Quartets Enable Statistically Consistent Estimation of Cell Lineage Trees Under an Unbiased Error and Missingness Model (Abstract)

Abstract

Cancer progression and treatment can be informed by reconstructing its evolutionary history from tumor cells [Lim et al., 2020]. Although many methods exist to estimate evolutionary trees (called phylogenies) from molecular sequences, traditional approaches assume the input data are error-free and the output tree is fully resolved. These assumptions are challenged in tumor phylogenetics because single-cell sequencing produces sparse, error-ridden data and because tumors evolve clonally [Jahn et al., 2016; Schwartz and Schäffer, 2017]. Here, we study the theoretical utility of methods based on quartets (four-leaf, unrooted phylogenetic trees) and triplets (three-leaf, rooted phylogenetic trees), in light of these barriers.

Quartets and triplets have long been used as the building blocks for reconstructing the evolutionary history of species [Wilkinson et al., 2005]. The reason triplet-based methods (e.g., MP-EST [Liu et al., 2010]) and quartet-based methods (e.g., ASTRAL [Mirarab et al., 2014]) have garnered such success in species phylogenetics is their good statistical properties under the Multi-Species Coalescent (MSC) model [Pamilo and Nei, 1988; Rannala and Yang, 2003]; see Allman et al. (2011) and Degnan (2006) for identifiability results under the MSC model for quartets and triplets, respectively.

Inspired by these efforts, we study the utility of quartets and triplets for estimating cell lineage trees under a popular tumor phylogenetics model [Jahn et al., 2016; Ross and Markowetz, 2016; Wu, 2019; Kizilkale et al., 2022] with two phases. First, mutations arise on a (highly unresolved) cell lineage tree according to the infinite sites model, and second, errors (false positives and false negatives) and missing values are introduced to the resulting mutation data in an unbiased fashion, mimicking data produced by single-cell sequencing protocols. This infinite sites plus unbiased error and missingness (IS+UEM) model generates mutations (rather than gene genealogies like the MSC model). However, a quartet (with leaves bijectively labeled by four cells) is implied by a mutation being present in two cells and absent from two cells [Molloy et al., 2021; Springer et al., 2019]; similarly, a triplet (on three cells) is implied by a mutation being present in two cells and absent from one cell.

Our main result is that under the IS+UEM, the most probable quartet identifies the unrooted model cell lineage tree on four cells, with a mild assumption: the probability of false negatives and the probability of false positives must not sum to one. Somewhat surprisingly, our identifiability result for quartets does not extend to triplets, with more restrictive assumptions being required for identifiability. These results motivate seeking an unrooted cell lineage tree such that the number of quartets shared between it and the input mutations is maximized. We prove an optimal solution to this problem is a consistent estimator of the unrooted cell lineage tree under the IS+UEM model; this guarantee includes the case where the model tree is highly unresolved, provided that tree error is defined as the number of false negative branches. We therefore conclude by outlining how quartet-based methods might be employed for tumor phylogenetics given other important challenges like copy number aberrations and doublets.