Pairtree

Pairtree infers the phylogeny underlying a cancer using genomic mutation data. Pairtree is particularly suited to settings with multiple tissue samples from each cancer, providing separate estimates of subclone frequency from each sample that constrain the set of consistent phylogenies. The algorithm consists of two phases:

1. Compute pairwise relation tensor over all mutations (or clusters of mutations). This provides the probability over each of four possible evolutionary relationships between each pair of mutations A and B.

2. Use the pairwise relation tensor to sample possible phylogenies, assigning a likelihood to each. As each phylogeny is sampled, Pairtree computes the subclone frequency of each mutation (or cluster of mutations) within the tree, balancing the need to fit the observed mutation data while still obeying tree constraints.

   The algorithm is described in Reconstructing complex cancer evolutionary histories from multiple bulk DNA samples using Pairtree (Wintersinger et al. (2022)).

   The practical aspects of using Pairtree are described in our STAR protocol paper Reconstructing cancer phylogenies using Pairtree, a clone tree reconstruction algorithm (Kulman et al. (2022)).

Table of Contents

1. Installation
2. Examples
3. Pairtree Executables
4. Pairtree Inputs
5. Clustering mutations
6. Pairtree outputs

Installing Pairtree

For more details on installation, please see here.

1. Install dependencies. Installation is usually easiest if you use Mambaforge, which includes a recent version of Python alongside the Mamba package manager. As Mamba is a faster reimplementation of Conda, you can alternatively use Conda or Miniconda.

   Refer to requirements.txt for the full list of dependencies. These can be installed in a new environment with the following commands. If you're using Anaconda instead of Mamba, replace mamba with conda below.

   To install all requirements using mamba (or conda), first clone the Pairtree repository and then create a new virtual environment with the dependencies.

    git clone https://github.com/jwintersinger/pairtree
    mamba create --name pairtree --file requirements.txt
   

   Or install the dependencies in your current environment:

    git clone https://github.com/jwintersinger/pairtree   
    conda create -n pairtree --file requirements.txt --yes
    conda activate pairtree
   

   Pairtree has only been tested on Linux systems, but should work on any UNIX-like OS (including macOS).

2. Download and build the C code required to fit subclone frequencies to the tree. This algorithm was published in Jia et al., and uses the authors' implementation with minor modifications.

    git clone https://github.com/jwintersinger/pairtree
    cd pairtree/lib
    git clone https://github.com/ethanumn/projectppm
    cd projectppm
    bash make.sh
   

3. Download plotting requirements. Unfortunately, not all plotting requirements Pairtree uses are available via conda. Instead, we recommend downloading them via pip. (optional)

    pip3 install -r plot_requirements.txt
   

Examples

After installing Pairtree, you can test your installation using provided example data.

    PTDIR=$HOME/path/to/pairtree
    cd $PTDIR/example
    mkdir results && cd results
    # Run Pairtree.
    $PTDIR/bin/pairtree --params $PTDIR/example/example.params.json $PTDIR/example/example.ssm example.results.npz
    # Plot results in an HTML file.
    $PTDIR/bin/plottree --runid example $PTDIR/example/example.ssm $PTDIR/example/example.params.json example.results.npz example.results.html
    # View the HTML file.
    open example.results.html

For a more detailed example of how to run Pairtree and interpret the results, please see our STAR Protocols paper Reconstructing cancer phylogenies using Pairtree, a clone tree reconstruction algorithm.

Pairtree executables

Pairtree provides several executable files to be used at different stages of your clone-tree-building pipeline. You can run each executable with the --help flag to get full usage instructions.

• bin/clustervars: cluster variants into subclones suitable for building clone trees with bin/pairtree. Please see the Clustering mutations section for full instructions.

• bin/plotvars: plot information about your variant clusters before proceeding with your Pairtree run. This step is optional. If --plot-relations is specified, the pairwise relations between subclones will be computed and plotted, but at a considerable computational cost.

• bin/pairtree: the main Pairtree executable, used for building clone trees. See the Input files and Tweaking Pairtree options sections for more information specific to this step.

• bin/plottree: plot results of your Pairtree run. This will plot a single clone tree, which by default will be the one with highest likelihood. See Interpreting Pairtree output for details.

• bin/summposterior: summarize the posterior distribution over clone trees from your Pairtree run by representing this distribution as a graph, and by showing the best individual tree samples. See Interpreting Pairtree output for details.

Input files

Pairtree requires two input files. See here for more details.

SSM file

Pairtree needs a .ssm (i.e., simple somatic mutation) file, which provides integer counts of the number of variant and total reads at each mutation locus in each tissue sample. The .ssm file is tab-delimited, with the first line serving as a header, and each subsequent line providing data for one mutation. You may wish to consult an example .ssm file.

The .ssm file contains the following fields:

• id: a unique ID used to identify the variant. The first mutation should be assigned ID s0, with subsequent variants assigned s1, s2, .... Generally, these should be contiguous, but this is not enforced, so you should be able to delete variants from this file without adjusting subsequent IDs. The s prefix indicates that the ID corresponds to an SSM.

• name: a text string used to label variants in Pairtree's output. This can be a gene name, genome position (e.g., 1_12345 may be used to represent that the variant occurred on chr1 at position 12345), or any other string. Spaces may be used, and uniqueness of names is not enforced (though duplicate names may lead to confusing output).

• var_reads: a comma-separated vector of how many whole-number-valued genomic reads at this mutation's locus corresponded to the variant allele in each tissue sample. Tissue samples may be provided in any order, so long as this order is consistent for all mutations. The names of the associated tissue samples are given in the .params.json file, detailed below. If running with only a single tissue sample, this field contains only a single scalar value for each mutation. If a variant has been called in only a subset of your tissue samples, consult the Imputing read counts section.

• total_reads: a comma-separated vector of the total number of whole-number-valued genomic reads observed at this mutation's locus in each tissue sample. Thus, the variant allele frequency (VAF) is given by var_reads / total_reads. If a variant has been called in only a subset of your tissue samples, consult the Imputing read counts section.

• var_read_prob: a comma-separted vector of probabilities \omega_{j1}, \omega_{j2}, ..., \omega_{js}, ..., \omega_{jS}, where S is the total number of tissue samples. The value \omega_{js} indicates the probabilty of observing a read from the variant allele for mutation j in a cell bearing the mutation. Thus, if mutation j occurred at a diploid locus, we should have \omega_{js} = 0.5. In a haploid cell (e.g., male sex chromosome), we should have \omega_{js} = 1.0. If a copy-number aberration (CNA) duplicated the reference allele in the lineage bearing mutation j prior to j occurring, there will be two reference alleles and a single variant allele in all cells bearing j, such that \omega_{js} = 0.3333. Thus, these values give Pairtree the ability to correct for the effect CNAs have on the relationship between VAF (i.e., the proportion of alleles bearing the mutation) and subclonal frequency (i.e., the proportion of cells bearing the mutation).

Parameters file

Pairtree also needs a .params.json file, which provides parameters for the Pairtree run. You may wish to consult an example .params.json file.

The .params.json file denotes three objects:

• samples (required): list of sample names, provided in the same order as entries in the var_reads and total_reads vectors in the .ssm file. Each sample name can be an arbitrary string.

• clusters (required): list-of-lists denoting variant clusters, each of which corresponds to a subclone. Each sub-list contains strings corresponding to the IDs of all variants that belong to a given cluster. All variants belonging to a cluster are assumed to share the same subclonal frequency in a given tissue sample, with the VAFs of each variant serving as a noisy estimate of this subclonal frequency after using var_read_prob to correct for ploidy. See the section on clustering mutations for instructions on how to use Pairtree to build clusters. Pairtree assumes that all variants within a cluster share the same subclonal frequencies across all cancer samples. Ploidy-corrected estimates of these frequencies can be obtained by dividing a variant's VAF by its variant read probability `omega.

• garbage (optional): list of IDs in the .ssm file that you do not want Pairtree to use in building your clone tree. Specifying variants as garbage can be useful when you want to exclude variants without having to remove them from the .ssm file, allowing you to easily re-introduce them to your clone tree in the future. Typically, you might remove a variant because you've determined that it violates the infinite sites assumption (ISA); because you think the associated read count data is likely erroneous; or because the variant was subject to a complex copy-number configuration that skews the relationship between VAF and subclonal frequency (e.g., the variant appears in different copy-number states in different subclones).

Imputing read counts for mutations missing in some samples

When some of your mutations appear in only some of your samples, you must impute the total read count for those mutations in the samples where they're not present. While the variant read count should clearly be zero in such cases, the value the total read count should take is unclear. This task is important, however, because it informs Pairtree how confident you are that the variant is indeed not present in the sample -- an instance where you have zero variant reads amongst ten total reads indicates considerably less confidence than if you have zero variant reads for 1000 total reads.

To impute missing read counts, the best option is to refer to the original sequencing data (e.g., the BAM file) to determine how many reads were present at the locus in the given sample. As this task can be arduous, you may alternatively wish to take the mean read count across samples for that genomic locus, or the mean read count for all loci in that sample. More complex strategies that try to correct for variable depth across loci (e.g., because of GC bias) or differing depths across samples (e.g., because some samples were more deeply sequenced than others) are also possible.

Clustering mutations

Pairtree requires mutations be clustered into subclones. For an example of how to use Pairtree's clustering utilities, please see here. These clusters should be provided to Pairtree in the .params.json file using the clusters array, described above. Pairtree works best with thirty or fewer subclones, but can build (less accurate) clone trees for 100 subclones or more.

To build clusters, you have three options.

1. Don't actually cluster mutations into subclones. That is, instead of building a clone tree, you can build a mutation tree instead in which each cluster contains only a single variant. You must still specify the clusters in your .params.json file, but you simply list a separate cluster for each variant. This works best for a small number of mutations (30 or fewer), and so is most suitable for sequencing data obtained from WES or targetted sequencing rather than WGS.

2. Use one of many published clustering methods, such as PyClone-VI. You must translate these methods' outputs into Pairtree's format, but this is typically easy.

3. Use Pairtree to cluster the variants.

Pairtree provides two clustering models. As Pairtree's focus is not on variant clustering, its clustering algorithms are less well-validated than its tree building, and so you may obtain better results with other algorithms. Nevertheless, Pairtree's authors have had reasonably good success with its cluster-building methods. Pairtree's two clustering models both use a Dirichlet process mixture model (DPMM), but differ in how they use the model.

You can use Pairtree to cluster variants through the bin/clustervars executable. Run with --help to obtain full usage instructions. Several options are of particular interest.

• ssm_fn: Input SSM file with mutations.

• in_params_fn: Input params file listing sample names and any existing garbage mutations.

• out_params_fn: Output params file with cluster assignments for each mutation (excluding those listed in garbage mutations).

• --model linfreq: This uses the default "lineage frequency" clustering model. This is the most computationally efficient choice, and is the most straightforwrd application of the DPMM to the clustering problem.

• --model pairwise: This uses the alternative "pairwise relationships" clustering model. In some instances, it may produce better results than linfreq. This model is typically slower to use, as it requires computing pairwise relationships between mutation pairs. For M mutations, there are (M choose 2) such pairs, and so this may become frustratingly slow for more than 1000 mutations.

• --prior <value>. This is used only with --model pairwise. This option specifies the prior probability for the coclustering relationship. The default of 0.25 corresponds to a uniform relationship over the four relationship types (i.e., coclustering, A is ancestral to B, B is ancestral to A, and A and B are on different branches). Higher values typically result in fewer, larger clusters. Valid values range between 0 and 1.

• --concentraton <value>. This is the log_10{alpha} used in the DPMM. Larger values typically result in more, smaller clusters.

Clustering results are sensitive to the model and parameters used. The best choice of parameters depends on the nature of your data, including the number of variants, read depth, and other characteristics. It is often prudent to try a range of different parameters, plot the results for each using the bin/plotvars executable, and select the clustering that best represents your data. The best clustering can depend on your application -- for some uses, you may want more granular clusters that produce a detailed tree with many subclones, while in other uses, you may want to collapse together similar subclones to create a simpler tree in which it is easier to discern major structural differences.

Interpreting and manipulating Pairtree output

See here for further details on interpretting Pairtree's outputs.

Description of the .results.npz file format

Here we provide a brief description of the output file (.npz) generated by Pairtree. The .npz file type is a zipped archive consisting of a set of key, value pairs. The numpy.load function can be used to load the contents of an .npz file (see the numpy.load docs).

When the pairtree program completes its execution it saves a .npz file with details of the trees it found, it's parameter setup, and the results for some of the important computations it performed. We detail below some of the most important data Pairtree stores in the .npz file:

• struct: a list of parents vectors sorted by the log-likelihood of how well the tree described by the parents vector fits the data. See the following source code to see how to generate an equivalent adjacency matrix from a parents vector.

• count: a list containing the number of times a tree (or equivalently a parents vector) was found during tree search. The number at index i in the count list corresponds to the number of times the tree at index i in the struct list was found during tree search.

• phi: a list of matrices, where the matrix at index i contains the tree constrained subclonal frequency fit to the tree at index i in the struct list. Each row of the phi matrix corresponds to a cluster, and each column corresponds to its tree constrained subclonal frequencing in a particular sample.

• llh: a list containing the tree log-likelihoods. The value at index i in the llh list corresponds to the log-likelihood of the tree at index i in the struct list.

• prob: a list containing the tree posterior probabilities. The value at index i in the prob list corresponds to the posterior probability of the tree at index i in the struct list.

It's important to note that all of these lists are sorted in descending order based on tree log-likelihood. Therefore, the data at index 0 in each of these lists corresponds to the data for the best fitting tree.

Understanding the html file format outputted by bin/plottree

The bin/plottree script uses the .npz file outputted by bin/pairtree to produce an html of plots and figures describing the results.

The title and description for each plot/figure are provided below.

• Tree: shows a graphical representation of the clonal evolution tree based on the variant clusters provided to bin/pairtree in the parameter file. These clusters are each labeled as their own subclone population (S0, S1, ...). There are three different additional values provided with the tree. tidx is the tree index of which tree from the .npz file is being displayed. nlglh is the negative log-likelihood of the tree. prob is posterior softmax probability of the tree.

• Population frequencies: shows the ratio of optimal subclone population frequencies when constrained by the proposed tree for each sample specified in the .params.json file provided to bin/pairtree. There are two additional values for each sample: Clone diversity index and Clone and mutation diversity index. The Clone diversity index is the entropy of subclone population frequency for a given sample. This is a value from 0 to 1, where 1 means a sample has many different subclone populations (and as a result, a higher entropy), and 0 means a sample has no diversity (or a single subclone, and a lower entropy). The Clone and mutation diversity index is the joint entropy of the subclone population frequency and the number of mutations present in a clone.

• Tree-constrained subclonal frequencies: shows the optimal subclone population frequency when constrained by the proposed tree.

• Data-implied subclonal frequencies: shows the implied subclone population frequency per sample based on the variant data provided.

• Interleaved subclonal frequencies: shows the tree-constrained subclonal frequencies, as well as the error between the tree-constrained subclonal frequencies and the data-implied subclonal frequencies for each subclone in a given sample. The Total Error value shown is the sum of all errors across subclones and samples.

• Diversity Indices: shows the Clone diversity index, Clone and mutation diversity index, and Shannon diversity index in bits for each sample. The Clone diversity index, and Clone and mutation diversity index are described above. The Shannon diversity index is the entropy of the proportion of mutations in each cluster.

• VAFs (corrected): shows a table of variant allele frequencies (VAFs) per sample for the following: tree-constrained subclones (Φ), Supervariants, Cluster members, and Garbage. The current selections for the table are highlighted in blue. The Φ values are the optimal tree-constrained subclones VAFs (or pseudo-variants denoted by P1, ... ,P_n). The Supervariants are the supervariants created for each cluster defined in the .params.json file provided to Pairtree. The Cluster members are the individual variants in each supervariant or cluster. The Garbage variants are those listed as garbage in the .params.json file provided to Pairtree. There is also a search bar which allows you to list comma delimited ID values that specify which IDs should be shown in the VAF table (e.g. s0,s1 will show only these two IDs if there is a match).

• ML relations: shows a matrix of the pairwise relationships between all of the different subclone populations. Each relationship is with respect to the subclone on the left hand side of the matrix, to the subclone listed on top of the matrix.

• garbage: shows a pairwise relationship matrix of which subclone populations have a garbage relationships.

• cocluster: shows a pairwise relationship matrix of which subclone populations are clustered together (or have a cocluster relationship).

• A_B: shows a pairwise relationship matrix of which subclone populations have an ancestral relationship to other subclone populations.

• B_A: shows a pairwise relationship matrix of which subclone populations have a descendant relationship to other subclone populations.

• diff_branches: shows a pairwise relationship matrix of which subclone populations are not on the same branch (i.e. are not ancestors or descendants of each other).

• Cluster stats: shows a table of statistics for each cluster. The columns are Cluster, Members, and Deviation. The Cluster column lists the ID number for the cluster or subclone. Members lists the number of variants in a cluster. Deviation lists the median absolute difference between the subclone frequency of the supervariant, and its related cluster of variants.

Exporting Pairtree output to other formats

• Use bin/summposterior to summarize the posterior distribution over trees and bin/plottree to plot all details corresponding to an individual tree. See the Pairtree executables section for details.

• To export Pairtree's visualizations from bin/plottree for use in publications, try SVG Crowbar. All figures Pairtree creates embedded in HTML web pages using the SVG figure format, which is suitable for use at arbitrarily high resolutions (including in print) because of its vector-based nature. Some of these figures are rendered with the Plotly plotting library, though most use custom code that ultimately invokes d3.js.

• Alternatively, to export the subset of Pairtree's visualizations that use Plotly without having to open the HTML plots in a web browser, try Kaleido, which is a Plotly-related library for exporting plots without using a browser (accomplished using an embedded version of the Chromium web browser). Most Plotly-related calls in Pairtree call plotly.io.to_html(fig, ...), which can be replaced by fig.write_image.

• Unfortunately, for the other plots that don't use Plotly (such as the tree structure and subclonal frequency plots), there is no straightforward means of rendering them without using a web browser. If SVG Crowbar fails or is otherwise not a viable option, try printing the web page containing the plots to a PDF file, then opening the PDF in Inkscape. The SVG plts on the web page should be stored using a vector representation in the PDF, allowing you to copy them from the PDF into a discrete SVG file where they can be manipulated as vector images.

• To export the data that bin/plottree visualizes for your own analysis, pass the --tree-json <filename>.json option to bin/plottree. This will export the data for a given tree to the <filename>.json file. For a tree with K nodes (i.e., the root node representing normal tissue and K-1 cancerous subclones) constructed from S cancer samples, some of the data included will be amongst the following.

  • phi: tree-constrained subclonal frequencies as KxS matrix in row-major format
  • phi_hat: data-implied subclonal frequencies as KxS matrix in row-major format
  • eta: population frequencies as KxS matrix in row-major format
  • llh: tree log likelihood in base e
  • nlglh: tree log likelihood normalized to number of subclones and cancer samples in base 2, allowing comparison of trees across datasets to determine how many bits you're paying per subclone per cancer sample
  • prob: posterior probability of tree
  • count: how many times this tree was sampled in Metropolis-Hastings
  • parents: K-1-length vector, where parents[i] provides the parent of node i+1
  • samples: names of cancer samples given in input (providing the names associated with each column of the phi, phi_hat, and eta matrices)
  • cdi, cmdi, sdi: S-length vectors providing the clonal dviersity index, clone and mutation diversity index, and Shannon diversity index for each sample. Refer to the Pairtree paper for definitions.

• If stderr is redirected to a file (e.g., via bin/pairtree ... 2> stderr.log), instead of rendering a progress bar, Pairtree will report its progress by writing to stderr a series of JSON objects, with one per line. An example line follows:

  {"desc": "Sampling trees", "count": 11457, "total": 12000, "unit": "tree", "started_at": "2021-11-01 02:45:37.226015", "timestamp": "2021-11-01 02:46:12.625482"}

Fixing incorrect ploidy

There are cases of incorrect ploidy that are unable to be detected using the pairwise framework. This can result in an implied subclonal frequency being greater than 1. This is often due to missed copy number calls, such as an uncalled loss of heterozygosity (LOH) event which removes the wildtype allele in a lineage and leaves only the variant allele. For an example of how to use the util/fix_bad_var_read_prob.py script, please see here.

In order to find these mutations such that they can be corrected, we provide the script util/fix_bad_var_read_prob.py. There are several options for this script which are of particular interest.

• --logbf-threshold: Logarithm of Bayes factor threshold at which the haploid model is accepted as more likely model than the model using the provided var_read_prob.
• --ignore-existing-garbage: Ignore any existing garbage variants listed in in_params_fn and test all variants. If not specified, any existing garbage variants will be kept as garbage and not tested again.
• --action (add_to_garbage, modify_var_read_prob): Action to take after script has completed. add_to_garbage will add the resulting 'bad_ssms' to the list of garbage samples in the params.json file provided. modify_var_read_prob will overwrite the var_read_prob in the .ssm file with the value provided by the --var-read-prob-alt option of this script.
• --var-read-prob-alt: Value that will be used to overwrite the var_read_prob of any garbage variants found if --action is set to modify_var_read_prob.
• in_ssm_fn: Input SSM file with mutations.
• in_params_fn: Input params file listing sample names and any existing garbage mutations.
• out_ssm_fn: Output SSM file with modified list of LOH mutations.
• out_params_fn: Output params file with modified list of garbage mutations.

The terminal output of this script will be in the following format:

num_bad_ssms: Number of variants which have an incorrect variant read probability estimate

bad_ssms: List of variant names which were found to have an incorrect variant read probability estimate

bad_samp_prop: Percent of variants out of all samples which were found to have an incorrect variant read probability estimate

bad_ssm_prop: Percent of variants which were found to have an incorrect variant read probability estimate in at least one sample

Removing garbage mutations

Pairtree relies on the infinite sites assumption (ISA) when converting mutation allele frequencies to subclonal frequencies, and building clone trees. Pairtree is also sensitive to other factors such as missed CNA calls, or technical noise which can skew the subclonal frequency conversion. Therefore, it is necessary to detect and remove such mutations before building the clone tree. Pairwise relationships between mutations reveal both ISA violations and other types of erroneous mutations, which we refer to collectively as garbage mutations.

In order to find these mutations such that they can be added to our list of garbage mutations, we provide the script bin/removegarbage. There are several options for this script which are of particular interest.

• --prior: Pairwise garbage prior probability. The default value of 0.2 represents a uniform prior over pairwise relationship types.
• --max-garb-prob: Maximum probability of a garbage relationship to permit for any pair in order for the algorithm to terminate.
• --ignore-existing-garbage: Ignore any existing garbage variants listed in in_params_fn and test all variants. If not specified, any existing garbage variants will be kept as garbage and not tested again.
• --pairwise-results: Filename to store pairwise evidence in, which allows you to try garbage removal with different parameters without having to repeat the pairwise computations.
• in_ssm_fn: Input SSM file with mutations.
• in_params_fn: Input params file listing sample names and any existing garbage mutations.
• out_params_fn: Output params file with modified list of garbage mutations.

Tweaking Pairtree options

Tweaking some of Pairtree's options can improve the quality of your results or reduce the method's runtime. The most relevant options are discussed below. For a full listing of tweakable Pairtree options, including algorithm hyperparameters, run pairtree --help.

Finding better trees by running multiple MCMC chains

Pairtree samples trees using MCMC chains. Like any optimization algorithm working with a non-convex objective, Pairtree's chains can become stuck in local optima. Running multiple MCMC chains helps avoid this, since each chain starts from a different point in space, and takes a different path through that space.

By default, Pairtree will run a separate MCMC chain for each CPU core (logical or physical) present. As these chains are run in parallel, using multiple chains does not increase runtime. For more details, please refer to the "Running computations in parallel to reduce runtime" section below.

Changing number of samples, burn-in, and thinning

Three options control the behaviour of each MCMC chain used to sample trees.

• Number of MCMC samples: by changing the --trees-per-chain option, you can make each chain sample more or fewer trees. The more samples each chain takes, the more likely it is that those samples will be from a good approximation of the true posterior distribution over trees permitted by your data. By default, Pairtree will take 3000 total samples per chain.

• Burn-in: each chain will take some number of samples to reach high-density regions of tree space, such that those trees come from a good approximation of the true posterior. The --burnin parameter controls what proportion of trees Pairtree will discard from the beginning of each chain, so as to avoid including poor-quality trees in your results. By default, Pairtree discards the first one-third of samples from each chain.

• Thinning: MCMC samples inherently exhibit auto-correlation, since each successive sample is based on a perturbation to the preceding one. We can reduce this effect by taking multiple samples for each one we record. You can change the --thinned-frac option to control this behaviour. By default, --thinned-frac=1, such that every sample taken is recorded. By changing this, for instance, to --thinned-frac=0.1, Pairtree will only record every tenth sample it takes. By recording only a subset of samples taken, the computational burden associated with processing the results (e.g., by computing summary statistics over the distribution of recorded trees) is reduced, alongside the storage burden of writing many samples to disk.

Running computations in parallel to reduce runtime

Pairtree can leverage multiple CPUs both when computing the pairwise relations tensor and when sampling trees. Exploiting parallelism when computing the tensor can drastically reduce Pairtree's runtime, since every pair can be computed independently of every other. Conversely, exploiting parallelism when sampling trees won't necessarily improve runtime, since trees are sampled using MCMC, which is an inherently serial process, While no single MCMC chain can be run in parallel, however, you can run multiple independent chains in parallel, as discussed in the previous section. This will help Pairtree avoid local optima and produce better-quality results.

By default, Pairtree will run in parallel on all present CPUs. (On hyperthreaded systems, this number will usually be inferred to be twice the number of physical CPU cores.) You can change this behaviour by specifying the --parallel=N option, causing Pairtree to use N parallel processes instead. When computing the pairwise tensor, Pairtree will compute all pairs in parallel, such that computing the full tensor using N parallel processes should only take 1/N as much time as computing with a single process. Later, when sampling trees, Pairtree will by default run as many independent MCMC chains as there are parallel processes. In this scenario, if you have N CPUs, running only a single chain will be no faster than running N chains, since each chain executes concurrently. However, in the N-chain case, you will obtain N times as many samples, improving your result quality.

You can explicitly specify --tree-chains=M to run M chains instead of the default, which is to run a separate chain for each parallel process. If M is greater than the number of parallel processes N, the chains will execute serially, increasing runtime. (Given the serial execution, you should expect that tree sampling with M > N will take ceiling(M / N) times as long as

Fitting subclone frequencies to trees

Critical to the Pairtree algorithm is the step of fitting subclone frequencies to trees. Beyond the user's interpretation of subclone frequencies for downstream analysis of Pairtree's results, the subclone frequencies will also affect tree structure -- the data likelihood for a tree is determined by how well the tree-constrained subclone frequencies fit the observed VAF data. We provide several different algorithms for computing these frequencies that strike different balances between computational speed and result accuracy.

• --phi-fitter=projection: By default, Pairtree uses the "Efficient Projection onto the Perfect Phylogeny Model" algorithm developed in Jia et al. This uses a Gaussian approximation of the binomial likelihood we're trying to maximize.

• --phi-fitter=rprop: Uses the rprop variant of gradient descent. Separate step sizes are maintained for each frequency scalar in each tissue sample. Those step sizes are increased when the direction of a step is consistent with the previous step taken, and reduced otherwise. Generally, this algorithm will produce more accurate results than projection (i.e., higher data likelihoods for the same tree), since it is directly optimizing the binomial objective rather than a Gaussian approximation, but at a higher computational cost. The extra computational burden becomes greater with more subclones (e.g., 30 or more), while being insignificant for small trees (e.g., with ten subclones or fewer).

• --phi-fitter=proj_rprop: First run the default projection algorithm, then refine its results using rprop. While this can produce better subclone frequency values at the cost of more computation, it has in testing demonstrated little benefit to accuracy.

Given the tradeoff between speed and accuracy, rprop is recommended for small trees (e.g., ten subclones or fewer), while projection is more suitable for larger trees. If Pairtree seems to be producing poor results with projection, try with rprop instead, as optimizing the exact objective rather than an approximation may produce better trees.

When running with --phi-fitter=rprop or --phi-fitter=proj_rprop, you can adjust the number of iterations the rprop algorithm uses by specifying the --phi-iterations=N parameter. Runtime of rprop will be proportional to this parameter, so smaller values will decrease runtime, at the potential cost of accuracy of subclone frequencies.

While other options are provided (--phi-fitter=graddesc_old and --phi-fitter=rprop_old), these remain only as artifacts and will be removed in future verisons, and so should not be used.