Maximum Likelihood Inference of Time-scaled Cell Lineage Trees with Mixed-type Missing Data

Recent dynamic lineage tracing technologies combine CRISPR-based genome editing with single-cell sequencing to track cell divisions during development. A key computational problem in dynamic lineage tracing is to infer a cell lineage tree from the measured CRISPR-induced mutations. Three features of dynamic lineage tracing data distinguish this problem from standard phylogenetic tree inference. First, the CRISPR-editing process modifies a genomic location exactly once. This non-modifiable property is not well described by the time-reversible models commonly used in phylogenetics. Second, as a consequence of non-modifiability, the number of mutations per time unit decreases over time. Third, CRISPR-based genome-editing and single-cell sequencing results in high rates of both heritable and non-heritable (dropout) missing data. To model these features, we introduce the Probabilistic Mixed-type Missing (PMM) model. We describe an algorithm, LAML (Lineage Analysis via Maximum Likelihood), to search for the maximum likelihood (ML) tree under the PMM model. LAML combines an Expectation Maximization (EM) algorithm with a heuristic tree search to jointly estimate tree topology, branch lengths and missing data parameters. We derive a closed-form solution for the M-step in the case of no heritable missing data, and a block coordinate ascent approach in the general case which is more efficient than the standard General Time Reversible (GTR) phylogenetic model. On simulated data, LAML infers more accurate tree topologies and branch lengths than existing methods, with greater advantages on datasets with higher ratios of heritable to non-heritable missing data. We show that LAML provides unbiased time-scaled estimates of branch lengths. In contrast, we demonstrate that maximum parsimony methods for lineage tracing data not only underestimate branch lengths, but also yield branch lengths which are not proportional to time, due to the nonlinear decay in the number of mutations on branches further from the root. On lineage tracing data from a mouse model of lung adenocarcinoma, we show that LAML infers phylogenetic distances that are more concordant with gene expression data compared to distances derived from maximum parsimony. The LAML tree topology is more plausible than existing published trees, with fewer total cell migrations between distant metastases and fewer reseeding events where cells migrate back to the primary tumor. Crucially, we identify three distinct time epochs of metastasis progression, which includes a burst of metastasis events to various anatomical sites during a single month.