to govern biological structure, function and dynamics [26]. Allometric relationships, characterized by power laws, are thought to be a general rule in biology, arising from the geometry of vascular networks and resourceexchange surfaces [27–29]. This rule should be incorporated into the QTL model to study the genetic origin of the covariation of trait plasticity over a series of environmental conditions and gain insight into how phenotypic plasticity has shaped the evolution of the diversity of biological form and function. The model is equipped to investigate the genetic basis of phenotypic plasticity. Given their important part in shaping phenotypic variation [30–32], epigenetic marks should be implemented into the model to unravel the genetic and epigenetic mechanisms for phenotypic plasticity. This implementation will help to better draw a comprehensive picture of the genetic machineries that cause the covariation of phenotypic plasticity among different traits. Powerful statistical methods are required to separate DNA sequence-based effects from epigenetic effects, quantify complex interactions between genetic and epigenetic factors [33], and evaluate their relative importance for phenotypic plasticity [32]. To chart a complete picture of genetic control for phenotypic plasticity, simultaneous analysis and modeling of all these factors are essential. This can be achieved by integrating variable selection approaches that can tackle the complexity of highdimensional data [34] into our model. As the study of phenotypic evolution is more relevant by merging it with development, phenotypic plasticity of developmental processes will be quickly a focus of evolutionary developmental biology [35].

Functional mapping, a dynamic model, derived to map development QTLs [13, 36, 37], can be integrated with the plasticity model, in a quest for identifying specific QTLs for developmental plasticity and its correlation among different traits. Studies of growth and plasticity through such integration could expand our knowledge of the range and origin of plasticity present in nature.

Key Points  Phenotypic plasticity has been regarded as an adaptive mechanism leading to species diversity and evolution.  Current theory on adaptive relevance of phenotypic plasticity has been mostly established from single traits, although the phenotype is subject to environmental modification as a constellation of traits that covary with each other.  We describe a QTL mapping model for studying the common genetic basis of phenotypic plasticity of different phenotypic traits.

Acknowledgements This work is partially supported by NSF/IOS-0923975, Changjiang Scholars Award, and ‘Thousand-person Plan’ Award.

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