Hypergraph factorization for multi-tissue gene expression imputation

Abstract

Integrating gene expression across tissues and cell types is crucial for understanding the coordinated biological mechanisms that drive disease and characterize homoeostasis. However, traditional multi-tissue integration methods either cannot handle uncollected tissues or rely on genotype information, which is often unavailable and subject to privacy concerns. Here we present HYFA (hypergraph factorization), a parameter-efficient graph representation learning approach for joint imputation of multi-tissue and cell-type gene expression. HYFA is genotype agnostic, supports a variable number of collected tissues per individual, and imposes strong inductive biases to leverage the shared regulatory architecture of tissues and genes. In performance comparison on Genotype–Tissue Expression project data, HYFA achieves superior performance over existing methods, especially when multiple reference tissues are available. The HYFA-imputed dataset can be used to identify replicable regulatory genetic variations (expression quantitative trait loci), with substantial gains over the original incomplete dataset. HYFA can accelerate the effective and scalable integration of tissue and cell-type transcriptome biorepositories.

Main

Sequencing technologies have enabled profiling of the transcriptome at tissue and single-cell resolutions, with great potential to unveil intra- and multi-tissue molecular phenomena such as cell signalling and disease mechanisms. Due to the invasiveness of the sampling process, gene expression is usually measured independently in easy-to-acquire tissues, leading to an incomplete picture of an individual’s physiological state and necessitating effective multi-tissue integration methodologies.

A question of fundamental biological importance is to what extent the transcriptomes of difficult-to-acquire tissues and cell types can be inferred from those of accessible ones^1,2. Due to their ease of collection, accessible tissues such as whole blood could have great utility for diagnosis and monitoring of pathophysiological conditions through metabolites, signalling molecules and other biomarkers, including possible transcriptome-level associations³. Moreover, all human somatic cells share the same genetic information, which may regulate expression in a context-dependent and temporal manner, partially explaining tissue- and cell-type-specific gene expression variation. Computational models that exploit these patterns could therefore be used to impute the transcriptomes of uncollected cell types and tissues, with potential to elucidate the biological mechanisms regulating a diverse range of developmental and physiological processes.

Multi-tissue imputation is a central problem in transcriptomics with broad implications for fundamental biological research and translational science. The methodological problem can powerfully influence downstream applications, including performing differential expression analysis, identifying regulatory mechanisms, determining co-expression networks and enabling drug target discovery. In practice, in experimental follow-up or clinical application, the task includes the special case of determining a good proxy or easily assayed system for causal tissues and cell types. Multi-tissue integration methods can also be applied to harmonize large collections of RNA-seq datasets from diverse institutions, consortia and studies⁴—each potentially affected by technical artifacts—and to characterize gene expression co-regulation across tissues. Reconstruction of unmeasured gene expression across a broad collection of tissues and cell types from available reference transcriptome panels may expand our understanding of the molecular origins of complex traits and of their context specificity.

Several methods have traditionally been employed to impute uncollected gene expression. Leveraging a surrogate tissue has been widely used in studies of biomarker discovery, diagnostics and expression quantitative trait loci (eQTLs), and in the development of model systems^5,6,7,8,9. Nonetheless, gene expression is known to be tissue and cell-type specific, limiting the utility of a proxy tissue. Other related studies impute tissue-specific gene expression from genetic information¹⁰. Wang et al.¹¹ propose a mixed-effects model to infer uncollected data in multiple tissues from eQTLs. Sul et al.¹² introduce a model termed Meta-Tissue, which aggregates information from multiple tissues to increase the statistical power of eQTL detection. However, these approaches do not model the complex nonlinear relationships between measured and unmeasured gene expression traits among tissues and cell types, and individual-level genetic information (for example, at eQTLs) is subject to privacy concerns and often unavailable.

Computationally, multi-tissue transcriptome imputation is challenging because the data dimensionality scales rapidly with the number of genes and tissues, often leading to overparameterized models. TEEBoT¹ addresses this issue by employing principal component analysis—a non-parametric dimensionality reduction method—to project the data into a low-dimensional manifold, followed by linear regression to predict target gene expression from the principal components. However, this technique does not account for nonlinear effects and can only handle a single reference tissue, that is, whole blood. Approaches such as standard multilayer perceptrons (MLPs) can exploit nonlinear patterns, but are massively overparameterized and computationally infeasible.

To address these challenges, we present HYFA (hypergraph factorization), a parameter-efficient graph representation learning approach for joint multi-tissue and cell-type gene expression imputation. HYFA represents multi-tissue gene expression in a hypergraph of individuals, metagenes and tissues, and learns factorized representations via a specialized message-passing neural network operating on the hypergraph. In contrast to existing methods, HYFA supports a variable number of reference tissues, increasing the statistical power over single-tissue approaches, and incorporates inductive biases to exploit the shared regulatory architecture of tissues and genes. In performance comparison, HYFA attains improved performance over TEEBoT and standard imputation methods across a broad range of tissues from the Genotype-Tissue Expression (GTEx) project (v8) (ref. ²). Through transfer learning on a paired single-nucleus RNA-seq dataset (GTEx-v9) (ref. ¹³), we further demonstrate the ability of HYFA to resolve cell-type signatures—average gene expression across cells for a given cell type, tissue and individual—from bulk gene expression. Thus, HYFA may provide a unifying transcriptomic methodology for multi-tissue imputation and cell-type deconvolution. In post-imputation analysis, application of eQTL mapping on the fully imputed GTEx data yields a substantial increase in number of detected replicable eQTLs. HYFA is publicly available at https://github.com/rvinas/HYFA.

Results

HYFA (hypergraph factorization)

We developed HYFA, a framework for inferring the transcriptomes of unmeasured tissues and cell types from bulk expression collected in a variable number of reference tissues (Fig. 1 and Methods). HYFA receives as input gene expression measurements collected from a set of reference tissues, as well as demographic information, and outputs gene expression values in a tissue of interest (for example uncollected). The first step of the workflow is to project the input gene expression into low-dimensional metagene representations^14,15 for every collected tissue. Each metagene summarizes abstract properties of groups of genes, for example sets of genes that tend to be expressed together¹⁶, that are relevant for the imputation task. In a second step, HYFA employs a custom message-passing neural network¹⁷ that operates on a 3-uniform hypergraph, yielding factorized individual, tissue and metagene representations. Finally, HYFA infers latent metagene values for the target tissue—a hyperedge-level prediction task—and maps these representations back to the original gene expression space. Through higher-order hyperedges (for example, a 4-uniform hypergraph), HYFA can also incorporate cell-type information and infer finer-grained cell-type-specific gene expression (Methods). Altogether, HYFA offers features to reuse knowledge across tissues and genes, capture nonlinear cross-tissue patterns of gene expression, learn rich representations of biological entities and account for variable numbers of reference tissues.

Fig. 1: Overview of HYFA.

a, HYFA processes gene expression from a number of collected tissues (for example, accessible tissues) and infers the transcriptomes of uncollected tissues. b, Workflow of HYFA. The model receives as input a variable number of gene expression samples ${{{{\mathbf{x}}}}}_{i}^{(k)}$ corresponding to the collected tissues $k\in {{{\mathcal{T}}}}(i)$ of a given individual i. The samples ${{{{\mathbf{x}}}}}_{i}^{(k)}$ are fed through an encoder that computes low-dimensional representations ${{{{\mathbf{e}}}}}_{ij}^{(k)}$ for each metagene j ∈ 1, …, M. A metagene is a latent, low-dimensional representation that captures certain gene expression patterns of the high-dimensional input sample. These representations are then used as hyperedge features in a message-passing neural network that operates on a hypergraph. In the hypergraph representation, each hyperedge labelled with ${{{{\mathbf{e}}}}}_{ij}^{(k)}$ connects an individual i with metagene j and tissue k if tissue k was collected for individual i, that is $k\in {{{\mathcal{T}}}}(i)$. Through message passing, HYFA learns factorized representations of individual, tissue and metagene nodes. To infer the gene expression of an uncollected tissue u of individual i, the corresponding factorized representations are fed through an MLP that predicts low-dimensional features ${{{{\mathbf{e}}}}}_{ij}^{(u)}$ for each metagene j ∈ 1, …, M. HYFA finally processes these latent representations through a decoder that recovers the uncollected gene expression sample ${\hat{{{{\mathbf{x}}}}}}_{i}^{(u)}$.

Characterization of cross-tissue relationships

Characterizing cross-tissue relationships at the transcriptome level can help elucidate coordinated gene regulation and expression, a fundamental phenomenon with direct implications for health homoeostasis, disease mechanisms and comorbidities^18,19,20. We trained HYFA on bulk gene expression from the GTEx project (GTEx-v8; Methods)² and assessed the cross-tissue gene expression predictability—measured using the Pearson correlation between the observed and the predicted gene expression across individuals—and quality of tissue embeddings (Fig. 2). Application of Uniform Manifold Approximation and Projection (UMAP)²¹ on the learnt tissue representations revealed strong clustering of biologically related tissues (Fig. 2a), including the gastrointestinal system (for example, oesophageal, stomach, colonic and intestinal tissues), the female reproductive tissues (that is, uterus, vagina and ovary) and the central nervous system (that is, the 13 brain tissues). For every pair of reference and target tissues in GTEx, we then computed the Pearson correlation coefficient ρ between the predicted and actual gene expression, averaged the scores across individuals and used a cutoff of ρ > 0.5 to depict the top pairwise associations (Fig. 2b and Extended Data Fig. 1). We observed connections between most GTEx tissues and whole blood, which suggests that blood-derived gene expression is highly informative on (patho)physiological processes in other tissues²². Notably, brain tissues and the pituitary gland were strongly associated with several tissues (ρ > 0.5), including gastrointestinal tissues (that is oesophagus, stomach and colon), the adrenal gland and skeletal muscle, which may account for known disease comorbidities.

Fig. 2: Analysis of cross-tissue relationships.

a–e, Colours are assigned to conform to the GTEx Consortium conventions. a, UMAP representation of the tissue embeddings learnt by HYFA. Note that human body systems cluster in the embedding space (for example, digestive system—stomach, small intestine, colon, oesophagus—and central nervous system). EBV, Epstein–Barr virus. b, Network of tissues depicting the predictability of target tissues with HYFA using the average per-sample ρ. The dimension of each node is proportional to its degree. Edges from reference to target tissues indicate an average ρ > 0.5. Interestingly, central nervous system tissues strongly correlate with several non-brain tissues such as gastrointestinal tissues and skeletal muscles. c, Top predicted genes in multiple brain regions with the oesophago-gastric junction as the reference tissue, ranked by average Pearson correlation. d, Common genes in the top 1,000 predicted genes for each brain tissue. e, Enriched Gene Ontology (GO) terms for the top shared genes at the intersection. The top predicted genes were enriched in signalling pathways (FDR < 0.05), consistent with studies reporting that gut microbes communicate to the central nervous system through endocrine and immune mechanisms. These results depict the cross-tissue associations and highlight the potential connection between the elements of the oesophago-gastric junction and the ciliary neurotrophic factor, which has been linked to the survival of neurons³³ and the control of body weight³⁵.

Imputation of gene expression from whole-blood transcriptome

Knowledge about tissue-specific patterns of gene expression can increase our understanding of disease biology, facilitate the development of diagnostic tools and improve patient subtyping^1,23, but most tissues are inaccessible or difficult to acquire. To address this challenge, we studied to what extent HYFA can recover tissue-specific gene expression from whole-blood transcriptomic measurements (Fig. 3). For each test individual with measured whole-blood gene expression, we predicted tissue-specific gene expression in the remaining collected tissues of the individual. We evaluated performance using the Pearson correlation between the inferred gene expression and the ground-truth samples. We observed strong prediction performance for oesophageal tissues (muscularis, ρ = 0.49; gastro, ρ = 0.46; mucosa, ρ = 0.36), heart tissues (left ventricle, ρ = 0.48; atrial, ρ = 0.46) and lung (ρ = 0.47), while Epstein Barr virus-transformed lymphocytes (ρ = 0.06), an accessible and renewable resource for functional genomics, was a notable outlier. We noted that the per-gene prediction scores followed smooth, unimodal distributions (Extended Data Fig. 2). The blood-imputed gene expression also predicted disease-relevant genes in the hard-to-access central nervous system (Extended Data Fig. 3). These include APP, PSEN1 and PSEN2, that is, the causal genes for autosomal dominant forms of early-onset Alzheimer’s disease²⁴, and Alzheimer’s disease genetic risk factors such as APOE²⁵. We compared our method with TEEBoT¹ (without expression single-nucleotide polymorphism information), which first projects the high-dimensional blood expression data into a low-dimensional space through principal component analysis (30 components; 75–80% explained variance) and then performs linear regression to predict the gene expression of the target tissue. Overall, TEEBoT and HYFA attained comparable scores when a single tissue (that is whole blood) was used as reference and both methods outperformed standard imputation approaches (mean imputation, blood surrogate and k-nearest neighbours; Fig. 3c).

Fig. 3: Performance comparison across gene expression imputation methods.

a,b, Per-tissue comparison between HYFA and TEEBoT when using whole blood (a) and all accessible tissues (whole blood, skin sun exposed, skin not sun exposed and adipose subcutaneous) (b) as reference. HYFA achieved superior Pearson correlation in 25 out of 48 target tissues when a single tissue was used as reference (a) and all target tissues when multiple reference tissues were considered (b). For under-represented target tissues (fewer than 25 individuals with source and target tissues in the test set), we considered all the validation and test individuals (translucent bars). We employed two-sided Mann–Whitney–Wilcoxon tests to compute P values (*1 × 10⁻² < P ≤ 5 × 10⁻², **1 × 10⁻³ < P ≤ 1 × 10⁻², ***1 × 10⁻⁴ < P ≤ 1 × 10⁻³, ****P ≤ 1 × 10⁻⁴). The top axis indicates the total number n of independent individuals for every target tissue. c,d, Prediction performance from whole-blood gene expression (n = 2,424 samples from 167 GTEx donors) (c) and accessible tissues as reference (n = 675 samples from 167 test GTEx donors) (d). Mean imputation replaces missing values with the feature averages. Blood surrogate utilizes gene expression in whole blood as a proxy for the target tissue. k-nearest neighbours (kNN) imputes missing features with the average of measured values across the k-nearest observations (k = 20). TEEBoT projects reference gene expression into a low-dimensional space with principal component analysis (30 components), followed by linear regression to predict target values. HYFA (all) employs information from all collected tissues of the individual. Boxes show quartiles, centrelines correspond to the median and whiskers depict the distribution range (1.5 times the interquartile range). Outliers outside the whiskers are shown as distinct points.

Multiple reference tissues improve performance

We hypothesized that using multiple tissues as reference would improve downstream imputation performance. To evaluate this, we selected individuals with measured gene expression both at the target tissue and four reference accessible tissues (whole blood, skin sun exposed, skin not sun exposed and adipose subcutaneous) and employed HYFA to impute target expression values (Fig. 3 and Extended Data Fig. 4). We discarded under-represented target tissues with fewer than 25 test individuals. Relative to using whole blood in isolation, using all accessible tissues as reference resulted in improved performance for 32 out of 38 target tissues (Extended Data Fig. 4). This particularly boosted imputation performance for oesophageal tissues (muscularis, Δρ = 0.068; gastro, Δρ = 0.061; mucosa, Δρ = 0.048), colonic tissues (transverse, Δρ = 0.065; sigmoid, Δρ = 0.056) and artery tibial (Δρ = 0.079). In contrast, performance for the pituitary gland (Δρ = −0.011), lung (Δρ = −0.003) and stomach (Δρ = −0.002) remained stable or dropped slightly. Moreover, the performance gap between HYFA and TEEBoT (trained on the set of complete multi-tissue samples) widened relative to the single-tissue scenario (Fig. 3 and Extended Data Fig. 5)—HYFA obtained better performance in all target tissues, with statistically significant improvements in 26 out of 38 tissues (two-sided Mann–Whitney–Wilcoxon P < 0.05). We attribute the improved scores to HYFA’s ability to process a variable number of reference tissues, reuse knowledge across tissues and capture nonlinear patterns.

Inference of cell-type signatures

We next investigated the potential of HYFA to predict cell-type-specific signatures—average gene expression across cells from a given cell type—in a given tissue of interest. We first selected GTEx donors with collected bulk (v8) and single-nucleus RNA-seq profiles (v9, Methods). Next, we trained HYFA to infer cell-type signatures from the multi-tissue bulk expression profiles. We evaluated performance using the observed (Fig. 4) and inferred library sizes (Supplementary Section K). To attenuate the small-data-size problem, we applied transfer learning on the model trained for the multi-tissue imputation task (Methods). We observed strong prediction performance (Pearson correlation ρ between log ground truth and log predicted signatures) for vascular endothelial cells (heart, ρ = 0.84; breast, ρ = 0.88; oesophagus muscularis, ρ = 0.68) and fibroblasts (heart, ρ = 0.84; breast, ρ = 0.89; oesophagus muscularis, ρ = 0.70). Strikingly, HYFA recovered the cell-type profiles of tissues that were never observed in the training set with high correlation (Fig. 4 and Supplementary Section K)—for example, skeletal muscle (vascular endothelial cells, ρ = 0.79; fibroblasts, ρ = 0.77; pericytes/smooth muscle cells, ρ = 0.68), demonstrating the benefits of the factorized tissue representations. Overall, our results highlight the potential of HYFA to impute unknown cell-type signatures even for tissues that were not considered in the original single-cell study. Additionally, our analyses point to promising downstream applications as single-cell RNA-seq datasets become larger in number of individuals (Supplementary Section N), including deconvolution and cell-type-specific eQTL mapping.

Fig. 4: Prediction of cell-type signatures.

HYFA imputes individual- and tissue-specific cell-type signatures from bulk multi-tissue gene expression. The scatter plots depict the Pearson correlation between the logarithmized ground truth and predicted signatures for N unseen individuals. To infer the signatures, we used the observed library size ${l}_{i}^{(k,q)}$ and number of cells ${n}_{i}^{(k,q)}$ (Methods). SMC, smooth muscle cell.

Multi-tissue imputation improves eQTL detection

The GTEx project has enabled the identification of numerous genetic associations with gene expression across a broad collection of tissues², also known as eQTLs²⁶. However, eQTL datasets are characterized by small sample sizes, especially for difficult-to-acquire tissues and cell types, reducing the statistical power to detect eQTLs²⁷. To address this problem, we employed HYFA to impute the transcript levels of every uncollected tissue for each individual in GTEx, yielding a complete gene expression dataset of 834 individuals and 49 tissues. We then performed eQTL mapping (Methods) on the original and imputed datasets and observed a substantial gain in the number of unique genes with detected eQTLs, the so-called eGenes (Fig. 5). Notably, this metric increased for tissues with low sample size (Spearman ρ = −0.83)—which are most likely to benefit from borrowing information across tissues with shared regulatory architecture. Kidney cortex displayed the largest gain in number of eGenes (from 215 to 12,557), while there was no increase observed for whole blood.

Fig. 5: HYFA’s imputed data improves eQTL discovery.

a, Number of unique genes with detected eQTLs (FDR < 0.1) on observed (circle) and full (observed plus imputed; rhombus) GTEx data. Note logarithmic scale of y axis. The eQTLs were mapped using Matrix eQTL^55,70 assuming an additive genotype effect on gene expression. Matrix eQTL conducts a test for each single nucleotide polymorphism (SNP)–gene pair and makes adjustments for multiple comparisons by computing the Benjamini–Hochberg FDR⁷¹. b, Fold increase in number of unique genes with mapped eQTLs (y axis) versus observed sample size (x axis). c, Histogram of replication P values among the HYFA-identified cis-eQTLs for whole blood (left) and brain prefrontal cortex (right). For replication, we used the independent eQTLGen Consortium (n > 30,000; ref. ²⁸) and PsychENCODE (n = 1,866; ref. ²⁹) eQTL datasets, respectively. d, Quantile–quantile plot showing the causal variants' association with gene expression in blood (left) and brain frontal cortex (right) in the HYFA-derived dataset using experimentally validated causal variant data from application of the Massively Parallel Reporter Assay dataset³¹. All statistical tests were two sided. HYFA’s imputed data substantially increase the number of identified associations with high replicability and strong enrichment of causal regulatory variants.

To assess the quality of the identified eQTLs from HYFA imputation, we conducted systematic replication analyses of (1) the whole-blood eQTL–eGene pairs, using the eQTLGen blood transcriptome dataset in more than 30,000 individuals²⁸, and (2) the frontal cortex eQTL–eGene pairs, using the PsychENCODE prefrontal cortex transcriptome dataset in 1,866 individuals²⁹. For each tissue, we quantified the replication rate for eQTL–eGene pairs using the π₁ statistic³⁰. Notably, we found a highly significant enrichment for low replication P values among the HYFA-derived eQTL–eGene pairs (Fig. 5), demonstrating strong reproducibility of the results. The replication rate π₁ was 0.80 for whole blood and 0.96 for frontal cortex. We also evaluated the extent to which the HYFA imputation could capture regulatory variants that directly modulate gene expression using experimentally validated causal variants from the Massively Parallel Reporter Assay dataset³¹. Notably, among the causal regulatory variants from this experimental assay, we found a highly significant enrichment for low P values among the HYFA-identified eQTLs in blood and in frontal cortex (Fig. 5). Thus, HYFA imputation enabled identification of biologically meaningful, replicable eQTL hits in the respective tissues. Our results generate a large catalogue of new tissue-specific eQTLs (Data availability), with potential to enhance our understanding of how regulatory variation mediates variation in complex traits, including disease susceptibility.

Brain–gut axis

The brain–gut axis is a bidirectional communication system of signalling pathways linking the central and enteric nervous systems. We investigated whether the transcriptomes of tissues from the gastrointestinal system are predictive of gene expression in brain tissues (Fig. 2 and Supplementary Section G). Overall, the top predicted genes were enriched in multiple signalling-related terms (for example cytokine receptor activity and interleukin-1 receptor activity), consistent with existing knowledge that gut microbes communicate with the central nervous system through signalling mechanisms³². Genes in the intersection were also notably enriched in the ciliary neurotrophic factor receptor activity, which plays an important role in neuron survival³³, enteric nervous system development³⁴ and body weight control³⁵.

HYFA-learned metagenes capture known biological pathways

A key feature of HYFA is that it reuses knowledge across tissues and metagenes, allowing exploitation of shared regulatory patterns. We explored whether HYFA’s inductive biases encourage learning of biologically relevant metagenes. To determine the extent to which metagene factors relate to known biological pathways, we applied gene set enrichment analysis (GSEA)³⁶ to the gene loadings of HYFA’s encoder (Methods). Similarly to ref. ³⁷, for a given query gene set, we calculated the maximum running sum of enrichment scores by descending the sorted list of gene loadings for every metagene and factor. We then computed pathway enrichment P values through a permutation test and employed the Benjamini–Hochberg method to correct for multiple testing independently for every metagene factor.

In total, we identified 18,683 statistically significant enrichments (false discovery rate, FDR < 0.05) of KEGG biological processes³⁸ (320 gene sets; Fig. 6) across all HYFA metagenes (n = 50) and factors (n = 98). Among the enriched terms, 2,109 corresponded to signalling pathways and 1,300 to pathways of neurodegeneration. We observed considerable overlap between several metagenes in terms of biologically related pathways: for example, factor 95 of metagene 11 had the lowest FDR for both Alzheimer’s disease (FDR < 0.001) and amyotrophic lateral sclerosis (FDR < 0.001) pathways. Enrichment analysis of TRRUST³⁹ transcription factors (TFs) further identified important regulators including GATA1 (known to regulate the development of red blood cells⁴⁰), SPI1 (which controls haematopoietic cell fate⁴¹), CEBPs (which play an important role in the differentiation of a range of cell types and the control of tissue-specific gene expression^42,43) and STAT1 (a member of the STAT protein family that drives the expression of many target genes⁴⁴). We also observed that the learnt HYFA factors recapitulate synergistic effects among the enriched TFs (Supplementary Section H and Extended Data Fig. 6). For example, GATA1 and SPI1, which were simultaneously enriched in 7 factors (FDR < 0.05), functionally antagonize each other through physical interaction⁴⁵. Similarly, IRF1 induces STAT1 activation via phosphorylation^44,46 and both TFs were enriched in 10 factors (FDR < 0.05), aligning with our enrichment analyses of GO biological process terms (Supplementary Section I and Extended Data Figs. 7 and 8). Altogether, our analyses suggest that HYFA-learned metagenes and factors are amenable to biological interpretation and capture information about known regulators of tissue-specific gene expression.

Fig. 6: Pathway enrichment analysis of metagene factors.

a, Manhattan plot of the GSEA results on the metagenes (n = 50) and factors (n = 98) learned by HYFA. The x axis represents metagenes (coloured bins) and each offset within the bin corresponds to a different factor. The y axis is the −log q value (FDR) from the GSEA permutation test, corrected for multiple testing via the Benjamini–Hochberg procedure. We identified 18,683 statistically significant enrichments (FDR < 0.05) of KEGG biological processes across all metagenes and factors. b, Total number of enriched terms for each type of pathway. c, FDR for pathways of neurodegeneration. For each pathway and metagene, we selected the factor with the lowest FDR and depicted statistically significant values (FDR < 0.05). Circle sizes are proportional to −log FDR values. Metagene 11 (factor 95) had the lowest FDR for both amyotrophic lateral sclerosis and Alzheimer’s disease. d, UMAP of latent values of metagene 11 for all spinal cord (amyotrophic lateral sclerosis: orange) and brain cortex (Alzheimer’s disease or dementia: orange) GTEx samples. e, Leading-edge subsets of top 15 enriched gene sets for factor 95 of metagene 11. NES, normalized enrichment score; SET, gene set. f,g, Enrichment plots for amyotrophic lateral sclerosis (f) and Alzheimer’s disease gene sets (g).

Discussion

Effective multi-tissue omics integration promises a system-wide view of human physiology, with potential to shed light on intra- and multi-tissue molecular phenomena. Such an approach challenges single-tissue and conventional integration techniques—often unable to model a variable number of tissues with sufficient statistical strength, necessitating the development of scalable, nonlinear and flexible methods. Here we developed HYFA, a parameter-efficient approach for joint multi-tissue and cell-type gene expression imputation, which imposes strong inductive biases to learn entity-independent relational semantics and demonstrates excellent imputation capabilities.

We performed extensive benchmarks on data from GTEx² (v8 and v9), the most comprehensive human transcriptome resource available, and evaluated imputation performance over a broad collection of tissues and cell types. In addition to standard transcriptome imputation approaches, we compared our method with TEEBoT¹, a linear method that predicts target gene expression from the principal components of the reference expression. In the single-tissue reference scenario, HYFA and TEEBoT attained comparable imputation performance, outperforming standard methods. In the multi-tissue reference scenario, HYFA consistently outperformed TEEBoT and standard approaches in all target tissues, demonstrating HYFA’s capabilities to borrow nonlinear information across a variable number of tissues and exploit shared molecular patterns.

In addition to imputing tissue-level transcriptomics, we investigated the ability of HYFA to predict cell-type-level gene expression from multi-tissue bulk expression measurements. Through transfer learning, we trained HYFA to infer cell-type signatures from a cohort of single-nucleus RNA-seq¹³ with matching GTEx-v8 donors. The inferred cell-type signatures exhibited a strong correlation with the ground truth despite the low sample size, indicating that HYFA’s latent representations are rich and amenable to knowledge transfer. Strikingly, HYFA also recovered cell-type profiles from tissues that were never observed at transfer time, pointing to HYFA’s ability to leverage gene expression programs underlying cell-type identity⁴⁷ even in tissues that were not considered in the original study¹³. HYFA may also be used to impute the expression of disease-related genes in a tissue of interest (Supplementary Section J).

In post-imputation analysis, we studied whether the imputed data improve eQTL discovery. We employed HYFA to impute the gene expression levels of every uncollected tissue in GTEx-v8, yielding a complete dataset, and performed eQTL mapping. Compared with the original dataset, we observed a substantial gain in number of genes with detected eQTLs, with kidney cortex showing the largest gain. The increase was highest for tissues with low sample sizes, which are the ones expected to benefit the most from knowledge sharing across tissues. Notably, HYFA’s detected eQTLs with their target eGenes could be replicated using independent, single-tissue transcriptome datasets that focus on depth, including the blood eQTLGen²⁸ and the brain frontal cortex PsychENCODE²⁹ datasets. Moreover, we found a substantial enrichment for experimentally validated causal variants from the Massively Parallel Reporter Assay³¹ dataset. Our results uncover a large number of previously undetected tissue-specific eQTLs and highlight the ability of HYFA to exploit shared regulatory information across tissues.

Finally, HYFA can provide insights on coordinated gene regulation and expression mechanisms across tissues. We analysed to what extent tissues from the gastrointestinal system are informative about gene expression in brain tissues—an important question that may shed light on the biology of the brain–gut axis—and identified enriched biological processes and molecular functions. Through GSEA³⁶, we observed, among the HYFA-learned metagenes, a substantial number of enriched pathways, TFs and known regulators of biological processes, opening the door to biological interpretations. Future work might also seek to impose stronger inductive bias to ensure that metagenes are identifiable and robust to batch effects.

We believe that HYFA, as a versatile graph representation learning framework, provides a novel methodology for effective integration of large-scale multi-tissue biorepositories. The hypergraph factorization framework is flexible (it supports k-uniform hypergraphs of arbitrary node types) and may find application beyond computational genomics.

Methods

Problem formulation

Suppose we have a transcriptomics dataset of N individuals/donors, T tissues and G genes. For each individual i ∈ {1, …, N}, let ${{{{\mathbf{X}}}}}_{i}\in {{\mathbb{R}}}^{T\times G}$ be the gene expression values in T tissues and define the donor’s demographic information by ${{{{\mathbf{u}}}}}_{i}\in {{\mathbb{R}}}^{C}$, where C is the number of covariates. Denote by ${{{{\mathbf{x}}}}}_{i}^{(k)}$ the kth entry of X_i, corresponding to the expression values of donor i measured in tissue k. For a given donor i, let ${{{\mathcal{T}}}}(i)$ represent the collection of tissues with measured expression values. These sets might vary across individuals. Let ${\tilde{\mathbf{X}}}_{i}\in {({\mathbb{R}}\cup \{* \})}^{T\times G}$ be the measured gene expression values, where * denotes unobserved, so that ${\tilde{\mathbf{x}}}_{i}^{(k)}={{{{\mathbf{x}}}}}_{i}^{(k)}$ if $k\in {{{\mathcal{T}}}}(i)$ and ${\tilde{\mathbf{x}}}_{i}^{(k)}=*$ otherwise. Our goal is to infer the uncollected values in ${\tilde{\mathbf{X}}}_{i}$ by modelling the distribution $p({{{\bf{X}}}}={{{{\mathbf{X}}}}}_{i}| {\tilde{\bf{X}}}={\tilde{\mathbf{X}}}_{i},{{{\bf{U}}}}={{{{\mathbf{u}}}}}_{i})$.

Multi-tissue model

An important challenge of modelling multi-tissue gene expression is that a different set of tissues might be collected for each individual. Moreover, the data dimensionality scales rapidly with the total number of tissues and genes. To address these problems, we represent the data in a hypergraph and develop a parameter-efficient neural network that operates on this hypergraph. Throughout, we make use of the concept of metagenes^14,15. Each metagene characterizes certain gene expression patterns and is defined as a linear combination of multiple genes^14,15.

Hypergraph representation

We represent the data in a hypergraph consisting of three types of node: donor, tissue and metagene nodes.

Mathematically, we define a hypergraph ${{{\mathcal{G}}}}=\{{{{{\mathcal{V}}}}}_{\mathrm{d}}\cup {{{{\mathcal{V}}}}}_{\mathrm{m}}\cup {{{{\mathcal{V}}}}}_{\mathrm{t}},{{{\mathcal{E}}}}\}$, where ${{{{\mathcal{V}}}}}_{\mathrm{d}}$ is a set of donor nodes, ${{{{\mathcal{V}}}}}_{\mathrm{m}}$ is a set of metagene nodes, ${{{{\mathcal{V}}}}}_{\mathrm{t}}$ is a set of tissue nodes and ${{{\mathcal{E}}}}$ is a set of multi-attributed hyperedges. Each hyperedge connects an individual i with a metagene j and a tissue k if $k\in {{{\mathcal{T}}}}(i)$, where ${{{\mathcal{T}}}}(i)$ are the collected tissues of individual i. The set of all hyperedges is defined as ${{{\mathcal{E}}}}=\{(i,j,k,{{{{\mathbf{e}}}}}_{ij}^{(k)})| (i,j,k)\in {{{{\mathcal{V}}}}}_{\mathrm{d}}\times {{{{\mathcal{V}}}}}_{\mathrm{m}}\times {{{{\mathcal{V}}}}}_{\mathrm{t}},k\in {{{\mathcal{T}}}}(i)\}$, where ${{{{\mathbf{e}}}}}_{ij}^{(k)}$ are hyperedge attributes that describe characteristics of the interacting nodes, that is features of metagene j in tissue k for individual i.

The hypergraph allows represention of data in a flexible way, generalizing the bipartite graph representation from ref. ⁴⁸. On the one hand, using a single metagene results in a bipartite graph where each edge connects an individual i with a tissue k. In this case, the edge attributes ${{{{\mathbf{e}}}}}_{i1}^{(k)}$ are derived from the gene expression ${{{{\mathbf{x}}}}}_{i}^{(k)}$ of individual i in tissue k. On the other hand, using multiple metagenes leads to a hypergraph where each individual i is connected to tissue k through multiple hyperedges. For example, it is possible to construct a hypergraph where genes and metagenes are related by a one-to-one correspondence, with hyperedge attributes ${{{{\mathbf{e}}}}}_{ij}^{(k)}$ derived directly from expression ${x}_{ij}^{(k)}$. The number of metagenes thus controls a spectrum of hypergraph representations and, as we shall see, can help alleviate the inherent oversquashing problem of graph neural networks.

Message-passing neural network

Given the hypergraph representation of the multi-tissue transcriptomics dataset, we now present a parameter-efficient graph neural network to learn donor, metagene and tissue embeddings, and infer the expression values of the unmeasured tissues. We start by computing hyperedge attributes from the multi-tissue expression data. Then, we initialize the embeddings of all nodes in the hypergraph, construct the message-passing neural network and define an inference model that builds on the latent node representations obtained via message passing.

Computing hyperedge attributes

We first reduce the dimensionality of the measured transcriptomics values. For every individual i and measured tissue k, we project the corresponding gene expression values ${{{{\mathbf{x}}}}}_{i}^{(k)}$ into low-dimensional metagene representations ${{{{\mathbf{e}}}}}_{ij}^{(k)}$:

$${{{{\mathbf{e}}}}}_{ij}^{(k)}=\,{{\mbox{ReLU}}}\left({{{{\mathbf{W}}}}}_{j}{{{{\mathbf{x}}}}}_{i}^{(k)}\right)\quad \forall j\in 1,\ldots ,M$$

(1)

where M, the number of metagenes, is a user-definable hyperparameter and W_j ∀j ∈ 1, …, M are learnable parameters. In addition to characterizing groups of functionally similar genes, employing metagenes reduces the number of messages being aggregated for each node, addressing the oversquashing problem of graph neural networks (Supplementary Section B).

Initial node embeddings

We initialize the node features of the individual ${{{{\mathcal{V}}}}}_{\mathrm{p}}$, metagene ${{{{\mathcal{V}}}}}_{\mathrm{m}}$ and tissue ${{{{\mathcal{V}}}}}_{\mathrm{t}}$ partitions with learnable parameters and available information. For metagene and tissue nodes, we use learnable embeddings as initial node values. The idea is that these weights, which will be approximated through gradient descent, should summarize relevant properties of each metagene and tissue. We initialize the node features of each individual with the available demographic information u_i of each individual i (we use age and sex). We encode sex as a binary value and age as a float normalized by 100 (for example, age 30 is encoded as 0.30). Importantly, this formulation allows transfer learning between sets of distinct donors.

Message-passing layer

We develop a custom graph neural network layer to compute latent donor embeddings by passing messages along the hypergraph. At each layer of the graph neural network, we perform message passing to iteratively refine the individual node embeddings. We do not update the tissue and metagene embeddings during message passing, in a similar vein to knowledge graph embeddings⁴⁹, because their node embeddings already consist of learnable weights that are updated through gradient descent. Sending messages to these nodes would also introduce a dependence between individual nodes and tissue and metagene features (and, by transitivity, dependences between individuals). However, if we foresee that unseen entities will be present in testing (for example, new tissue types), our approach can be extended by initializing their node features with constant values and introducing node-type-specific message-passing equations.

Mathematically, let $\{{{{{\mathbf{h}}}}}_{1}^{\mathrm{d}},\ldots ,{{{{\mathbf{h}}}}}_{N}^{\mathrm{d}}\},\{{{{{\mathbf{h}}}}}_{1}^{\mathrm{m}},\ldots ,{{{{\mathbf{h}}}}}_{M}^{\mathrm{m}}\}$ and $\{{{{{\mathbf{h}}}}}_{1}^{\mathrm{t}},\ldots ,{{{{\mathbf{h}}}}}_{T}^{\mathrm{t}}\}$ be the donor, metagene and tissue node embeddings, respectively. At each layer of the graph neural network, we compute refined individual embeddings $\{{\hat{{{{\mathbf{h}}}}}}_{1}^{\mathrm{d}},\ldots ,{\hat{{{{\mathbf{h}}}}}}_{N}^{\mathrm{d}}\}$ as follows:

$$\begin{array}{ll}{\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}={\phi }_{\mathrm{h}}\left({{{{\mathbf{h}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{m}}}}}_{i}\right),\quad {{{{\mathbf{m}}}}}_{i}=\mathop{\sum }\limits_{j=1}^{M}\mathop{\sum}\limits_{k\in {{{\mathcal{T}}}}(i)}{\phi }_{\mathrm{a}}\left({{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}},{{{{\mathbf{m}}}}}_{ijk}\right),\\ {{{{\mathbf{m}}}}}_{ijk}={\phi }_{\mathrm{e}}\left({{{{\mathbf{h}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}},{{{{\mathbf{e}}}}}_{ij}^{(k)}\right),\end{array}$$

(2)

where the functions ϕ_e and ϕ_h are edge and node operations that we model as MLPs, and ϕ_a is a function that determines the aggregation behaviour. In its simplest form, choosing ${\phi }_{\mathrm{a}}\left({{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}},{{{{\mathbf{m}}}}}_{ijk}\right)=\frac{1}{M| {{{\mathcal{T}}}}(i)| }{{{{\mathbf{m}}}}}_{ijk}$ results in average aggregation. We analyse the time complexity of the message-passing layer in Supplementary Section A. Optionally, we can stack several message-passing layers to increase the expressivity of the model.

The architecture is flexible and may be extended as follows.

• Incorporation of information about the individual embeddings ${{{{\mathbf{h}}}}}_{i}^{\mathrm{d}}$ into the aggregation mechanism ϕ_a.

• Incorporation of target tissue embeddings ${{{{\mathbf{h}}}}}_{u}^{\mathrm{t}}$, for a given target tissue u, into the aggregation mechanism ϕ_a.

• Update hyperedge attributes ${{{{\mathbf{e}}}}}_{ij}^{(k)}$ at every layer.

Aggregation mechanism

In practice, the proposed hypergraph neural network suffers from a bottleneck. In the aggregation step, the number of messages being aggregated is $M| {{{\mathcal{T}}}}(i)|$ for each individual i. In the worst case, when all genes are used as metagenes (that is, M = G; it is estimated that humans have around G ≈ 25,000 protein-coding genes), this leads to serious oversquashing—large amounts of information are compressed into fixed-length vectors⁵⁰. Fortunately, choosing a small number of metagenes reduces the dimensionality of the original transcriptomics values, which in turn alleviates the oversquashing and scalability problems. We perform an ablation study on the number of metagenes and message-passing architectures in Supplementary Section B. To further attenuate oversquashing, we propose an attention-based aggregation mechanism ϕ_a that weighs metagenes according to their relevance in each tissue:

$$\begin{array}{ll}{\phi }_{\mathrm{a}}\left({{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}},{{{{\mathbf{m}}}}}_{ijk}\right)={\alpha }_{jk}{{{{\mathbf{m}}}}}_{ijk},\quad {\alpha }_{jk}=\frac{\exp \left[e\left({{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}\right)\right]}{{\sum }_{v}\exp \left[e\left({{{{\mathbf{h}}}}}_{v}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}\right)\right]},\\ e\left({{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}\right)={{{{\bf{a}}}}}^{\mathrm{T}}\,{{\mbox{LeakyReLU}}}\left({{{\bf{W}}}}\left[{{{{\mathbf{h}}}}}_{j}^{\mathrm{m}}| | {{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}\right]\right),\end{array}$$

where ∣∣ is the concatenation operation and a and W are learnable parameters. The proposed attention mechanism, which closely follows the neighbour aggregation method of graph attention networks^51,52, computes dynamic weighting coefficients that prioritize messages originating from important metagenes. Optionally, we can leverage multiple heads⁵³ to learn multiple modes of interaction and increase the expressivity of the model.

Hypergraph model

The hypergraph model, which we define as f, computes latent individual embeddings ${\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}$ from incomplete multi-tissue expression profiles as ${\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}=f({\tilde{\mathbf{X}}}_{i},{{{{\mathbf{u}}}}}_{i})$.

Downstream imputation tasks

The resulting donor representations ${\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}$ summarize information about a variable number of tissue types collected for donor i, in addition to demographic information. We leverage these embeddings for two downstream tasks: inference of gene expression in uncollected tissues and prediction of cell-type signatures.

Inference of gene expression in uncollected tissues

Prediction of the transcriptomic measurements ${\hat{{{{\mathbf{x}}}}}}_{i}^{(k)}$ of a tissue k (for example, uncollected) is achieved by first recovering the latent metagene values ${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}$ for all metagenes j ∈ 1, …, M, a hyperedge-level prediction task, and then decoding the gene expression values from the predicted metagene representations ${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}$ with an appropriate probabilistic model.

Prediction of hyperedge attributes

To predict the latent metagene attributes ${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}$ for all j ∈ 1, …, M, we employ an MLP that operates on the factorized metagene ${{{{\mathbf{h}}}}}_{j}^{\mathrm{m}}$ and tissue representations ${{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}$ as well as the latent variables ${\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}$ of individual i:

$${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}=\,{{\mbox{MLP}}}\left({\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}}\right),$$

where the MLP is shared for all combinations of metagenes, individuals and tissues.

Negative-binomial imputation model

For raw count data, we use a negative-binomial likelihood. To decode the gene expression values for a tissue k of individual i, we define the probabilistic model $p({{{{\mathbf{x}}}}}_{i}^{(k)}| {\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k)$:

$$\begin{array}{ll} p\left({\mathbf{x}}_{i}^{(k)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{\mathbf{u}}_{i},k\right)\right.=\mathop{\prod }\limits_{j}^{G}p\left({x}_{ij}^{(k)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{\mathbf{u}}_{i},j,k\right)\right., \\ p\left({x}_{ij}^{(k)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},j,k \right)\right.=\,{{\mbox{NB}}}\,\left({x}_{ij}^{(k)};{\mu }_{ij}^{(k)},{\theta }_{ij}^{(k)}\right),\end{array}$$

where NB is a negative-binomial distribution. The mean ${\mu }_{ij}^{(k)}$ and dispersion ${\theta }_{ij}^{(k)}$ parameters of this distribution are computed as follows:

$$\begin{array}{ll}{{{{\mathbf{\mu }}}}}_{i}^{(k)}={l}_{i}^{(k)}{{{{\mathbf{s}}}}}_{i}^{(k)},\quad {{{{\mathbf{s}}}}}_{i}^{(k)}={{{\rm{softmax}}}}\left({{{{\mathbf{W}}}}}_{s}{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}+{{{{\mathbf{b}}}}}_{s}\right),\\ {{{{\mathbf{\theta }}}}}_{i}^{(k)}=\exp \left({{{{\mathbf{W}}}}}_{\theta }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}+{{{{\mathbf{b}}}}}_{\theta }\right),\quad {\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}=\,{{\mbox{MLP}}}\,\left({\bigg\Vert }_{j = 1}^{M}{\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}\right),\end{array}$$

where ${{{{\bf{s}}}}}_{i}^{(k)}$ are mean gene-wise proportions, W_s, W_θ, b_s and b_θ are learnable parameters and ${l}_{i}^{(k)}$ is the library size, which is modelled with a log-normal distribution

$$\log {l}_{i}^{(k)} \!\sim\! {{\mathcal{N}}}\left({l}_{i}^{(k)};{\nu }_{i}^{(k)},{\omega }_{i}^{(k)}\right),\quad {\nu }_{i}^{(k)}\!=\!{{{{\mathbf{W}}}}}_{\nu }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}\!+\!{{{{\mathbf{b}}}}}_{\nu },\quad {\omega }_{i}^{(k)}\!=\!\exp \left({{{{\mathbf{W}}}}}_{\omega }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}\!+\!{{{{\mathbf{b}}}}}_{\omega }\right),$$

where W_ν, W_ω, b_ν and b_ω are learnable parameters. Optionally, we can use the observed library size.

Gaussian imputation model

For normalized gene expression data (that is, inverse normal transformed data), we use the Gaussian likelihood

$$\begin{array}{ll} p\left({{{{\mathbf{x}}}}}_{i}^{(k)}\left| {\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k\right)\right.=\mathop{\prod }\limits_{j}^{G}p\left({x}_{ij}^{(k)}\left| {\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},j,k\right)\right.,\\ p\left({x}_{ij}^{(k)}\left| {\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},j,k\right)\right.={{{\mathcal{N}}}}\left({x}_{ij}^{(k)};{\mu }_{ij}^{(k)},{{\sigma }^{2}}_{ij}^{(k)}\right),\end{array}$$

where the mean ${\mu }_{ij}^{(k)}$ and s.d. ${\sigma }_{ij}^{(k)}$ are computed as follows:

$$\begin{array}{ll}{{{{\mathbf{\upmu }}}}}_{i}^{(k)}={{{{\mathbf{W}}}}}_{\mu }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}+{{{{\mathbf{b}}}}}_{\mu },\quad {{{{\mathbf{\upsigma }}}}}_{i}^{(k)}=\,{{\mbox{softplus}}}\left({{{{\mathbf{W}}}}}_{\sigma }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}+{{{{\mathbf{b}}}}}_{\sigma }\right),\\ {\hat{{{{\mathbf{e}}}}}}_{i}^{(k)}=\,{{\mbox{MLP}}}\left({\bigg\Vert }_{j = 1}^{M}{\hat{{{{\mathbf{e}}}}}}_{ij}^{(k)}\right),\end{array}$$

W_μ, W_σ, b_μ and b_σ are learnable parameters and softplus(x) = log[1 + exp(x)].

Optimization

We optimize the model to maximize the imputation performance on a dynamic subset of observed tissues, that is, tissues that are masked out in training, similarly to ref. ⁵⁴. For each individual i, we randomly select a subset ${{{\mathcal{C}}}}\subset {{{\mathcal{T}}}}(i)$ of pseudo-observed tissues and treat the remaining tissues ${{{\mathcal{U}}}}={{{\mathcal{T}}}}(i)-{{{\mathcal{C}}}}$ as unobserved (pseudo-missing). We then compute the individual embeddings ${\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}}$ using the gene expression of pseudo-observed tissues ${{{\mathcal{C}}}}$ and minimize the loss:

$${{{\mathcal{L}}}}({\tilde{\mathbf{X}}}_{i},{{{{\mathbf{u}}}}}_{i},{{{\mathcal{C}}}},{{{\mathcal{U}}}})=-\frac{1}{| {{{\mathcal{U}}}}| }\mathop{\sum}\limits_{k\in {{{\mathcal{U}}}}}\log p\left({{{{\mathbf{x}}}}}_{i}^{(k)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k\right),\right.$$

which corresponds to the average negative log likelihood across pseudo-missing tissues. Importantly, the pseudo-mask mechanism generates different sets of pseudo-missing tissues for each individual, effectively enlarging the number of training examples and regularizing our model. We summarize the training algorithm in Supplementary Section D.

Inference of gene expression from uncollected tissues

At test time, we infer the gene expression values ${\hat{{{{\mathbf{x}}}}}}_{i}^{(v)}$ of an uncollected tissue v from a given donor i via the mean, that is ${\hat{{{{\mathbf{x}}}}}}_{i}^{(v)}={{{{\mathbf{\mu }}}}}_{i}^{(v)}$. Alternatively, we can draw random samples from the conditional predictive distribution $p({{{{\mathbf{x}}}}}_{i}^{(k)}| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k)$.

Prediction of cell-type signatures

We next consider the problem of imputing cell-type signatures in a tissue of interest. We define a cell-type signature as the sum of gene expression profiles across cells of a given cell type in a certain tissue. Formally, let ${{{{\mathbf{x}}}}}_{i}^{(k,q)}$ be the gene expression signature of cell type q in a tissue of interest k of individual i. Our goal is to infer ${{{{\mathbf{x}}}}}_{i}^{(k,q)}$ from the multi-tissue gene expression measurements ${\tilde{{{{\mathbf{X}}}}}}_{i}$. To achieve this, we first compute the hyperedge features of a hypergraph consisting of four-node hyperedges and then infer the corresponding signatures with a zero-inflated model.

Prediction of hyperedge attributes

We consider a hypergraph where each hyperedge groups an individual, a tissue, a metagene and a cell-type node. For all metagenes j ∈ 1, …, M, we compute latent hyperedge attributes ${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k,q)}$ for a cell type q in a tissue of interest k of individual i as follows:

$${\hat{{{{\mathbf{e}}}}}}_{ij}^{(k,q)}=\,{{\mbox{MLP}}}\left({\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{h}}}}}_{j}^{\mathrm{m}},{{{{\mathbf{h}}}}}_{k}^{\mathrm{t}},{{{{\mathbf{h}}}}}_{q}^{\mathrm{c}}\right),$$

where ${{{{\mathbf{h}}}}}_{q}^{\mathrm{c}}$ are parameters specific to each unique cell type q and the MLP is shared for all combinations of metagenes, individuals, tissues and cell types.

Zero-inflated model

We employ the following probabilistic model:

$$\begin{array}{ll} p\left({{{{\mathbf{x}}}}}_{i}^{(k,q)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k,q\right)\right.=\mathop{\prod }\limits_{j}^{G}p\left({x}_{ij}^{(k,q)}\left| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},j,k,q\right)\right.,\\ p\left({{{{\mathbf{x}}}}}_{i}^{(k,q)}\left| {\hat{{{{\mathbf{h}}}}}}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},j,k,q\right)\right.=\,{{\mbox{ZINB}}}\left({x}_{ij}^{(k,q)};{\mu }_{ij}^{(k,q)},{\theta }_{ij}^{(k,q)},{\pi }_{ij}^{(k,q)}\right),\end{array}$$

where ZINB is a zero-inflated negative-binomial distribution. The mean ${\mu }_{ij}^{(k,q)}$, dispersion ${\theta }_{ij}^{(k,q)}$ and dropout probability ${\pi }_{ij}^{(k,q)}$ parameters are computed as

$$\begin{array}{ll}{{{{\mathbf{\mu }}}}}_{i}^{(k,q)}={n}_{i}^{(k,q)}{l}_{i}^{(k,q)}\,{{{\rm{softmax}}}}\left({{{{\mathbf{W}}}}}_{s}{\hat{{{{\mathbf{e}}}}}}_{i}^{(k,q)}+{{{{\mathbf{b}}}}}_{s}\right),\\ {{{{\mathbf{\theta }}}}}_{i}^{(k,q)}=\exp \left({{{{\mathbf{W}}}}}_{\theta }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k,q)}+{{{{\mathbf{b}}}}}_{\theta }\right),\quad {{{{\mathbf{\pi }}}}}_{i}^{(k,q)}=\sigma \left({{{{\mathbf{W}}}}}_{\pi }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k,q)}+{{{{\mathbf{b}}}}}_{\pi }\right),\end{array}$$

where W_s, W_θ, W_π, b_s, b_θ and b_π are learnable parameters, ${n}_{i}^{(k,q)}$ is the number of cells in the signature and ${l}_{i}^{(k,q)}$ is their average library size. In training, we set ${n}_{i}^{(k,q)}$ to match the ground-truth number of cells. At test time, the number of cells ${n}_{i}^{(k,q)}$ is user definable. We model ${l}_{i}^{(k,q)}$ with a log-normal distribution

$$\begin{array}{ll} \log {l}_{i}^{(k,q)} \sim {{{\mathcal{N}}}}\left({l}_{i}^{(k,q)};{\nu }_{i}^{(k,q)},{\omega }_{i}^{(k,q)}\right),\quad {\nu }_{i}^{(k,q)}={{{{\mathbf{W}}}}}_{\nu }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k,q)}+{{{{\mathbf{b}}}}}_{\nu },\\ {\omega }_{i}^{(k,q)}=\exp \left({{{{\mathbf{W}}}}}_{\omega }{\hat{{{{\mathbf{e}}}}}}_{i}^{(k,q)}+{{{{\mathbf{b}}}}}_{\omega }\right).\end{array}$$

Optionally, we can use the observed library size.

Optimization

Single-cell transcriptomic studies typically measure single-cell gene expression for a limited number of individuals, tissues and cell types, so aggregating single-cell profiles per individual, tissue and cell type often results in small sample sizes. To address this challenge, we apply transfer learning by pretraining f on the multi-tissue imputation task and then fine-tuning the parameters of the signature inference module on the cell-type signature profiles. Concretely, we minimize the loss:

$${{{\mathcal{L}}}}\left({{{{\mathbf{x}}}}}_{i}^{(k,q)},{\tilde{\mathbf{X}}}_{i},{{{{\mathbf{u}}}}}_{i},k,q\right)=-\log p\left({{{{\mathbf{x}}}}}_{i}^{(k,q)}| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k,q\right),$$

which corresponds to the negative log likelihood of the observed cell-type signatures.

Inference of uncollected gene expression

To infer the signature of a cell type q in a certain tissue v of interest, we first compute the latent individual embeddings ${\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}}$ from the multi-tissue profiles ${\tilde{\mathbf{X}}}_{i}$ and then compute the mean of the distribution $p({{{{\mathbf{x}}}}}_{i}^{(k,q)}| {\hat{\mathbf{h}}}{\,}_{i}^{\mathrm{d}},{{{{\mathbf{u}}}}}_{i},k,q)$ as ${{{{\mathbf{\mu }}}}}_{i}^{(k,q)}(1-{{{{\mathbf{\uppi }}}}}_{i}^{(k,q)})$. Alternatively, we can draw random samples from that distribution.

eQTL mapping

The breadth of tissues in the GTEx-v8 collection enabled us to comprehensively evaluate the extent to which eQTL discovery could be improved through the HYFA-imputed transcriptome data. We mapped eQTLs that act in cis to the target gene (cis-eQTLs), using all single nucleotide polymorphisms within ±1 megabase pairs of the transcription start site of each gene. For the imputed and the original (incomplete) datasets, we considered single nucleotide polymorphisms significantly associated with gene expression, at FDR ≤ 0.10. We applied the same GTEx eQTL mapping pipeline, as previously described⁵⁵, to the imputed and original datasets to quantify the gain in eQTL discovery from the HYFA-imputed dataset.

Pathway enrichment analysis

Similarly to ref. ³⁷, we employed GSEA³⁶ to relate HYFA’s metagene factors to known biological pathways. This is advantageous to over-representation analysis, which requires selecting an arbitrary cutoff to select enriched genes. GSEA, instead, computes a running sum of enrichment scores by descending a sorted gene list^36,37.

We applied GSEA to the gene loadings in HYFA’s encoder. Specifically, let ${{{{\mathbf{W}}}}}_{j}\in {{\mathbb{R}}}^{F\times G}$ be the gene loadings for metagene j, where F is the number of factors (that is number of hyperedge attributes) and G is the number of genes (equation (1)). For every factor in W_j, we employed blitzGSEA⁵⁶ to calculate the running sum of enrichment scores by descending the gene list sorted by the factor’s gene loadings. The enrichment score for a query gene set is the maximum difference between ${p}_{\mathrm{hit}}({{{\mathcal{S}}}},i)$ and ${p}_{\mathrm{miss}}({{{\mathcal{S}}}},i)$ (ref. ³⁷), where ${p}_{\mathrm{hit}}({{{\mathcal{S}}}},i)$ is the proportion of genes in ${{{\mathcal{S}}}}$ weighted by their gene loadings up to gene index i in the sorted list³⁷. We then calculated pathway enrichment P values through a permutation test (with n = 100 trials) by randomly shuffling the gene list. We employed the Benjamini–Hochberg method to correct for multiple testing.

GTEx bulk and single-nucleus RNA-seq data processing

The GTEx dataset is a public resource that has generated a broad collection of gene expression data collected from a diverse set of human tissues². We downloaded the data from the GTEx portal (Data availability). After the processing step, the GTEx-v8 dataset consisted of 15,197 samples (49 tissues, 834 donors) and 12,557 genes. The dataset was randomly split into 500 training, 167 validation and 167 testing donors. Each donor had an average of 18.22 collected tissues. The processing steps are described below.

Normalized bulk transcriptomics (GTEx-v8)

Following the GTEx eQTL discovery pipeline (https://github.com/broadinstitute/gtex-pipeline/tree/master/qtl), we processed the data as follows.

• 1. Discard under-represented tissues (n = 5), namely bladder, cervix (ectocervix, endocervix), fallopian tube and kidney (medulla).

• 2. Select set of overlapping protein-coding genes across all tissues.

• 3. Discard donors with only one collected tissue (n = 4).

• 4. Select genes on the basis of expression thresholds of ≥0.1 transcripts per kilobase million in ≥20% of samples and ≥6 reads (unnormalized) in ≥20% of samples.

• 5. Normalize read counts across samples using the trimmed mean of M values method⁵⁷.

• 6. Apply inverse normal transformation to the expression values for each gene.

Cell-type signatures from a paired snRNA-seq dataset (GTEx-v9)

We downloaded paired snRNA-seq data for 16 GTEx individuals¹³ (Data availability) collected in eight GTEx tissues, namely skeletal muscle, breast, oesophagus (mucosa, muscularis), heart, lung, prostate and skin. We split these individuals into training, validation and testing donors according to the GTEx-v8 split. We processed the data as follows.

• 1. Select set of overlapping genes between bulk RNA-seq (GTEx-v9) and paired snRNA-seq dataset¹³.

• 2. Select top 3,000 variable genes using the Scanpy function scanpy.pp.highly_variable_genes with flavour setting seurat_v3 (refs. ^58,59).

• 3. Discard under-represented cell types occurring in fewer than 10 tissue–individual combinations.

• 4. Aggregate (that is sum) read counts by individual, tissue and (broad) cell type. This resulted in a dataset of 226 unique signatures, of which 135 belong to matching GTEx-v8 individuals.

Implementation and reproducibility

We report the selected hyperparameters in Supplementary Section B. HYFA is implemented in Python⁶⁰. Our framework and implementation are flexible (that is, we support k-uniform hypergraphs), may be integrated in other bioinformatics pipelines and may be useful for other applications in different domains. We used PyTorch⁶¹ to implement the model and Scanpy⁵⁸ to process the gene expression data. We performed hyperparameter optimization with wandb⁶². We employed blitzGSEA⁵⁶ for pathway enrichment analysis. We also used NumPy⁶³, scikit-learn⁶⁴, pandas⁶⁵, matplotlib⁶⁶, seaborn⁶⁷ and statannotations⁶⁸. Figure 1 was created with BioRender.com.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.