Group symmetries in the lineage trees of single cells

Fate maps on lineage trees have been a defining conceptual framework for understanding development in multicellular organisms. Yet the heterogeneity that is increasingly observed in single-cell lineage data across a variety of living systems has been difficult to reconcile with traditional notions of fate determination and differentiation. Examination of the group symmetries of a binary tree has revealed a set of orthogonal components for describing variation in a lineage. These provide a natural way to aggregate tree-structured data and suggest a statistical definition of fate determination and differentiation that allows for noisy variation from each division and at each generation. This new type of harmonic analysis has been applied to data on CD8+ T cell lineages. As benchmarks, the analysis has been applied to previously-published data on C. Elegans, a lineage with clear determination stages, and to a simulated branching process, which has none. The results demonstrate how group representation theory can be used to improve inference in, and extract scientific meaning from, complex structured data.