Fig. 1

Comparison of the simplicial complex model to other approaches for interpreting tumor genomic data. A simple hypothetical evolutionary tree model (left) describing possible progression pathways of a tumor from an initially healthy cell (1), to an early progression state (2), which then diverges into two subtypes distinguished by two possible late progression states, (3) or (4). If each tumor represents evolution along one of the two subtypes then we would expect plotting many tumors in a low-dimensional representation of gene expression space to yield a point cloud describing mixtures of cell types (1,2,3) or (1,2,4), resulting in a geometric structure consisting of two triangular point clouds joined at an edge (middle). Conventional clustering analysis such as k-medioids with the appropriate distance metric separates this structure into two clusters (top right) that conflate the shared cell states (1,2) and thus provide poor representations of differences in underlying cell populations (3) and (4). Prior approaches to unmixing, such as Tolliver et al., [23] with the correct parameters and under sufficient noise constraints, reconstruct four populations interpreting each tumor as a mixture of all four, introducing high error because the resulting tetrahedral simplex (center right) is poorly populated with data points. The proposed simplicial complex approach here explicitly searches for an overall point cloud that is composed of lower-dimensional subsimplices (bottom right), potentially providing better resolution of mixture compositions in individual tumors and direct ability to infer aspects of the evolutionary process from the geometric structure