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Inferring clonal evolution of tumors from single nucleotide somatic mutations

Wei Jiao, Shankar Vembu, Amit G. Deshwar, Lincoln Stein, Quaid Morris

arXiv:1210.3384v4cs.LGq-bio.PEq-bio.QMstat.ML

TL;DR

The paper addresses when tumor evolutionary histories can be reconstructed from SNV frequencies and introduces PhyloSub for automated inference. PhyloSub estimates subclonal phylogenies and genotypes with Bayesian uncertainty modeling, producing inferences that agree with single-cell and expert-driven reconstructions in evaluated datasets.

  • Problem

    Automated reconstruction of tumor subclonal evolutionary histories from SNV frequencies was unavailable, and the conditions enabling unique reconstruction had not been described.

  • Method

    PhyloSub uses a nonparametric Bayesian model with a Dirichlet process prior, MCMC inference, and latent trees linking SNVs to subclonal lineages.

  • Results

    PhyloSub inferred linear and branching subclonal lineages, with trees agreeing with single-cell assays and an expert-driven semi-manual reconstruction procedure.

  • Takeaways & Limitations

    SNV frequencies can support tumor lineage reconstruction in some cases, while multiple samples substantially constrain the possible evolutionary histories.

  • Takeaways & Limitations

    PhyloSub’s ability to detect lineages depends on lineage frequency, the number of defining SNVs, sequencing depth, and copy-number uncertainty.

Abstract

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High-throughput sequencing allows the detection and quantification of frequencies of somatic single nucleotide variants (SNV) in heterogeneous tumor cell populations. In some cases, the evolutionary history and population frequency of the subclonal lineages of tumor cells present in the sample can be reconstructed from these SNV frequency measurements. However, automated methods to do this reconstruction are not available and the conditions under which reconstruction is possible have not been described. We describe the conditions under which the evolutionary history can be uniquely reconstructed from SNV frequencies from single or multiple samples from the tumor population and we introduce a new statistical model, PhyloSub, that infers the phylogeny and genotype of the major subclonal lineages represented in the population of cancer cells. It uses a Bayesian nonparametric prior over trees that groups SNVs into major subclonal lineages and automatically estimates the number of lineages and their ancestry. We sample from the joint posterior distribution over trees to identify evolutionary histories and cell population frequencies that have the highest probability of generating the observed SNV frequency data. When multiple phylogenies are consistent with a given set of SNV frequencies, PhyloSub represents the uncertainty in the tumor phylogeny using a partial order plot. Experiments on a simulated dataset and two real datasets comprising tumor samples from acute myeloid leukemia and chronic lymphocytic leukemia patients demonstrate that PhyloSub can infer both linear (or chain) and branching lineages and its inferences are in good agreement with ground truth, where it is available.

Background

Tumors contain genetically diverse subclonal populations, and SNV frequencies can sometimes reveal their evolutionary history. PhyloSub combines topological rules with Bayesian inference to reconstruct tumor phylogenies while representing uncertainty.

  • Background: Tumors often contain multiple genetically diverse subclonal populations whose evolutionary history can sometimes be reconstructed from mutation population frequencies.Deep targeted sequencing improves the accuracy of estimated SNV frequencies used for reconstruction.
  • PhyloSub: The model uses Bayesian inference and MCMC sampling to represent uncertainty from measurement noise and multiple phylogenies consistent with observed SNV frequencies.A Dirichlet process prior groups SNVs into major subclonal lineages.
  • Model assumptions: The infinite sites assumption implies that cells containing a descendant SNV also contain its ancestral SNVs, so ancestral frequencies are at least as large.The model assumes each SNV appears once and does not revert to its original state.
  • Topological constraint rules: A single sample can leave linear and branching phylogenies ambiguous, but the sum rule and crossing rule can make the topology uniquely identifiable.The sum rule uses frequency sums within one sample, whereas the crossing rule compares descendant frequencies across multiple samples.
  • PhyloSub: PhyloSub models read counts and copy-number status to estimate major subclonal lineages, their genotypes, evolutionary relationships, and population proportions.Its probabilistic model associates SNVs with latent subclonal lineages and estimates the number of such lineages.

Results and Discussion

PhyloSub models subclonal lineages and their evolutionary relationships from SNV read counts, while accounting for uncertainty in tree structure and frequency estimates. Across simulated and tumor datasets, it recovered supported linear and branching structures, with uncertainty varying by sample number and data informativeness.

  • PhyloSub model: PhyloSub represents each subclonal lineage as a tree node with an associated SNV genotype and cell-population frequency.Its model estimates the number of major lineages, their mutational profiles, and their proportions from SNV read counts and copy-number status.
  • PhyloSub model: The Dirichlet process prior uses α0, γ, and λ to control tree size, height, and sibling structure, while hyperparameters are sampled during MCMC.The simulation established ranges intended to represent diverse tree structures and reduce prior bias in real-data analyses.
  • Simulation: At approximately 1,000X read depth, PhyloSub recovered clusters with SNV frequencies correlated above 0.99 with baseline values; at approximately 200X, nearby clusters sometimes merged.The reported frequency correlation remained above 0.96 at approximately 200X despite the slight performance degradation.
  • AML datasets: For SU048, PhyloSub and single-cell data agreed that TET2-E1357stop occurred early, but the remaining structure had substantial uncertainty and no strong linear-versus-branching evidence.The TET2-E1357stop genotype had posterior probability 0.81, whereas the two-lineage genotype reported by the single-cell assay had probability 0.06.
  • AML datasets: For SU070, PhyloSub nearly reconstructed the single-cell result, assigning posterior probability above 0.96 to two of three subclonal lineages but misordering two SNVs.The inferred order of CXorf36 and TET2-T1884A differed from single-cell data, plausibly reflecting sequencing bias or experimental error.
  • CLL datasets: With multiple samples, posterior uncertainty was greatly reduced: PhyloSub exactly matched the Schuh et al. trees for CLL077 and CLL003 and preferred splitting cluster A for CLL006.For CLL006, the split had higher complete-data log likelihood than Schuh et al.’s chain structure; the CLL dataset lacked ground truth.

Conclusions

PhyloSub infers subclonal phylogenies from single and multiple tumor samples while representing uncertainty when SNV frequencies do not uniquely determine ancestry. Its performance agrees with single-cell and expert-driven reconstructions, but depends on frequency estimation quality and may scale poorly to many SNVs.

  • Conclusions: PhyloSub uses a nonparametric Bayesian tree prior and structural constraints to infer relationships among subclones from single or multiple tumor samples.It also represents posterior phylogenetic uncertainty with partial order plots when frequencies are insufficient for unique reconstruction.
  • Conclusions: PhyloSub’s inferred lineage trees agreed well with single-cell assays for single-sample data and expert-driven semi-manual reconstructions for multiple-sample data.
  • Conclusions: Multiple tumor samples highly constrain the lineages consistent with SNV frequency data, while some linear lineages can be detected from a single high-cellularity sample.
  • Conclusions: Detection depends on lineage frequency relative to descendants, the number of defining SNVs, sequencing depth, and uncertainty about SNV copy number.For single-SNV lineages, frequency uncertainty must be smaller than the subclonal population frequency; additional defining SNVs can relax this constraint.
  • Conclusions: Tumors with many subclonal lineages defined by few SNVs may be difficult to reconstruct unless SNV frequencies are estimated very accurately.
  • Conclusions: Scaling to orders of magnitude more SNVs may cause the Markov chain to mix slowly or become trapped in local minima.

Dirichlet process mixture models

Dirichlet process mixture models extend finite mixture models to an unbounded number of components, avoiding the need to select the number of clusters in advance. Stick-breaking weights govern cluster assignments, while the concentration parameter influences the number of occupied clusters.

  • Dirichlet process mixture models: A finite Bayesian mixture model assigns objects to K clusters through mixing weights, component parameters, and component-specific data distributions.
  • Dirichlet process mixture models: Replacing the finite Dirichlet prior with a Dirichlet process permits infinitely many mixture components.The base distribution H generates component parameters, while α controls clustering behavior.
  • Dirichlet process mixture models: The stick-breaking process generates mixture weights by recursively breaking a unit-length stick according to beta variates.The concentration parameter α controls the number of clusters, with larger values producing more clusters.
  • Dirichlet process mixture models: The resulting infinite mixture model samples weights from GEM(α), cluster assignments from those weights, component parameters from H, and observations from their assigned components.

Tree-structured stick-breaking process

The tree-structured stick-breaking process extends flat stick-breaking into a rooted relational clustering model. Separate processes allocate mass across descendants and among sibling branches, while depth- and width-related concentration parameters shape the tree.

  • Tree-structured stick-breaking process: Interleaving two stick-breaking processes connects clusters into a rooted tree rather than leaving them independent.
  • Tree-structured stick-breaking process: Nodes are indexed by sequences of positive integers, with the empty sequence representing the root and extensions representing children.
  • Tree-structured stick-breaking process: Beta-distributed ν values allocate mass between each node and its descendants, while ψ values determine the probability of selecting particular child sequences.
  • Tree-structured stick-breaking process: The construction makes mixing weights sum to one, with α controlling tree height and γ controlling tree width.The depth-dependent concentration is defined as α(j) = λ^jα0.

PhyloSub model

PhyloSub models SNV allelic counts with a tree-structured stick-breaking prior so mutations can be grouped into subclonal lineages while preserving evolutionary relationships. MCMC jointly infers the tree, SNV frequencies, assignments, and related parameters under clonal constraints.

  • PhyloSub model: The allelic-count model represents reference and variant read counts, sequencing error probabilities, variant genotypes, and SNV population frequencies.The SNV frequency is the fraction of cells belonging to the variant population.
  • PhyloSub model: The posterior allelic-count calculation includes genotype-specific Dirichlet compound multinomial terms and the Gamma function.
  • PhyloSub model: The observation model uses a Dirichlet process to group mutations occurring at the same SNV population frequency without selecting the number of groups beforehand.
  • PhyloSub model: PhyloSub replaces the flat Dirichlet process with a tree-structured stick-breaking prior to model rooted clonal evolution and enforce evolutionary constraints when adding nodes.
  • PhyloSub model: MCMC inference jointly samples cluster assignments, stick lengths, hyperparameters, and SNV population frequencies to infer the tree structure.Gibbs sampling generates posterior samples of the SNV population frequencies.

Sampling SNV population frequencies

PhyloSub samples auxiliary weights and derives SNV population frequencies while enforcing clonal evolutionary constraints. A Metropolis-Hastings procedure samples the posterior over weights, and tree traversal generates frequency samples.

  • Constraint construction: SNV population frequencies are constrained so each non-leaf frequency is at least the sum of its children's frequencies.Auxiliary weights are introduced to enforce this constraint while summing descendant weights into node frequencies.
  • Constraint construction: Auxiliary weights sum to one and are assigned using a root value of one or Uniform(0, ηpar(v)) draws for non-root nodes.When a new node is added, its sampled weight is subtracted from its parent's remaining weight.
  • Frequency generation: Node frequencies are computed from auxiliary weights by adding each node's weight to the weights of its descendants.The resulting construction ensures parent mutation frequencies are at least the sum of all child frequencies.
  • Frequency generation: Algorithm 1 generates weight-frequency pairs for every tree node by breadth-first traversal.The algorithm takes a rooted tree as input and outputs the auxiliary weights and SNV population frequencies.
  • Posterior sampling: Metropolis-Hastings samples the posterior distribution of auxiliary weights using an asymmetric Dirichlet proposal.Its inputs are the current rooted tree, a scaling factor σ, and the iteration count; accepted samples yield auxiliary weights and SNV frequencies.

Extension to multiple tumor samples

PhyloSub extends to multiple tumor samples by sharing one tree-structured prior across samples while enforcing evolutionary constraints within each sample. A global Metropolis-Hastings move then uses the joint auxiliary-weight distribution.

  • Shared model: The multiple-sample model shares a tree-structured stick-breaking process prior across all tumor samples.Its graphical model represents allelic counts from multiple samples with shared tree structure.
  • Sample-specific quantities: For each sample, read counts define reference and variant alleles, variant-cell fractions, and corresponding auxiliary weights.The variant-cell fraction is the SNV population frequency at a locus for that sample.
  • Sampling procedure: The model enforces clonal evolutionary constraints separately for each tumor sample before applying a global Metropolis-Hastings move.The global move is based on the product of the sample-specific auxiliary-weight distributions.

Partial order plot

PhyloSub summarizes posterior uncertainty over tumor trees with a partial order plot. The plot uses SNVs as nodes, edge thickness for posterior ancestry frequency, and border colors for posthoc lineage clusters.

  • Purpose: A partial order plot summarizes and visualizes trees from all posterior MCMC samples.It represents uncertainty when multiple phylogenies are consistent with the observed data.
  • Encoding: SNVs, rather than SNV clusters, are the nodes of the partial order plot.Directed-edge thickness records how often one SNV's lineage is the parent of another's across MCMC samples.
  • Encoding: Directed-edge thickness is proportional to the fraction of MCMC samples supporting the corresponding parent-child ancestry.Border colors indicate posthoc subclonal lineage clusters assigned using correlation clustering.
  • Caveat: The color-based clustering summarizes assignments across samples but does not represent uncertainty in cluster membership.The clustering algorithm uses a symmetric co-clustering matrix comparing same-cluster and different-cluster assignments.

MCMC settings

The experiments use fixed MCMC and Metropolis-Hastings settings, select among random initializations using likelihood diagnostics, and monitor convergence with CODA.

  • Sampling settings: 5,000 MCMC iterations are used with a 100-sample burn-in in all experiments.The Metropolis-Hastings sampler also uses 5,000 iterations.
  • Sampling settings: The Dirichlet proposal scaling factor is fixed at σ = 100.The same fixed scaling factor is used for the Metropolis-Hastings procedure.
  • Diagnostics: A single run is selected from multiple random initializations using the complete-data likelihood trace and autocorrelation function.All 5,000 post-burn-in samples are retained without thinning for partial order plots.
  • Diagnostics: The CODA R package is used to monitor sampler convergence.Likelihood traces and autocorrelation plots after burn-in are reported for AML and CLL experiments.

Datasets and inputs to PhyloSub

The experiments use datasets whose inputs to PhyloSub are provided in Additional file 2, including the set of observations used by the model.

  • All experimental datasets and details of the inputs to PhyloSub are provided in Additional file 2.The inputs include a set of observations represented with notation beginning {(a_i, d_i, μ_r …).
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