ABSTRACT
A time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics but little is known about the parameter space of such objects. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the NNI graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs, including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by extending the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Although time values greatly increase the number of possible trees, we show that 1-neighborhood sizes remain linear, allowing for efficient local exploration and construction of these graphs. We also obtain upper bounds on the r-neighborhood sizes of these graphs, including a smaller bound than was previously known for NNI.
Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of prominent open problems for those applications. In particular, we conjecture that the split theorem applies to shortest paths in time-tree graphs, a property not shared in the general NNI case.
Footnotes
AG was supported by Marsden Fund of the Royal Society of New Zealand, grant UOA1324, through the Centre for Computational Evolution, the University of Auckland, where this work has been carried out. This work was done in part while AG was visiting Fred Hutchinson Cancer Research Center.
CW and FM were funded by National Science Foundation awards 1223057 and 1564137.
CW is a Simons Foundation Fellow of the Life Sciences Research Foundation.
FM supported in part by a Faculty Scholar grant from the Howard Hughes Medical Institute and the Simons Foundation.