Quantifying concordant genetic effects of de novo mutations on multiple disorders
Abstract
Exome sequencing on tens of thousands of parentproband trios has identified numerous deleterious de novo mutations (DNMs) and implicated risk genes for many disorders. Recent studies have suggested shared genes and pathways are enriched for DNMs across multiple disorders. However, existing analytic strategies only focus on genes that reach statistical significance for multiple disorders and require large trio samples in each study. As a result, these methods are not able to characterize the full landscape of genetic sharing due to polygenicity and incomplete penetrance. In this work, we introduce EncoreDNM, a novel statistical framework to quantify shared genetic effects between two disorders characterized by concordant enrichment of DNMs in the exome. EncoreDNM makes use of exomewide, summarylevel DNM data, including genes that do not reach statistical significance in singledisorder analysis, to evaluate the overall and annotationpartitioned genetic sharing between two disorders. Applying EncoreDNM to DNM data of nine disorders, we identified abundant pairwise enrichment correlations, especially in genes intolerant to pathogenic mutations and genes highly expressed in fetal tissues. These results suggest that EncoreDNM improves current analytic approaches and may have broad applications in DNM studies.
Editor's evaluation
Lu et al. provide a powerful statistical method that measures excess sharing of de novo mutations between pairs of disorders. This method extends the concept of 'genetic correlation' to disorders caused by denovo mutations, measuring the correlation in excess denovo mutations in genomewide genes for different classes of mutations. The authors apply the method to nine disorders including a developmental disorder, autism spectrum disorder, congenital heart disease, schizophrenia, and intellectual disability, finding a statistically significant overlap between 12 pairs of disorders in de novo mutations that cause a loss of gene function. This method will be of interest to researchers working on disorders caused by denovo mutations.
https://doi.org/10.7554/eLife.75551.sa0Introduction
De novo mutations (DNMs) can be highly deleterious and provide important insights into the genetic cause for disease (Veltman and Brunner, 2012). As the cost of sequencing continues to drop, wholeexome sequencing (WES) studies conducted on tens of thousands of family trios have pinpointed numerous risk genes for a variety of disorders (Lelieveld et al., 2016; Kaplanis et al., 2020; Satterstrom et al., 2020). In addition, accumulating evidence suggests that risk genes enriched for pathogenic DNMs may be shared by multiple disorders (Hoischen et al., 2014; Fromer et al., 2014; Homsy et al., 2015; Li et al., 2016; Nguyen et al., 2020). These shared genes could reveal biological pathways that play prominent roles in disease etiology and shed light on clinically heterogeneous yet genetically related diseases (Homsy et al., 2015; Li et al., 2016; Nguyen et al., 2020).
Most efforts to identify shared risk genes directly compare genes that are significantly associated with each disorder (Nguyen et al., 2017; Willsey et al., 2018). There have been some successes with this approach in identifying shared genes and pathways (e.g. chromatin modifiers) underlying developmental disorder (DD), autism spectrum disorder (ASD), and congenital heart disease (CHD), thanks to the large trio samples in these studies (Kaplanis et al., 2020; Satterstrom et al., 2020; Jin et al., 2017), whereas findings in smaller studies remain suggestive (Allen et al., 2013; Jin et al., 2020b). Even in the largest studies to date, statistical power remains moderate for risk genes with weaker effects (Kaplanis et al., 2020; Howrigan et al., 2020). It is estimated that more than 1000 haploinsufficient genes contributing to developmental disorder risk have not yet achieved statistical significance in large WES studies (Kaplanis et al., 2020). Therefore, analytic approaches that only account for top significant genes cannot capture the full landscape of genetic sharing in multiple disorders. Recently, a Bayesian framework named mTADA was proposed to jointly analyze DNM data of two diseases and improve risk gene mapping (Nguyen et al., 2020). Although mTADA produces estimates for the proportion of shared risk genes, the statistical property of these parameter estimates has not been studied. There is a pressing need for powerful, robust, and interpretable methods that quantify concordant DNM association patterns for multiple disorders using exomewide DNM counts.
Recent advances in estimating genetic correlations using summary data from genomewide association studies (GWAS) may provide a blueprint for approaching this problem in DNM research (Zhang et al., 2021a). Modeling ‘omnigenic’ associations as independent random effects, linear mixedeffects models leverage genomewide association profiles to quantify the correlation between additive genetic components of multiple complex traits (Lee et al., 2012; BulikSullivan et al., 2015; Lu et al., 2017; Ning et al., 2020). These methods have identified ubiquitous genetic correlations across many human traits and revealed significant and robust genetic correlations that could not be inferred from significant GWAS associations alone (Shi et al., 2017; Brainstorm, 2018; Guo et al., 2021; Zhang et al., 2021b).
Here, we introduce EncoreDNM (Enrichment correlation estimator for De Novo Mutations), a novel statistical framework that leverages exomewide DNM counts, including genes that do not reach exomewide statistical significance in singledisorder analysis, to estimate concordant DNM associations between disorders. EncoreDNM uses a generalized linear mixedeffects model to quantify the occurrence of DNMs while accounting for de novo mutability of each gene and technical inconsistencies between studies. We demonstrate the performance of EncoreDNM through extensive simulations and analyses of DNM data of nine disorders.
Results
Method overview
DNM counts in the exome deviate from the null (i.e. expected counts based on de novo mutability) when mutations play a role in disease etiology. Disease risk genes will show enrichment for deleterious DNMs in probands and nonrisk genes may be slightly depleted for DNM counts. Our goal is to estimate the correlation of such deviation between two disorders, which we refer to as the DNM enrichment correlation. More specifically, we use a pair of mixedeffects Poisson regression models (Munkin and Trivedi, 1999) to quantify the occurrence of DNMs in two studies.
Here, ${Y}_{i1},{Y}_{i2}$ are the DNM counts for the ith gene and ${N}_{1},{N}_{2}$ are the number of parentproband trios in two studies, respectively. The log Poisson rates of DNM occurrence are decomposed into three components: the elevation component, the background component, and the deviation component. The elevation component ${\beta}_{k}$ ($k=\mathrm{1,2}$) is a fixed effect term adjusting for systematic, exomewide bias in DNM counts. One example of such bias is the batch effect caused by different sequencing and variant calling pipelines in two studies. The elevation parameter ${\beta}_{k}$ tends to be positive when DNMs are overcalled with higher sensitivity and negative when DNMs are undercalled with higher specificity (Wei et al., 2015). The background component $\mathrm{l}\mathrm{o}\mathrm{g}\left(2{N}_{k}{m}_{i}\right)$ is a genespecific fixed effect that reflects the expected mutation counts determined by the genomic sequence context under the null (Samocha et al., 2014). ${m}_{i}$ is the de novo mutability for the ith gene, and $2{N}_{1}{m}_{i}$ and $2{N}_{2}{m}_{i}$ are the expected DNM counts in the ith gene under the null in two studies. The deviation component ${\varphi}_{ik}$ is a genespecific random effect that quantifies the deviation of DNM profile from what is expected under the null (i.e. no risk genes for the disorder). ${\varphi}_{i1}$ and ${\varphi}_{i2}$ follow a multivariate normal distribution with dispersion parameters ${\sigma}_{1}$ and ${\sigma}_{2}$ and a correlation $\rho $. A larger value of the dispersion parameter ${\sigma}_{k}$ indicates a more substantial deviation from the null. That is, DNM counts show strong enrichment in some genes and depletion in other genes compared to the expectation based on de novo mutability. A smaller value of ${\sigma}_{k}$ suggests that the DNM count data is well in line with what is expected based on the null model. DNM enrichment correlation is denoted by $\rho $ and is our main parameter of interest. It quantifies the concordance of DNM burden in two disorders.
Parameters in this model can be estimated using a Monte Carlo maximum likelihood estimation (MLE) procedure. Standard errors of the estimates are obtained through inversion of the observed Fisher information matrix. In practice, we use annotated DNM data as input and fit mixedeffects Poisson models for each variant class separately: loss of function (LoF), deleterious missense (Dmis, defined as MetaSVMdeleterious), tolerable missense (Tmis, defined as MetaSVMtolerable), and synonymous (Figure 1). More details about model setup and parameter estimation are discussed in Materials and methods.
Simulation results
We conducted simulations to assess the parameter estimation performance of EncoreDNM in various settings. We focused on two variant classes, that is, Tmis and LoF variants, since they have the highest and lowest median mutabilities in the exome. We used EncoreDNM to estimate the elevation parameter $\beta $, dispersion parameter $\sigma $, and enrichment correlation $\rho $ (Materials and methods). Under various parameter settings, EncoreDNM always provided unbiased estimation of the parameters (Figure 2 and Figure 2—figure supplements 1–2). Furthermore, the 95% Wald confidence intervals achieved coverage rates close to 95% under all simulation settings, demonstrating the effectiveness of EncoreDNM to provide accurate statistical inference.
Next, we compared the performance of EncoreDNM with mTADA (Nguyen et al., 2020), a Bayesian framework that estimates the proportion of shared risk genes for two disorders. First, we simulated DNM data under the mixedeffects Poisson model. We evaluated two methods across a range of combinations of elevation parameter, dispersion parameter, and sample size for two disorders. The false positive rates for our method were wellcalibrated in all parameter settings, but mTADA produced false positive findings when the observed DNM counts were relatively small (e.g. due to reduced elevation or dispersion parameters or a lower sample size; Figure 3a). We also assessed the statistical power of two approaches under a baseline setting where false positives for both methods were controlled. As enrichment correlation increased, EncoreDNM achieved universally greater statistical power compared to mTADA (Figure 3b).
To ensure a fair comparison, we also considered a misspecified model setting where we randomly distributed the total DNM counts for each disorder into all genes with an enrichment in causal genes (Materials and methods). EncoreDNM showed wellcontrolled false positive rate across all simulation settings, whereas severe inflation of false positives arose for mTADA when the total mutation count, the proportion of probands that can be explained by DNMs, or the sample size were small (Figure 3c). Furthermore, we compared the statistical power of two methods under this model in a baseline setting where false positive rate was controlled. EncoreDNM showed higher statistical power compared to mTADA as the fraction of shared causal genes increased (Figure 3d).
Pervasive enrichment correlation of damaging DNMs among developmental disorders
We applied EncoreDNM to DNM data of nine disorders (Supplementary file 1STable 1; Materials and methods): developmental disorder (n=31,058; number of trios; Kaplanis et al., 2020), autism spectrum disorder (n=6430; Satterstrom et al., 2020), schizophrenia (SCZ; n=2772; Howrigan et al., 2020), congenital heart disease (n=2645; Jin et al., 2017), intellectual disability (ID; n=820; Lelieveld et al., 2016), Tourette disorder (TD; n=484) Willsey et al., 2017, epileptic encephalopathies (EP; n=264; Allen et al., 2013), cerebral palsy (CP; n=250; Jin et al., 2020b), and congenital hydrocephalus (CH; n=232; Jin et al., 2020a). In addition, we also included 1789 trios comprising healthy parents and unaffected siblings of autism probands as controls (Krumm et al., 2015).
We first performed singletrait analysis under the mixedeffects Poisson model for each disorder. The estimated elevation parameters (i.e. $\beta $) were negative for almost all disorders and variant classes (Figure 4a), with LoF variants showing particularly lower parameter estimates. This may be explained by more stringent quality control in LoF variant calling (Jin et al., 2017) and potential survival bias (Lek et al., 2016). It is also consistent with a depletion of LoF DNMs in healthy control trios (Homsy et al., 2015). The dispersion parameter estimates (i.e. $\sigma $) were higher for LoF variants than other variant classes (Figure 4b), which is consistent with our expectation that LoF variants have stronger effects on disease risk and should show a larger deviation from the null mutation rate in disease probands. We also compared the goodness of fit of our proposed mixedeffects Poisson model to a simpler fixedeffects model without the deviation component (Materials and methods). The expected distribution of recurrent DNM counts showed substantial and statistically significant improvement under the mixedeffects Poisson model (Figure 4c–f, Figure 4—figure supplement 1, and Supplementary file 1STable 2).
Next, we estimated pairwise DNM enrichment correlations for 9 disorders. In total, we identified 25 pairs of disorders with significant correlations at a false discovery rate (FDR) cutoff of 0.05 (Figure 5 and Figure 5—figure supplement 1), including 12 significant correlations for LoF variants, 7 for Dmis variants, 5 for Tmis variants, and only 1 significant correlation for synonymous variants. Notably, all significant correlations are positive (Supplementary file 1STable 3). No significant correlation was identified between any disorder and healthy controls (Figure 5—figure supplement 2). This is consistent with our expectation, since DNMs in the control groups will distribute proportionally according to the de novo mutability without showing enrichment in certain genes. The number of identified significant correlations for each disorder was proportional to the sample size in each study (Spearman correlation = 0.70) with controls being a notable outlier (Figure 5—figure supplement 3).
We identified highly concordant and significant LoF DNM enrichment among developmental disorder, autism, intellectual disability, and congenital heart disease, which is consistent with previous reports (Li et al., 2016; Nguyen et al., 2020; Nguyen et al., 2017; Hormozdiari et al., 2015). Schizophrenia shows highly significant LoF correlations with developmental disorder (p=2.0e3) and intellectual disability (3.7e5). The positive enrichment correlation between autism and cerebral palsy in LoF variants ($\rho $=0.81, p=3.3e3) is consistent with their cooccurrence (Christensen et al., 2014). The high enrichment correlation between intellectual disability and cerebral palsy in LoF variants ($\rho $=0.68, p=1.0e4) is consistent with the associations between intellectual disability and motor or nonmotor abnormalities caused by cerebral palsy (Reid et al., 2018). A previous study also suggested significant genetic sharing of intellectual disability and cerebral palsy by overlapping genes harboring rare damaging variants (Jin et al., 2020b). Here, we obtained consistent results after accounting for de novo mutabilities and potential confounding bias.
Some significant correlations identified in our analysis are consistent with phenotypic associations in epidemiological studies, but have not been reported using genetic data to the extent of our knowledge. For example, the LoF enrichment correlation between congenital heart disease and cerebral palsy ($\rho $=0.88, p=1.7e3) is consistent with findings that reduced supply of oxygenated blood in fetal brain due to cardiac malformations may be a risk factor for cerebral palsy (Garne et al., 2008). The enrichment correlation between intellectual disability and congenital hydrocephalus in LoF variants ($\rho $=0.63, p=2.4e3) is consistent with lower intellectual performance in a proportion of children with congenital hydrocephalus (Lumenta and Skotarczak, 1995).
Genes showing pathogenic DNMs in multiple disorders may shed light on the mechanisms underlying enrichment correlations (Supplementary file 1STable 4). We identified five genes (CTNNB1, NBEA, POGZ, SPRED2, and KMT2C) with LoF DNMs in five different disorders and 21 genes had LoF DNMs in four disorders (Supplementary file 1STable 5). These 26 genes with LoF variants in at least four disorders were significantly enriched for 63 gene ontology (GO) terms with FDR <0.05 (Supplementary file 1STable 6). Chromatin organization (p=7.8e11), nucleoplasm (p=2.8e10), chromosome organization (p=6.8e10), histone methyltransferase complex (p=1.4e9), and positive regulation of gene expression (p=2.2e9) were the most significantly enriched GO terms. One notable example consistently included in these gene sets is CTNNB1 (Figure 5—figure supplement 4). It encodes $\beta $catenin, is one of the only two genes reaching genomewide significance in a recent WES study for cerebral palsy (Jin et al., 2020b), and also harbors multiple LoF variants in developmental disorder, intellectual disability, autism, and congenital heart disease. It is a fundamental component of the canonical Wnt signaling pathway which is known to confer genetic risk for autism (O’Roak et al., 2012). Genes with recurrent damaging DNMs in multiple disorders also revealed shared biological function across these disorders (Rees et al., 2021). We identified 30 recurrent crossdisorder LoF mutations that were not recurrent in developmental disorder alone (Supplementary file 1). FBXO11, encoding the Fbox only protein 31, shows two recurrent p.Ser831fs LoF variants in autism and congenital hydrocephalus (Figure 5—figure supplement 5; p=1.9e3; Materials and methods). The Fbox protein constitutes a substraterecognition component of the SCF (SKP1cullinFbox) complex, an E3ubiquitin ligase complex responsible for ubiquitination and proteasomal degradation (Cardozo and Pagano, 2004). DNMs in FBXO11 have been previously implicated in severe intellectual disability individuals with autistic behavior problem (Jansen et al., 2019) and neurodevelopmental disorder (Gregor et al., 2018).
For comparison, we also applied mTADA to the same nine disorders and control trios. In total, mTADA identified 117 disorder pairs with significant genetic sharings at an FDR cutoff of 0.05 (Supplementary file 1STable 8 and Figure 5—figure supplement 6). Notably, we identified significant synonymous DNM correlations for all 36 disorder pairs and between all disorders and healthy controls (Figure 5—figure supplement 7). These results are consistent with the simulation results and suggest a substantially inflated false positive rate in mTADA.
Partitioning DNM enrichment correlation by gene set
To gain biological insights into the shared genetic architecture of nine disorders, we repeated EncoreDNM correlation analysis in several gene sets. First, we defined genes with high/low probability of intolerance to LoF variants using pLI scores (Karczewski et al., 2020), and identified genes with high/low brain expression (HBE/LBE) (Werling et al., 2020; Materials and methods; Supplementary file 1STable 9). We identified 11 and 12 disorder pairs showing significant enrichment correlations for LoF DNMs in highpLI genes and HBE genes, respectively (Figure 6a–b). We observed fewer significant correlations for Dmis and Tmis variants in these gene sets (Figure 6—figure supplements 1–2). All identified significant correlations were positive (Supplementary file 1STables 10 11). No significant correlations were identified for synonymous variants (Figure 6—figure supplements 1–2) or between disorders and controls (Figure 6—figure supplements 3–4).
We observed a clear enrichment of significant correlations in diseaserelevant gene sets. Overall, highpLI genes showed substantially stronger correlations across disorders than genes with low pLI (onesided KolmogorovSmirnov test; p=2.3e6). Similarly, enrichment correlations were stronger in HBE genes than in LBE genes (p=8.8e7). Among the 11 disorder pairs showing significant enrichment correlations in highpLI genes, two pairs, that is, autismschizophrenia ($\rho $=0.68, p=2.4e3) and developmental disordercongenital hydrocephalus ($\rho $=0.43, p=1.5e3), were not identified in the exomewide analysis. We also identified four novel disorder pairs with significant correlations in HBE genes, including developmental disordercerebral palsy ($\rho $=0.80, p=9.5e5), developmental disordercongenital hydrocephalus ($\rho $=0.67, p=1.4e3), autismcongenital hydrocephalus ($\rho $=0.82, p=4.7e4), and schizophreniaepileptic encephalopathies ($\rho $=0.66, p=2.0e3). These novel enrichment correlations are consistent with known comorbidities between these disorders (Kielinen et al., 2004; Kilincaslan and Mukaddes, 2009) and findings based on significant risk genes (Li et al., 2016; Jin et al., 2020a; Kume et al., 1998; Cao and Wu, 2015).
Furthermore, we estimated DNM enrichment correlations in genes with high/low expression in mouse developing heart (HHE/LHE) (Homsy et al., 2015; Materials and methods; Supplementary file 1STable 9). We identified 9 significant enrichment correlations for LoF variants in HHE genes (Figure 6c). Strength of enrichment correlations did not show a significant difference between HHE and LHE genes (p=0.846), possibly due to a lack of cardiac disorders in our analysis. Finally, we estimated enrichment correlations between congenital heart disease and other disorders in known pathways for congenital heart disease (Zaidi and Brueckner, 2017; Materials and methods; Supplementary file 1STable 9). We identified five significant correlations for LoF variants (Figure 6d), including a novel correlation between congenital heart disease and Tourette disorder ($\rho $=0.93, p=3.3e9). Of note, arrhythmia caused by congenital heart disease is a known risk factor for Tourette disorder (Gulisano et al., 2011). In these analyses, all significant enrichment correlations were positive (Supplementary file 1) and other variant classes showed generally weaker correlations than LoF variants (Figure 6—figure supplements 5–6). We did not observe significant correlations in these gene sets between disorders and controls (Figure 6—figure supplements 7–8).
Discussion
In this paper, we introduced EncoreDNM, a novel statistical framework to quantify correlated DNM enrichment between two disorders. Through extensive simulations and analyses of DNM data for nine disorders, we demonstrated that our proposed mixedeffects Poisson regression model provides unbiased parameter estimates, shows wellcontrolled false positive rate, and is robust to exomewide technical biases. Leveraging exomewide DNM counts and genomic contextbased mutability data, EncoreDNM achieves superior fit for real DNM datasets compared to simpler models and provides statistically powerful and computationally efficient estimation of DNM enrichment correlation. Further, EncoreDNM can quantify concordant genetic effects for userdefined variant classes within prespecified gene sets, thus is suitable for exploring diverse types of hypotheses and can provide crucial biological insights into the shared genetic etiology in multiple disorders. In comparison, the Bayesian approach implemented in mTADA can produce false positives findings, especially when the DNM count is low, possibility due to the overestimated proportion of risk genes. We still observed inflation in false positive rates under a more stringent significance cutoff or using posterior probability threshold strategy (Supplementary file 1STables 1417).
Multitrait analyses of GWAS data have revealed shared genetic architecture among many neuropsychiatric traits (Brainstorm, 2018; Lee et al., 2013; Gratten et al., 2014; Abdellaoui and Verweij, 2021). These findings have led to the identification of pleiotropic variants, genes, and hub genomic regions underlying many traits and have revealed multiple psychopathological factors jointly affecting human neurological phenotypes (Lee, 2019; Wang et al., 2015). Although emerging evidence suggests that causal DNMs underlying several disorders with wellpowered studies (e.g. congenital heart disease and neurodevelopmental disorders; Homsy et al., 2015) may be shared, our understanding of the extent and the mechanism underlying such sharing remains incomplete. Applied to DNM data for nine disorders, EncoreDNM identified pervasive enrichment correlations of DNMs. We observed particularly strong correlations in pathogenic variant classes (e.g. LoF and Dmis variants) and diseaserelevant genes (e.g. genes with high pLI and genes highly expressed in relevant tissues). Genes underlying these correlations were significantly enriched in pathways involved in chromatin organization and modification and gene expression regulation. The DNM correlations were substantially attenuated in genes with lower expression and genes with frequent occurrences of LoF variants in the population. A similar attenuation was observed in less pathogenic variant classes (e.g., synonymous variants). Further, no significant correlations were identified between any disorder and healthy controls. We also compared DNM enrichment correlations of five disorders with genetic correlations estimated from GWAS summary statistics (Supplementary file 1STable 18). We had consistent findings from GWAS and DNM data (Spearman correlation = 0.70; Figure 5—figure supplement 8 and Supplementary file 1STable 19). These results lay the groundwork for future investigations of pleiotropic mechanisms of DNMs.
Our study has some limitations. First, EncoreDNM assumes probands from different input studies to be independent. In rare cases when two studies have overlapping proband samples, enrichment correlation estimates may be inflated and must be interpreted with caution. Second, genetic correlation methods based on GWAS summary data provided key motivations for the mixedeffects Poisson regression model in our study. Built upon genetic correlations, a plethora of methods have been developed in the GWAS literature to jointly model more than two GWAS (Turley et al., 2018), identify and quantify common factors underlying multiple traits (Grotzinger et al., 2019; Grotzinger et al., 2020), estimate causal effects among different traits (Pickrell et al., 2016), and identify pleiotropic genomic regions through hypothesisfree scans (Guo et al., 2021). Future directions of EncoreDNM include using enrichment correlation to improve gene discovery, learning the directional effects and the causal structure underlying multiple disorders, and dynamically searching for gene sets and annotation classes with shared genetic effects without prespecifying the hypothesis.
Taken together, we provide a new analytic approach to an important problem in DNM studies. We believe EncoreDNM improves the statistical rigor in multidisorder DNM modeling and opens up many interesting future directions in both method development and followup analyses in WES studies. As trio sample size in WES studies continues to grow, EncoreDNM will have broad applications and can greatly benefit DNM research.
Materials and methods
Statistical model
Request a detailed protocolFor a single study, we assume that DNM counts in a given variant class (for example, synonymous variants) follow a mixedeffects Poisson model:
where ${Y}_{i}$ is the DNM count in the ith gene, $N$ is the number of trios, ${m}_{i}$ is the de novo mutability for the ith gene (for example, mutation rate per chromosome per generation) which is known a priori (Samocha et al., 2014), and $G$ is the total number of genes in the study. The elevation parameter $\beta $ quantifies the global elevation of mutation rate compared to mutability estimates based on genomic sequence alone. Genespecific deviation from expected DNM rate is quantified by random effect ${\varphi}_{i}$ with a dispersion parameter $\sigma $. Here, the ${\varphi}_{i}$ are assumed to be independent across different genes, in which case the observed DNM counts of different genes are independent. There is no constraint on the value of $\beta $, and the dispersion parameter $\sigma $ can be any positive value.
Next, we describe how we expand this model to quantify the shared genetics of two disorders. We adopt a flexible Poissonlognormal mixture framework that can accommodate both overdispersion and correlation (Munkin and Trivedi, 1999). We assume DNM counts in a given variant class for two diseases follow:
where ${Y}_{i1},{Y}_{i2}$ are the DNM counts for the ith gene and ${N}_{1},{N}_{2}$ are the trio sizes in two studies, respectively. Similar to the singletrait model, ${m}_{i}$ is the mutability for the ith gene. ${\beta}_{1},{\beta}_{2}$ are the elevation parameters, and ${\varphi}_{i1},{\varphi}_{i2}$ are the genespecific random effects with dispersion parameters ${\sigma}_{1},{\sigma}_{2}$ , for two disorders respectively. $\rho $ is the enrichment correlation which quantifies the concordance of the genespecific DNM burden between two disorders. Here, ${\beta}_{1},{\beta}_{2},{\sigma}_{1},{\sigma}_{2},\rho $ are unknown parameters. The gene specific effects for two disorders are assumed to be independent for different genes. We also assume that there is no shared sample for two disorders, in which case ${Y}_{i1}$ is independent with ${Y}_{i2}$ given $\left[\begin{array}{c}{\lambda}_{i1}\\ {\lambda}_{i2}\end{array}\right]$ .
Parameter estimation
Request a detailed protocolWe implement an MLE procedure to estimate unknown parameters. For singletrait analysis, the loglikelihood function can be expressed as follows:
where $\mathit{Y}={\left[{Y}_{1},\dots ,{Y}_{G}\right]}^{T}$ , ${\lambda}_{i}=2N{m}_{i}\mathrm{exp}(\beta +{\varphi}_{i})$ , $C=\sum _{i=1}^{G}\mathrm{log}\left({Y}_{i}!\right)$ , and $f\left({\varphi}_{i}\right)=\frac{1}{\sqrt{2\pi}\sigma}\mathrm{exp}\left(\frac{{\varphi}_{i}^{2}}{2{\sigma}^{2}}\right)$ . Note that there is no closed form for the integral in the loglikelihood function. Therefore, we use Monte Carlo integration to evaluate the loglikelihood function. Let ${\varphi}_{ij}=\sigma {\xi}_{ij}$ , where the ${\xi}_{ij}$ are independently and identically distributed random variables following a standard normal distribution. We have
where ${\lambda}_{ij}=2N{m}_{i}\mathrm{exp}(\beta +\sigma {\xi}_{ij})$ , and $M$ is the Monte Carlo sample size which is set to be 1,000. Then, we could obtain the MLE of $\beta ,\sigma $ through maximization of ${l}^{\prime}(\beta ,\sigma \mathit{Y})$. We obtain the standard error of the MLE through inversion of the observed Fisher information matrix. However, when the DNM count is small, the Fisher information may be noninvertible and the parameter vector is not numerically identifiable. In this case, we employ groupwise jackknife using 100 randomly partitioned gene groups to obtain standard errors for parameter estimates. This approach produces consistent standard errors compared to the Fisher information approach (Figure 5—figure supplement 9).
The estimation procedure can be generalized to multitrait analysis. Loglikelihood function can be expressed as follows:
where ${\mathit{Y}}_{1}={\left[{Y}_{11},\dots ,{Y}_{G1}\right]}^{T}$ , ${\mathit{Y}}_{2}={\left[{Y}_{12},\dots ,{Y}_{G2}\right]}^{T}$ , ${\lambda}_{i1}=2{N}_{1}{m}_{i}\mathrm{exp}({\beta}_{1}+{\varphi}_{i1})$ , ${\lambda}_{i2}=2{N}_{2}{m}_{i}\mathrm{exp}({\beta}_{2}+{\varphi}_{i2})$ , $C=\sum _{i=1}^{G}\left[\mathrm{log}\left({Y}_{i1}!\right)+\mathrm{log}\left({Y}_{i2}!\right)\right]$ , and $f\left({\varphi}_{i1},{\varphi}_{i2}\right)=\frac{1}{2\pi {\sigma}_{1}{\sigma}_{2}\sqrt{1{\rho}^{2}}}\mathrm{exp}\left[\frac{1}{2\sqrt{1{\rho}^{2}}}\left(\frac{{\varphi}_{i1}^{2}}{{\sigma}_{1}^{2}}+\frac{{\varphi}_{i2}^{2}}{{\sigma}_{2}^{2}}\frac{2\rho {\varphi}_{i1}{\varphi}_{i2}}{{\sigma}_{1}{\sigma}_{2}}\right)\right]$ . We use Monte Carlo integration to evaluate the loglikelihood function. Let ${\varphi}_{i1j}={\sigma}_{1}{\xi}_{i1j}$ and ${\varphi}_{i2j}={\sigma}_{2}\left(\rho {\xi}_{i1j}+\sqrt{1{\rho}^{2}}{\xi}_{i2j}\right)$ , where the ${\xi}_{i1j}$ and ${\xi}_{i2j}$ are independently and identically distributed random variables following a standard normal distribution. We have
where ${\lambda}_{i1j}=2{N}_{1}{m}_{i}\mathrm{exp}({\beta}_{1}+{\sigma}_{1}{\xi}_{i1j})$ and ${\lambda}_{i2j}=2{N}_{2}{m}_{i}\mathrm{exp}\left[{\beta}_{2}+{\sigma}_{2}\left(\rho {\xi}_{i1j}+\sqrt{1{\rho}^{2}}{\xi}_{i2j}\right)\right]$ . Then, we obtain the MLE of ${\beta}_{1},{\beta}_{2},{\sigma}_{1},{\sigma}_{2},\rho $ through maximization of $l`\left({\beta}_{1},{\beta}_{2},{\sigma}_{1},{\sigma}_{2},\rho {\mathit{Y}}_{1},{\mathit{Y}}_{2}\right)$ . Standard error of MLE can be obtained either through inversion of the observed Fisher information matrix or groupwise jackknife if noninvertibility issue occurs.
Computation time
Request a detailed protocolAnalysis of a typical pair of disorders with 18,000 genes takes about 10 min on a 2.5 GHz cluster with 1 core.
DNM data and variant annotation
Request a detailed protocolWe obtained DNM data from published studies (Supplementary file 1STable 1). DNM data for epileptic encephalopathies from the original release (Allen et al., 2013) were not in an editable format and were instead collected from denovodb (Turner et al., 2017). We used ANNOVAR (Wang et al., 2010) to annotate all DNMs. Synonymous variants were determined based on the ‘synonymous SNV’ annotation in ANNOVAR; Variants with ‘startloss’, ‘stopgain’, ‘stoploss’, ‘splicing’, ‘frameshift insertion’, ‘frameshift deletion’, or ‘frameshift substitution’ annotations were classified as LoF; Dmis variants were defined as nonsynonymous SNVs predicted to be deleterious by MetaSVM Dong et al., 2015; nonsynonymous SNVs predicted to be tolerable by MetaSVM were classified as Tmis. Other DNMs which did not fall into these categories were removed from the analysis. For each variant class, we estimated the mutability of each gene using a sequencebased mutation model (Samocha et al., 2014) while adjusting for the sequencing coverage factor based on control trios as previously described (Jin et al., 2017; Supplementary file 1STable 20). We included 18,454 autosomal proteincoding genes in our analysis. TTN was removed due to its substantially larger size.
Description and implementation of mTADA
Request a detailed protocolThe method mTADA employs a Bayesian framework and estimates the proportion of shared risk genes. Specifically, mTADA assigns all genes into four groups: genes that are not relevant for either disorder, risk genes for the first disorder alone, risk genes for the second disorder alone, and risk genes shared by both disorders. The proportion of these groups are parametrized as ${\pi}_{0},{\pi}_{1},{\pi}_{2},{\pi}_{3}$ , respectively. In particular, parameter ${\pi}_{3}$ quantifies the extent of genetic sharing between two disorders, with a larger value indicating stronger genetic overlap (Nguyen et al., 2020). The 95% credible interval constructed through MCMC is used to measure the uncertainty in ${\pi}_{3}$ estimates.
The software mTADA requires the following parameters as inputs: proportion of risk genes (${\pi}_{1}^{S},{\pi}_{2}^{S}$), mean relative risks (${\stackrel{}{\gamma}}_{1}^{S},{\stackrel{}{\gamma}}_{2}^{S}$), and dispersion parameters (${\stackrel{}{\beta}}_{1}^{S},{\stackrel{}{\beta}}_{2}^{S}$) for both disorders. We used extTADA (Nguyen et al., 2017 )to estimate these parameters as suggested by the mTADA paper (Nguyen et al., 2020). mTADA reported the estimated proportion of shared risk genes ${\pi}_{3}$ (posterior mode of ${\pi}_{3}$) and its corresponding 95% credible interval $[\mathrm{L}\mathrm{B},\mathrm{}\mathrm{U}\mathrm{B}]$. We considered ${\pi}_{1}^{S}*{\pi}_{2}^{S}$ as the expected proportion of shared risk genes, and there is significant genetic sharing between two disorders when $\mathrm{L}\mathrm{B}>{\pi}_{1}^{S}\ast {\pi}_{2}^{S}$ . We quantify statistical evidence for genetic sharing by comparing the posterior distribution of ${\pi}_{3}$ with ${\pi}_{1}^{S}*{\pi}_{2}^{S}$ ,
where ${\pi}_{3}^{i}$ is the ith MCMC iteration sample, ${N}_{MCMC}$ is the number of iterations, and $I\left(\right)$ is the indicator function. This is also equivalent to performing twosided inference using posterior probability P $\left({\pi}_{3}>{\pi}_{1}^{S}\ast {\pi}_{2}^{S}\right)$ . Number of MCMC chain was set as 2 and number of iterations was set as 10,000.
Simulation settings
Request a detailed protocolWe assessed the performance of EncoreDNM under the mixedeffects Poisson model. We performed simulations for two variant classes: Tmis and LoF variants, which have the largest and the smallest median mutability values across all genes. First, we performed singletrait simulations to assess estimation precision of elevation parameter $\beta $ and dispersion parameter $\sigma $. We set the true values of $\beta $ to be −0.5,–0.25, and 0, and the true values of $\sigma $ to be 0.5, 0.75, and 1. These values were chosen based on the estimated parameters in real DNM data analyses and ensured simulation settings to be realistic. Next, we performed simulations for crosstrait analysis to assess estimation precision of enrichment correlation $\rho $, whose true values were set to be 0, 0.2, 0.4, 0.6, and 0.8. Sample size for each disorder was set to be 5000. Coverage rate was calculated as the percentage of simulations that the 95% Wald confidence interval covered the true parameter value. Each parameter setting was repeated 100 times.
We also carried out simulations to compare the performance of EncoreDNM and mTADA. False positive rate and statistical power for EncoreDNM were calculated as the proportion of simulation repeats that pvalue for enrichment correlation $\rho $ was smaller than 0.05. and the proportion of simulation repeats that pvalue for estimated proportion of shared risk genes ${\pi}_{3}$ was smaller than 0.05 was used for mTADA. We aggregated all variant classes together, so mutability for each gene was determined as the sum of mutabilities across four variant classes (i.e. LoF, Dmis, Tmis, and synonymous).
First, we simulated DNM data under the mixedeffects Poisson model. To see whether two methods would produce false positive findings, we performed simulations under the null hypothesis that the enrichment correlation $\rho $ is zero. We compared two methods under a range of parameter combinations of ($\beta ,\sigma ,N$) for both disorders: (–0.25, 0.75, 5000) for the baseline setting, (–1, 0.75, 5000) for a setting with small $\beta $, (–0.25, 0.5, 5000) for a setting with small $\sigma $, and (–0.25, 0.75, 1000) for a setting with small sample size. We also assessed the statistical power of two methods under the alternative hypothesis. True value of enrichment correlation $\rho $ was set to be 0.05, 0.1, 0.15, and 0.2. In the power analysis, parameters ($\beta ,\sigma ,N$) were fixed at (–0.25, 0.75, 5000) as in the baseline setting when both methods had wellcontrolled false positive rate.
To ensure a fair comparison, we also compared EncoreDNM and mTADA under a multinomial model, which is different from the data generation processes for the two approaches. For each disorder ($k=\mathrm{1,2}$), we randomly selected causal genes of proportion ${\pi}_{k}^{S}$ . A proportion (i.e. ${\pi}_{3}$) of causal genes overlap between two disorders. We assumed that the total DNM count to follow a Poisson distribution: ${C}_{k}~\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left({u}_{k}\mathrm{*}2{N}_{k}\sum _{i=1}^{G}{m}_{i}\right)$, where ${u}_{k}$ represents an elevation factor to represent systematic bias in the data. Let ${\mathbf{Y}}_{k}$ denote the vector of DNMs counts in the exome, $\mathit{m}$ denote the vector of mutability values for all genes, and ${\mathit{m}}_{causal,k}$ denote the vector of mutability with values set to be 0 for noncausal genes of disorder $k$. We assumed that a proportion ${p}_{k}$ of the probands could be attributed to DNMs burden in causal genes, and $1{p}_{k}$ of the probands obtained DNMs by chance:
To check whether false positive findings could arise, we performed simulations under the null hypothesis that ${\pi}_{3}={\pi}_{1}^{S}*{\pi}_{2}^{S}$ across a range of parameter combinations of ($u,p,N$) for both disorders: (0.95, 0.25, 5000) for the baseline setting, (0.75, 0.25, 5000) for a setting with small $u$ (i.e., reduced total mutation count), (0.95, 0.15, 5000) for a setting with small $p$ (fewer probands explained by DNMs), and (0.95, 0.25, 1000) for a setting with smaller sample size. ${\pi}_{1}^{S}$ and ${\pi}_{2}^{S}$ were set as 0.1. We also assessed the statistical power of two methods under the alternative hypothesis that $\pi}_{3}>{\pi}_{1}^{S}\ast {\pi}_{2}^{S$ . In power analysis, ($u,p,N$) were fixed at (0.95, 0.25, 5000) as in the baseline setting when false positive rate for both methods were wellcalibrated.
Comparison to the fixedeffects Poisson model
Request a detailed protocolFor singletrait analysis, the fixedeffects Poisson model assumes that
Note that the fixedeffects Poisson model is a special case of our proposed mixedeffects Poisson model when $\sigma =0$. We compared the two models using likelihood ratio test. Under the null hypothesis that $\sigma =0$, $2\left({l}_{alt}{l}_{null}\right)\sim \frac{1}{2}{\chi}_{1}^{2}$ asymptotically, where ${l}_{alt}$ and ${l}_{null}$ represent the log likelihood of the fitted mixedeffects and fixedeffects Poisson models respectively.
Recurrent genes and DNMs
Request a detailed protocolWe used FUMA (Watanabe et al., 2017) to perform GO enrichment analysis for genes harboring LoF DNMs in multiple disorders. Due to potential sample overlap between the studies of developmental disorder (Kaplanis et al., 2020) and intellectual disability (Lelieveld et al., 2016), we excluded intellectual disability from the analysis of recurrent DNMs. We calculated the probability of observing two identical DNMs in two disorders using a Monte Carlo simulation method. For each disorder, we simulated exomewide DNMs profile from a multinomial distribution, where the size was fixed at the observed DNM count and the perbase mutation probability was determined by the trinucleotide base context. We repeated the simulation procedure 100,000 times to evaluate the significance of recurrent DNMs. Lollipop plots for recurrent mutations were generated using MutationMapper on the cBio Cancer Genomics Portal (Cerami et al., 2012).
Implementation of crosstrait LD score regression
Request a detailed protocolWe used crosstrait LDSC (BulikSullivan et al., 2015) to estimate genetic correlations between disorders. LD scores were computed using European samples from the 1000 Genomes Project Phase 3 data (Auton et al., 2015). Only HapMap 3 SNPs were used as observations in the explanatory variable with the mergealleles flag. Intercepts were not constrained in the analyses.
Estimating enrichment correlation in gene sets
Request a detailed protocolGenes with a high/low probability of intolerance to LoF variants (highpLI/lowpLI) were defined as the 4,614 genes in the upper/lower quartiles of pLI scores (Karczewski et al., 2020). Genes with high/low brain expression (HBE/LBE) were defined as the 4,614 genes in the upper/lower quartiles of expression in the human fetal brain (Werling et al., 2020). Genes with high/low heart expression (HHE/LHE) were defined as the 4,614 genes in the upper/lower quartiles of expression in the developing heart of embryonic mouse (Zaidi et al., 2013). Five biological pathways have been reported to be involved in congenital heart disease: chromatin remodeling, Notch signaling, cilia function, sarcomere structure and function, and RAS signaling (Zaidi and Brueckner, 2017). We extracted 1730 unique genes that belong to these five pathways from the gene ontology database (Ashburner et al., 2000) and referred to the union set as CHDrelated genes. We repeated EncoreDNM enrichment correlation analysis in these gene sets. Onesided KolmogorovSmirnov test was used to assess the statistical difference between enrichment correlation signal strength in different gene sets.
URLs
GWAS summary statistics data of autism spectrum disorder, schizophrenia, and Tourette disorder were downloaded on the PGC website, https://www.med.unc.edu/pgc/downloadresults/; Summary statistics of cognitive performance were downloaded on the SSGAC website, https://thessgac.com/; Summary statistics of epilepsy were downloaded on the epiGAD website, https://www.epigad.org/; pLI scores were downloaded from gnomAD v3.1 repository https://gnomad.broadinstitute.org/downloads; mTADA, https://github.com/hoangtn/mTADA, Nguyen et al., 2021; denovodb, https://denovodb.gs.washington.edu/denovodb/; MutationMapper on cBioPortal, https://www.cbioportal.org/mutation_mapper; LDSC, https://github.com/bulik/ldsc; Schorsch, 2020.
Code availability
Request a detailed protocolEncoreDNM software is available at https://github.com/ghm17/EncoreDNM; Guo, 2022.
Data availability
The current manuscript is a computational study, so no data have been generated for this manuscript.
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Decision letter

Alexander YoungReviewing Editor; University of California, Los Angeles, United States

Molly PrzeworskiSenior Editor; Columbia University, United States
Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "Quantifying concordant genetic effects of de novo mutations on multiple disorders" for consideration by eLife. Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Molly Przeworski as the Senior Editor. The reviewers have opted to remain anonymous.
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
1) The manuscript needs to explain better the different approaches taken by encoreDNM and mTADA, and the corresponding strengths and weaknesses. The authors focus on power in frequentist hypothesis testing for nonzero genetic overlap between disorders, but mTADA takes a Bayesian approach. Clarification is needed here on whether the statistical basis of the comparison is fair.
2) Greater clarification of the model fitting procedure is needed: how the denovo mutability parameter is used and the computational complexity of optimizing the likelihood function.
3) The heatmaps that present the results of the empirical analysis show only statistical significance levels. Since the authors' method's primary parameter is the correlation parameter, it would be helpful to display estimates of this parameter in the main text and to focus discussion on that parameter more and on statistical significance less.
Reviewer #1 (Recommendations for the authors):
Here, I have the following comments.
1. Authors proposed a mixedeffect Poisson model with overdispersion. I have a few questions regarding this model. First, it is not clear whether the de novo mutability m_i is a parameter or a given estimation in the model. Authors mentioned on line 110 that the background component is a genespecific fixed effect but in the Methods they do not treat it as a parameter in the section of parameter estimation. Second, to treat overdispersion in Poisson model, negative binomial distribution is usually applied. Here, authors added an overdispersion variable \phi_i in the model. It will be helpful to discuss on this point. Third, a MCMC algorithm was used to solve the EncoreDNM model. How is the computational efficiency? On line 431, authors used the inversion of Fisher information matrix to obtain the covariance for parameters. Is there any noninvertibility problem here?
2. In the data analysis, authors used Figure 4cf to suggest better fit of mixedeffect model. It is better to use a table to summarize the goodness of fit test instead of using bar plots in Supplementary Figure 3. On line 197, before reading the whole paragraph, it is not clear at the beginning whether singletrait analysis refers to mixedeffect or fixedeffect. Next, authors discovered that no significant correlation as identified between any disorders and control groups. Discussions about this should be helpful.
3. In Figure 5, panel b and c can be simplified with upper triangle for LoF and lower triangle for synoymous. Moreover, the heatmaps here only reflect the significance of correlations but no information regarding the correlation magnitude itself. It would be helpful to include this in the same figure. This point could be applied to other heatmaps in the supplement. In addition, Figure 5b shows more significant correlations for diseases with larger sample sizes. This is consistent with common sense. So including correlation magnitudes in this figure would be more informative.
Reviewer #2 (Recommendations for the authors):
1. I felt the introduction and discussion lacked a more indepth comparison between EncoreDNM and other similar methods (specifically mTADA). In particular, what are the differences in parameters between these two methods and why do you see such an increase in false positives (Figure 3) for mTADA over EncoreDNM?
2. L66: The technical wording of this statement could be improved. The paper cited was commenting on remaining haploinsufficient genes yet to be discovered. I also do not believe "undetected" to be the correct word here as this refers to a statistical test for enrichment rather than a clinical association. There are several genes (e.g. in Kaplanis et al. Supp Table 2) that are known to cause developmental disorders clinically but do not pass p. value thresholds suggesting a statistical enrichment in studies like DDD.
3. In Figure 1, is there a vertical line missing? On the left, there are squares representing genes, but there is also one rectangle.
4. L103117: Could the derivation of the dispersion parameter Φ be explained in better layterms somewhere? I understand that it is attempting to quantify the random nature of DNM counts one expects when sequencing a subset of individuals with a given disorder, but the maths in the methods section is a bit impenetrable for somebody who is not wellversed in calculus. This is especially crucial considering the major role it plays in interpreting the various results presented, and in comparison, to mTADA. It will be difficult for some readers of a journal with a broad range of topics such as eLife to understand how this parameter, particularly σ, was estimated.
5. L108: Further to point 4, what are reasonable values of the parameters used to fit your model? Additionally, the authors state that β "tends to be larger… and smaller…" under various conditions, but based on Figure 3, it appears that it shifts between positive and negative?
6. L151: The word "could estimate" is loaded and represents an opinion and should be removed in favour of "estimates".
7. L184187: The excessive use of acronyms for various disorders makes for difficult reading. While I am familiar with acronyms for developmental disorders, autism spectrum, and schizophrenia because these are my field, I constantly had to refer back to the definitions is it possible to just use the actual disorder names throughout the manuscript where possible?
8. Figure 3: I think it is relevant to put the various simulated parameters that constitute Φ (e.g. σ, ρ) into context with actual data as presented in Figure 4. i.e. do the simulated parameters used here represent reasonable assumptions of these values and are they within the expectations of mTADA? Could the range of parameters have an outsized effect on the differences between mTADA and EncoreDNM? Or does it have to do with p. value thresholding (see below). Furthermore, I would appreciate it if xaxis labels included the actual parameter setting rather than the less descriptive "small" and plots A and C had labels which described which parameters were fixed in those respective analyses.
9. L246: "i.e." should be removed. There are five genes and you listed all five.
10. L267272: Are the authors certain the thresholding on the mTADA results is reasonable? Whilst the FDR cutoff the authors have applied may suggest "significance", the p. values for the mTADA synonymous variant analysis seem very similar across all disorder combinations. Furthermore, the general pattern of p. values for LoFs seems similar to that shown for EncoreDNM. I suspect that the correlation between p. values of EncoreDNM and mTADA will be high and one could generate similar results by adjusting significance thresholds independently for each tool.
Additionally, as mTADA is based on a Bayesian approach, shouldn't the authors threshold based on posterior probabilities rather than p. values? I am unsure if comparing the π values from the different mTADA results is a valid approach as described in the methods? How do the authors conclusions change when using thresholds based on those the authors of mTADA suggest (e.g. PP > 0.8)?
11. Figure 6: Could a visual cue/text be added to differentiate between the analyses in the upper and lower triangles?
12. What is the overall rank of genes identified between different disorders when comparing between mTADA and EncoreDNM? Could the authors plot relative p/PP values for genes identified to be significantly enriched between disorders?
13. L228229: Do not use the terms "hints at correlation" or "correlate strongly". It either does or does not meet your p. value threshold and/or correlate.
14. L257260: Are the recurrent LoF mutations found in developmental disorders any different than those already identified by Kaplanis et al.? If so, how do your results increase understanding of the mechanisms underlying developmental disorders? I would perhaps shift the focus of this section to identifying recurrent mutations across disorders (and perhaps cite/refer to Rees et al. in Nature Communications; PMID: 34504065).
15. L274280: I do not feel the LD score analysis constitutes a new analysis and should be moved to the discussion. A similar result was recently published in by Abdellaoui and Verweij in Nature Human Behaviour (PMID: 33986517) that covers many of these traits.
16. Throughout: Could the term "Type I Error" be avoided? I would prefer falsediscovery/falsepositive be used as it is a much clearer term and immediately recognisable by the majority of readers.
17. L365366: I do not think identifying genedisease associations is a goal of EncoreDNM so I do not consider it a limitation of the study.
18. L584: I think the hyperlink is broken? I was able to find the repository for EncoreDNM in ghm17's github account, so this was not a major issue.
https://doi.org/10.7554/eLife.75551.sa1Author response
Essential revisions:
1) The manuscript needs to explain better the different approaches taken by encoreDNM and mTADA, and the corresponding strengths and weaknesses. The authors focus on power in frequentist hypothesis testing for nonzero genetic overlap between disorders, but mTADA takes a Bayesian approach. Clarification is needed here on whether the statistical basis of the comparison is fair.
2) Greater clarification of the model fitting procedure is needed: how the denovo mutability parameter is used and the computational complexity of optimizing the likelihood function.
3) The heatmaps that present the results of the empirical analysis show only statistical significance levels. Since the authors' method's primary parameter is the correlation parameter, it would be helpful to display estimates of this parameter in the main text and to focus discussion on that parameter more and on statistical significance less.
We really appreciate the thoughtful and constructive comments from the editor and both reviewers. In this revision, we have provided additional justifications for the comparison between EncoreDNM and mTADA, and clarified statistical details in the model fitting and parameter estimation procedure. We also present the enrichment correlation estimates in addition to significance levels in the updated heatmaps as suggested. The new analyses have produced highly consistent results compared to our initial submission and have strengthened the manuscript. We provide details of these analyses in the pointbypoint response below.
Reviewer #1 (Recommendations for the authors):
Here, I have the following comments.
1. Authors proposed a mixedeffect Poisson model with overdispersion. I have a few questions regarding this model. First, it is not clear whether the de novo mutability m_i is a parameter or a given estimation in the model. Authors mentioned on line 110 that the background component is a genespecific fixed effect but in the Methods they do not treat it as a parameter in the section of parameter estimation.
Thank you for the comment. The de novo mutability $m}_{i$ is given a priori rather than being a parameter to be estimated. There is extensive literature on estimating de novo mutability from genomic sequence context. In this paper, we used mutability values estimated from a trinucleotidebased model proposed by Samocha and colleagues^{1}.
Second, to treat overdispersion in Poisson model, negative binomial distribution is usually applied. Here, authors added an overdispersion variable \phi_i in the model. It will be helpful to discuss on this point.
We thank the reviewer for pointing this out. It is true that negative binomial distribution as a Poissonγ mixture is commonly used for handling overdispersion in count data. This is because Poisson distribution and γ distribution are a conjugate pair, and their compound model can be expressed in closed form. However, the main goal of our study is to quantify the shared genetic component of two disorders, and there is no trivial way to parametrize the bivariate extension of negative binomial distribution for this purpose. Marshall and Olkin proposed a bivariate Poissonγ mixture (negative binomial) model with a restriction that both count variables share the same γ distribution component^{2}. The mixture model has closed form solution but assumes the dispersion of two count variables to be identical and limits the unconditional correlation of two count variables to be positive.
In this paper, we adopted a Poissonlognormal mixture framework, which is another commonly used Poisson mixture model that accommodates overdispersion (see Chapter 4.2 in Cameron, A.C. et al.^{3}). Unlike the Poissonγ mixture, this model does not have the computational convenience of having closedform solutions. Instead, we implemented the Monte Carlo integration method to calculate the likelihood which is computationally more intensive but still obtained accurate and robust results. A major reason why we chose to use the Poissonlognormal mixture is that it can be easily generalized to bivariate count model by replacing its univariate normal component with a bivariate normal distribution. This flexible bivariate count model was initially proposed by Munkin and Trivedi in econometrics, and has been demonstrated to accommodate both overdispersion and correlation, which suits our main goal very well^{4}. In general, finding a computationally simpler approach to parametrize correlation in bivariate count data remains an interesting methodological question, but we are content with the excellent empirical performance of the Poissonlognormal mixture model in our analyses. We have added some clarifications and discussions into the Methodsstatistical model section of our revised manuscript.
Third, a MCMC algorithm was used to solve the EncoreDNM model. How is the computational efficiency?
Our Monte Carlo integration method is computationally efficient. Analysis of a typical trait pair with 18,000 genes takes about 10 minutes on a 2.5GHz cluster with 1 core. We have added these details into the Methodscomputation time section in the revised manuscript.
On line 431, authors used the inversion of Fisher information matrix to obtain the covariance for parameters. Is there any noninvertibility problem here?
Thank you for raising this important point. We double checked our previous analyses and found that the noninvertibility issue indeed occurred when the DNM count is small (mostly happened when analyzing synonymous variants). In this revision, we newly implemented a groupwise jackknife method to obtain standard errors for parameter estimates when the Fisher information matrix is noninvertable. More specifically, we randomly partitioned all genes into 100 groups with equal size. Each time, we left one group out and estimated the parameters using the DNM data of the remaining genes. We then repeated the procedure 100 times and calculated the jackknife standard errors. This approach produced similar standard error estimates compared to the Fisher information approach in our analysis, as illustrated in Figure 5—figure supplement 9 in the revised manuscript. We have also incorporated the new method details into the Methodsparameter estimation section in our revised manuscript.
2. In the data analysis, authors used Figure 4cf to suggest better fit of mixedeffect model. It is better to use a table to summarize the goodness of fit test instead of using bar plots in Supplementary Figure 3.
Thank you for your great suggestion. We have added Supplementary Table 2 in our revised manuscript to show the results of likelihood ratio tests.
On line 197, before reading the whole paragraph, it is not clear at the beginning whether singletrait analysis refers to mixedeffect or fixedeffect.
Thank you for your comment. We have added the word “under the mixedeffects Poisson model” for clarification.
Next, authors discovered that no significant correlation as identified between any disorders and control groups. Discussions about this should be helpful.
For disorders, DNMs will be enriched in risk genes and slightly depleted in nonrisk genes. For control groups (these are healthy siblings recruited in a study for autism), DNMs are expected to distribute proportionally according to the de novo mutability (determined by the genomic sequence context) without showing enrichment in certain genes. Therefore, we expect the enrichment correlation, characterized by concordant enrichment of DNMs in the exome, to be near zero between disorders and the control group. Our results are consistent with our expectation. We have added some related discussions into the Results section of our revised manuscript.
3. In Figure 5, panel b and c can be simplified with upper triangle for LoF and lower triangle for synoymous. Moreover, the heatmaps here only reflect the significance of correlations but no information regarding the correlation magnitude itself. It would be helpful to include this in the same figure. This point could be applied to other heatmaps in the supplement. In addition, Figure 5b shows more significant correlations for diseases with larger sample sizes. This is consistent with common sense. So including correlation magnitudes in this figure would be more informative.
Thank you for this great suggestion. We have incorporated the enrichment correlation estimates into heatmaps (Figures 56). Figure 6—figure supplements 18 in the revised manuscript have also been updated accordingly. We also included the updated Figure 5 below for your convenience.
Reviewer #2 (Recommendations for the authors):
1. I felt the introduction and discussion lacked a more indepth comparison between EncoreDNM and other similar methods (specifically mTADA). In particular, what are the differences in parameters between these two methods and why do you see such an increase in false positives (Figure 3) for mTADA over EncoreDNM?
Thank you for the comment. To quantify shared genetic effects between two disorders, EncoreDNM assumes a mixedeffects Poisson model and estimates the correlation of deviation components across two disorders, whereas mTADA employs a Bayesian framework and estimates the proportion of shared risk genes. We have provided statistical details of EncoreDNM in the Methods section of our manuscript. Here, we briefly introduce mTADA.
The mTADA method assigns all genes into four groups: genes that are not relevant for either disorder, risk genes for the first disorder alone, risk genes for the second disorder alone, and risk genes shared by both disorders. The proportion of these groups are parametrized as $\pi}_{0},{\pi}_{1},{\pi}_{2},{\pi}_{3$, respectively. In particular, parameter $\pi}_{3$ quantifies the extent of genetic sharing between two disorders, with a larger value indicating stronger genetic overlap (for example, see Figures 4ab in the mTADA paper^{5}). The 95% credible interval constructed through MCMC is used to measure the uncertainty in $\pi}_{3$ estimates.
Through extensive simulations and analysis of nine disorders, we demonstrated that EncoreDNM provides accurate statistical inference, but mTADA can produce false positives findings when following the authorrecommended procedure, especially when the DNM count is small. One possible reason is that when there is not sufficient DNM counts, mTADA tends to overestimate $\pi}_{3$. Nguyen et al. also reported this phenomenon in their simulation settings with small mean relative risks^{5} (see Supplementary Figure 3 in the mTADA paper). Further, although the inflation of false positive rate for mTADA may be alleviated by inducing a more stringent significance threshold, researchers will not know how to select such a threshold in practice. We will provide more details on this in our response to Comment #10 below. Related discussions have also been incorporated into the revised manuscript.
2. L66: The technical wording of this statement could be improved. The paper cited was commenting on remaining haploinsufficient genes yet to be discovered. I also do not believe "undetected" to be the correct word here as this refers to a statistical test for enrichment rather than a clinical association. There are several genes (e.g. in Kaplanis et al. Supp Table 2) that are known to cause developmental disorders clinically but do not pass p. value thresholds suggesting a statistical enrichment in studies like DDD.
We have replaced the word “more than 1,000 genes associated with DD remain undetected” with “more than 1,000 haploinsufficient genes contributing to DD risk have not yet achieved statistical significance”.
3. In Figure 1, is there a vertical line missing? On the left, there are squares representing genes, but there is also one rectangle.
We use squares to represent different genes. The rectangle in the middle represents many other genes that are omitted due to limited space.
4. L103117: Could the derivation of the dispersion parameter Φ be explained in better layterms somewhere? I understand that it is attempting to quantify the random nature of DNM counts one expects when sequencing a subset of individuals with a given disorder, but the maths in the methods section is a bit impenetrable for somebody who is not wellversed in calculus. This is especially crucial considering the major role it plays in interpreting the various results presented, and in comparison, to mTADA. It will be difficult for some readers of a journal with a broad range of topics such as eLife to understand how this parameter, particularly σ, was estimated.
We appreciate the suggestion. The main goal of EncoreDNM is to quantify correlated DNM enrichment between two disorders. The statistical framework is parametrized as follows. $\left[\begin{array}{c}{Y}_{i1}\\ </p><p>{Y}_{i2}\\ </p><p>\end{array}\right]\sim Poisson\left(\left[\begin{array}{c}</p><p>{\lambda}_{i1}\\ </p><p>{\lambda}_{i2}\\ </p><p>\end{array}\right]\right),$$\mathrm{log}\left(\left[\begin{array}{c}{\lambda}_{i1}\\ </p><p>{\lambda}_{i2}\\ </p><p>\end{array}\right]\right)=\left[\begin{array}{c}</p><p>{\beta}_{1}\\ </p><p>{\beta}_{2}\\ </p><p>\end{array}\right]+\mathrm{log}\left(\left[\begin{array}{c}</p><p>2{N}_{1}{m}_{i}\\ </p><p>2{N}_{2}{m}_{i}\\ </p><p>\end{array}\right]\right)+\left[\begin{array}{c}</p><p>{\varphi}_{i1}\\ </p><p>{\varphi}_{i2}\\ </p><p>\end{array}\right],$$\left[\begin{array}{c}{\varphi}_{i1}\\ </p><p>{\varphi}_{i2}\\ </p><p>\end{array}\right]\sim \text{MVN}\left(\left[\begin{array}{c}</p><p>0\\ </p><p>0\\ </p><p>\end{array}\right],\left[\begin{array}{cc}</p><p>{\sigma}_{1}^{2}& \rho {\sigma}_{1}{\sigma}_{2}\\ </p><p>\rho {\sigma}_{1}{\sigma}_{2}& {\sigma}_{2}^{2}\\ </p><p>\end{array}\right]\right).$We have described the statistical details of this framework in the Methods section of our manuscript. Briefly, in this model, DNM rate is affected by the elevation component $\beta}_{k$, the background component $log(2{N}_{k}{m}_{i})$, and the deviation component $\varphi}_{\text{ik}$. The component in question, $\varphi}_{\text{ik}$, is modeled as a random effect that follows a bivariate normal distribution. More specifically, $\varphi$ quantifies the degree to which DNM counts look different from what we expect to see under the null (i.e., no risk genes for the disorder). A larger value of the dispersion parameter $\sigma$ indicates a more substantial deviation from the null. That is, DNM counts show strong enrichment in some genes and depletion in other genes compared to the expectation based on de novo mutability. If $\sigma $ has a small value, it means the DNM count data is largely consistent with the null. Parameter $\rho$ further allows such deviation of two disorders to be correlated and is a key parameter of interest in our framework. We have added some clarifications into the Resultsmethod overview section in the revised manuscript.
5. L108: Further to point 4, what are reasonable values of the parameters used to fit your model? Additionally, the authors state that β "tends to be larger… and smaller…" under various conditions, but based on Figure 3, it appears that it shifts between positive and negative?
In our mixedeffects Poisson regression model, there is no constraint on what value the elevation parameter $\beta$ can be. A positive value of $\beta$ represents overcalling DNMs while a negative value represents undercalling. The dispersion parameter $\sigma$ can be any positive value. A larger $\sigma$ indicates that DNM counts show a strong deviation compared to the expectation rate determined by de novo mutability. The enrichment correlation $\rho$ quantifies the concordance of DNM enrichments between two disorders and can be any value between 1 and 1. Applying EncoreDNM to nine disorders, we found that the estimates of $\beta$ were almost always negative across variant classes, which may be explained by strict quality control in DNM calling pipelines. We also note that the empirical estimates of these parameters are affected by noise in the data, especially when the sample size is small.
6. L151: The word "could estimate" is loaded and represents an opinion and should be removed in favour of "estimates".
Thank you for the suggestion. We have replaced the word "could estimate" with “estimates”.
7. L184187: The excessive use of acronyms for various disorders makes for difficult reading. While I am familiar with acronyms for developmental disorders, autism spectrum, and schizophrenia because these are my field, I constantly had to refer back to the definitions is it possible to just use the actual disorder names throughout the manuscript where possible?
We have replaced the acronyms with the actual names of disorders throughout the revised manuscript.
8. Figure 3: I think it is relevant to put the various simulated parameters that constitute Φ (e.g. σ, ρ) into context with actual data as presented in Figure 4. i.e. do the simulated parameters used here represent reasonable assumptions of these values and are they within the expectations of mTADA? Could the range of parameters have an outsized effect on the differences between mTADA and EncoreDNM? Or does it have to do with p. value thresholding (see below). Furthermore, I would appreciate it if xaxis labels included the actual parameter setting rather than the less descriptive "small" and plots A and C had labels which described which parameters were fixed in those respective analyses.
In the real DNM data analysis using EncoreDNM, the mean(SD) of the estimated $\beta$ and $\pi}_{3$ were 0.54(0.25) and 0.99(0.36), respectively. In our simulations, $\beta$ was chosen as 1, 0.5, or 0, and $\sigma$ was chosen as 0.5, 0.75, or 1. Therefore the parameter values used in simulations would be reasonable and not have outsized effects on the performance of two methods. We have also added the parameter values into the xaxis labels of Figure 3.
9. L246: "i.e." should be removed. There are five genes and you listed all five.
We have removed “i.e.” from the sentence.
10. L267272: Are the authors certain the thresholding on the mTADA results is reasonable? Whilst the FDR cutoff the authors have applied may suggest "significance", the p. values for the mTADA synonymous variant analysis seem very similar across all disorder combinations. Furthermore, the general pattern of p. values for LoFs seems similar to that shown for EncoreDNM. I suspect that the correlation between p. values of EncoreDNM and mTADA will be high and one could generate similar results by adjusting significance thresholds independently for each tool.
Additionally, as mTADA is based on a Bayesian approach, shouldn't the authors threshold based on posterior probabilities rather than p. values? I am unsure if comparing the π values from the different mTADA results is a valid approach as described in the methods? How do the authors conclusions change when using thresholds based on those the authors of mTADA suggest (e.g. PP > 0.8)?
This is an important point, and we appreciate the comment. In the paper that introduced the mTADA approach, Nguyen et al. used the proportion of shared risk genes $\pi}_{3$ to assess genetic overlaps between disorders^{5}. They argued that a larger value of $\pi}_{3$ compared to 0 indicates stronger genetic sharing (for example, see Figures 4ab in the mTADA paper^{5}), and importantly, used credible intervals for $\pi}_{3$ to quantify the imprecision in their estimates. It is true that a PP cutoff of 0.8 was used to identify disease risk genes but this was not the approach for studying genetic overlaps in their paper.
To compare the performance of mTADA with EncoreDNM, we followed Nguyen et al. and performed posterior inference for $\pi}_{3$. We used the same 95% credible interval approach to quantify the imprecision in $\pi}_{3$ estimates, and compared it with $\pi}_{1}^{S}\ast {\pi}_{2}^{S$ which quantifies the expected proportion of shared risk genes if two disorders are genetically independent. Here, $\pi}_{1}^{S$ and $\pi}_{2}^{S$ are the estimated proportions of risk genes for two disorders respectively. We compared the posterior distribution of $\pi}_{3$ with $\pi}_{1}^{S}\ast {\pi}_{2}^{S$ and used the following metric to quantify the statistical evidence:
p = $2\ast \frac{\sum _{i=1}^{10000}I\left({\pi}_{3}^{i}<{\pi}_{1}^{S}\ast {\pi}_{2}^{S}\right)}{10000}$.
Here, $\pi}_{3}^{i$ is the $i$th MCMC iteration sample and $I()$ is the indicator function. If the lower bound of the 95% credible interval for $\pi}_{3$ exactly equals $\pi}_{1}^{S}\ast {\pi}_{2}^{S$, then the corresponding p will be 0.05. This is also equivalent to using a (twosided) 0.95 posterior probability cutoff on P($\pi}_{3}>{\pi}_{1}^{S}\ast {\pi}_{2}^{S$) to claim statistical significance. Importantly, we once again highlight an adjustment we made in the posterior inference. Instead of comparing $\pi}_{3$ with 0 (this is what Nguyen et al. did in the mTADA paper), comparing with the expected proportion $\pi}_{1}^{S}\ast {\pi}_{2}^{S$ will lead to more conservative inference results for $\pi}_{3$ since $\pi}_{1}^{S}\ast {\pi}_{2}^{S$ is always greater than 0. Therefore, the posterior inference approach we used for mTADA is largely based on but statistically more conservative than what Nguyen et al. used for analyzing shared genetics between disorders.
Further, we investigated whether varying the significance threshold for mTADA would substantially change its performance in simulations under a mixedeffects Poisson regression model. At the significance cutoff 0.05, mTADA produced substantial proportion of false positive findings in the small $\beta$, small $\sigma$, and small $N$ settings (Supplementary Table 14). Under a more stringent significance cutoff of 0.01, mTADA still produced a substantial inflation in false positive rates when $\beta$ and $\sigma$ are small. We also employed a multinomial model instead of Poisson regression to simulate DNM counts. We obtained consistent results (Supplementary Table 15).
Although it is not the analytic approach used in the mTADA paper, we also investigated an alternative strategy which makes inference based on whether two disorders share at least one risk gene with PP>0.8. This strategy produced substantial inflation in false positive rates in the baseline setting (Supplementary Tables 1617).
The discussions above have been incorporated into the revised manuscript.
11. Figure 6: Could a visual cue/text be added to differentiate between the analyses in the upper and lower triangles?
Thank you for the constructive suggestion. We have added the text for different gene sets in Figure 6. As suggested by Comment 8 from reviewer #1, we depicted correlation estimates rather than significance levels. Figure 6—figure supplements 18 have also been updated accordingly.
12. What is the overall rank of genes identified between different disorders when comparing between mTADA and EncoreDNM? Could the authors plot relative p/PP values for genes identified to be significantly enriched between disorders?
EncoreDNM does not prioritize disease risk genes, but instead estimates the enrichment correlation which quantifies concordant DNM effects between two disorders. This is conceptually similar to genetic correlation which can be estimated from GWAS data. Similarly, genetic correlation quantifies the overall shared additive genetic components between two traits but does not prioritize specific SNP associations. Therefore, we are not able to compare the rank of genes between mTADA and EncoreDNM.
13. L228229: Do not use the terms "hints at correlation" or "correlate strongly". It either does or does not meet your p. value threshold and/or correlate.
We have deleted the statement to avoid confusion.
14. L257260: Are the recurrent LoF mutations found in developmental disorders any different than those already identified by Kaplanis et al.? If so, how do your results increase understanding of the mechanisms underlying developmental disorders? I would perhaps shift the focus of this section to identifying recurrent mutations across disorders (and perhaps cite/refer to Rees et al. in Nature Communications; PMID: 34504065).
This is a great suggestion. We identified 30 recurrent crossdisorder LoF mutations that were not recurrent in developmental disorder alone (see Supplementary Table 7 ). We have now shifted the focus of this section to crossdisorder findings. In particular, we have highlighted the gene FBXO11 that shows recurrent LoF variants in autism and congenital hydrocephalus as an example. The Rees et al. paper has also been added as a reference in our revised manuscript.
15. L274280: I do not feel the LD score analysis constitutes a new analysis and should be moved to the discussion. A similar result was recently published in by Abdellaoui and Verweij in Nature Human Behaviour (PMID: 33986517) that covers many of these traits.
We have moved the LD score regression analysis into the Discussion section.
16. Throughout: Could the term "Type I Error" be avoided? I would prefer falsediscovery/falsepositive be used as it is a much clearer term and immediately recognisable by the majority of readers.
We have replaced the word “typeI error” with “false positive rate” throughout the manuscript.
17. L365366: I do not think identifying genedisease associations is a goal of EncoreDNM so I do not consider it a limitation of the study.
We have removed this point from the Discussion section.
18. L584: I think the hyperlink is broken? I was able to find the repository for EncoreDNM in ghm17's github account, so this was not a major issue.
Thank you for pointing this out. The hyperlink has been fixed.
References
1. Samocha, K.E. et al. A framework for the interpretation of de novo mutation in human disease. Nature genetics 46, 944950 (2014).
2. Marshall, A.W. & Olkin, I. Multivariate distributions generated from mixtures of convolution and product families. Lecture NotesMonograph Series , 371393 (1990).
3. Cameron, A.C. & Trivedi, P.K. Regression analysis of count data, (Cambridge university press, 2013).
4. Munkin, M.K. & Trivedi, P.K. Simulated maximum likelihood estimation of multivariate mixed‐Poisson regression models, with application. The Econometrics Journal 2, 2948 (1999).
5. Nguyen, T.H. et al. mTADA is a framework for identifying risk genes from de novo
mutations in multiple traits. Nature Communications 11, 2929 (2020).
https://doi.org/10.7554/eLife.75551.sa2Article and author information
Author details
Funding
National Science Foundation of China (No. 12071243)
 Lin Hou
Shanghai Municipal Science and Technology Major Project (No. 2017SHZDZX01)
 Lin Hou
Wisconsin Alumni Research Foundation
 Qiongshi Lu
Waisman Center pilot grant program at University of WisconsinMadison
 Qiongshi Lu
National Institutes of Health (No. R03HD100883 and R01GM134005)
 Hongyu Zhao
National Science Foundation (DMS 1902903)
 Hongyu Zhao
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
LH acknowledges research support from the National Science Foundation of China (Grant No. 12071243) and Shanghai Municipal Science and Technology Major Project (Grant No. 2017SHZDZX01). QL acknowledges research support from the University of WisconsinMadison Office of the Chancellor and the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation and the Waisman Center pilot grant program at University of WisconsinMadison. HZ acknowledges research support from the National Institutes of Health (Grant No. R03HD100883 and R01GM134005) and the National Science Foundation (DMS 1902903).
Senior Editor
 Molly Przeworski, Columbia University, United States
Reviewing Editor
 Alexander Young, University of California, Los Angeles, United States
Publication history
 Preprint posted: June 14, 2021 (view preprint)
 Received: November 14, 2021
 Accepted: June 1, 2022
 Accepted Manuscript published: June 6, 2022 (version 1)
 Version of Record published: June 22, 2022 (version 2)
Copyright
© 2022, Guo et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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