Interpretable spherical geometry of single-cell state transitions from dominant principal components

Abstract

Single-cell RNA-seq atlases are commonly explored with nonlinear embeddings that preserve neighborhoods but provide limited coordinate-level interpretation. We asked whether projecting the dominant principal components (PCs) of single-cell gene expression onto a unit sphere would yield an interpretable coordinate system. SPHERE-PCA L2-normalizes the first three PC coordinates, aligns a biologically defined root to the north pole, and represents each cell by three coordinates: root-aligned geodesic distance (θ), angular position (ϕ), and pre-projection radial magnitude (r). Across developmental and disease-associated datasets, this representation reveals structured spherical geometry, ranging from near-great-circle trajectories to multi-arc manifolds. In developmental atlases, root-aligned geodesic distance increases as CytoTRACE-inferred stemness decreases, while gene-coordinate analyses separate programs associated with angular position from those associated with radial magnitude. Fixed-loading perturbations decompose each gene’s effect on cell position into progression, branch- or state-position, and radial activity components. SPHERE-PCA therefore provides a deterministic, loading-preserving coordinate framework for interpreting dominant transcriptomic variance and establishing a transparent geometric coordinate framework for perturbation analysis and virtual-cell models. Significance Statement Single-cell atlases are often interpreted through nonlinear maps whose axes are visually powerful but difficult to explain. This study shows that projecting the first three principal components onto a unit sphere yields an interpretable coordinate system for single-cell data. Across developmental atlases, the resulting coordinates expose distance from a biological root, angular branch or state position, and pre-projection radial magnitude, which capture progression-associated, state-associated, and activity-associated variation. Because the method preserves principal component loadings, gene-associated displacements can be decomposed along the same coordinates. SPHERE-PCA provides a transparent geometric layer on classical dimension reduction and a coordinate scaffold for future models of cell-state change such as virtual-cell.