Spatially Varying Gene Regulatory Networks via Bayesian Nonparametric Covariate-Dependent Directed Cyclic Graphical Models

Spatial transcriptomics technologies enable the measurement of gene expression with spatial context, providing opportunities to understand how gene regulatory networks vary across tissue regions. However, existing graphical models focus primarily on undirected graphs or directed acyclic graphs, limiting their ability to capture feedback loops that are prevalent in gene regulation. Moreover, ensuring the so-called stability condition of cyclic graphs, while allowing graph structures to vary continuously with spatial covariates, presents significant statistical and computational challenges. We propose BNP-DCGx, a Bayesian nonparametric approach for learning spatially varying gene regulatory networks via covariate-dependent directed cyclic graphical models. Our method introduces a covariate-dependent random partition as an intermediary layer in a hierarchical model, which discretizes the covariate space into clusters with cluster-specific stable directed cyclic graphs. Through partition averaging, we obtain smoothly varying graph structures over space while maintaining theoretical guarantees of stability. We develop an efficient parallel tempered Markov chain Monte Carlo algorithm for posterior inference and demonstrate through simulations that our method accurately recovers both piecewise constant and continuously varying graph structures. Application to spatial transcriptomics data from human dorsolateral prefrontal cortex reveals spatially varying regulatory networks with feedback loops, identifies potential cell subtypes within established cell types based on distinct regulatory mechanisms, and provides new insights into spatial organization of gene regulation in brain tissue.

💡 Research Summary

This paper introduces a novel Bayesian nonparametric statistical model named BNP-DCGx (Bayesian Nonparametric Covariate-Dependent Directed Cyclic Graphical Model) for inferring spatially varying gene regulatory networks (svGRNs) from spatial transcriptomics data. The core challenge addressed is learning directed cyclic graphs (which can capture essential biological feedback loops) whose structures vary smoothly with spatial covariates (e.g., tissue coordinates), all while ensuring the graphs satisfy a “stability condition” – a constraint on the eigenvalues of the graph’s adjacency matrix required for causal interpretability.

The authors propose an innovative hierarchical modeling framework that cleverly circumvents the direct, intractable problem of defining a function B(X) that is both flexible and always stable. Instead, they introduce an intermediate layer: a covariate-dependent random partition of the data points (cells) based on their spatial locations. This partition discretizes the continuous covariate space into clusters. Within each cluster ℓ, a cluster-specific, stable directed cyclic graph (DCG) parameterized by a coefficient matrix B_ℓ is learned. The stability condition is enforced directly and manageably on each finite set of B_ℓ matrices. To recover a smoothly varying graph structure over space, the model employs “partition averaging,” marginalizing over the random partitions. Theoretically, this guarantees that the implied graph B(X) at any spatial location X is stable.

Methodologically, the model builds upon a Structural Equation Model (SEM). To ensure model identifiability—avoiding the issue where different DCGs produce the same observational distribution—the error terms are modeled using Laplace distributions, leveraging theory from Independent Component Analysis (ICA). The authors develop an efficient parallel tempered Markov Chain Monte Carlo (MCMC) algorithm for posterior inference, which helps avoid local optima.

Simulation studies demonstrate that BNP-DCGx accurately recovers both piecewise constant graph structures (where distinct spatial regions have different networks) and continuously varying graph structures. It outperforms existing methods designed for undirected or directed acyclic graphs.

The method is applied to a real spatial transcriptomics dataset from the human dorsolateral prefrontal cortex (DLPFC). The analysis reveals that gene regulatory networks, particularly those involving feedback loops, exhibit significant spatial variation across different cortical layers, even within the same broad cell type (e.g., neurons). Furthermore, the results suggest the existence of potential neuronal subtypes characterized by distinct regulatory mechanisms, offering new biological insights into the spatial organization of gene regulation in the brain. This work represents a significant advance in the analysis of spatial transcriptomics, providing a powerful tool for uncovering context-dependent gene regulatory dynamics.