Supplementary materials.

Emerging ultra-low coverage single-cell DNA sequencing (scDNA-seq) technologies have enabled high resolution evolutionary studies of copy number aberrations (CNAs) within tumors. While these sequencing technologies are well suited for identifying CNAs due to the uniformity of sequencing coverage, the sparsity of coverage poses challenges for the study of single-nucleotide variants (SNVs). In order to maximize the utility of increasingly available ultra-low coverage scDNA-seq data and obtain a comprehensive understanding of tumor evolution, it is important to also analyze the evolution of SNVs from the same set of tumor cells. We present Phertilizer, a method to infer a clonal tree from ultra-low coverage scDNA-seq data of a tumor. Based on a probabilistic model, our method recursively partitions the data by identifying key evolutionary events in the history of the tumor. We demonstrate the performance of Phertilizer on simulated data as well as on two real datasets, finding that Phertilizer effectively utilizes the copy-number signal inherent in the data to more accurately uncover clonal structure and genotypes compared to previous methods.</div

Phertilizer infers a clonal tree <i>T</i>, clonal genotypes Y and a cell clustering <i>ϕ</i> given ultra-low coverage single-cell sequencing data.

(a) A tumor consists of clones with distinct genotypes. (b) Ultra-low coverage scDNA-seq produces total read counts and variant read counts for n cells and m SNV loci, and low dimension embedding of binned read counts . (c) Given maximum copy number c and sequencing error probability α, Phertilizer infers a clonal tree T, clonal genotypes Y and cell clustering ϕ with maximum posterior probability .</p

Phertilizer outperforms Baseline and SBMClone on simulated data.

We show aggregated results for n ∈ {1000, 2000} cells, k ∈ {5, 9} clones and coverage g = 0.05×. (a) Example of SNV accuracy, ancestral pair recall (APR), clustered pair recall (CPR), and incomparable pair recall (IPR) metrics. See Section B.1.4 in S1 Appendix for the corresponding example for cell metrics and genotype similarity. (b) Phertilizer outperforms Baseline +SCITE and SBMClone +SCITE in APR and IPR for both SNVs and cells. Although competing methods rank higher in CPR, overall Phertilizer performs the best considering the accuracy. (c) Phertilizer more accurately recovers clonal genotypes than competing methods.</p

Phertilizer solves the CTI problem by enumerating clonal trees using three elementary operations in a recursive fashion.

(a) Each operation yields a new clonal tree T′ by extending a leaf vj of the previous clonal tree T and reassigning its SNVs Δ(yj) and cells ϕ(j). The resulting clonal genotypes Y′ and cell clustering ϕ′ are constrained as depicted: (b) Linear, (c) Branching, and (d) Identity.</p

Phertilizer infers clonal tree for breast cancer tumor TN1.

(a) The tree inferred by Phertilizer with numbers of SNVs labeled beside the edges, and numbers of cells labeled beneath the leaves. Cancer-related genes are labeled next to the SNVs (‘*’: stop-gain variant). (b) A mapping between Phertilizer’s cell clusters and the Minussi et al. [7] superclones. (c) The cell mutational burden (CMB) comparison between cells within (blue) and outside of (red) each clade in the inferred clonal tree.</p

Phertilizer improves upon the SNV phylogeny previously inferred by Laks et al. [6].

(a) The clonal tree inferred by Laks et al. [6] with edges labeled by SNV gains and cell numbers shown below leaf nodes. (b) The clonal tree inferred by Phertilizer with edges labeled by SNV gains and cell numbers shown below leaf nodes. Cancer-related genes are labeled next to the SNVs in (a) and (b) with ‘*’ indicating a stop-gain variant. (c) Mapping between the Laks et al. [6] cell placement and Phertilizer cell placement. (d-e) Cell mutational burden (CMB) comparison between cells within (blue) and outside (red) of each clade in the inferred Laks et al. [6] and Phertilizer clonal tree.</p

MOESM1 of Complexity and algorithms for copy-number evolution problems

Additional file 1. The appendix contains the proofs omitted from the main text, the ILP formulation for the Copy-Number Triplet Problem, the complete ILP formulation for the Copy-Number Tree Problem, and additional details about the results