Nested tree space: a geometric framework for co-phylogeny
TLDR
Introduces σ-space, a CAT(0) geometric framework for nested phylogenetic trees, enabling analysis of co-evolutionary systems.
Key contributions
- Introduces σ-space, a novel geometric framework for fully nested ultrametric phylogenetic trees.
- Constructs σ-space as a cubical complex, generalizing the existing τ-space for co-phylogeny.
- Proves σ-space is CAT(0), ensuring unique geodesics and well-defined Fréchet means.
- Characterizes admissible tree orderings via binary nesting sequences and details its geometric structure.
Why it matters
This framework provides a robust mathematical tool for studying co-evolutionary systems. The CAT(0) property is significant for statistical analysis and comparison of co-phylogenies, offering new avenues for research in evolutionary biology.
Original Abstract
Nested (or reconciled) phylogenetic trees model co-evolutionary systems in which one evolutionary history is embedded within another. We introduce a geometric framework for such systems by defining $σ$-space, a moduli space of fully nested ultrametric phylogenetic trees with a fixed leaf map. Generalizing the $τ$-space of Gavryushkin and Drummond, $σ$-space is constructed as a cubical complex parametrised by nested ranked tree topologies and inter-event time coordinates of the combined host and parasite speciation events. We characterise admissible orderings via binary \textit{nesting sequences} and organise them into a natural poset. We show that $σ$-space is contractible and satisfies Gromov's cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fréchet means. We further describe its geometric structure, including boundary strata corresponding to cospeciation events, and relate it to products of ultrametric tree spaces via natural forgetful maps.
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