BACKGROUND: Time-series scRNA-seq data have opened a door to elucidate cell differentiation, and in this context, the optimal transport theory has been attracting much attention. However, there remain critical issues in interpretability and computational cost. RESULTS: We present scEGOT, a comprehensive framework for single-cell trajectory inference, as a generative model with high interpretability and low computational cost. Applied to the human primordial germ cell-like cell (PGCLC) induction system, scEGOT identified the PGCLC progenitor population and bifurcation time of segregation. Our analysis shows TFAP2A is insufficient for identifying PGCLC progenitors, requiring NKX1-2. Additionally, MESP1 and GATA6 are also crucial for PGCLC/somatic cell segregation. CONCLUSIONS: These findings shed light on the mechanism that segregates PGCLC from somatic lineages. Notably, not limited to scRNA-seq, scEGOT's versatility can extend to general single-cell data like scATAC-seq, and hence has the potential to revolutionize our understanding of such datasets and, thereby also, developmental biology.

Node2vec is a graph embedding method that learns a vector representation for each node of a weighted graph while seeking to preserve relative proximity and global structure. Numerical experiments suggest Node2vec struggles to recreate the topology of the input graph. To resolve this we introduce a topological loss term to be added to the training loss of Node2vec which tries to align the persistence diagram (PD) of the resulting embedding as closely as possible to that of the input graph. Following results in computational optimal transport, we carefully adapt entropic regularization to PD metrics, allowing us to measure the discrepancy between PDs in a differentiable way. Our modified loss function can then be minimized through gradient descent to reconstruct both the geometry and the topology of the input graph. We showcase the benefits of this approach using demonstrative synthetic examples.

Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy topological properties. This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA. The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method allows a fast approximation technique. The method is applied into practical data on proteins and oxide glasses, and the results show the advantage of our method compared to other relevant methods on persistence diagrams.