SpaOTsc model

SpaOTsc constructs a mapping between the n cells in scRNA-seq data and the m positions in spatial data by solving an optimal transport problem²⁰ given three dissimilarity/distance matrices, M∈Rn×m for the gene expression dissimilarity between cells and locations, Dsc∈Rn×n for the gene expression dissimilarity among cells, and Dspa∈Rm×m for the distances among spatial locations. The optimal transport plan γ* is obtained by solving

where ω₁,ω₂ are weight vectors and L measures the difference between scaled dissimilarities/distances. The first term quantifies the major transport cost, the second penalty term promotes weight conservation (unbalanced transport)²¹, and the last term preserves the distance within datasets through the mapping (structured transport)²². The spatial cell–cell distance D^sc is then computed based on γ* using the optimal transport distance:

One can carry out three major tasks immediately after obtaining γ* and D^sc: (1) prediction of spatial gene expression at the ith position by ∑jγj,i*gj/∑jγj,i* where g∈Rn is the expression vector for a gene in scRNA-seq data; (2) identification of spatially localized cell subclusters by distance-based clustering using D^sc within each previously identified cluster; and (3) visualization of scRNA-seq data constrained by cell–cell distances using the distance matrix D^sc.

The intercellular gene–gene regulatory information flow is inferred by using partial information decomposition^28,29,34. We estimate how much unique information about a gene (target gene) can be provided by another gene (source gene) in its spatial neighborhood through the calculation of the accumulated unique information:

where G_tar is the variable for target gene expression in the cells, G~src is the variable for source gene expression in η-neighborhoods of cells whose observation is estimated using D^sc, and G is a collection of genes with high intracellular correlation with the target gene. The unique information Unq_X(Z;Y) measures how much unique information Y provides about Z in addition to X.

For the case of intercellular signaling with known ligands, receptors, and their downstream genes, we use random forest models^58,59 to infer the spatial distance of signaling. The ligand expressions of cells in a neighborhood of distance of η, denoted as L~η, together with other genes highly correlated to a downstream target gene of the ligand–receptor interaction are used as features to fit a random forest model outputting the target gene. The receptor expressions are used as sample weights. The η under which L~η has the highest feature importance is considered to be the spatial distance of this signaling.

Knowing the ligands, receptors and downstream genes involved in intercellular signaling and D^sc, we then infer cell–cell communication by solving another optimal transport problem. First, the source distribution over the cells ω_L is constructed to be proportional to the expression of ligand gene. Next a destination distribution ω_D is constructed based on the expression of receptors and downstream genes to represent the probability of a cell to receive the signal. A cell highly expressing receptors with downstream genes consistent with the up-/down-regulation relationships (low expression of down-regulated genes and high expression of up-regulated genes) is assigned with a high probability. With this information we solve the following optimal transport problem

The optimal transport plan γS* is interpreted as likelihood of cell–cell communications, e.g. its ijth element describes how likely cell j receives signal from cell i. When spatial distances for signaling are available, we can simply adjust the cost matrix D^sc by setting entries greater than this distance to a large number to enforce a spatial constraint on communications identification. When a spatial constraint is applied, long-distance connections will be eliminated and new short connections may emerge (Supplementary Figs. ²¹, ²²).