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

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 ({theta}), angular position ({phi}), 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.