Bayesian Birth-Death Model Reconstructs Cancer Cell Trees From Copy Number Profiles

A computational research team has introduced NestedBD, a Bayesian phylogenetic framework that reconstructs the evolutionary trees of cancer cells from single-cell copy number profiles using a continuous-time birth-death process. Reported in Algorithms for Molecular Biology by researchers Yushu Liu, Mohammadamin Edrisi, Zhi Yan, Huw A. Ogilvie, and Luay Nakhleh, the method models somatic copy number aberrations as discrete birth and death events occurring across genomic regions. Evaluated on synthetic benchmarks and biological datasets, NestedBD inferred tree topologies, lineage division times, and branch-specific mutation rates with greater fidelity than traditional phylogenetic tools. The software operates as an open-source package within the BEAST 2 environment, providing an analytical bridge between raw single-cell genomic measurements and the temporal dynamics that drive intra-tumor heterogeneity during cancer initiation, progression, and metastatic dissemination.

Copy number aberrations, or CNAs, represent widespread somatic mutations that delete or amplify genomic segments, frequently driving oncogenesis through the amplification of oncogenes or the loss of tumor suppressor genes. Unlike germline copy number variations used in population genetics, somatic CNAs differ substantially in size, genomic distribution, and rate of occurrence across individual malignancies. These alterations can encompass entire chromosomes or whole genomes, though focal events frequently span thousands or fewer base pairs. Over time, the iterative accumulation of distinct CNA signatures generates intra-tumor heterogeneity, establishing sub-clonal cell populations that respond differently to therapeutic pressure. Single-cell DNA sequencing technologies have emerged as a primary platform for resolving this cellular variation by producing sequencing reads from individual cells; however, translating these noisy single-cell profiles into accurate, time-calibrated evolutionary histories has remained an unresolved challenge in computational cancer biology.

Phylogenetic reconstruction of single-cell copy number profiles is typically constrained by computational complexity and statistical assumptions regarding genomic structure. Genomic analyses routinely segment chromosomes into non-overlapping bins of fixed nucleotide lengths, but because biological CNA events naturally span multiple contiguous bins or overlap historically, copy number states between adjacent bins are fundamentally dependent. While specialized tools such as MEDICC attempt to capture these structural dependencies directly, their combinatorial requirements become computationally intractable when applied to growing single-cell datasets. NestedBD addresses this trade-off by adopting an independent-bins assumption, testing whether treating bins independently enables tractable probabilistic modeling without degrading topological accuracy. By decoupling bin dependencies, the framework focuses computational power on joint statistical estimation of tree topologies, evolutionary branch durations, and varying mutation rates across expanding cellular populations.

At the core of NestedBD is a discrete-state, continuous-time birth-death process that governs transitions between copy number states over evolutionary time. For each genomic bin, the system defines state transitions where an amplification represents a birth event occurring at rate lambda and a deletion represents a death event occurring at rate mu. Because each existing genomic copy can independently undergo duplication or loss, the instantaneous transition rates scale directly with the current copy number integer state. When a bin reaches a copy number of zero, the transition rate becomes zero, preventing amplification or deletion from complete gene loss. In the absence of prior empirical estimates for individual rates, the baseline formulation sets the birth and death parameters equal to a shared constant rate, yielding closed-form conditional transition probabilities across branches. The model bounds the allowable state space by a user-specified ceiling, set by default to nine copies to balance computational efficiency with the biological range observed in human cancer cells.

To account for technical noise inherent in single-cell DNA sequencing platforms, NestedBD couples its evolutionary transition core with an explicit observational error model. Inferred copy number calls derived from single-cell reads inevitably contain estimation errors, which the authors model by treating observed bin states as normally distributed variables centered on true underlying integer copy numbers with a constant error variance. Inference is conducted via Markov chain Monte Carlo sampling over a joint posterior distribution encompassing tree topology, divergence times, error variance, and molecular clock parameters. The tree topology follows a two-parameter birth-death prior parameterized by diversification rate and extinction fraction, rooted at a known diploid common ancestor. To capture rate heterogeneity among distinct cellular lineages, mutation rates across branches follow a random local clock model, where lineages either inherit parental mutation rates or draw new rates from a gamma distribution under a Poisson prior on rate-shift events.

Beyond inferring tree topologies and relative divergence times, NestedBD implements a dynamic programming algorithm adapted to correct noisy copy number profiles at the leaves while reconstructing ancestral profiles across internal nodes. Operating on the inferred phylogeny, the algorithm maximizes the joint likelihood of observed single-cell binned profiles under the birth-death transition matrix and normal error distribution. For each terminal leaf, the method evaluates the likelihood of every candidate integer state given the estimated sequencing read profile and branch length to its parent, selecting the optimal integer assignment. This simultaneous leaf correction and ancestral state reconstruction addresses single-cell profiling artifacts, allowing investigators to trace the exact chronological acquisition of amplifications and deletions from the initial diploid root through intermediate evolutionary branching points to terminal single cells.

The authors validated NestedBD using comprehensive synthetic datasets generated with an updated evolutionary simulator incorporating non-uniform CNA distributions, overlapping alterations, and realistic error profiles. Synthetic trees containing 20 and 50 single cells were parameterized with event multipliers of 90, 125, and 250 to represent low, medium, and high CNA frequencies, where more than 90 percent of simulated alterations overlapped prior events. Synthetic sequencing reads were aligned using BWA and profiled with Ginkgo, yielding estimated copy number profiles with a median error rate of 0.24. NestedBD was benchmarked against maximum parsimony, neighbor-joining using Hamming distances, and Lazac, a specialized copy number phylogeny method utilizing ZCNT distances. Posterior distributions were generated using five-chain coupled MCMC executed for 80 million iterations, discarding the initial 20 percent as burn-in to extract maximum clade credibility consensus trees with median node heights.

Benchmarking evaluations demonstrated that NestedBD consistently recovered true evolutionary branches with higher accuracy across all simulated event frequencies and tree sizes. Because identical cell profiles can produce arbitrary binary resolutions that disproportionately distort Robinson-Foulds metrics, performance was evaluated by grouping branch lengths into deciles and calculating the true positive recovery rate. As anticipated across all methods, longer phylogenetic branches exhibited higher recovery rates than shorter branches, and overall accuracy declined in larger 50-cell trees. However, while maximum parsimony and neighbor-joining suffered notable accuracy declines under high event densities due to complex overlapping alterations, NestedBD maintained stable topological accuracy. Lazac demonstrated resilience to bin dependencies through its distance metric, but NestedBD achieved the highest topological recovery under the most complex simulated scenarios, confirming that the independent-bins assumption remains robust even when synthetically violated.

Branch length evaluations revealed that NestedBD uniquely provided calibrated estimates of relative evolutionary time between lineage divisions while accurately estimating total mutational burdens. Assessed through ordinary least squares scaling against ground truth node heights, the median inferred node heights closely tracked true divergence points, with true heights consistently falling within estimated 95 percent highest posterior density intervals. When converted to mutation counts by scaling branch lengths by branch-specific mutation rates, NestedBD produced normalized pairwise leaf distances with lower Euclidean errors than those obtained directly from neighbor-joining or parsimony. Furthermore, the dynamic programming correction module significantly reduced copy number estimation errors at tree leaves, outperforming initial Ginkgo profiles under both true and inferred tree topologies, with the greatest error reductions observed in high-frequency event regimes characterized by recurrent locus alterations.

The investigative team extended NestedBD to biological sequencing data from two colorectal cancer patients, designated CRC01 and CRC04, to evaluate its capacity to resolve real-world clinical tumor histories. For patient CRC01, analysis of 20 cells sampled from primary tumor tissue and liver metastasis revealed a deep trunk separating the normal diploid ancestor from the common tumor ancestor, a structural feature consistent with punctuated tumor evolution. The tree captured four initial driver alterations on this stem, including mutations affecting the SRC oncogene, which is linked to colon cancer progression and hepatic metastasis. Subsequent branching traced the divergence of the metastatic lineage following the acquisition of a MAP2K4 mutation, with a downstream sub-lineage acquiring MAP2K1 and SMAD3 alterations alongside elevated local mutation rates. Conversely, cells remaining at the primary tumor site accrued UBR5, RAD21, and RSPO2 alterations, exhibiting distinct rate accelerations that parsimony and neighbor-joining failed to separate cleanly.

In patient CRC04, comprising 50 cells from primary tumor and lymph node metastasis sites, NestedBD resolved nine clonal alterations along the trunk lineage, prominently featuring the canonical initiator gene APC. An early primary tumor sub-lineage subsequently acquired six alterations encompassing AKT1, BCL9L, B2M, FBXW7, MUTYH, and RSPO2, accompanied by a marked elevation in mutation rate. A subset of these altered cells seeded the lymph node metastasis, maintaining accelerated evolutionary rates, while a distinct primary tumor lineage accumulated alterations in SRC, SALL4, BAX, and SMAD3 under substantially lower mutation rates. In contrast to NestedBD, maximum parsimony incorrectly placed primary tumor cells as descendants of the metastatic lymph node lineage, while neighbor-joining inferred branch lengths that distorted evolutionary timing by placing metastatic cells closer to the diploid root. The authors noted that while NestedBD assumes pre-estimated binned profiles and single-time-point sampling, future algorithmic expansions directly processing raw sequencing reads will further refine cancer phylogenetic modeling.

Subject of Research: Bayesian phylogenetic inference of single-cell cancer evolutionary trees from copy number aberrations

Article Title: NestedBD: Bayesian inference of phylogenetic trees from single-cell copy number profiles under a birth-death model

Article References: Liu, Y., Edrisi, M., Yan, Z., A Ogilvie, H., & Nakhleh, L. (2024). NestedBD: Bayesian inference of phylogenetic trees from single-cell copy number profiles under a birth-death model. Algorithms for Molecular Biology, 19(1), Article 18. https://doi.org/10.1186/s13015-024-00264-4

Image Credits: AI Generated

DOI: 10.1186/s13015-024-00264-4

Keywords: NestedBD, copy number aberrations, single-cell DNA sequencing, Bayesian phylogenetics, birth-death process, BEAST 2, intra-tumor heterogeneity, Markov chain Monte Carlo, colorectal cancer, branch lengths, random local clock, phylogenetic trees, ancestral reconstruction, Ginkgo, punctuated evolution, somatic mutations, maximum clade credibility