Fig. 1

Defining and validating short-range and long-range interactions. a The illustration of short-range and long-range interactions. b The workflow of defining short-range and long-range interactions. First, generate spatial ligand and receptor gene distributions from ST data and calculate the Wasserstein distance between them. Next, perform permutation test on the Wasserstein distance to get the interaction’s spatial tendency and its confidence for filtering short-range and long-range interactions. d_real: the actual Wasserstein distance between ligand and receptor gene distributions; d_simulation: the Wasserstein distance between two permuted ligand and receptor gene distributions; d_ratio: the ratio of actual Wasserstein distance and average permuted Wasserstein distance, indicating the spatial tendency of interaction; one-sided P-value: indicate the confidence of interaction’s spatial tendency. c The actual numbers of short-range, medium-range, and long-range interactions in each sample. Color for interaction tendency type, green: short-range; yellow: medium; red: long-range. PDAC: pancreatic ductal adenocarcinoma; SCC: squamous cell carcinoma. d The interaction type proportion in short-range and long-range interactions. Color for interaction type, blue: secreted signaling type interaction; orange: cell-cell contact type interaction. e The GO analysis results of short-range (left) and long-range (right) interactions’ ligands in the sample P2_rep2 in the SCC dataset