Fig. 1: The main framework of HALO.

A Top: The causal diagram of individual gene expression and its corresponding peaks, in both coupled (left) and decoupled (right) cases. Bottom: the causal diagram of relations between scRNA-seq and scATAC-seq data on the representation level, for decoupled (left) and coupled (right) cases. θ are functional parameters as a function of time for RNA (R) or ATAC (A) data, as well as coupled (c) or decoupled (d). B Architecture for representation learning within a causal regularized variational autoencoder (VAE) framework. From the jointly profiled scRNA-seq (blue) and scATAC-seq (red) data, latent representations are learned and can then be used via interpretable decoder to determine important genes and peaks. For the ATAC modality, the latent Z^A is divided into \(\left[{{{\bf{Z}}}}_{d}^{A}{{\mathbf{,\,}}}{{{\bf{Z}}}}_{c}^{A}\right]\), representing the decoupled and coupled latent representations, respectively. Similarly, the RNA modality’s latent representations comprise \({{{\bf{Z}}}}_{d}^{R}\) (decoupled) and \({{{\bf{Z}}}}_{c}^{R}\) (coupled). The decoupled representations \({{{\bf{Z}}}}_{d}^{A}\) and \({{{\bf{Z}}}}_{d}^{R}\) adhere to the decoupled causal constraints Δ_decouple, whereas the coupled representations, \({{{\bf{Z}}}}_{c}^{A},\,{{{\bf{Z}}}}_{c}^{R}\) conform to the coupled causal constraints Δ_couple. C Illustration of the gene-peak level analysis process. Initially, genes and ATAC peaks within specified proximities are matched using non-negative binomial regression, linking gene expression to neighboring ATAC peaks. Subsequently, we compute decouple and couple scores to categorize gene-peaks as either decoupled or coupled. Finally, we employ the Granger causality test to identify distal regulatory relationships between peaks and genes, uncovering potential mechanisms of genetic regulation. Created with elements from BioRender, Jia, M. (2025) https://BioRender.com/jcqqg5n.