Characterizing and comparing phylogenies from their Laplacian spectrum

Phylogenetic trees are central to many areas of biology, ranging from population genetics and epidemiology to microbiology, ecology, and macroevolution. The ability to summarize properties of trees, compare different trees, and identify distinct modes of division within trees is essential to all these research areas. But despite wide-ranging applications, there currently exists no common, comprehensive framework for such analyses. Here we present a graph-theoretical approach that provides such a framework. We show how to construct the spectral density profile of a phylogenetic tree from its Laplacian graph. Using ultrametric simulated trees as well as non-ultrametric empirical trees, we demonstrate that the spectral density successfully identifies various properties of the trees and clusters them into meaningful groups. Finally, we illustrate how the eigengap can identify modes of division within a given tree. As phylogenetic data continue to accumulate and to be integrated into various areas of the life sciences, we expect that this spectral graph-theoretical framework to phylogenetics will have powerful and long-lasting applications.

系统发育树（phylogenetic tree）是诸多生物学研究领域的核心支撑，其应用范围涵盖种群遗传学、流行病学、微生物学、生态学与宏演化等方向。对系统发育树的属性进行归纳总结、比较不同树结构的差异，以及识别树内部的差异化分化模式，是所有上述研究领域的必要前提。尽管此类分析的应用场景极为广泛，但目前尚未存在通用且全面的分析框架。本文提出一种图论方法以构建此类分析框架，我们展示了如何基于拉普拉斯图（Laplacian graph）生成系统发育树的谱密度分布。通过结合超度量模拟树与非超度量经验树两类样本，我们验证了谱密度可有效识别各类系统发育树的多种核心属性，并能将其聚类为具有生物学意义的类群。最后，我们阐释了特征间隙（eigengap）可用于识别指定系统发育树内部的分化模式。随着系统发育数据持续积累并不断整合进入生命科学的各个研究领域，我们相信这套面向系统发育学的谱图论框架将具备强大且持久的应用价值。