PHLOWER leverages single-cell multimodal data to infer complex, multi-branching cell differentiation trajectories – Nature Methods

TITLE: PHLOWER: Unraveling Cellular Development Paths with Topological Data Analysis

Decoding Cell Differentiation with Advanced Mathematical Frameworks

In the rapidly evolving field of single-cell biology, researchers face the complex challenge of mapping how cells transition from progenitor states to specialized types. PHLOWER (Potential of Hodge-Laplacian Organization for Watershed-based Embedding and Reconstruction) represents a groundbreaking computational approach that leverages sophisticated mathematical principles from topology and geometry to infer multi-branching cell differentiation trajectories. This method moves beyond traditional trajectory inference tools by incorporating Hodge Laplacian theory to provide unprecedented insights into cellular development processes.

The Mathematical Foundation: Hodge Laplacian and Simplicial Complexes

At its core, PHLOWER utilizes discrete Hodge Laplacian (HL) operators applied to simplicial complexes—mathematical structures that represent relationships between cells in various dimensions. The system represents single-cell data as a simplicial complex consisting of:

• Nodes (0-simplices): Individual cells within the dataset
• Edges (1-simplices): Potential differentiation events between cells
• Triangles (2-simplices): Higher-order relationships connecting multiple cells

The first-order HL describes how edges relate to each other through both nodes (lower-adjacency) and triangular faces (upper-adjacency). This framework enables the decomposition of cellular flows into gradient-free, curl-free, and harmonic components, similar to the classical Helmholtz decomposition used in vector calculus. The harmonic components are particularly significant as they capture the essential topological features—specifically “holes” in the data structure—that correspond to distinct differentiation trajectories.

From Raw Data to Biological Insights: The PHLOWER Pipeline

The PHLOWER analytical workflow begins with processing single-cell data, which can include multimodal sequencing (such as simultaneous ATAC and RNA sequencing) or traditional unimodal scRNA-seq data. The process involves several sophisticated computational stages:, as earlier coverage

Graph Construction and Pseudotime Estimation
PHLOWER first constructs a graph representation of the single-cell data by estimating a joint embedding using methods like MOJITOO for multimodal integration or principal component analysis for unimodal data. The system then employs diffusion maps and Gaussian kernel estimation to represent cellular relationships, followed by pseudotime estimation—a computational method that orders cells along developmental trajectories based on their transcriptional similarity.

Simplicial Complex Formation
Using the graph structure and pseudotime information, PHLOWER applies Delaunay triangulation to create a network of triangles between cells. The system identifies terminal differentiated cells and root progenitor cells based on pseudotime values, then strategically adds artificial edges between these endpoints. This crucial step creates topological “holes” in the simplicial complex that correspond to actual differentiation pathways in the biological system.

Harmonic Analysis and Trajectory Embedding
PHLOWER computes the harmonic eigenvectors of the normalized first-order HL associated with the simplicial complex. Eigenvectors with zero eigenvalues—the harmonic eigenvectors—precisely delineate the holes in the complex that represent cell differentiation trajectories. By sampling trajectories as edge flows and computing their dot products with harmonic eigenvectors, PHLOWER creates a trajectory embedding space where clustering analysis reveals major trajectory groups.

Advanced Applications and Future Directions

While PHLOWER currently focuses on the harmonic components of the Hodge decomposition, future developments aim to incorporate additional components such as curl and gradient terms. This expansion would enable the analysis of more complex cell differentiation structures, including acyclic graphs. The methodology also shows promise for application to three-dimensional molecular data, such as protein-ligand prediction or three-dimensional spatial transcriptomics, where higher-order structures like tetrahedrons could reveal additional geometric features.

Biological Validation and Regulatory Insights
PHLOWER outputs comprehensive differentiation trees, cell branch associations, and refined pseudotime estimates. The system also identifies branch-specific regulatory factors by detecting transcription factors with specific expression patterns along differentiation paths. This capability provides researchers with not only the roadmap of cellular development but also the key molecular drivers controlling these processes.

Technical Innovations and Computational Advantages

PHLOWER represents a significant advancement over previous methods that used HL primarily for visualization purposes. Unlike earlier approaches, PHLOWER can infer complete differentiation trees and allocate cells along these trees with precision. The method’s ability to transform complex single-cell data into mathematically robust trajectory models makes it particularly valuable for studying developmental biology, cancer progression, and cellular response to therapeutic interventions.

The integration of topological data analysis with single-cell biology opens new avenues for understanding cellular decision-making processes at unprecedented resolution. As single-cell technologies continue to advance, tools like PHLOWER will become increasingly essential for extracting meaningful biological insights from the complex, high-dimensional data generated by modern sequencing platforms.

References

• https://github.com/dynverse/dynbenchmark/
• https://cdn.10xgenomics.com/image/upload/v1666737555/support-documents/CG0003…

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