Geometry-preserving vector field reconstruction of high-dimensional cell-state dynamics using ddHodge.

The differentiation potency of cells is governed by dynamic changes in gene expression, which can be inferred from single-cell RNA sequencing (scRNA-seq) data. While velocity-based approaches have been used to analyze cell state changes as vector fields, extracting acceleration (change of change) information remains challenging because of the sparsity and high-dimensionality of the data. Here, we develop ddHodge, a framework based on Hodge decomposition for precise vector-field reconstruction. ddHodge accurately recovers all basic components of the vector field, namely, the gradient, curl, and divergence, including the acceleration of the cell state, as second-order derivatives, even from biased and sparse samples. Furthermore, we extend the method to approximate high-dimensional gene expression dynamics on lower-dimensional data manifolds. By applying ddHodge to scRNA-seq data from mouse embryogenesis, we reveal that the gene expression dynamics during development follow a gradient system shaped by potential landscapes, which has not previously been validated with real data. Furthermore, we quantify differentiation potency as cell state stability on the basis of the divergence and identify key genes that drive potency. Our general computational framework for analyzing complex biological systems can elucidate cell fate decisions in developmental processes.