Modeling latent flows on single-cell data using the Hodge decomposition

Single-cell analysis is a powerful technique used to identify a specific cell population of interest during differentiation, aging, or oncogenesis. Individual cells occupy a particular transient state in the cell cycle, circadian rhythm, or during cell death. An appealing concept of pseudo-time trajectory analysis of single-cell RNA sequencing data was proposed in the software Monocle, and several methods of trajectory analysis have since been published to date. These aim to infer the ordering of cells and enable the tracing of gene expression profile trajectories in cell differentiation and reprogramming. However, the methods are restricted in terms of time structure because of the pre-specified structure of trajectories (linear, branched, tree or cyclic) which contrasts with the mixed state of single cells.\n\nHere, we propose a technique to extract underlying flows in single-cell data based on the Hodge decomposition (HD). HD is a theorem of vector fields on a manifold which guarantees that any given flow can decompose into three types of orthogonal component: gradient-flow (acyclic), curl-, and harmonic-flow (cyclic). HD is generalized on a simplicial complex (graph) and the discretized HD has only a weak assumption that the graph is directed. Therefore, in principle, HD can extract flows from any mixture of tree and cyclic time flows of observed cells. The decomposed flows provide intuitive interpretations about complex flow because of their linearity and orthogonality. Thus, each extracted flow can be focused on separately with no need to consider crosstalk.\n\nWe developed ddhodge software, which aims to model the underlying flow structure that implies unobserved time or causal relations in the hodge-podge collection of data points. We demonstrated that the mathematical framework of HD is suitable to reconstruct a sparse graph representation of diffusion process as a candidate model of differentiation while preserving the divergence of the original fully-connected graph. The preserved divergence can be used as an indicator of the source and sink cells in the observed population. A sparse graph representation of the diffusion process transforms data analysis of the non-linear structure embedded in the high-dimensional space of single-cell data into inspection of the visible flow using graph algorithms. Hence, ddhodge is a suitable toolkit to visualize, inspect, and subsequently interpret large data sets including, but not limited to, high-throughput measurements of biological data.\n\nThe beta version of ddhodge R package is available at: https://github.com/kazumits/ddhodge