An Information Geometry approach to model topological trajectories and Gene Expression Radius from UMAP geometry.

Comprehending the correlation between gene expression dynamics and cellular identity continues to be a paramount concern in the realm of single cell biology. In this study, we present a groundbreaking computational and mathematical methodology that amalgamates information geometry, fuzzy topology, and UMAP analysis to model gene expression landscapes derived from single cell RNA sequencing data.

We conceptualize gene expression data as a fuzzy topological space, wherein interactions between expression points are dictated by probabilistic distributions inspired by manifold learning techniques like UMAP. Within this paradigm, we establish an information geometric structure through a Fisher metric derived from these distributions, facilitating the calculation of geodesic trajectories that encapsulate cellular differentiation processes.

A principal contribution of our research is the derivation of analytical conditions, articulated as expression radius formulas, which characterize local neighborhoods in gene expression space. These conditions facilitate the identification of genes linked to stem cell states and predictions pertaining to transitional cell types in subsequent investigations.

Upon applying our proposed framework to single cell datasets, we uncover biologically significant gene sets enriched in crucial regulatory pathways and transcription factors, substantiating the potential of our approach to unveil hidden structure in intricate gene expression data. Our findings imply that the integration of differential geometry with statistical learning theory presents a potent framework for modeling genotype and phenotype relationships and cellular state transitions, with prospective ramifications for precision medicine and systems biology.