Fixed effect procedure

When the environmental conditions are controlled, environments can be a priori classified into several groups. One can then assume the marker effect to be stable within each group but different from one group to another.

We consider that the environments are classified into J distinct groups. Since the same panel of varieties—or overlapping panels—are used in all environments, the Z-scores cannot be assumed to be mutually independent. The model is updated as follows: where the incidence matrix X of size K × J is such that: is the vector containing the group-specific marker effects, and Σ_m is the inter-environment correlation matrix of size K × K. If allelic effects are assumed to be stable across all environments (i.e. J = 1), the null hypothesis reduces to H₀ : {μ_m = 0} as in the FE procedure described in S1 Text.

The parameters to be estimated are the inter-environment correlation matrix Σ_m and the marker effects within groups . We assume the inter-environment correlation matrix to be common to all markers, i.e. Σ_m = Σ. Considering only the M₀ markers under H₀, i.e. the markers having no effect in any environment, the correlation between the z-scores of two environments k and k′ can be estimated as follows: (1)

As the list of markers under H₀ is unknown, a filtering step for markers with a high probability of being under H₁ is needed. Different filtering approaches exist, and we present two of them. The first one consists of considering only the markers m whose p-values p_mk are higher than a certain threshold fixed by the user, in each environment k ∈ 1, …, K [34].

The second filtering approach that we propose consists, for each environment k, in estimating the distribution of the random variables Φ⁻¹(p_mk), m ∈ 1, …, M, where Φ⁻¹ is the inverse distribution function of the normal distribution . By definition, the distribution to be estimated is a mixture between a distribution corresponding to the markers under H₀ and a second unknown distribution corresponding to those under H₁. This mixture distribution can be inferred by a kernel method. Then, the filtering consists of considering only the markers m whose a posteriori probabilities of being under H₁ are lower than a certain threshold [35]. For the applications detailed in Section Results, the filtering method was performed with a threshold set at 0.6.

Given an estimate of Σ, the within-group marker effects can be inferred by the empirical means of the z-scores of each group: where and n_j is the number of experiments in group j.

The association of each marker m can be tested as follows:

Alternatively, one may test whether the marker m has different effects across groups of environments: (2)

More generally, one may test H₀ : {Cμ_m = 0} for any contrast matrix C of size Q × J, where Q ∈ {1, …, J} is the number of linear constraints on the marker effects to be tested. Under the null, where V_C = CDX^TΣXDC^T. Therefore: which leads to the following test statistic:

The p-value is given by: with t_m the observed value of T_m, and Φ_χ²(Q) the cumulative distribution function of the χ²(Q) distribution.

Random effect procedure

In the case of uncontrolled environmental conditions, the heterogeneity of the QTL effects across environments can be accounted for through a random marker effect.

The model is updated as follows: where and Λ_m are the variance and the correlation matrix associated with the random marker effect, respectively.

As for matrix Σ, we assume the random effect correlation matrix to be common to all markers, i.e. Λ_m = Λ. Furthermore, since Σ quantifies the similarity between environments, it provides some a priori knowledge about the similarities of allelic effects at the scale of a marker. Consequently, it will be assumed that Λ = Σ.

The correlation matrix Σ is inferred using an estimator defined in Eq (1). The effect of the marker and the between environment variance are inferred by maximum likelihood inference, yielding

The test for the association of the marker m corresponds to: and can be performed using a likelihood ratio test. Designating l₀ and l₁ as the likelihood of Z_m under H₀ and H₁, respectively, the test statistic is 2(l₁ − l₀) and its associated distribution under H₀ is a mixture of Chi-square distributions [36]:

Meta-regression test

In MET studies, trials can be characterized through some quantitative environmental covariates (e.g. temperature or evapotranspiration). One can then aim to identify markers whose effects are correlated to a given environmental covariate. The covariate is denoted by X, and X is assumed to be centred. For each marker m, one can test:

The covariance of interest can be inferred by its empirical counterpart, leading to the following test statistic: that follows a distribution under the null hypothesis.

The p-value is given by: where Φ stands for the standard Gaussian cumulative distribution function.

Multiple test control

In order to control Type I errors, we apply the local score approach developed by [37]. The local score approach detects significant regions in a genome sequence by accumulating single marker p-values while controlling the FDR. This approach depends on the choice of the threshold ξ (in log₁₀ scale) below which small p-values are accumulated. The significant genomic regions are computed chromosome by chromosome, and a significance threshold for the FDR control is associated with each chromosome (see [37, 38] for details). In practice, the value of ξ should be selected between the average and the maximum of the set of −log₁₀(p- values). In our analyses the observed average −log₁₀(p_m) ranged from 0.5 to 0.9, and the maximum value from 10 to 19, leading to a value of ξ set to 3 or 4.

Simulation framework

The performance of the proposed meta-analysis methodology was assessed through simulations. We used the genotypic data obtained from the study of [39], along with the environmental data and the experimental design described in [40]. These datasets are publicly available at https://doi.org/10.15454/AEC4BN and https://doi.org/10.15454/IASSTN, respectively.

The dataset includes 247 F1 hybrids resulting from crossing 247 diverse dent maize inbred lines with the same flint line. These hybrids were genotyped at 506,460 positions (SNPs) and phenotyped in 22 different environments. The phenotypic data were generated by simulating QTLs and a GxE genetic background. We considered four different types of QTLs:

1. Fixed effect QTLs: corresponds to QTL with a constant effect across all environments.
2. Completely random effect QTLs: corresponds to QTL whose effects are randomly drawn in each environment.
3. Random effects based on the environment correlations QTLs: correspond to QTL whose effects are drawn in each environment in a normal distribution whose covariance matrix reflects the similarities between environments, as measured from the environmental data.
4. Fixed effect depending on an environmental covariate QTLs: corresponds to QTLs whose effects are proportional to a (centred and scaled) environmental covariate.

In our simulation setting, QTL types were considered separately: 50 simulation runs were conducted for each type. In a given simulation run, phenotypic data were generated based on 12 QTLs, which were categorized into three minor allele frequency (MAF) groups: low [0.2, 0.25], medium [0.3, 0.35], and high [0.4, 0.45], with four QTLs assigned to each MAF category. These QTLs accounted for 44% of the total genetic variance, while the remaining 56% was attributed to the genetic background. The heritability in each environment was fixed at 0.5. QTL and background marker positions were randomly assigned in each run, with two chromosomes randomly designated as containing no QTLs, referred to as H₀ chromosomes hereafter. For each simulation run, GWAS were performed per environment using FastLMM [10]. See S2 Text for a complete description of the data simulation procedure. The resulting sets of 22 GWAS summary statistics were analyzed using the metaGE random-effect (RE) meta-analysis model for the simulations involving completely random-effect QTLs and random-effect QTLs based on environmental correlations. For simulations involving fixed-effect QTL, we applied the metaGE fixed-effect (FE) meta-analysis model. Meta-regression analyses were conducted on the three environmental covariates (Tmax, Tnight, and Psi) that were used to simulate environmental covariate-regressed QTL effects.

Additionally, the METAL procedure [31] was applied, using the FE model for fixed-effect QTL simulations and the RE model for others. The mash procedure [32] was also applied, using both canonical and data-driven covariance matrices to capture a wide range of effect patterns for the random-effect QTL and the covariate-regressed QTL simulations. In the fixed-effect QTL simulation, a specific covariance matrix effect with entries equal to one (corresponding to a shared effect for all environments) was used. When applying the mash procedure the correlation matrix was estimated from the data using a z-score threshold, as recommended in the original publication [32]. The threshold was set at 3.5, as this value resulted in a significantly reduced number of false positives for the procedure compared to the by-default threshold. For both the METAL and metaGE procedures, the local score procedure with ξ = 4, a nominal false discovery rate (FDR) of 0.05, and a peak threshold at 10 were used to control for multiple testing and identify putative QTL regions. As the mash procedure does not provide p-values, we followed the recommended global null test approach, considering an association significant if the local false sign rate [41] was less than 0.05 in at least one GWAS (see [32] for details). Significant regions (or markers, in the case of the mash procedure) located within 1 Mb of each other were merged into contiguous regions not exceeding 20 Mb.

False Discovery Rate (FDR) was evaluated in two different ways. A first FDR estimate (hereafter referred to as FDR_chr) was computed based on the two H₀ chromosomes as the percentage of simulation runs exhibiting a significant region on these chromosomes. A second FDR estimate was computed at the whole genome level as the percentage of significant local score peaks that were located outside windows of 5, 10 or 20 Mb centred on QTL positions. QTL detection power was evaluated by calculating the true positive rate, defined as the proportion of simulated QTLs correctly identified (i.e. power = Number of QTLs detected/Number of simulated QTLs). A QTL was considered correctly detected if at least one significant peak was located within a window of 5, 10, or 20 Mb centered on the QTL position. Note that for the metaGE meta-regression tests, QTLs used to calculate power were limited to those associated with the corresponding covariate, i.e., four per covariate.

Application to MET GWAS analysis

The performance of the metaGE, METAL and mash procedures are illustrated across four datasets selected to represent classical MET experiments for G × E interactions association studies in plant quantitative genetics. These application cases cover controlled and partially/not controlled experimental designs along with an application to multi-parent MET population. Descriptions of the datasets used in these analyses are provided below.

• Arabidopsis dataset [42]: GWAS analyses were performed on the local mapping population TOU-A, with a panel of 195 whole-genome sequenced accessions evaluated in six controlled micro-habitats (combinations of three soils × presence/absence of inter-specific competition, noted A to F). Each accession was phenotyped for bolting time and had genotypic information for 981,278 SNPs (after quality control and at a minor allele frequency threshold of 0.07). The trials were conducted under controlled stress conditions, with Arabidopsis being grown in competition with the weed Poa annua in environments B, D, F and without competition in environments A, C and E.
• Maize dataset [4]: GWAS analyses were performed on a panel of 244 maize dent lines evaluated as hybrids with a common parental line (a usual practice in maize genetics) in 22 environments (combinations of location × year × treatment). Each line was genotyped at 602,356 SNPs (after quality control) and phenotyped for grain yield (GY) in the 22 environments. Environmental indices were calculated as the average or the sum of one environmental variable in a period over 20 days at 20°C encompassing flowering time of the reference hybrid. The environmental variables used to calculate these indices are the soil water potential (Psi), the maximum temperature (Tmax), the night temperature (Tnight), the cumulated global radiation (Rad), the leaf to leaf Vapour Pressure Difference (VPDmax) and the Evapotranspiration (ET0). In addition, the mean night temperature during the grain filling period (Tnight.Fill) was calculated.
• EU-NAM Flint [43, 44]: A panel consisting of 11 biparental populations was obtained from crosses between UH007 and 11 peripheral parents representative of the Northern European maize diversity. In each population, double haploid lines were produced and genotyped at 5,263 SNPs (following quality control). All populations were evaluated for biomass dry matter yield at four locations: La Coruna, Roggenstein, Einbeck, and Ploudaniel. Three populations with fewer than 30 progenies were excluded from the present analysis.
• Wheat dataset [45]: GWAS analyses were performed on a panel of 210 wheat lines phenotyped for grain yield in 16 environments (combinations of location × year × treatment). Lines were genotyped at 108,410 SNPs (after quality control) and phenotyped for grain yield. This application is presented in S3 Text and S8–S10 Figs.

For the Arabidopsis, Maize, and Wheat datasets, we applied a local score approach to control for multiple testing, setting ξ = 3 and the nominal FDR at 0.05 for both metaGE and METAL results. Due to the lower genotypic resolution in the EU-NAM Flint dataset, which is incompatible with the local score method, we used the adaptive Benjamini-Hochberg correction procedure [46] with a nominal FDR of 0.05. The mash procedure was applied as in the simulation study, using either both canonical and data-driven covariance matrices to capture a wide range of effect patterns, or a specific covariance matrix effect with entries equal to one which corresponds to a shared effect for all environments. The mash correlation matrix was estimated from the data using a z-score threshold of 3.5. The global null test was used, with a control of the local false sign rate at 0.05 [32]. All datasets used are publicly available (see the Data Availability section).

Simulation results

Fig 1A illustrates the results obtained from the metaGE RE procedure on a single simulation run corresponding to the “completely random-effect” QTL type. The red vertical dotted lines indicate the true locations of the 12 simulated QTLs, while the blue vertical regions represent the putative QTLs identified by the procedure. Overall, the metaGE RE procedure successfully identified 7 of the 12 QTLs within a 5 Mb detection window. Fig 1B focuses on chromosome 3, which illustrates the characteristic behaviour of the local score procedure. The region containing the simulated QTL was initially tagged by ten significant local score peaks, along with several smaller residual peaks. After the merging and thresholding steps, the number of peaks was reduced to three. The peak falling within the 5 Mb QTL detection window is indicated by the red horizontal line at the bottom of the figure. The two remaining peaks, labelled as “FP” (false positives) on the graph, were classified as false positives for the 5 Mb window and as true positives when evaluated against detection windows of 10 Mb and 20 Mb, respectively. Among the QTLs undetected by the metaGE RE procedure, three corresponded to QTLs with a low MAF between 0.20 and 0.25, and two corresponded to QTLs with a medium MAF between 0.30 and 0.35. On this same dataset, the mash procedure [32] identified 65 significant regions after merging the significant SNPs located within 1 Mb. Of these, 53 were outside the 5 Mb QTL detection windows, yielding a false discovery rate of 0.81. The remaining 13 regions fell within the detection windows of 10 QTLs. The METAL procedure [31] resulted in 14 significant local score peaks after the merging and thresholding steps, all being outside the 5 Mb QTL detection windows. These results are representative of the ones obtained for all the simulation runs, regardless of the simulated QTL type.

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Fig 1. Results of the metaGE RE procedure applied to a simulation run corresponding to the “completely random-effect” QTL type.

(A) Local scores along the chromosomes. The red vertical dotted lines correspond to the localization of the 12 simulated QTLs. The range of values on the y-axis has been bounded from 0–15 to highlight minor QTL. The blue boxes represent the significant zones identified. (B) Same representation restricted to a QTL region on chromosome 3. The red horizontal bottom line corresponds to the 5Mb QTL detection windows. FP stands for False Positive.

https://doi.org/10.1371/journal.pgen.1011553.g001

Table 1 displays the FDR_chr and the average FDR and power for each meta-analysis procedure across different QTL types, evaluated with 5 Mb, 10 Mb, and 20 Mb detection windows. The FDR_chr was lower than 0.05 for the different metaGE MA procedures. For the two competing methods, the FDR_chr was systematically higher than the nominal level, ranging from 0.14 to 0.88 for mash and close to 1 in most scenarios for METAL.

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Table 1. Performance over 50 simulation runs for each simulation scenario, using a 5, 10 or 20 Mb detection window size.

https://doi.org/10.1371/journal.pgen.1011553.t001

At the whole genome level, the FDR was high, ranging from 0.1 to 0.18 for metaGE to more than 0.93 for METAL when using the most stringent detection window size (5Mb). The whole genome FDR decreased with respect to the detection window size, as expected, but remained above 0.84 for METAL and 0.23 for mash even when considering the largest detection window (20Mb).

Detection powers were around 0.09, 0.39 and 0.37 for the metaGE FE model, RE model and meta-regression tests, respectively. Overall, both METAL and mash procedures displayed higher detection power than metaGE across most QTL types, but their prohibitive False Positive levels undermined the reliability of the detected associations. Note that despite its higher False Positive rate, METAL achieved the lowest detection power for the completely random-effect QTLs and QTLs regressed on environmental covariates. One could also observe a strong impact of the Minor Allele Frequency (MAF) on the detection power for the metaGE procedure, as illustrated in Table 2. Specifically, the detection power for QTLs with high MAF was three times greater than that for those with low MAF (e.g. from 0.04 to 0.14 in the case of metaGE_FE).

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Table 2. Average detection power for the different minor allele frequency levels over the 50 simulation runs, using a 5 detection window size.

https://doi.org/10.1371/journal.pgen.1011553.t002

Although the detection power performance of the meta-regression (MR) tests reported in Table 1 only accounts for QTLs specifically correlated to the corresponding covariate, it may happen that a meta-regression test detects QTLs associated with another covariate that the one under study. Table 3 displays the QTLs detected by the 3 MR tests. The MR test associated with the Tnight covariate detected 34.5% of its target QTLs over the 50 simulation runs. It also detected 5.5% of the Tmax target QTLs but did not detect any of the Psi target QTLs. This can be directly related to the correlations between the different environmental covariates: the correlation between Tnight and Tmax was 0.71, whereas the correlation between Tnight and Psi was -0.11.

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Table 3. Percentage of QTLs detected by each metaGE meta-regression test across 50 simulation runs.

https://doi.org/10.1371/journal.pgen.1011553.t003

Genetic response to competition in Arabidopsis

We consider the Arabidopsis dataset from [42], which includes six GWAS analyses aimed at identifying associations with bolting time across six controlled micro-habitats, labelled A through F. Inter-specific competition was present in environments B, D, and F, while environments A, C, and E were competition-free.

Three different meta-analysis procedures were used to analyze the six GWAS summary statistics: the metaGE FE procedure, the mash procedure [32] using only the matrix with all ones as covariance matrix, corresponding to a shared effect for all environments, and the METAL procedure. Fig 2 shows the resulting histograms and Q-Q plots of the p-value distributions from the METAL procedure and the metaGE FE method for quality control purposes. The METAL q-q plot clearly exhibits an inflated proportion of low p-values, with more than 165,000 p-values under 0.01, compared to the ≈10,000 that one would expect under the full H₀ hypothesis. This inflated proportion results in declaring more than 15% of the markers as associated after an FDR correction. In comparison, the metaGE FE procedure effectively controls the type I error rate, as demonstrated by the p-value distribution. It led to the identification of 191 SNPs clustered into 61 QTLs. The mash procedure identified 401 SNPs, with the corresponding peaks mostly overlapping the ones identified with the metaGE procedure (S1 Fig).

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Fig 2. P-value distributions of METAL and metaGE FE procedures applied to Arabidopsis dataset.

(A) Histogram of the METAL p-values. (B) Histogram of the metaGE FE p-values. (C) QQ-plot of the -log10(p-values) of METAL in red and metaGE in green. The observed -log10(p-values) are compared to the expected quantiles generated by the uniform null distribution.

https://doi.org/10.1371/journal.pgen.1011553.g002

Out of the 61 putative QTLs that were identified through the metaGE FE procedure, 51 were significant in at least one individual GWAS (see S3 Text for details). Following the methods of [47], the list of significant SNPs identified by the metaGE method was further corroborated by examining the enrichment ratio for a priori candidate genes linked to bolting time. The markers identified by the metaGE approach were found to have a significantly higher enrichment (enrichment = 4.13) than what would be expected at random ([q_0.05; q_0.95] = [0.066; 3.2]).

We then applied the metaGE contrasted FE testing procedure described in the Methods section, Eq (2) with a group effect corresponding to the presence or absence of competition with Poa annua to detect markers with contrasted allelic effects based on the presence or absence of competition. The metaGE contrasted FE procedure identified 221 SNPs located in 72 QTLs (Fig 3A) and covering 160 candidate genes that showed significant enrichment for three Gene Ontology terms (MapMan functional annotation, [48]), i.e. ‘development’ (P = 8.866e-03), ‘cell’ (P = 1.522e-03) and ‘tetrapyrrole synthesis’ (P = 0.020). Interestingly, the two latter MapMan processes were also detected as enriched when subjecting the local mapping population TOU-A to the presence of three weed species in greenhouse conditions [49].

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Fig 3. Results of the metaGE FE procedure applied to the Arabidopsis dataset to detect markers with competition contrasted effects.

(A) Local scores along the chromosomes. The blue boxes represent the significant zones identified. (B) Z-scores of two QTL regions located on chromosome 5 (QTL5_22.0 a,d QTL5_22.1), with markers in rows and environments in columns. A, C and E correspond to the 3 environments without competition.

https://doi.org/10.1371/journal.pgen.1011553.g003

Fig 3B represents the z-scores of two top QTLs found on chromosome 5 involving 22 and 9 markers, respectively (local score of 8.69 and 9.1). The identified markers clearly exhibited a contrasted marker effect profile: when the effects were positive in one of the two groups of environments, they were negative or null in the other. All the 22 SNPs of the second QTL (QTL5_22.1) are located in the AtCNGC4 genomic region that is well-known to affect floral transition [50, 51].

Importantly, as the metaGE contrasted FE procedure aimed to identify markers with contrasted (i.e. unstable) effects across the two sets of environments, 71 out of the 72 QTLs identified by the contrasted FE procedure were new candidates that were not detected by the standard FE procedure.

Genetic response to drought in maize

We consider the Maize dataset of [4], where GWAS analyses were performed to identify association with grain yield in 22 environments (combinations of location × year × treatment).

The metaGE RE procedure detailed in Materials and methods section was applied to perform the joint analysis of the 22 per environment GWAS summary statistics. In total, 52 genomic regions were identified, of which 14 correspond to QTLs also detected in the original publication [4]. The three putative QTLs with the most significant association peaks (Fig 4A) were located on chromosomes 3 (QTL3_120.0) (local score = 38), 6 (QTL6_20.3) (local score = 415) and 7 (QTL7_41.4) (local score = 18, not detected in the original publication). The z-score heatmaps corresponding to QTL6_20.3 and QTL7_41.4 are displayed in Fig 4B. The heatmap of QTL6_20.3 revealed a cluster of six environments (Cra12R, Cam12R, Cra12W, Mur13R, Mur13W and Cam12W) with strong similar marker effects that were found to be characterized by severe heatwaves with high night temperatures (close to 22°C) and high maximal temperatures (above 36°C) together with high evaporative demand during the day (3.6 KPa). The QTL6_20.3 individual GWAS p-values were all lower than 1e-6 in the six aforementioned environments, confirming the strong association signal detected by the RE procedure. In contrast, the QTL7_41.4 showed a moderate positive effect across nearly half of the environments—individual GWAS p-values were lower than 1e-2 in ten environments out of 22—and was found to be significant in only two individual GWAS.

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Fig 4. Results of the metaGE procedures applied to the Maize dataset.

(A) Local score obtained from the metaGE RE procedure along the chromosomes. The range of values on the y-axis has been bounded from 0–20 to highlight minor QTLs. The blue boxes represent the significant zones identified. (B) Z-scores of two QTLs located on chromosomes 6 and 7 (QTL6_20.3 and QTL7_41.4), with markers in rows and environments in columns. Only the 100 top significant SNPs out of 169 composing the QTL6_20.3 have been displayed. The six environments highlighted in blue correspond to environments characterized by severe heatwaves (night temperature close to 22°C and maximal temperature above 36°C). (C) Local score obtained from the meta-regression test for the evapotranspiration along the chromosomes. The range of values on the y-axis has been bounded from 0–20 to highlight minor QTL. The blue boxes represent the significant zones identified. (D) Z-scores as a function of the evapotranspiration, for the top significant marker of QTL2_162.5 detected with the meta-regression procedure (Maize dataset, marker AX-91538480).

https://doi.org/10.1371/journal.pgen.1011553.g004

On the same dataset, METAL detected 40% of the markers as significant (S2 Fig). The results of the mash procedure were similar to the ones obtained with metaGE, except for chromosome 9 where mash declared 909 markers covering most of the chromosome (from 13.4Mb to 154.4Mb) as significant (S3 Fig). See S3 Text for details.

We performed a genome-wide detection of markers whose effect variations were correlated with environmental variables by running the meta-regression test procedure described in Materials and methods section. The environmental variables considered were the evapotranspiration, the mean night temperature during the flowering period (Tnight) and the mean night temperature during the grain filling period (Tnight.Fill).

Regarding evapotranspiration, the meta-regression test identified 14 QTLs located on chromosomes 2 and 9 (Fig 4C). Allelic effects on grain yield of one of the most significant markers (marker AX-91538480, QTL2_153.8) substantially changed with potential evapotranspiration (Fig 4D).

For the mean night temperature during the flowering period, the meta-regression test identified 21 QTLs, with the main association corresponding to a genomic region located at less than 0.6 Mb from QTL6_20.3 (S4(A) Fig). This finding corroborates the results previously obtained in [4].

Regarding the night temperature during the grain filling period, the meta-regression test identified 15 QTLs across six chromosomes. In particular, the allelic effects on GY of one of the most significant markers located on chromosome 9 (marker AX-91123283, QTL9_28.6) changed dramatically according to night temperature during the grain filling period, with positive effects on cool nights and negative effects on hot nights during the grain filling period (S4(B) Fig). See S3 Text for an analysis regarding night temperatures during the grain filling period.

Extension to Multi-Parental Population MET experiments

Unlike previous datasets involving an association panel, the present dataset consists of a multi-parent crossing design, with each bi-parental progeny being phenotyped in four locations (La Coruna, Roggenstein, Einbeck and Ploudaniel). In this context, we extend the notion of environment to the combination of one sub-population and one location.

To estimate the allelic effect per parent and environment, GWAS analyses were performed on each combination of cross and location, resulting in 32 individual analyses. The metaGE RE procedure detailed in Materials and methods section was applied to perform the joint analysis of the 32 individual GWAS summary statistics. In total, 16 QTLs were identified, highlighting some very significant association peaks, especially on chromosomes 1 (QTL1_117.6) and 6 (QTL6_84.2). These two QTLs were also identified in the publication of Garin et al. [33]. The allelic effect of QTL1_117.6 was almost consistent across all populations except in F2 (S7 Fig). QTL6_84.2 showed an interesting genetic effect series since an ancestral allele inherited by parents D152, F03802, F2, F283, UH006, and DK105 [33] had a strong negative effect, primarily in the environment TUM (Fig 5A).

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Fig 5. Results of the metaGE RE procedure applied to the EU-NAM Flint dataset.

(A) Allelic effects of QTL6_84.2 (EU-NAM Flint, marker PZE.106101278) through locations and sub-populations (B) Allelic effects of QTL5_23.9 (EU-NAM Flint, marker PZE.105012387) through locations and sub-populations. A combination of location × sub-population is highlighted where individual GWAS p-values were below 0.01.

https://doi.org/10.1371/journal.pgen.1011553.g005

While the analysis of Garin et al. [33] was limited to the study of two of the four locations, the present analysis included all locations. In this study, we revealed ten new QTLs not detected in the initial analysis, five of which corresponded to QTLs, which showed effect inversion among populations. For example, QTL5_23.9 showed a positive effect for the population F03802 and a negative effect for the population F64 (Fig 5B). We also noticed that 3 of them were associated with the flowering time QTL detected from the same materials in Giraud et al. [52]. Flowering time has a simpler genetic determinism than grain yield and is one of its main drivers, with negative, null or positive correlations according to environmental conditions [4]. The EU-NAM Flint dataset was also analyzed with the METAL and mash procedures. Similar to the previous application cases, METAL led to an inflated detection rate (≈ 30% of the markers, S5 Fig) while mash detected 278 markers, including the ones corresponding to the three main peaks identified by metaGE, S6 Fig. See S3 Text for details.

Computational efficiency

Table 4 displays the computational times for the metaGE, METAL and mash procedures in simulated data and real datasets. In the simulation studies, all procedures run the analysis in less than 3 minutes, except for the mash procedure using canonical and data-driven covariance matrices that required 16.6 minutes.

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Table 4. Computational time for the meta-analysis procedures across different datasets.

https://doi.org/10.1371/journal.pgen.1011553.t004

The analysis of the different real datasets was handled in less than 3 minutes using the metaGE procedure, even for datasets characterized by a large number of environments and/or markers. Note that for metaGE, most of the required computational time corresponds to the inference of the inter-environment correlation matrix, a step that needs to be run only once as it is common to both the RE and FE models and all subsequent testing procedures.

Comparison to existing methods

Hereafter, we provide a typology of methodological requirements that are key for MET analysis and, more generally, GxE applications. We discuss how our developments to address these are connected to the many MA procedures developed in the last decade motivated by other fields of application, mainly in the context of human genetics.

A by-default feature of most popular MA procedures [31, 53] is the assumption of independence between GWAS. However, as shown by the behaviour of METAL in our simulation studies and real data applications, ignoring dependencies when applying MA on MET analysis significantly inflates the Type I error rate, with FDR exceeding 0.84 across all simulated scenarios. We hypothesize that this assumption of independence is a key reason why MA has seen limited application in MET and GxE analyses. Some recent works have extended the standard MA model to account for dependency between GWAS results. In [54], the authors proposed a method to handle potential overlapping samples by computing correlations based on information on shared subjects. In the context of pleiotropy analysis, where multiple traits measured on the same panel are jointly analyzed, different strategies have been proposed to infer correlations between summary statistics, either based on LD-score regression [55–57] or directly from the z-scores [32, 34, 58, 59]. The metaGE procedure used to estimate the correlation matrix between environments aligns with these approaches. It can be related to the inference procedure of the correlation matrix between traits described in [32, 34] but with a different strategy for filtering H₁ markers. In those methods, the filtering is achieved by directly applying thresholds to p-values or z-scores. In contrast, the metaGE filtering approach relies on the posterior probabilities of markers being under H₁ as obtained from the procedure detailed in [35].

Most GxE experiments are characterized by complex designs that aim to compare two or more environmental scenarios. These scenarios may include the presence/absence of competitors (e.g. adventices) or pathogens, abiotic stress (e.g. drought or nitrogen stress), or contrasted climatic conditions (Temperature gradient). Environments may then be a priori sorted into groups according to the environmental scenario, with the aim of comparing the marker effect stability between groups. However, standard meta-analysis methods like METAL assume that marker effects are consistent across all GWAS studies, which complicates the detection of unstable or environment-dependent markers, as shown in our simulations. In contrast, the mash method can model different effect patterns but cannot account for patterns influenced by environmental factors, which is essential for studying genotype-environment interactions. Alternatively, a wide range of MA procedures have been developed in contexts other than association genetics to jointly analyze summary statistics from independent studies [60]. In particular, subgroup MA or tests of contrast [61, 62] and meta-regression [63, 64] can be used to explore the relationship between the summary statistics and a study-level categorial or quantitative covariate, respectively. Recently, in [65], the subgroup MA was applied to gestational diabetes mellitus association studies to investigate the relationship between different confounding factors or demographic characteristics of the study populations and the marker effect estimates. Additionally, [66] developed a MA of independent GWAS results that explores all possible subsets of association studies and identifies the subset with the strongest association signal. However, to our knowledge, these MA procedures were not available in the context of non-independent studies.

GWAS-MA confers gain in power over individual GWAS

GWAS-MA has proven to yield significant gains in power over individual preliminary analyzes while effectively controlling for false positives [21]. The observed gain in detection power applies to the MET context as well. The metaGE MA approach revealed interesting new QTLs even in regions where the test statistics did not pass the nominal significance threshold in most individual environments. For example, the genomic region corresponding to the third highest local score in the Maize dataset (QTL7_41.4, local score = 18) was found to be significant in only two environments out of 22 and was not detected in the original study [4]. This region harbours QTLs controlling plant growth rate and final biomass in water deficit conditions [67]. Similarly, in the Wheat dataset, none of the QTLs identified by the metaGE RE method were found to be significant in any environment. Therefore, the joint analysis of individual GWAS is suitable for complex traits (such as yield) whose genetic variations are usually due to many QTLs with minor effects that might go undetected in single-environment analyses.

Interpreting variability in QTL effects across environmental conditions

The MA extensions proposed here facilitate the assessment of the stability or variation in QTL effects across environmental conditions, which is of key interest in detecting alleles conferring specific adaptive features. As an illustration, the analysis of the Arabidopsis dataset highlighted a major region on chromosome 5 (QTL5_22.0), with an effect sign that switched according to the presence or absence of competition with Poa annua. Arabidopsis QTL5.220 is located in the AtCNGC4 genomic region that is well-known to affect floral transition [50, 51]. In addition, AtCNGC4 impairs plant immunity [51, 68], which is in line with the negative effect of competitive interactions on pathogen defense to the benefit of plant development, such as with floral transition [69].

Similar results were found in the maize analysis, where environmental conditions were not controlled a priori. The analysis of allelic effects of QTL6_20.3 detected with the RE model highlighted a group of six environments (Fig 4C). A subsequent analysis showed that these environments were characterized by severe heat waves at night. Consistently, QTL6_20.3 colocalized with a QTL affecting grain yield in the “Po valley” in which maize is currently irrigated but subjected to high temperatures during summer [70]. QTL6_20.3 also overlaps with a large 2.4 Mbp present/absent variant harbouring dozens of genes—including one encoding an ABA-induced protein in response to water deficit [4]. These genes were further shown to be associated with environmental adaptation to high temperatures and to have undergone strong selection during both domestication and improvement [71].

Beyond a posteriori interpretation, environmental variables can also be incorporated as a covariate in the GWAS-MA model for a whole genome scan of the response of QTL effects. Such a relationship between marker effects and environmental variables was investigated in [4] for the Maize dataset. However, this initial analysis was restricted to markers found to be significantly associated in at least one environment due to the prohibitive computational time required to perform a genome-wide analysis. To address this limitation, we proposed a meta-regression test designed to identify QTLs whose effect variations correlate with a specified environmental covariate. The performance of the meta-regression test was evaluated through a simulation study, which demonstrated proper FDR control and high detection power. The simulation study also pointed out that when two environmental covariates are highly correlated, the meta-regression test applied to the first covariate may detect QTLs whose effect variations are linked to the second covariate. Although this finding does not impact the performance of the procedure, it emphasizes that careful interpretation is necessary in cases of high correlation between environmental covariates.

We performed the genome-wide analysis of the Maize dataset with the metaGE meta-regression procedure using three covariates: evapotranspiration, mean night temperature during the flowering and grain filling periods. This analysis led to the identification of 14, 20, and 14 new regions, respectively. Fig 4D illustrates how the effect of a QTL located on chromosome 2 (pos154) varies linearly from negative to positive effects according to evapotranspiration. This QTL colocalizes with the QTL of expression of aquaporins (eQTL of PIP2.2 ez and eQTL of PIP2.1 ez). These channel proteins facilitate water transport between cells and impact water use efficiency (WUE) and stomatal conductance [72]. These two physiological adaptive traits are highly sensitive to environmental conditions [67]. A second region was detected on chromosome 9 (pos125) using the same environmental covariate. It colocalizes with a QTL for plant growth (PG) rate and a QTL for WUE under water deficit conditions. Both PG and WUE are traits that are highly sensitive to evaporative demand. Lastly, a third QTL, also located on chromosome 9 (pos135), colocalizes with a QTL of plant growth sensitivity to soil water potential, consistent with previously observed effects [67]. None of these genomic regions on chromosomes 2 and 9 were detected in the initial publication.

Benefits of working with summary statistics

An attractive feature of MA procedures is their ability to cope with unbalanced/incomplete data—without the need for further data imputation or additional computational overhead. The procedures presented in this article rely on summary statistics (i.e. p-values and effect signs) obtained from per-environment GWAS, which enables the avoidance of re-scaling if traits were measured on different scales. The use of summary statistics also makes the addition/removal of a given environment (based e.g. on post hoc quality control) and the update of the results straightforward.

Regarding the genotypic information, MA procedures can be used to easily handle cases where summary statistics are missing for some markers in a series of environments. This may occur e.g. when different technologies or sequencing depths were used in the individual experiments. Furthermore, in plant genetics, multi-parental population analyses are pervasive and also yield missing summary statistics, as different sets of markers may be monomorphic in different parental subpopulations. Our method is still applicable in such cases, as illustrated with the EU-NAM example, where inversions of allelic effects were identified for some subpopulations (Fig 5). These inversions may correspond to a genetic background effect (i.e. to conditional epistasis, see for instance [73]), or alternatively to an allelic series with several haplotypes presenting a range of effects, with the SNP allele contrasting central haplotypes with extreme opposite haplotypes.

In human genetics, meta-analysis methods have facilitated collaborations and data sharing between different consortia while preserving individual data confidentiality, as only summary statistics are required for the integrated association analysis. As an illustration, the recent Global Biobank Meta-analysis Initiative [74] gathered data from 24 BioBanks covering 2.2 million research participants, resulting in large-scale genetic association studies with improved detection power. We strongly believe that the meta-analysis procedures presented in this paper will open the way to similar consortia in plant breeding by allowing private companies to share GWAS results without the need to share individual plant genotypic or phenotypic information.

GWAS-MA is an efficient and versatile tool for GxE analysis

While several approaches have been proposed to model the GxE interactions in different contexts [14–17, 33], many of them require an amount of computational time that may become prohibitive when dealing with either large scale genomic datasets or a large number of environments. In contrast, the MA procedure introduced here is computationally efficient, processing large datasets, such as 22 environments and approximately 600,000 markers, in a few minutes (Table 4).

In addition, the metaGE procedures can be applied to various MET designs such as controlled experiments, uncontrolled or partially controlled experiments and all kinds of multi-parent population METs widely used in modern breeding programs. Our methodology integrates several tools for the characterization of environments. The contrast test associated with the FE procedure enables the identification of QTLs specific to some subsets of environments, i.e., whose behaviour interacts with the environmental characterization defined by the classification. The classification definition is very flexible and can rely on any qualitative covariate, such as known control/stress conditions or sub-populations in the case of an MPP experiment. The meta-regression test allows the detection of QTL, whose variations are correlated to any quantitative environmental covariate. Searching for such relationships between marker effects and environmental characteristics is a key issue in plant genetics. It has also been widely investigated in the context of animal and human MA studies, where dedicated procedures have been developed to handle environmental covariates measured at the individual level [75, 76]. Our meta-regression procedure represents a new contribution to account for environmental covariates measured at the environment level and complements the existing MA toolbox. All the testing procedures presented here, along with the FE and RE procedures, are implemented in the metaGE R package available on the CRAN repository.

In recent years, a number of public initiatives involving thousands of individuals and evaluated in dozens of well-characterized environments have been developed, such as the Genomes To Fields project for maize [77] or the elite yield trial nurseries from CIMMYT’s bread wheat breeding program [78, 79]. METs are one of the central elements of a breeding program [80]. Screening such datasets for genome-wide associations requires the development of scalable and flexible methodological tools. In this context, GxE Meta-Analysis can be routinely applied in most breeding programs, and we strongly believe that it will represent a methodology of choice in the future to address the many challenges of modern GWAS.