ARVCF increased BMI. Similarly, increased DGCR6 expression causally reduced mean platelet volume, in line with the corresponding CNV effect. Furthermore, cross-trait multivariable Mendelian randomization (MVMR) suggested a predominant role of genuine  pleiotropy in the CNV region. Our findings show that within the general population, 22q11.2 CNVs are associated with traits previously linked to genes in the region, and duplications and deletions act upon traits in different fashions. We also showed that gain or loss of distinct segments within 22q11.2 may impact a trait under different association models. Our results have provided new insights to help further the understanding of the complex 22q11.2 region.While extensively studied in clinical cohorts, the phenotypic consequences of 22q11.2 copy-number variants (CNVs) in the general population remain understudied. To address this gap, we performed a phenome-wide association scan in 405,324 unrelated UK Biobank (UKBB) participants by using CNV calls from genotyping array. We mapped 236 Human Phenotype Ontology terms linked to any of the 90 genes encompassed by the region to 170 UKBB traits and assessed the association between these traits and the copy-number state of 504 genotyping array probes in the region. We found significant associations for eight continuous and nine binary traits associated under different models . The causal effect of the expression level of 22q11.2 genes on associated traits was assessed through transcriptome-wide Mendelian randomization (TWMR), revealing that increased expression of  Exploring the phenome-wide impact of 22q11.2 CNVs in the general population revealed variable effects of deletions and duplications of specific segments in the region. Mendelian randomization found the effect of the expression level of certain 22q11.2 genes to be concordant with the estimated CNV effect on the same trait. Specifically, deletions within the ∼3 Mb segment from LCRA to LCRD represent the main cause of the 22q11.2 deletion syndrome  remain more elusive. Most of what is known is based on studies of a few individuals or families, but the findings indicate pleiotropy and variable consequences, similar to the deletion. Some features, such as heart defects, velopharyngeal insufficiency, and neurodevelopmental and psychiatric disorders, are shared with the 22q11.2DS.,Studies in clinical cohorts have investigated the phenotypic consequences of the 22q11.2 deletion, which include cardiac defects; facial and palate alterations; immunodeficiencies; endocrine, genitourinary, and gastrointestinal alterations;231200), caused by SNVs in GP1BB (MIM: 138720),609528) syndrome, caused by SNVs in SNAP29 (MIM: 604202).Finally, rare single-nucleotide variants (SNVs) in genes encompassed by the region have been linked to various disorders, such as Bernard-Soulier syndrome , a volunteer-based cohort from the general UK adult population.http://genome.ucsc.edu/cgi-bin/hgTables?command=start).We defined the 22q11.2 region as chr22: 18,630,000–21,910,000 on the basis of the human genome reference build GRCh37/hg19 in order to encompass LCRs from A to D. The 90 NCBI RefSeq genes contained in the region were downloaded from the UCSC Table Browser . Overall, 24 out of 90 genes in the 22q11.2 region, all protein coding, were associated with at least one HPO term, yielding 631 associated HPO terms.Phenotypes linked to the 22q11.2 region’s genetic content were identified with the Human Phenotype Ontology (HPO) mapping,https://www.ebi.ac.uk/spot/oxo/) to map HPO terms to International Classification of Diseases, 10th Revision (ICD-10) codes, followed by manual curation and grouping of ICD-10 codes into broader phenotypes when appropriate according to the Phecode map.To map HPO terms to binary UKBB traits, we used two complementary approaches. First, we used the online tool EMBL-EBI Ontology Xref Service (OxO) (https://hpo.jax.org/app/) to obtain the webpage’s results for each query. Results were filtered for HPO terms of interest, i.e., 631 terms linked to 22q11.2 genes. With this approach, 18 UKBB continuous traits were obtained from 18 HPO terms .We converted probe-level matrices to PLINK binary file sets, where copy-number states were encoded to accommodate analysis according to four different association models: duplication-only, deletion-only, mirror, and U-shape models . The dup2 ≥ 0.999) to at least ten other probes, allowing us to reduce the multiple testing burden while ensuring that selected probes adequately capture the CNV landscape of the region.Probes with high genotype missingness (>5%) were excluded, resulting in 864 CNV proxy probes spanning chr22: 18,630,000–21,910,000. We retained 504 CNV proxy probes that are highly correlated (reff-probes = 6). We used the same approach to account for correlation among 18 continuous (Neff-continuous = 16) and 152 binary traits (Neff-binary = 113). This resulted in 774 effective tests (Neff = Neff-probes × (Neff-continuous + Neff-binary)), setting the threshold for significance at p ≤ 0.05/774 = 6.5 × 10−5.The number of effective probes  was calculated according to Gao et al.2, sex, genotyping batch, and principal components (PCs) 1–40. Associations between the copy number (CN) of selected probes and normalized covariate-corrected traits were performed in PLINK v.2.0 according to all four association models with linear regression, as previously described.−5) were retained.The 18 selected continuous traits were inverse normal transformed and corrected for covariates: age, age2, sex, genotyping batch, and PCs 1–40 that were significantly associated with the trait (p ≤ 0.05) were selected with logistic regression in R. Associations between the CN of selected probes and 152 binary selected traits were performed in PLINK v.2.0 according to all four association models with logistic regression and correcting for trait-specific selected covariates. Significant associations (p ≤ 6.5 × 10−5) were retained.For each trait, covariates among age, ageThe number of independent signals per trait and association model was determined by stepwise conditional analysis,Due to the low frequency of CNVs within the 22q11.2 region, alternative tests were performed to ensure the confidence of significant associations. For significant associations with continuous traits, we performed a Wilcoxon rank-sum test as a sensitivity analysis to assess agreement with linear regression. Significant associations with binary traits were retained only when confirmed by at least one of two approaches: (1) Fisher’s exact test (p ≤ 0.005) for the duplication-only, deletion-only, and U-shape models and Cochran-Armitage test (p ≤ 0.0005) for the mirror model; (2) linear regression (p ≤ 0.005) of the inverse-normal-quantile-transformed trait residuals obtained from the logistic regression model of the binary outcome on the selected covariates.a: unrelated traits have lower association p values with the focal gene than related ones). We calculated the number of effective tests  and each association model were compared between traits in the two groups with a one-sided Wilcoxon rank-sum test . Given their genetic effect sizes on these two quantities, a causal effect of the exposure on the outcome can be estimated with two-sample Mendelian randomization (MR). Genetic effect sizes on transcript levels originate from whole blood expression quantitative trait loci (eQTLs) provided by the eQTLGen consortium .http://www.nealelab.is/uk-biobank/; Pan-UKBB team: https://pan.ukbb.broadinstitute.org) (TWMR was conducted as previously describedute.org) . Prior tTWMR results were used for validation of the mirror model associations. It is expected that TWMR and mirror model effects are directionally concordant, i.e., increase/decrease in copy number has the same direction of effect on a trait as an increase/decrease in gene expression. For this purpose, nominally significant (p < 0.05) TWMR effects were retained and their direction was compared to the direction of the probe with the smallest nominally significant p value (p < 0.05) in the mirror association model for the corresponding gene  and Pan-UKBB (https://pan.ukbb.broadinstitute.org)  MR for all exposure-outcome combinations . Second, all exposures with nominally significant IVW causal effect estimates for a given outcome were included in an MVMR analysis as exposures. To reduce bias due to potential reverse causation, we performed Steiger filtering in all MR analyses (p < 5 × 10−3).We performed MVMR to assess the causal relationship between significantly associated traits and compute a phenotype network. IVs were obtained from Neale Lab UKBB (ute.org)  GWAS sumexpected outcome) if that outcome is a downstream result of the exposure trait as suggested by the MVMR analysis . βexpected outcome was determined as βexposure × βIVW, where βexposure is the effect size of the best probe in the mirror model for each exposure  and βIVW is the causal estimate for each exposure-outcome pair obtained from IVW MR. We then compared βexpected outcome with the observed CNV effect on the outcome trait (βobserved outcome) obtained from the mirror association model.MVMR established the causal relationships among assessed traits by using genetic variants as IVs. To infer whether the pleiotropic effect of CNVs is vertical (indirect) or horizontal (genuine), we estimated what would be the expected CNV effect on the outcome trait   had a higher disease burden than individuals that are copy neutral within this region (mean number of diagnoses = 8), we compared the reported number of ICD-10 codes and identified no statistical difference (two-sided Wilcoxon rank-sum test: p= 0.053) B.CNVs were classified according to their localization as defined by LCRA-D. Between LCRs A and B, duplications were identified at a frequency of 0.01% and deletions at 0.002%; CNVs from LCR A to D had a frequency of 0.06% and 0.001% for duplications and deletions, respectively; from LCR B to D, duplications had a frequency of 0.002% and no deletions were identified; between LCRs C and D, duplications were identified at a frequency of 0.04% while deletions were identified at 0.008%. CNVs that did not fall into these categories were considered as atypical and had a frequency of 0.16% for both duplications and deletions A.To account for all CNVs and bypass issues related to breakpoint variability, CNV calls were converted into probe-by-sample matrices for the CNV association scan. Probe-level CNV frequency after excluding LCRA probes  ranged between 0.004% and 0.1% and 0.001% and 0.01% for duplications and deletions, respectively C.CNV association scan revealed significant links for eight continuous  and nine2, p = 4.9 × 10−10) throughout LCRA to LCRD (ARVCF (MIM: 602269) increases BMI , concordantly with the positive association found by the mirror CNV association scan  was found associated under the U-shape model , and the strongest association occurred in the LCRA to LCRB region (−15) and deletion-only  models, providing further evidence of a “true mirror” effect, despite the deletion effect’s being slightly stronger than the duplication one (DGCR6 (MIM: 601279) expression causally reduces MPV  . As expected, MVMR showed bidirectional causality between both traits, highlighting the challenges on interpreting their association separately.Mean platelet volume (MPV) was found associated under the mirror model  C. It is −7), while the deletion-only model was the only significant one at the distal portion between LCRC and LCRD   A. Given  × 10−6) A to insp × 10−6) B and 4C, × 10−6) D.Figure 109350), which was found to be associated under the duplication-only model  and had a stronger association occurring in the LCRA to LCRB region (Given the low number of deletion carriers affected by binary outcomes (0–3 carriers) , associaB region A, indicaB region B.Figure −9) to show stronger association with the corresponding UKBB continuous trait. For the binary traits, however, only two out of 19 assessed genes were significant in the mirror model, which does not indicate an enrichment (p = 0.07).For continuous traits, six out of eight assessed genes were found to have significantly greater association p values for the group of unrelated traits compared to the group of linked traits for all association models (see “ARVCF and DGCR6) causally affects two associated traits (BMI and MPV), TWMR results were also used to reinforce reliability of CNV associations. We evaluated concordance in the direction of effect sizes from nominally significant (p < 0.05) results of the mirror CNV association scan and nominally significant (p < 0.05) TWMR results  against the effect observed in the association scan (βobserved outcome) to examine the concordance in effect direction  by using the IVW MR method. MVMR was then applied on these 48 combinations and 17 trait-pairs were significant after Bonferroni correction (p < 0.05/289 = 0.0002) A. Most tirection ”. This a,,,,,Most of our knowledge regarding the impact of CNVs in the 22q11.2 region in the general population stems from genome-wide studies.LZTR1 ) and other venous embolism and thrombosis , that were both associated under the duplication model in the distal region between LCRC and LCRD, which harbors these genes.Assessed traits linked to 22q11.2 genes have been previously identified in different contexts including the 22q11.2 deletion and duplication syndromes, clinical conditions caused by variants in a single gene, and complex conditions associated with the locus . TherefoOur enrichment analysis showed that for continuous—but not binary—traits, leveraging the HPO database for trait selection was an effective approach. This observation may stem from the fact that continuous traits are better powered and more accurate than binary traits, which may ignore underlying continuous phenomenon. In addition, because association p values for binary traits are closer to the multiple testing threshold, we expect weaker enrichment p values.601665) (BMI > 30) has been previously described in adults with 22q11.2DS.,ARVCF—a gene whose product is part of the catenin family and is involved in protein-protein interactions at adherent junctions—on BMI. Recently, a rare ARVCF missense variant of unknown significance has been identified in an individual with early-onset severe obesity,ARVCF may play an important role in the etiology of obesity.Our results validated several known associations and furthermore shed light on traits that have not yet been extensively studied in the context of 22q11.2 CNVs. For instance, gastroesophageal reflux disease is not a vastly explored clinical feature in 22q11.2 deletion or duplication syndromes. While LCRA to LCRD duplications have been previously associated with this trait in the UKBB cohort,313900) is a well-known clinical hallmark in 22q11.2DSGP1BB represents a top candidate to explain the observed platelet phenotypes as bi-allelic loss of function variants in the gene are responsible for Bernard-Soulier syndrome, a platelet disorder, and inclusion of GP1BB in the deleted region has been implicated in reduction of platelet count levels in 22q11.2DS-affected individuals.GP1BB could not be assessed by TWMR analysis, which instead revealed a causal effect of DGCR6 differential expression on MPV. While DGCR6’s function is not yet clearly defined, it has been implicated in regulating other genes in the 22q11.2 region,Besides validating the link between CNVs in the 22q11.2 region and platelet count,,,Usage of four different association models allowed for the identification of deletion-specific effects  as well as traits in which duplications and deletions act in the same or in opposite directions. By performing association scans at the probe level, we also showed that gain or loss of distinct segments within 22q11.2 may impact a trait following different association models, as was seen for height. Short stature has been identified for the 22q11.2DSA drawback of studying pathogenic CNVs in a general population such as the UKBB is that the number of affected participants is low, as carriers of 22q11.2 CNVs with larger phenotypic impact are less likely to participate, a phenomenon often described as the “healthy volunteer” selection bias.While the absolute number of CNV carriers considered in our study is still larger than the sample size of some clinical cohorts, these individuals tend to exhibit milder phenotypes. This hampers statistical power to detect associations, especially for binary outcomes for which trait definition through grouping of ICD-10 codes is imperfect and arbitrary and case number can be extremely low. We offer corroborating evidence of our findings’ reliability by performing sensitivity analyses and examining the concordance of CNV findings with TWMR effects. Importantly, effects observed in our study are potentially smaller than the ones observed in clinical cohorts,We found that 22q11.2 CNVs affect traits compatible with clinical manifestations seen in the genomic disorders within the general population. The probe-level association scan revealed that dosage of different segments within the 22q11.2 region may impact a trait through different mechanisms, as illustrated with height. Besides, yielding further insights into the complex 22q11.2 region, our study provides a framework that can be adapted to study the phenotypic consequences of other clinically relevant genomic regions.


Heterobasidion annosum is a severe problem in boreal Scots pine. Dissecting the features of disease resistance is generally an essential step in resistance breeding in plants and forest trees. In this study, we explored inherent resistance factors of Scots pine against H. annosum. A total of 236 families consisting of 85 full-sib (FS), 35 half-sib population mix (HSpm), and 116 half-sib (HS) families of Scots pine seedlings were inoculated with a H. annosum isolate. We sampled needle tissues before inoculation for terpene measurements and RNA sequencing. Based on the lesion area, the extremes of 12 resistant and 12 susceptible families were selected for further analyses. Necrotic lesions resulting from fungal infection were in a weak to moderate relationship with the plant height. Monoterpenes were the principal terpene compounds observed in Scots pine seedlings. Concentrations of 3-carene were significantly higher in pine genotypes inherently resistant compared with susceptible seedlings. By contrast, susceptible genotypes had significantly higher proportions of α-pinene. Gene ontology analysis of differential expressed transcripts (DETs) revealed that response to biotic factors was enriched in resistant seedlings. Functional characterization of individual DETs revealed that higher expression of transcripts involved in response to abiotic stress was common in susceptible genotypes. This observation was supported by the annotation of hub genes in a key module that was significantly correlated with the lesion trait through weighted gene co-expression network analysis (WGCNA) of 16 HS and HSpm samples. These findings contribute to our understanding of constitutive resistance factors of Scots pine against Heterobasidion root and stem rot diseases.Root and stem rot caused by  Heterobasidion parviporum and Heterobasidion annosum within a species complex Heterobasidion annosum sensu lato (Fr.) Bref. are necrotrophic pathogens causing severe diseases in forest trees in the boreal region. The preferential host of H. parviporum is Norway spruce [Picea abies (L.) H. Karst.], whereas H. annosum attacks preferentially Scots pine (Pinus sylvestris L.) and also other species, including Norway spruce and common juniper (Juniperus communis L.)  using RNA sequencing (RNA-seq) data were carried out to investigate defense mechanisms underlying inherent resistance. Moreover, we analyzed individual transcripts according to their roles in signaling pathways that are central for crosstalk between biotic and abiotic stress responses. We aimed to explore constitutive variation in different resistant phenotypes.A transcriptomic analysis offers substantial information on gene expression on a genomic scale and assists researchers in intensely exploring the molecular cues of tree resistance  consisting of 85 full-sib (FS) families, 35 half-sib population mix (HSpm) families, and 116 half-sib (HS) families  previously shown to induce strong necrotic lesions was cultured at 20°C on malt extract–sawdust agar plates (MEA-S) containing 2% Scots pine sawdust, 2% malt extract, and 1.8% agar for 2 weeks before inoculation. In July 2019, inoculations were conducted on the stem at ca 15 cm above the greenhouse table according to the following steps: surface sterilization of the stem with 70% ethanol, drilling a hole with the puncher of 2 mm in diameter, and filling the punched hole with MEA-S agar plugs pre-colonized by H. annosum. Control treatments received mock inoculation after wounding. Both treatments were sealed with Parafilm M . At 3 months post-inoculation, the height of each plant was measured from the bottom to the top branches. At 4 months post-inoculation, sapwood lesions of all plants were measured. Lesion size was the sum of vertical and horizontal lesions as described by Keriö et al.  followed by Tukey's multiple comparisons at 0.05 confidence level. Pearson correlation coefficients and p values between lesion size, lesion area, and height based on family means by treatments were computed using the corr. test function in the package psych in R 4.1.0 . The variance components were derived from two mixed models where Treatment was a fixed factor and female (HS) or female and male parents (FS) were the random factors. The GCA referred either to the effect of the female parent (HS) or to the average effect of female and male parents (FS). The SCA was an interaction of a specific female–male combination (only estimable from the full-sib data). The analysis was performed using the ASReml software  < 0.05 and fold change (FC) > 2. Gene ontology (GO) enrichment of DETs was analyzed using the R package clusterProfiler  and 6.3 mm2 (SE 0.09 mm2), respectively, with significant differences at p < 0.001 for the comparison between the two treatments in each family type. Variations in lesion size, lesion area, and plant height were detected among families and data of the above traits were normally distributed  to central (4.9 mm2) and north (6.3 mm2) parts of Finland  and triphosphate tunnel metalloenzyme 2 (TTM2), as well as abundant DETs associated with abiotic stress responses  samples. After filtering, 20,618 transcripts were used to construct the network to find key modules of highly correlated genes. Samples from the same family type, having a similar lesion area clustered well , where rH. annosum varied among genotypes. Lesions observed in wounded control samples implied that defense responses could be triggered by mock inoculation , growth traits including diameter and height were positively correlated with lesion length in the inner bark and fungal growth  together with the ethylene response factor (ERF) and the TF function in response to necrotrophs  can be found in the article/FA and ML designed the study. ML performed greenhouse experiments and extracted RNA. MH estimated the heritability. RG and MK conducted terpene detection. ML and KW analyzed the data and drafted the manuscript. FA conceived the study and helped to draft the manuscript. All authors edited and revised the manuscript.The study was supported by the research funding from Academy of Finland (Grant No. 307580) to FA, scholarship from China Scholarship Council (Grant No. 201607960002), and Niemi Foundation (Grant No. 20200006) to ML.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


N = 991) showed improved prediction accuracy without considering age such as AUC of 0.830 for no vs. any hair loss. Because of the large number of markers and the large independent datasets used for the different analytical steps, the newly presented genetic prediction models are the most reliable ones currently available for MPB or any other human appearance trait.Genetic prediction of male pattern baldness (MPB) is important in science and society. Previous genetic MPB prediction models were limited by sparse marker coverage, small sample size, and/or data dependency in the different analytical steps. Here, we present novel models for genetic prediction of MPB based on a large set of markers and large independent subsample sets drawn among 187,435 European subjects. We selected 117 SNP predictors within 85 distinct loci from a list of 270 previously MPB-associated SNPs in 55,573 males of the UK Biobank Study (UKBB). Based on these 117 SNPs with and without age as additional predictor, we trained, by use of different methods, prediction models in a non-overlapping subset of 104,694 UKBB males and tested them in a non-overlapping subset of 26,177 UKBB males. Estimates of prediction accuracy were similar between methods with AUC ranges of 0.725–0.728 for severe, 0.631–0.635 for moderate, 0.598–0.602 for slight, and 0.708–0.711 for no hair loss with age, and slightly lower without, while prediction of any versus no hair loss gave 0.690–0.711 with age and slightly lower without. External validation in an early-onset enriched MPB dataset from the Bonn Study ( Male pattern baldness (MPB), or androgenetic alopecia, is a highly heritable, age independent appearance trait , 2. MPB A key requirement for the development of accurate genetic prediction models for genetically complex traits like MPB is the availability of sufficiently large number of DNA predictors. To date, four sufficiently powered GWASs for MPB have identified hundreds of MPB-associated SNPs, available for MPB prediction modelling. Heilmann-Heimbach et al.  reportedA key requirement for developing reliable genetic prediction models for MPB and any other trait are sufficiently large and independent datasets needed for the different analytical steps. The genetic prediction models by Hagenaars et al. , which wAiming to overcome limitations of previously reported models, we developed new genetic prediction models for MPB based on a large number of DNA markers and four large independent datasets for the four analytical steps i.e., i) 55,573 UKBB samples for prediction marker ascertainment, ii) 104,694 UKBB samples for prediction model training, iii) 26,177 UKBB samples for prediction model testing, and iv) 991 samples from the early-onset MPB enriched Bonn Study for external model validation.http://www.ukbiobank.ac.uk) is a large, population-based genetic epidemiology cohort of volunteers living in the UK. In response to UK Biobank question 2395  any hair loss) BLR models in the model training subset of 104,694 UKBB males. Subsequently, these models were validated in the independent model testing subset of 26,177 UKBB males and prediction accuracy parameters were estimated  MLR models as well as 2-category (no The 4-category MLR models without considering age predicted with considerably higher accuracies for severe (AUC = 0.718) and no hair loss 0.702) than for moderate (0.607) and slight (0.574) hair loss . Due to differences in genotyping platforms and quality control analyses, 10 of the 117 SNPs (Supplementary Table In the Bonn Study, the 107-SNP models that did not include age as predictor, achieved considerably higher prediction accuracy estimates compared to UKBB i.e., 0.830 in Bonn vs. 0.696 in UKBB for no hair loss, 0.775 vs. 0.712 for severe hair loss, and 0.830 vs. 0.696 for any vs. no hair loss. For the slight hair loss phenotype, the AUC in Bonn was slightly higher than in UKBB with 0.599 vs. 0.573, while for moderate hair loss it was slightly lower with 0.591 vs. 0.602 Table .For the 107-SNP models that included age as predictor, we achieved considerably less accurate prediction performances in the Bonn Study compared to UKBB Table  i.e., AUReliable genetic predictions for MPB can contribute to different disciplines of scientific research and science applications, including in medicine, evolutionary biology, anthropology, human history, and forensics. Previously reported genetic prediction models for MPB were limited by sparsity of genetic marker coverage, low-powered sample size, and non-independence of data used in the different analytical steps of prediction modelling. Here, we have sought to overcome these previous limitations by developing and validating new genetic MPB prediction models based on a large number of DNA predictors and large and fully independent datasets in the different analytical steps. These new models achieve more accurate and more reliable genetic MPB prediction than previous models.Previously, Marcińska et al.  applied For another previous genetic MPB prediction model based on 11 SNPs from only 7 genetic loci plus age, Liu et al.  reportedThe recently reported genetic prediction model for MPB by Hagenaars et al. based on 331 SNPs , includiOur study also demonstrates that it still is a long way towards highly accurate genetic prediction of MPB such as already available for eye colour with AUCs of 0.94 , 4. As nIn summary, we introduce new genetic prediction models for MPB that overcome limitations of previous models by including a large number of genetic predictors and utilizing large and independent datasets for the different analytical steps. Therefore our new models can be considered the most reliable genetic prediction models available to date for MPB and any other human appearance trait.Supplementary Information


Genetic studies of complex traits often show disparities in estimated heritability depending on the method used, whether by genomic associations or twin and family studies. We present a simulation of individual genomes with dynamic environmental conditions to consider how linear and nonlinear effects, gene-by-environment interactions, and gene-by-environment correlations may work together to govern the long-term development of complex traits and affect estimates of heritability from common methods. Our simulation studies demonstrate that the genetic effects estimated by genome wide association studies in unrelated individuals are inadequate to characterize gene-by-environment interaction, while including related individuals in genome-wide complex trait analysis (GCTA) allows gene-by-environment interactions to be recovered in the heritability. These theoretical findings provide an explanation for the “missing heritability” problem and bridge the conceptual gap between the most common findings of GCTA and twin studies. Future studies may use the simulation model to test hypotheses about phenotypic complexity either in an exploratory way or by replicating well-established observations of specific phenotypes. Although it has been clear for one hundred years that differences in genotype are correlated with differences in complex human behavior e.g., , the bioA second theme of this paper is the role that nonlinearity and nonadditivity play in estimates of heritability. In the presence of gene–gene interactions and gene-by-environment interactions, the conventional notion of heritability loses its focus. Broadly, pairs of identical twins are more similar than pairs of fraternal twins for almost every behavioral phenotype that has ever been studied . More ge1 However, even though twin studies estimate substantial non-shared environmental variance components, these latent components cannot generally be decomposed into detectable causal effects of the actual environmental events they comprise. When The causal role of the environment in the development of behavior is, if anything, even more intractable than that of genetics. Some behavior geneticists have expressed doubt about the very existence of systematic environmental effects that apply equally to all people e.g., . In twinThe phenotype is attracted to the genes (additive genetic effects).The phenotype is attracted to the environment .The environment is attracted to the phenotype (gene-by-environment correlation).The relative strength of each gene depends on the location of the environment (gene-by-environment interaction).As a means of understanding the disconnect between reliable variance components and inscrutable causal processes, The final position of the phenotype converged on a location, which was recorded and analyzed in terms of the genotype and the original position of the environment. This simple system reproduced many of the findings of classical twin studies. Twin simulations with identical genotypes produced highly similar outcomes. “DZ” twins – who shared half of their gene locations in the two-dimensional space – showed some similarity but much less than their “MZ” twin counterparts. In general, outcomes were much more predictable from the static genotype than from the dynamic environment. In fact, much of the variability in the simulation was not predictable by either the genes or the environment; the outcome was chaotic and fractal in nature.In the years since the publication of those simulations, sequencing of the human genome has been completed, and genetic technology based on measured DNA has risen to prominence, to some extent supplanting methods based on genetic relatedness among family members. These technological advances led to an expectation that relatively straightforward biological pathways linking genes to behavior would eventually be discovered e.g., , but sucall DNA variants may plausibly be associated with complex outcomes. Moreover, causal pathways between genetic variants and phenotypic outcomes are highly complex, largely unknown, and have been shown to be mediated by environmental processes. Genetic associations within sibling pairs . Thus, P(ig = 0) = π2, P(ig = 1) = 2π(1 − π), and P(ig = 2) = (1 − π)2. For simplicity, the minor allele frequency π was the same for all loci. Generating allelic values as independent draws from a categorical distribution has some correspondence to the true underlying biological processes, but lacks other known aspects of the biology. In particular, there is no linkage, linkage disequilibrium, or population structure in the simulated genome.The model begins by generating genomes as vectors of allelic values i to its attractor strength Λi. Although Λi is not itself the effect found by regressing the phenotype on SNPi, it ultimately produces that effect by defining how strongly SNPi attracts the phenotype toward a particular point in the phenotype space, as will be defined later.ψ determines the shape of the distribution of SNP effect sizes, with larger values of ψ concentrating the total effect over fewer SNPs. At ψ = 0, all SNPs have the same effect size. Dividing by Ng ensures that the SNP attractor strength distribution defined by Λi is invariant with respect to the number of SNPs. We chose the exponential function for the distribution of SNP effect sizes because it instantiates key characteristics of the current thinking about polygenicity and SNP effects  that changes over time t depending on the SNP’s proximity to dynamic environment h.h,i(t) is 1 for SNPs with G×E effects. The denominator of Γh,i(t) is a function of the “distance” between the SNP and the dynamic environment. The exponent ξ defines the G×E activation curve, or how quickly G×E effects increase as dynamic environment h approaches an interacting SNP i. Setting ξ = 2 yields an inverse square law of attraction similar to gravitational attraction proportional to the inverse of the squared distance between the SNP and the dynamic environment. The notion of proximity or distance used here is abstract and represents the degree to which an environmental factor is in any sense present to the individual, rather than any literal, physical distance. For example, certain individuals may have a heightened genetic liability toward alcohol addiction that does not actively affect their alcoholism or drinking behavior phenotypes until alcohol is initially consumed. The moment at which alcohol is consumed may be regarded in our simulation as a point of close proximity between the dynamic environment representing alcohol and the SNP effect representing the liability.Each SNP ξ of the distance between the SNP and the dynamic environmental factors.The G×E that occurs in our model differs from G×E that occurs in some other settings e.g., , Ch. 9. p(t) and the dynamic environmental factors dh(t) are solutions of these differential equations. In ig − p(t). This term instantiates the first rule that the phenotype is attracted to the genes. The second term gives the G×E and has time-varying effects modified by the gene-environment distance related to h,iΓ(t). This term instantiates the fourth rule that the relative strength of each gene depends on the location of the environment. The third and fourth terms are proportional to the distance between the phenotype and the dynamic and static environmental factors, respectively. This term instantiates the second rule that the phenotype is attracted to the environment. Finally, the fifth term gives viscous friction for the phenotype that resists all movement. This term does not instantiate any rule listed previously, and its role will be discussed later. The third rule is not instantiated in p(t) around its space of possible values is modeled in terms of acceleration p(t), its velocity dh(t), the static SNPS ig and the static environmental factors ej. The total additive genetic effect is given by α, and the total G×E effects is given by γ. δ and ∈ give the effect size of the dynamic and static environmental factors, respectively.The phenotype-gene-environment system is modeled as as a system of second-order, nonlinear differential equations. The trajectories of the expressed phenotype gi in the phenotypic space is simply the number of minor alleles . The linear component of the attraction strength to that locus is then given by ʌi, whereas the G×E component is given by Γh,i(t). The gene-by-environment interaction, with strength given by Γh,t, is defined by attraction to a randomly chosen locus in the phenotypic space, lh,i  in i and dynamic environmental factors h:The locus of each SNP effect α and γ, respectively.Normalizing each set of effect sizes is required to parameterize the model in terms of the total additive genetic effects and G×E on the phenotype, dh(t), in turn, is attracted toward the phenotype with strength hβ:Each dynamic environment gth βh:d¨h(t)=βζp and ζd, in both the phenotypic and environment equations above are included to induce long-term convergence of the phenotype and dynamic environment to fixed values. In physics, damping terms are used to model the dissipation of energy due to friction, and we use this as an analogy for the effects of aging on developmental processes. For example, most of the intraindividual changes in measures of personality occur before age 30  were not included in the model. Substantively, this is justified by the infrequency of common environment main effects in most phenotypes, but also allowed us to simplify our examination to just the effects of gene-by-environment interaction. Hence, no main effects of C were simulated.For simplicity, common environment main effects  are given in To gain insights from the proposed model, we conduct four simulation studies. Many simulation studies in behavior genetics begin with assuming the true, data-generating model is closely related to the model used to analyze data, usually variance component models and regression models. We diverge from this path. We begin with the dynamical system model discussed in the previous sections rather than a statistical model. We intend the dynamical system model to have some basic plausibility and correspondence to an abstract description of the true mechanisms which generate gene-environment-phenotype data. We then explore the output from the dynamical system model for its implications on the conventional data-analytic statistical models. The approach is similar to that recently outlined by Our first study explores under what conditions our model allows individual gene effects to be discovered. In overview, we generate data from our model under three conditions and analyze these data using standard methodology developed for genome-wide association studies (GWAS). The goal is to correctly recover the linear genetic effects in the presence of varying amounts of G×E.ψ as in γ from γ = α = 0.05. In condition (b), the G×E effect strength is slightly larger than the additive genetic effect strength: γ = 0.20, α = 0.05. In condition (c), the G×E effect strength is much larger than the linear effect strength: γ = 0.35, α = 0.05. These three conditions create models that are (a) almost linear, (b) moderately nonlinear, or (c) highly nonlinear.We generate data on 10,000 individuals each with 1,000 genes. All individuals are generated independently, with identity-by-state (IBS) relatedness occurring by chance. The true, linear gene effect sizes follow an exponential distribution as in i for all possible locations using ordinary least squares regression. From the regressions, we obtain the slopes, squared correlations (R2), and p-values for each of the 1,000 genes. Each of these three outcomes is shown for each of the three conditions in We analyze the data with standard GWAS methods. In particular, we predict the phenotype at the last time point from the gene at index location p-value  is graphed against the gene index to create a pseudo Manhattan plot. We call the first row of 4 a pseudo Manhattan plot because there is no true sense of gene location in our simulated genome. Genes that are statistically significant after a Bonferroni correction are highlighted as being above the dashed line. In the “almost linear” condition, 26 genes have Bonferroni-corrected significant effects, and the genes with the largest true effects are the ones that are observed empirically to have the largest effects. In the “moderately nonlinear” condition, only 4 genes have Bonferroni-corrected significant effects. In the “highly nonlinear” condition, no genes have Bonferroni-corrected significant effects. So, in the almost linear condition, the linear part of the true gene effects can be recovered, but nonlinearity destroys the ability to discover the linear gene effects.3 As the strength of the nonlinear effect increases, the ability to find statistically significant effects for genes decreases.In the first row of R2) is plotted against the gene index. The cumulative R2 is computed by summing the R2 from each of the separate simple linear regressions. Therefore, it is theoretically possible for the cumulative variance to exceed the total phenotypic variance due to correlated predictors being entered as predictors in separate regressions with a common outcome variable.4 However, even in the almost linear condition, the cumulative R2 never exceeds 0.40 in these simulated data because the true gene effects are independent. Across all conditions, the first few genes have the largest effects and thus increase the cumulative R2 at a steep rate, whereas the later genes have smaller effects and thus increase the cumulative R2 at a shallow rate. As the condition changes from almost linear to highly nonlinear, the cumulative R2 decreases. The same linear gene effects have different R2 values depending on the strength of the nonlinearity. Hence, the nonlinearity destroys the ability of the linear gene effects to explain phenotypic variance. Whereas the first row of In the second row of In the third row of Taken together, study 1 finds that when the true situation is primarily governed by linear gene effects, then the typical gene-finding methods in GWAS do indeed find the correct, linear gene effects. However, as nonlinearity increases, the true, linear gene effects that are present are no longer discoverable. These findings hold regardless of the metric used for gene finding: Manhattan plot, cumulative variance explained by genes, or QQ plot of the SNP effects. Given the results of study 1 for GWAS, we expect similar findings for risk prediction using polygenic risk scores and for heritability estimation using genome-wide complex trait analysis. However, to eliminate the possibility of some unexpected results, we still conduct their analyses.In study 2 we use the same data generated in study 1, but for a new purpose. Instead of attempting to find the genes that have the largest influences on the phenotype, we are now attempting to explain as much variability in the phenotype as possible while using only the SNPs. The method creates a linear regression model with the phenotype as the outcome and some – but usually not all – of the SNPs as predictors. The regression model outputs a “risk” score for the phenotype  based on many, linear gene effects. Thus, the predicted phenotype is called a polygenic risk score (PRS).The PRS differs from the GWAS in several important ways. First, GWAS is often a preliminary step to PRS regression that determines which and how many SNPs are used to create the PRS. Second, multiple SNP effects are used simultaneously to predict the phenotype in PRS regression, whereas GWAS exclusively considers one SNP at a time. Third, the evaluation of a PRS often uses a hold-out sample: that is, the PRS model is trained on one part of a data set, but is then evaluated on a separate part of the data set that was not used to fit the model. This cross-validation is critical to PRS evaluation. In study 2, we evaluate the utility of PRS under the same nonlinearity conditions as study 1 using the same data, but now with the addition of multiple regression and cross-validation.R2 for a PRS created by adding SNP predictors cumulatively in order from most statistically significant to least statistically significant. Thus, the first PRS uses the single SNP with the smallest p-value; the second PRS uses the two SNPs with smallest p-values; and so on until the final PRS uses all available SNPs. The gray points indicate the R2 for the in-sample data that trained the model, thus being vulnerable to capitalization on chance. The red points are Bonferroni-corrected statistically significant. The black line indicates the out-of-sample data, thus being a measure of predictive quality that is less susceptible to capitalization on chance.R2 decreases. The peak R2 in the almost linear condition is slightly more than 0.20 which matches the findings observed in many studies. In the moderately nonlinear and highly nonlinear conditions, the in-sample R2 follows a similar pattern to the almost linear condition: rapid initial increase, followed by an asymptote. However, the out-of-sample R2 never shows any appreciable predictive utility. Note that the shared vertical scale across conditions obscures some of the nuances for the moderately nonlinear and highly nonlinear conditions.In the almost linear condition, there is clear predictive utility to the PRS for the first several SNPs. However, after the first 50 SNPs, the out-of-sample γ increases, all predictive quality of the PRS is eliminated.The findings for study 2 replicate those of study 1. In the almost linear condition, a useful PRS is possible to construct. The predictive utility is limited by the small amount of nonlinearity that is present, but a high-quality PRS can explain around 20 percent of the variance in the phenotype using only a linear combination of the SNPs. However, as the nonlinear effect strength a priori between pairs of unrelated individuals, which are usually between plus and minus 0.025. This correlation structure gives the IBS genetic relatedness between people.In study 3, we simulated data to determine the impact of G×E on the ability to recover the heritability of a phenotype from a sample of people with measured genomes. In assessing heritability, we want to know how much variability can be explained by all the gene effects together. We report narrow sense heritability as a value between 0 and 1, the proportion of phenotypic variance accounted for by the linear, additive effect of genes. Methods for estimating heritability from samples of known relatives have been well-known for one hundred years e.g., , but theGCTA differs from GWAS and PRS in several important ways. Unlike GWAS and PRS, GCTA does not seek particular genes that influence the phenotype. Rather, GCTA estimates the proportion of variability “explained” by genes without any effort to determine the loci or magnitude of particular effects. Moreover, GCTA does not predict particular phenotypes or evaluate the “risk” of developing a phenotypic outcome. Instead, GCTA addresses the question of the total, additive genetic contribution to phenotypic variation.δ, d0) were randomly generated and unique to each individual. The G×E effect size (γ) was gradually increased from 0, in which case genetic effects were entirely additive and the GCTA model was correctly specified, to 0.2, simulating strong, unmodeled G×E. The total additive genetic attractor strength . Environmental dynamics  is about 80 percent, but as G×E increases, the estimated heritability approaches its lower limit of zero. The change in heritability cannot be explained by the addition of environmental variance alone. That is, the proportion of the additive genetic variance cannot be explained by simply increasing the total variance . As In our fourth and final study, we simulated data to explore how the degree of genetic relatedness among pairs of participants in a sample impacts the estimates of heritability when an interaction between genes and the shared environment is present. A well-established finding is that the heritability estimates from twin and family designs are often larger than those found using GCTA with nominally unrelated people. The deficit in the estimated heritability among unrelated people is called missing heritability, and holds regardless of the phenotype. We use the same GCTA method in study 4 as study 3, but instead of varying the G×E effect strength, we vary the degree of relatedness among pairs of simulated people. The GCTA then uses both within-family and between-family genetic correlations. The within-family correlation, or relatedness, varies by condition across typical familial values . The between-family relatedness is for unrelated individuals and is always near zero .a priori relatedness given by the zygosity of the relatives, such as a genetic correlation of 1.0 in monozygotic twins and 0.5 in dizygotic twins, versus empirical genetic correlations in unrelated individuals, (2) the size of the family, which is usually two in twin and family designs, but thousands in the modern measured genome designs, and (3) how relatedness is measured, which is usually average amount of segregating genes shared in twin and family designs , but the measured correlation between SNPs in modern molecular designs . Hence, missing heritability could arise as a function of pair relatedness in the sample or as a function of differences between IBD and IBS relatedness  can be seen as a special case of the modern measured genome designs of nominally unrelated people . The difWe propose that one source of missing heritability specifically occurs because shared genes interact with shared environmental factors . If an A–C interaction was present but was omitted from the fitted model, the effect of the genes could be systematically incorrect  slightly differed across families .In the study 4 simulation, the sample consisted of 1000 simulated pairs of relatives, with related individuals sharing both the initial values and dynamics of their environmental influences, representing familial environment. Related genomes were generated by first producing two random, independent genomes, then randomly replacing the specified proportion  of alleles in one person’s genome with the relative’s values at the same loci. No environmental attributes  or genes were shared between pairs. The GCTA estimates of heritability used empirical, IBS estimates of both the correlations within and between pairs, rather than including the relatedness γ = 0.2. All other parameters remained the same as those in study 3, with the exception that dynamic environmental factors were constrained to have the same dynamic properties and gene pairings among related twins. The degree of relatedness among the pairs varied across conditions: 0 (unrelated), 0.125 (first cousins), 0.25 , 0.5 (DZ twins or full siblings), and 1.0 (MZ twins). In each condition, the sample consisted entirely of people with only one type of relatedness. Thus, the first condition has only first cousins; the second condition has only half siblings; and so on. The goal was to see the effect of the degree of relatedness on estimated heritability in the presence of gene-by-environment interaction when related individuals also shared the environmental component of the interaction. The model contained no specific sources of linear shared environmental effects (C), as is typically the case for GCTA studies of unrelated individuals.The gene-by-environment effect size was fixed at the highest value from the study 3 simulation, Twenty years ago, the senior author demonstrated that a simple dynamic simulation of genes and environment could reproduce some of the more perplexing findings of twin studies. Here, we have shown that a more sophisticated version of the same basic principles can reproduce many of the findings that have occurred in complex genetics since that time, especially as regards analysis of measured DNA in place of indirect inferences from twins. We have proposed an abstract dynamical systems model of phenotypic development involving gene-by-environment interaction and gene-by-environment correlation. Model simulations were used to produce phenotypic distributions under various conditions, which could then be used to test theoretical and methodological questions about nonlinear development.In the first and second set of simulations, we considered recovery of SNP effects in GWAS and the utility of corresponding polygenic risk scores in the presence of gene-by-environment interaction. In a linear model with unrelated individuals and thus no shared environmental effects, GxE contributes uniquely to each individual due to the dependence on environmental factors. Adding G×E to the simulation not only introduced effects that could not be subsequently recovered, but also demonstrated masking of otherwise recoverable linear SNP effects.In the third and fourth set of simulations, we considered recovery of total heritability estimated with GCTA in the presence of gene-by-environment interaction and varying degrees of relatedness in the sample. G×E decreased GCTA-based estimates of heritability to near zero even when the simulated genetic effects accounted for 80% of the total variance. However, some portion of the genetic effect can still be recovered in GCTA when the sample includes many pairs or groups of related individuals that share not only genes but environmental factors. This result gives one possible explanation for why twin studies produce larger estimates of heritability overall than GCTA or polygenic risk scores from GWAS.In terms of latent variance components, notated A for additive genetic, C for common or shared environmental, and E for unique environmental variance, an interaction of genes with shared environment would be an A–C interaction, whereas the more common concept of individual-specific G×E is better described as an A–E interaction. The presence of a true, underlying A–C interaction inflates estimates of A in the classical twin model, but not in models where participants are unrelated, because it depends on the existence of shared environmental factors. However, A–C interaction does not necessitate a main effect from C. An A–C interaction with no main effects may be understood as a shared environmental factor of which the effect is conditional entirely upon particular alleles. In our simulations, no static C-component was simulated, but A–C interaction took the form of a dynamic environmental attractor shared between twins. The effects of particular genes were enhanced by proximity to the dynamic environment, and MZ twins were more likely to share genes that interacted with the dynamic environment. In this way, unmodeled A–C interaction with no main C effect is one possible cause of the disparity in heritability estimates between twin studies, GCTA, and via PRS.The A–C interaction term can, in principle, be included in a twin and family modeling framework and is statistically identified in several designs, including the commonplace twins reared together and apart design. −1(C), in which case the additive genetic contribution is scaled between 0 and 100% by the common environmental factor, or A2C, in which case the effects of the common environmental factor are largest when one falls close to either tail of the additive genetic liability distribution. For most purposes the A–C product adequately represents the property that additional effects on the phenotype arise only as a joint function of both factors.A variance-components representation of gene-by-environment interactions dispenses with the chaotic details of exactly how the phenotype is nonlinearly governed by combined genes and environment and refers only the expected distribution of the phenotype taken at some developmental terminus or as a time-invariant abstraction. Other, more specific nonlinear components could be considered possible: for example, an interaction between A and logitThe consequence of these considerations is that the original hope of DNA-based complex genetics, that of finding individual genes or SNPs with statistical main effects of sufficient magnitude to form the basis of biological processes leading from the genome to high-level phenotypes, will under a reasonable set of parameters be mostly impossible. It is important to note that the simulations we have performed, while complex in comparison to simple linear models, almost certainly underestimate the complexity of actual human development by many orders of magnitude. If one imagines the kinds of “regulatory networks” that might underlie a trait like human personality, and multiply them by individual-level G×E interactions in the course of behavioral development, one can quickly reach a level of complexity that will not be rescued by ever-larger sample sizes or more sophisticated sequencing.In all of our simulation studies, genes had a strong causal influence on the phenotype. However, modern molecular behavioral genetics techniques only found the gene effects when the dynamics were mostly linear . We could not find the gene effects using GWAS, PRS, or GCTA even though we knew the effects were present. In classical twin and family designs it is well-known that some nonlinear genetic effects  attenuate linear genetic effect sizes . These sOur study 4 produced one possible explanation for the phenomena of missing heritability: estimates of heritability were higher in samples of more closely related people than in samples of less related people. This simulation study suggests that missing heritability can result from unmodeled A–C interaction. Specifically, people who are closely related may also share many of the same environmental influences, and consequently, may both experience the same gene-by-environment interactions. The kind of heritability detected due to A–C interaction in our simulation is not, however, the same additive source of variation expected by a GWAS or GCTA, and may be regarded as a separate kind of heritability altogether.It may be argued that GCTA and other linear methods do perform their intended task correctly by only estimating the additive, linear contributions of genes, whereas the missing heritability found in twin models reflects violations of their core assumptions. At its root, the missing heritability issue invokes the question of how heritability is best characterized. For certain purposes, it may not be satisfactory or relevant to define heritability such that it conditions on transient, unpredictable, and mutable environmental factors. If heritability is specifically posed in the context of fixed, immutable sources of variation in the population, then additive linear effects, though potentially very small, are still of central interest. Where additive effects are consistently very small, the utility of such a definition becomes less clear. We argue that when the goal of research is to understand the specific relations between genes and phenotypic outcomes, overly general concepts like heritability are far less useful than specific theories about gene-by-environment complexity.It is an historical irony that the field of complex trait genetics has seen little input from the science of complexity. Human development is certainly a complex system in the colloquial sense of a system that is complicated. However, Developmental studies of behaviorally complex traits commonly find that the relative influence of genes is greatest later in life , due in We developed the mathematical model from several basic principles in biology and behavior genetics. However, the model lacks full fidelity to some known aspects of biology. In particular, the model lacks any representation of chromosomes, inheritance, and consequently linkage and linkage disequilibrium (see p(t)) may be momentary affect, eating behavior, number of drinks or cigarettes, respectively, while the frequency and amplitude of their fluctuations represent clinically meaningful complex traits, subject to the underlying genetic and environmental architecture.For our simulations, we defined “the phenotype” as the value of the time-varying phenotype upon convergence after a sufficient amount of time has passed. This results in relatively simple analysis of the outputs in terms of phenotypic variance components, as all periodic or chaotic variability leading up to that point can be disregarded. However, other questions may be considered or framed using different phenotypic definitions. If the phenotype is defined to be a particular value in the trajectory before long-term convergence, then such periodic or chaotic movement of the dynamic environment behaves as a source of unique environmental variance. Other definitions are possible, such as frequency bands from a spectral analysis of the phenotype or descriptive statistics such as the mean and standard deviation of the total distribution of each person’s time series. For example, the parameters of genetic and environmental complexity in our model may be adjusted to reproduce observed episodic fluctuations observed in bipolar disorder, binging behavior associated with eating disorders, alcohol, or substance abuse. In each of these, the phenotypic variable  did not rely on random noise, (2) represented only key, well-known concepts in behavioral genetics, and (3) represented such concepts with minimal unnecessary complexity.Other representations of the same concepts would be possible with stochastic simulations. Such simulations would have involved stochastic differential equations (SDEs) instead of the ordinary differential equations (ODEs) used in our model. Some benefit of simulating with SDEs include known analytic derivations of the total series variance and prevention of identical outcomes due to identical starting conditions. However, our simulations demonstrate that the key concepts of behavioral genetics can be represented without random noise, although the complex, deterministic variations in time series here are necessarily subsumed in noise components of corresponding cross-sectional variance-components models.Our simulation serves mainly as a proof of concept and was not intended to replicate any specific phenotypic findings. Our dynamic environmental factors, for example, are particularly simple, taking the form of smooth sinusoidal functions. Many real complex traits exhibit much greater environmental and developmental complexity, and much smaller estimated heritabilities. By keeping the model relatively simple, we found that even a small set of nonlinear factors can create dramatic problems for finding causal effects. Using unrealistic genetic effects allowed us to demonstrate a general principle with minimal unnecessary computation that nonetheless can be scaled to more realistic effects.The simulation model can be configured to replicate empirical findings in specific phenotypes, including additive genetic and environmental variance components, nonlinearities, and temporal or developmental patterns at various timescales. If some or all of this information has been determined previously, then the simulation may be used to infer other aspects of the phenotypic model that are difficult or impossible to estimate empirically, either due to a lack of data, determination of covariates or endophenotypes, or because models with necessary components like A–C interaction are difficult or impossible to statistically identify . We have produced an online graphical interface for the model and invite phenotypic domain experts to explore configurations of the model that best represent known phenotypic etiology .ϵ, δ) and the damping parameters of each , as these are represented in models of continuous-time vector autoregression. As this modeling approach is primarily theoretical and differs from conventional data-driven fitting and estimation procedures, we did not design the model for parameter identification. Hence, fitting the simulated trajectories to data would be difficult. If appropriate, momentary phenotypic data were available in abundance over the full developmental timeline, then some parameters of the model could be estimated approximately using continuous-time state-space model techniques. Two such parameters include the attraction between the phenotype and environment (Genetic studies of complex traits often show disparities in estimated heritability depending on the method used, whether by modeling twins, GWAS and associated PRS, or by GCTA. We developed a simulation of individual genomes and dynamic environmental conditions to consider how linear effects, gene-by-environment interactions, and gene-by-environment correlations may work together to govern the long-term development of complex traits. Our simulation studies demonstrate ways complexity obscures linear, narrow sense heritability in conventional modeling strategies and produces a more nuanced, nonlinear understanding of genetic effects. The genetic effects estimated by GWAS and GCTA in unrelated individuals were inadequate to characterize the true, data-generating gene-by-environment model.Inclusion of related individuals in GCTA resulted in high heritability estimates when highly related individuals shared a common dynamic environment and the genes with which it interacts. The resulting estimated heritability represents genetic sources of phenotypic covariance that are distinct from, and can actually mask, additive genetic variance. Researchers are thus likely to underestimate the roles of both genetic and common environmental factors in phenotypic variation when only modeling unrelated individuals from independent environments.Future studies may use the simulation model to test hypotheses about phenotypic complexity either in a general way or by replicating well-established observations of specific phenotypes.


Second, for each subset, TPs representative of the new SC breeding lines were determined based on genetic relatedness using the ‘STPGA’ R package. Third, for each TP, phenotypic values and SNP data were incorporated into the ‘rrBLUP’ mixed models for generation of GEBVs of YLD, TW, HD and PH. Using a five-fold cross-validation strategy, an average accuracy of r = 0.42 was obtained for yield between all TPs. The validation performed with 58 SC elite breeding lines resulted in an accuracy of r = 0.62 when the TP included complete historical data. Lastly, QTL-by-environment interaction for 18 major effect genes across three geographic regions was examined. Lines harboring major QTL in the absence of disease could potentially underperform , whereas it is advantageous to express a major QTL under biotic pressure . This study highlights the importance of genomics-enabled breeding and multi-institutional partnerships to accelerate cultivar development.With the rapid generation and preservation of both genomic and phenotypic information for many genotypes within crops and across locations, emerging breeding programs have a valuable opportunity to leverage these resources to 1) establish the most appropriate genetic foundation at program inception and 2) implement robust genomic prediction platforms that can effectively select future breeding lines. Integrating genomics-enabled Triticum aestivum L.) is a major cereal crop worldwide as its production ranks third in the US (49.7 million tonnes) and globally (895.2 million tonnes) behind maize and soybean. Wheat has a high production value of US$8.7 billion in the United States and $188.1 billion globally (FAOSTAT 2022). The effects of climate change, including warming temperatures, variable precipitation and more frequent extreme weather events  is becoming a valuable technology for modern crop breeding programs, and its implementation for cultivar development has been shown to accelerate the rate of genetic gain by shortening the breeding cycle and/or increasing selection accuracy . Genomicr = 0.50 in France   , agronomce (FHB)  and qualce (FHB) . Lastly,ce (FHB) .This study was conducted to understand how new or reestablished breeding programs should leverage existing historical genomic, and multi-environment and multi-trait phenotype data of elite breeding lines. The Clemson University winter wheat breeding program was reestablished in 2017 and used as a case study to understand the foundational genetics and requirements to maximize predictive ability of genomic models to successfully develop cultivars adapted to a target population of environments. To accomplish this, historical genotypic and phenotypic information for advanced soft red winter (SRW) wheat breeding lines, evaluated as part of the Southeastern University small grains (SunGrains) breeding alliance, was used to predict YLD, TW, HD and PH using optimized TPs for a set of untested SC-derived breeding lines. Two validation strategies were completed to assess and compare fitted models’ prediction accuracy. Finally, QTL-by-environment (QE) interaction analysis was completed using 18 major effect QTL to identify whether there was a favorable effect on yield across three major testing regions. The use of comprehensive datasets and genomic models have great value to securing the needed increases in genetic gain and enhance the efficiency of cultivar development.Annually, advanced SRW wheat lines entered into the Gulf Atlantic wheat nursery (GAWN) and advanced wheat nursery (SunWheat) are evaluated across the greater southeastern US, which is coordinated by SunGrains and partnering public wheat breeding programs. The SunGrains cooperative includes seven land-grant university breeding programs  having very strong collaborations for field evaluation and unfettered distribution of adapted germplasm and data exchange. As part of this historical cooperative, a total of 1,285 lines were tested in 19 locations  from 200−1), TW (kg hl−1), HD (Julian days) and PH (cm) was 17,645, 14,942, 11,092 and 8,678, respectively. The number of replications ranged from one to three, depending on location-year combination.Historical phenotypic data consisted of a multi-location, multi-year and multi-trait dataset generated and maintained by SunGrains. A total of 1,285 elite SRW wheat breeding lines were tested in two regional nurseries (GAWN and SunWheat) in 19 trial locations in the southeastern US . The num‘princomp’, ‘cor’ and ‘corrplot’ packages in R; second, biplots showing the relationship among environments  plots from Pearson’s correlation matrix using the ronments  were obts , the following linear model  was fittage in R  to estimYijk represents the phenotypic observation of genotype i in environment j and replication k, µ is the overall mean; Gi is the effect of genotype i, Ej is the effect of environment (location-year combination) j; Rk(j) the effect of replication k nested in environment j; GEij the G × E interaction between genotype i and environment j; and eijk the residual effect associated with genotype i in environment j and replication k. All terms except genotype were set as random effects. Genotype was defined as fixed effect  with the ‘H2cal’ function of the ‘inti’ R package  new SC breeding lines were used to identify the best TP using each of the subsets of trial locations . The bes package  was utilt to 400 , and otht to 400 . The fir‘ggplot’ package in R (pheatmap’ package (prcomp’ package and plotted with ‘ggplot’.The frequency and percentage (%) of breeding lines by breeding program selected by STPGA was calculated for each TP and normalized by number of lines by program. A stacked barplot was displayed with age in R . A heatm package , and PCAGenomic best linear unbiased prediction (GBLUP) mixed models were fitted to estimate GEBVs for YLD, TW, TW and PH with the following equation:y represents the vector of BLUEs for each genotype; X and Z represent the design matrices for fixed and random effects, respectively; β is the vector of fixed effects; μ is the vector for random genotypic effects; and e is a vector of residuals  and B (lower-left) for correctly classified lines, and C (lower-right) and D (upper-left) for wrongly categorized genotypes.Two types of validation were implemented to assess and compare each model’s prediction ability using each TP for each trait. First, five-fold cross-validation (CV) is a procedure that randomly divides the TP into five groups of approximately equal size (20%). One random group is masked and GEBVs are calculated for the masked set using the remaining four folds (80% of lines) . After c‘mixed.solve’ function of the R ‘rrBLUP’ package as previously described. With the aim to select a subset of lines as the foundation for the breeding program as well as first year advanced field testing, one hundred and five superior lines were identified based on GEBVs for YLD, and 20 low-ranked lines were identified for comparison. Advanced field evaluation for these lines is in progress (data not shown).Furthermore, GBLUP mixed models were fitted to estimate GEBVs for the 998 breeding lines developed by the Clemson University breeding program. Using the EMMs and BLUPs of the TP with highest prediction accuracy, predictions of YLD, TW, HD and PH were performed with the Fusarium graminearum) resistance, leaf rust (Puccinia triticina) resistance, stem rust (P. graminis) resistance, stripe rust (P. striiformis) resistance, Hessian fly (Mayetiola destructor) resistance, powdery mildew (Blumeria graminis) resistance, septoria nodorum blotch (Parastagonospora nodorum) susceptibility, or photoperiod sensitivity  to generate composite calls for major effect genes at the Eastern Regional Genotyping Small Grains Laboratory . Over the course of 14 years (2008–2021), 4,426 breeding lines were tested for 75 molecular markers associated with disease/pest resistance, photoperiod, vernalization, dwarfing, grain texture and kernel color . For thisitivity . In addiHistorical phenotype data from 1,285 breeding lines was used to assess the effect on agronomic traits when the expression of a major effect QTL differs under different environmental pressure  across production locations . The his‘lmer’ of ‘lme4’ package in R:For each of the 18 major genes, only absent/present calls were considered for analysis by discarding heterozygous, null, and failed calls. The following linear mixed model was calculated to test the significance of QTL-by-environment (region/origin) interactions for YLD, TW and HD using the function Yijk represents the phenotypic observation of genotype i in environment j and replication k, µ the overall mean; Gi is the effect of genotype i, Ej is the effect of environment (location-year combination) j; Rk(j) the effect of replication k nested in environment j; GEij the G x E interaction between genotype i and environment j; QE the QTL-by-environment interaction effect; and eijk the residual effect associated with genotype i in environment j and replication k. QE effect was considered fixed and all the remaining effects were considered random.Where ‘emmeans’ function in R for the fixed effect of the interaction between major gene (absent/present) and region . To estimate the significant difference at 0.05, p-values were calculated using pairwise comparisons between groups with the option ‘pairwise’ and adjusted with the Tukey correction method. Plots for EMMs and p-values were displayed with ‘ggplot’ in R. The same analysis was performed for YLD using the classification of locations by origin to assess for genetic background bias.Using the mixed model, EMMs were calculated using the r = 1) between BLUPs calculated with ‘coef’ and ‘ranef’ R functions, hereinafter only results using BLUPs estimated with ‘coef’ function are presented.The historical phenotypic dataset was used to calculate biplots and PC plots showing the relationship among environments . These p2) using the Cullis method was moderate to high for YLD (r2= 0.56), TW (r2= 0.74), HD (r2= 0.85) and PH (r2= 0.83).Broad-sense heritability  where fewer lines were selected due to a lower number of available entries. The normalized frequency of lines selected by breeding program showed that overall representation of breeding programs within TPs was LA (µ = 13.9%), NC (µ = 13.8%), AR (µ = 13.3%), GA (µ = 13.2%), TX (µ = 12.9%), FL (µ = 12.5%), VA (µ = 11.7%), and SC (8.8%). After normalizing the number of lines included from each program by the total lines available from each program , it was apparent that STPGA included approximately one-third of individuals from every program, with a mean over all 22 sets ranging from a high of 36.3% (LA) to a low of 24% (SC)  . The hea24% (SC) . The Sun24% (SC) . STPGA s24% (SC) , in blueopheevii .r = 0.42 was observed across the 22 TPs when using EMMs as the observed data for YLD. When using BLUPs, the average prediction accuracy was reduced to r = 0.31. For TW, mean prediction accuracies ranged from r = 0.26 to 0.32 when using BLUPs and EMMs, respectively. Prediction accuracies oscillated from r = 0.42 to 0.45 for HD and PH  and EMMs of observed YLD data. The four-quadrant plot for this comparison, where 69% (40 of 58) of breeding lines were accurately categorized into the proper quadrant (A and B) and 31% of the breeding lines were categorized in quadrants C and D (r = 0.59 was obtained by comparing the GEBVs (predicted with the BLUPs of the combined TP), versus the EMMs of YLD. In this case, 74.1% of the breeding lines with high or low observed YLD were categorized in quadrants A and B, whereas 25.9% of the breeding lines fell into quadrants C and D  (−1) were selected for field testing. This set of lines had predicted values of 73.8–74.6 kg hl−1, 100–104 Julian days and 82.9–88.9 cm for TW, HD and PH, respectively (−1) were included for comparison. These lines had predicted values for TW, HD and PH of 72.6–73.5 kg hl−1, 103–105 Julian days and 84.9–86.8 cm, respectively   was used = 0.62) . Based oectively , in red.p-values  were also calculated based on origin of the breeding lines  , along wp-values  for nineng lines .For each of the studied QTL , lines tFive genes that are relevant for target population of environments across the southeastern US were more rigorously assessed individually for YLD trends within testing regions and by breeding program :The FHB resistance gene Fhb1 , which fFhb_1B , an addiThis third FHB resistance QTL under study  was deriVS:2A translocation segment derived from Aegilops ventricosa , and outperformed predictions made with TPs with reduced data. The complete dataset not only included more high-quality data for predictions, but also TPs selected from a historical pool of lines tested in multiple years and geographic regions aids in capturing a broader range of environmental conditions when compared to newly generated, population-specific TPs. In this case study, it was apparent that the historical phenotypic dataset using all data (combined GAWN + SunWheat over 14 years) effectively captured environments that were representative of the collection of locations in 2021 where the 58 SC breeding lines were tested. Specifically, grain yield GEBVs for 40 of the 58 lines (69%) used for validation correctly grouped with observed data  were observed when using training data of 318 lines collected over 11 years at six locations in France (r2 = 0.56) is highly affected by genotype-by-environment interactions  produced the highest prediction accuracy for grain yield  toxin contamination, with FHB not known to severely hinder yield unless present at epidemic levels . For H13, the QTL that confers strong resistance to the local biotype L Hessian fly, was present at much lower frequency (11%). Because of the tendency of Hessian fly biotype L to be more frequent and impactful along the Atlantic Coastal Plain  utilizing a combination of historical phenotype data and genome-wide SNP markers to build a reliable GS model for predicting best lines for a target population of environments, and 2) identifying major effect QTL using existing PCR-based marker reports that were favorable, within the context of region and biotic pressure. This study highlights the importance of cooperative efforts between breeding programs that share a target population of environments to not only perform extensive multi-environment field trials but also to compile genotypic and phenotypic datasets that are key to enhancing genetic gain through robust genomic prediction models.


Sparse regularized regression methods are now widely used in genome-wide association studies (GWAS) to address the multiple testing burden that limits discovery of potentially important predictors. Linear mixed models (LMMs) have become an attractive alternative to principal components (PCs) adjustment to account for population structure and relatedness in high-dimensional penalized models. However, their use in binary trait GWAS rely on the invalid assumption that the residual variance does not depend on the estimated regression coefficients. Moreover, LMMs use a single spectral decomposition of the covariance matrix of the responses, which is no longer possible in generalized linear mixed models (GLMMs).pglmm, a penalized GLMM that allows to simultaneously select genetic markers and estimate their effects, accounting for between-individual correlations and binary nature of the trait. We develop a computationally efficient algorithm based on penalized quasi-likelihood estimation that allows to scale regularized mixed models on high-dimensional binary trait GWAS. We show through simulations that when the dimensionality of the relatedness matrix is high, penalized LMM and logistic regression with PC adjustment fail to select important predictors, and have inferior prediction accuracy compared to pglmm. Further, we demonstrate through the analysis of two polygenic binary traits in a subset of 6731 related individuals from the UK Biobank data with 320K SNPs that our method can achieve higher predictive performance, while also selecting fewer predictors than a sparse regularized logistic lasso with PC adjustment.We introduce a new method called PenalizedGLMM.jl is publicly available on github: https://github.com/julstpierre/PenalizedGLMM.Our Julia package Bioinformatics online. Genome-wide association studies (GWAS) have led to the identification of hundreds of common genetic variants, or single nucleotide polymorphisms (SNPs), associated with complex traits  can control for the confounding effect due to population structure by including the top eigenvectors of a genetic similarity matrix (GSM) as fixed effects in the regression model  estimation with sparsity inducing regularization to perform selection of fixed and/or random effects in generalized linear mixed model (GLMM) , and the parameter vector W vary slowly with the conditional mean, we drop the first term in λ is a non-negative regularization parameter, and jv is a penalty factor for the jth predictor. This two-step approach is known as the P3D (population parameters previously determined) method  GWAS data of binary traits, for which we can use a lower bound on W. Hence, for other distributions, further work is needed to address these computational limitations for application to high-dimensional GWAS data. All calculations and algorithmic steps are detailed in anderformed . Althougpglmm versus a GLM with PC adjustment.It is often of interest in genetic association studies to make predictions on a new set of individuals, e.g. the genetic risk of developing a disease for a binary response or the expected outcome in the case of a continuous response. In what follows, we compare how predictions are obtained using 2.2.1n subjects that are used to fit the GLMM n the working linear mixed modelsn individuals with predictor set Suppose a single variance component is needed such that n × n matrix of PCs obtained from the spectral decomposition of the GSM for training subjects.The predictions are based on the conditional expectation 2.2.2r columns of n × r design matrix for the first r PCs and W is the diagonal matrix of GLM weights. Recall that Another approach to control for population structure and/or subjects’ relatedness is to use the first pglmm use a projection of the training PCs on the testing set to predict new responses, but with different coefficients for the projected PCs. For the former, the estimated coefficients for the first r projected PCs in pglmm, the estimated coefficients for all projected PCs are also obtained by iteratively solving GLS on the partial working residuals i the ith eigenvalue of V that is associated with the ith PC.By comparing pglmm shrinks PCs coefficients proportionally to their corresponding eigenvalues in a smooth way, while the fixed effect GLM uses a thresholding approach; the first r predictors with larger eigenvalues are kept intact, and the others are completely removed. This implies that the confounding effect from population structure and/or relatedness on the phenotype is fully captured by the first r PCs. As we show in simulations, departure from this assumption may lead to higher false-positive rates and decrease prediction accuracy.Hence, R package ggmix . We performed a total of 50 replications for two simulation scenarios, drawing anew genotypes and simulated traits. Values for all simulation parameters are presented in We evaluated the performance of our proposed method against that of a lasso LMM, using the 2.3.1bnpsd package in R , validation (30%) and test (30%) sets, ensuring all related individuals were assigned into the same set. For cross-validation, the full lasso solution path was fitted on the training set, and the regularization parameter was obtained on the model which maximized AUC on the validation set. We also evaluated the performance of our proposed method when using AIC as a model selection criterion. Again, we compared methods performance on the basis of TPR, AUC on the test sets and RMSE. In addition, we compared each model selection approach on the total number of predictors selected and on the model precision, which is defined as the proportion of selected predictors that are true positives.2.3.3S be the set of candidate causal SNPs, with jβ were generated from a Gaussian distribution https://nealelab.github.io/UKBB_ldsc/h2_summary_20002_1111.html), again under strong random polygenic effects  on two highly heritable binary traits, asthma  and high cholesterol . We present demographics and number of cases for both analyses in pglmm, i.e. a model where we include age, sex and genotyping platform as fixed effects, and one random effect with variance–covariance proportional to the GSM. For both gBLUP and pglmm, we did not include any PC since kinship is accounted for by the random effect. Finally, we also fitted a logistic lasso in which the top 10 PCs were included as unpenalized covariates in addition to age, sex and genotyping platform (glmnetPC). To evaluate the predictive performance of the compared methods in independent subjects, we randomly split the subjects in training (80%) and test (20%) sets for a total of 40 times. For each of the 40 replications, the full lasso solution path was fitted on the training set only. For pglmm, the regularization parameter (λ) was selected to minimize the AIC on the training data. For glmnetPC, the regularization parameter was obtained by minimizing the deviance using 10-fold cross-validation on the training data. We compared mean prediction accuracy on the test sets as well as the median number of predictors included in all models.To better understand the contribution of the PRS for predicting asthma and high cholesterol, we fitted for each trait a null model with only age, sex, genotyping platform and the first 10 PCs as fixed effects. Since for highly polygenic traits, it is generally considered that there are a large number of predictors with small to moderate effects , each with 64GB of RAM. As implemented in the glmnet package and other high-dimensional sparse regression methods, we use sequential strong rules for solving the lasso problem such that most of the predictors are discarded from the optimization problem at each iteration . For each replication, we randomly selected 1% of the predictors to be causal. Simulations were performed on a single core of an AMD Rome 7532 (2.40 GHz) with 64 GB of RAM. Results for the computational efficiency of all methods for different sample sizes and number of predictors are reported in pglmm ranged between 5.3 min .In this additional simulation scenario, we compared the computational efficiency of all methods. We considered a grid of values for the sample size pglmm based on regularized PQL estimation, for selecting important predictors and estimating their effects in high-dimensional GWAS data, accounting for population structure, close relatedness and binary nature of the trait. By simulating random genotypes from the BN-PSD admixture model for 10 or 20 subpopulations with 1D geography or independent subpopulations, we showed that pglmm was markedly better than a logistic lasso with PC adjustment when the number of subpopulations was greater than the number of PCs included. We also showed that a lasso LMM was unable to estimate predictor effects with accuracy for binary responses, which greatly decreased its predictive performance. Performance assessment was based on TPR of selected predictors, RMSE of estimated effects and AUC of predictions. These results strongly advocate for using methods that explicitly account for the binary nature of the trait while effectively controlling for population structure and relatedness in genetic studies.We have introduced a new method called pglmm effectively led to sparser models with higher precision and prediction accuracy than a lasso LMM and a logistic lasso model with or without PC adjustment. We also demonstrated that using AIC as a model selection strategy led to similar prediction performance than cross-validation, with even sparser models. Using the same dataset, we illustrated the potential advantages of pglmm over a logistic lasso with PC adjustment in a real data application for constructing a PRS on two highly heritable binary traits. Although these analyses have limited power compared to using all UK Biobank subjects of White British ancestry, it is often the case that researchers may be limited to relatively smaller datasets. For these cases, it is of primary importance to avoid discarding samples based on relatedness and properly account for the possible correlation between observations. Thus, we used this reduced sample from the UK Biobank to demonstrate the potential advantages of using penalized multivariable GLMMs in smaller datasets where subjects’ relatedness might be an important confounder.In the second simulation scenario, we used real genotype data from a subset of related individuals from the UK Biobank data to simulate binary responses, and showed that pglmm compared to a logistic lasso with PC adjustment is the computational cost of performing multiple matrix calculations that comes from incorporating a GSM to account for population structure and relatedness between individuals. These computations are clearly too prohibitive for application to large cohorts such as the full UK Biobank with a total of 500K samples. Solutions to explore in order to increase computation speed and decrease memory usage would be the use of conjugate gradient methods with a diagonal preconditioner matrix, as proposed by A limitation of In this study, we focused solely on the lasso as a regularization penalty for the genetic markers effects. However, it is known that estimated effects by lasso will have large biases because the resulting shrinkage is constant irrespective of the magnitude of the effects. Alternative regularizations like the Smoothly Clipped Absolute Deviation (SCAD)  and Minibtad063_Supplementary_DataClick here for additional data file.