Perfect Phylogeny Research Articles

The Bayesian Phylogenetic Bootstrap, Application to Short Trees and Branches.

Felsenstein's bootstrap is the most commonly used method to measure branch support in phylogenetics. Current sequencing technologies can result in massive sampling of taxa (e.g. SARS-CoV-2). In this case, the sequences are very similar, the trees are short, and the branches correspond to a small number of mutations (possibly 0). Nevertheless, these trees contain a strong signal, with unresolved parts but a low rate of false branches. With such data, Felsenstein's bootstrap is not satisfactory. Due to the frequentist nature of bootstrap sampling, the expected support of a branch corresponding to a single mutation is ∼63%, even though it is highly likely to be correct. Here we propose a Bayesian version of the phylogenetic bootstrap in which sites are assigned uninformative prior probabilities. The branch support can then be interpreted as a posterior probability. We do not view the alignment as a small subsample of a large sample of sites, but rather as containing all available information (e.g., as with complete viral genomes, which are becoming routine). We give formulas for expected supports under the assumption of perfect phylogeny, in both the frequentist and Bayesian frameworks, where a branch corresponding to a single mutation now has an expected support of ∼90%. Simulations show that these theoretical results are robust to realistic data. Analyses on low-homoplasy viral and non-viral datasets show that Bayesian bootstrap support is easier to interpret, with high supports for branches very likely to be correct. As homoplasy increases, the two supports become closer and strongly correlated.

Joint inference of cell lineage and mitochondrial evolution from single-cell sequencing data.

Eukaryotic cells contain organelles called mitochondria that have their own genome. Most cells contain thousands of mitochondria which replicate, even in nondividing cells, by means of a relatively error-prone process resulting in somatic mutations in their genome. Because of the higher mutation rate compared to the nuclear genome, mitochondrial mutations have been used to track cellular lineage, particularly using single-cell sequencing that measures mitochondrial mutations in individual cells. However, existing methods to infer the cell lineage tree from mitochondrial mutations do not model "heteroplasmy," which is the presence of multiple mitochondrial clones with distinct sets of mutations in an individual cell. Single-cell sequencing data thus provide a mixture of the mitochondrial clones in individual cells, with the ancestral relationships between these clones described by a mitochondrial clone tree. While deconvolution of somatic mutations from a mixture of evolutionarily related genomes has been extensively studied in the context of bulk sequencing of cancer tumor samples, the problem of mitochondrial deconvolution has the additional constraint that the mitochondrial clone tree must be concordant with the cell lineage tree. We formalize the problem of inferring a concordant pair of a mitochondrial clone tree and a cell lineage tree from single-cell sequencing data as the Nested Perfect Phylogeny Mixture (NPPM) problem. We derive a combinatorial characterization of the solutions to the NPPM problem, and formulate an algorithm, MERLIN, to solve this problem exactly using a mixed integer linear program. We show on simulated data that MERLIN outperforms existing methods that do not model mitochondrial heteroplasmy nor the concordance between the mitochondrial clone tree and the cell lineage tree. We use MERLIN to analyze single-cell whole-genome sequencing data of 5220 cells of a gastric cancer cell line and show that MERLIN infers a more biologically plausible cell lineage tree and mitochondrial clone tree compared to existing methods. https://github.com/raphael-group/MERLIN.

Three Metaheuristic Approaches for Tumor Phylogeny Inference: An Experimental Comparison

Being able to infer the clonal evolution and progression of cancer makes it possible to devise targeted therapies to treat the disease. As discussed in several studies, understanding the history of accumulation and the evolution of mutations during cancer progression is of key importance when devising treatment strategies. Given the importance of the task, many methods for phylogeny reconstructions have been developed over the years, mostly employing probabilistic frameworks. Our goal was to explore different methods to take on this phylogeny inference problem; therefore, we devised and implemented three different metaheuristic approaches—Particle Swarm Optimization (PSO), Genetic Programming (GP) and Variable Neighbourhood Search (VNS)—under the Perfect Phylogeny and the Dollo-k evolutionary models. We adapted the algorithms to be applied to this specific context, specifically to a tree-based search space, and proposed six different experimental settings, in increasing order of difficulty, to test the novel methods amongst themselves and against a state-of-the-art method. Of the three, the PSO shows particularly promising results and is comparable to published tools, even at this exploratory stage. Thus, we foresee great improvements if alternative definitions of distance and velocity in a tree space, capable of better handling such non-Euclidean search spaces, are devised in future works.

Enumeration of binary trees compatible with a perfect phylogeny

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy—a rooted, binary, timed tree, with tips corresponding to individual sequences. Under the infinitely-many-sites mutation model, mutations are randomly superimposed along the branches of the genealogy, so that every mutation occurs at a chromosomal site that has not previously mutated; if a mutation occurs at an interior branch, then all individuals descending from that branch carry the mutation. The implication is that observed patterns of molecular variation from this model impose combinatorial constraints on the hidden state space of genealogies. In particular, observed molecular variation can be represented in the form of a perfect phylogeny, a tree structure that fully encodes the mutational differences among sequences. For a sample of n sequences, a perfect phylogeny might not possess n distinct leaves, and hence might be compatible with many possible binary tree structures that could describe the evolutionary relationships among the n sequences. Here, we investigate enumerative properties of the set of binary ranked and unranked tree shapes that are compatible with a perfect phylogeny, and hence, the binary ranked and unranked tree shapes conditioned on an observed pattern of mutations under the infinitely-many-sites mutation model. We provide a recursive enumeration of these shapes. We consider both perfect phylogenies that can be represented as binary and those that are multifurcating. The results have implications for computational aspects of the statistical inference of evolutionary parameters that underlie sets of molecular sequences.

Strong inequalities and a branch-and-price algorithm for the convex recoloring problem

A coloring of the vertices of a connected graph is convex if each color class induces a connected subgraph. We address the convex recoloring problem: Given a connected graph G and a coloring of its vertices, recolor a minimum number of vertices of G so that the resulting coloring is convex. The convex recoloring problem (or CR, for short) was firstly motivated by applications on perfect phylogenies, and it is a hard computational problem even restricted to paths. In this work, we introduce a polytope based on connected subgraphs for CR and propose a class of strong inequalities with righthand side ranging from 1 to the number of colors. Those with righthand side one generalize and strengthen the inequalities of the corresponding formulation and comprise all facet-defining inequalities on binary coefficients when G is a tree. We report on computational experiments for an application on mobile networks to evaluate the potential of the proposed inequalities to reduce the integrality gap. Finally, we propose a branch-and-price algorithm to solve CR on trees and present the results of experiments on several classes of instances, including phylogenetic trees. The experiments indicate that our approach outperforms the fastest solving method in the literature.

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

Accuracy in Near-Perfect Virus Phylogenies.

Phylogenetic trees from real-world data often include short edges with very few substitutions per site, which can lead to partially resolved trees and poor accuracy. Theory indicates that the number of sites needed to accurately reconstruct a fully resolved tree grows at a rate proportional to the inverse square of the length of the shortest edge. However, when inferred trees are partially resolved due to short edges, “accuracy” should be defined as the rate of discovering false splits (clades on a rooted tree) relative to the actual number found. Thus, accuracy can be high even if short edges are common. Specifically, in a “near-perfect” parameter space in which trees are large, the tree length ξ (the sum of all edge lengths) is small, and rate variation is minimal, the expected false positive rate is less than ξ∕3; the exact value depends on tree shape and sequence length. This expected false positive rate is far below the false negative rate for small ξ and often well below 5% even when some assumptions are relaxed. We show this result analytically for maximum parsimony and explore its extension to maximum likelihood using theory and simulations. For hypothesis testing, we show that measures of split “support” that rely on bootstrap resampling consistently imply weaker support than that implied by the false positive rates in near-perfect trees. The near-perfect parameter space closely fits several empirical studies of human virus diversification during outbreaks and epidemics, including Ebolavirus, Zika virus, and SARS-CoV-2, reflecting low substitution rates relative to high transmission/sampling rates in these viruses.

Combinatorial perspectives on Dollo-k characters in phylogenetics

Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-k characters, i.e. traits that are gained at most once and lost precisely k times throughout evolution. We first introduce an algorithm based on the notion of spanning subtrees for finding a Dollo-k labelling for a given character and a given tree in linear time. We then compare persistent characters (consisting of the union of Dollo-0 and Dollo-1 characters) and general Dollo-k characters. While it is known that there is a strong connection between Fitch parsimony and persistent characters, we show that Dollo parsimony and Fitch parsimony are in general very different. Moreover, while it is known that there is a direct relationship between the number of persistent characters and the Sackin index of a tree, a popular index of tree balance, we show that this relationship does not generalize to Dollo-k characters. In fact, determining the number of Dollo-k characters for a given tree is much more involved than counting persistent characters, and we end this manuscript by introducing a recursive approach for the former. This approach leads to a polynomial time algorithm for counting the number of Dollo-k characters, and both this algorithm as well as the algorithm for computing Dollo-k labellings are publicly available in the Babel package for BEAST 2.

Distinguishing linear and branched evolution given single-cell DNA sequencing data of tumors

BackgroundCancer arises from an evolutionary process where somatic mutations give rise to clonal expansions. Reconstructing this evolutionary process is useful for treatment decision-making as well as understanding evolutionary patterns across patients and cancer types. In particular, classifying a tumor’s evolutionary process as either linear or branched and understanding what cancer types and which patients have each of these trajectories could provide useful insights for both clinicians and researchers. While comprehensive cancer phylogeny inference from single-cell DNA sequencing data is challenging due to limitations with current sequencing technology and the complexity of the resulting problem, current data might provide sufficient signal to accurately classify a tumor’s evolutionary history as either linear or branched.ResultsWe introduce the Linear Perfect Phylogeny Flipping (LPPF) problem as a means of testing two alternative hypotheses for the pattern of evolution, which we prove to be NP-hard. We develop Phyolin, which uses constraint programming to solve the LPPF problem. Through both in silico experiments and real data application, we demonstrate the performance of our method, outperforming a competing machine learning approach.ConclusionPhyolin is an accurate, easy to use and fast method for classifying an evolutionary trajectory as linear or branched given a tumor’s single-cell DNA sequencing data.

Tumor Phylogeny Topology Inference via Deep Learning.

SummaryPrincipled computational approaches for tumor phylogeny reconstruction via single-cell sequencing typically aim to build the most likely perfect phylogeny tree from the noisy genotype matrix – which represents genotype calls of single cells. This problem is NP-hard, and as a result, existing approaches aim to solve relatively small instances of it through combinatorial optimization techniques or Bayesian inference. As expected, even when the goal is to infer basic topological features of the tumor phylogeny, rather than reconstructing the topology entirely, these approaches could be prohibitively slow. In this paper, we introduce fast deep learning solutions to the problems of inferring whether the most likely tree has a linear (chain) or branching topology and whether a perfect phylogeny is feasible from a given genotype matrix. We also present a reinforcement learning approach for reconstructing the most likely tumor phylogeny. This preliminary work demonstrates that data-driven approaches can reconstruct key features of tumor evolution.

PhISCS-BnB: a fast branch and bound algorithm for the perfect tumor phylogeny reconstruction problem.

Recent advances in single-cell sequencing (SCS) offer an unprecedented insight into tumor emergence and evolution. Principled approaches to tumor phylogeny reconstruction via SCS data are typically based on general computational methods for solving an integer linear program, or a constraint satisfaction program, which, although guaranteeing convergence to the most likely solution, are very slow. Others based on Monte Carlo Markov Chain or alternative heuristics not only offer no such guarantee, but also are not faster in practice. As a result, novel methods that can scale up to handle the size and noise characteristics of emerging SCS data are highly desirable to fully utilize this technology. We introduce PhISCS-BnB (phylogeny inference using SCS via branch and bound), a branch and bound algorithm to compute the most likely perfect phylogeny on an input genotype matrix extracted from an SCS dataset. PhISCS-BnB not only offers an optimality guarantee, but is also 10-100 times faster than the best available methods on simulated tumor SCS data. We also applied PhISCS-BnB on a recently published large melanoma dataset derived from the sublineages of a cell line involving 20 clones with 2367 mutations, which returned the optimal tumor phylogeny in <4 h. The resulting phylogeny agrees with and extends the published results by providing a more detailed picture on the clonal evolution of the tumor. https://github.com/algo-cancer/PhISCS-BnB. Supplementary data are available at Bioinformatics online.

Combinatorial views on persistent characters in phylogenetics

The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well-known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one 0→1 transition followed by at most one 1→0 transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize persistent characters in terms of the first phase of the famous Fitch algorithm. Afterwards we focus on the number of persistent characters for a given phylogenetic tree. We show that this number solely depends on the balance of the tree. To be precise, we develop a formula for counting the number of persistent characters for a given phylogenetic tree based on an index of tree balance, namely the Sackin index. Lastly, we consider the question of how many (carefully chosen) binary characters together with their persistence status are needed to uniquely determine a phylogenetic tree and provide an upper bound for the number of characters needed.

Finding Most Compatible Phylogenetic Trees over Multi-State Characters

The reconstruction of the evolutionary tree of a set of species based on qualitative attributes is a central problem in phylogenetics. In the NP-hard perfect phylogeny problem the input is a set of taxa (species) and characters (attributes) on them, and the task is to find an evolutionary tree that describes the evolution of the taxa so that each character state evolves only once. However, in practical situations a perfect phylogeny rarely exists, motivating the maximum compatibility problem of finding the largest subset of characters admitting a perfect phylogeny. Various declarative approaches, based on applying integer programming (IP), answer set programming (ASP) and pseudo-Boolean optimization (PBO) solvers, have been proposed for maximum compatibility. In this work we develop a new hybrid approach to solving maximum compatibility for multi-state characters, making use of both declarative optimization techniques (specifically maximum satisfiability, MaxSAT) and an adaptation of the Bouchitt'e-Todinca approach to triangulation-based graph optimization problems. Empirically our approach outperforms in scalability the earlier proposed approaches w.r.t. various parameters underlying the problem.

No background in biology is assumed

No background in biology is assumed

Clustering the biological networks using shortest path.

Abstract The aim of graph clustering is to partition the given graph into subgraphs, and then assign the subgraphs to overlapping or non-overlapping clusters. In this process, densely connected subgraphs fall into a single cluster, thereby maximizing the intra-cluster similarity. Here we propose a clustering method based on the shortest path between the nodes. We propose a 2- partition in a few graph classes namely outerplanar graphs, graphs with simplicial vertices, chordal graphs, distance hereditary graphs and cographs. The solution to the chordal and outerplanar graphs can be found in linear time. Chordal graphs are widely used in perfect phylogeny related research whereas cographs are used to model the orthology relations. Hence a clustering method based on shortest path in chordal graphs and cographs may have significant impact in the research related to orthologous genes as well as phylogenetic relations.

PerfectphyloR: An R package for reconstructing perfect phylogenies

BackgroundA perfect phylogeny is a rooted binary tree that recursively partitions sequences. The nested partitions of a perfect phylogeny provide insight into the pattern of ancestry of genetic sequence data. For example, sequences may cluster together in a partition indicating that they arise from a common ancestral haplotype.ResultsWe present an R package perfectphyloR to reconstruct the local perfect phylogenies underlying a sample of binary sequences. The package enables users to associate the reconstructed partitions with a user-defined partition. We describe and demonstrate the major functionality of the package.ConclusionThe perfectphyloR package should be of use to researchers seeking insight into the ancestral structure of their sequence data. The reconstructed partitions have many applications, including the mapping of trait-influencing variants.

PhISCS: a combinatorial approach for subperfect tumor phylogeny reconstruction via integrative use of single-cell and bulk sequencing data.

Available computational methods for tumor phylogeny inference via single-cell sequencing (SCS) data typically aim to identify the most likely perfect phylogeny tree satisfying the infinite sites assumption (ISA). However, the limitations of SCS technologies including frequent allele dropout and variable sequence coverage may prohibit a perfect phylogeny. In addition, ISA violations are commonly observed in tumor phylogenies due to the loss of heterozygosity, deletions, and convergent evolution. In order to address such limitations, we introduce the optimal subperfect phylogeny problem which asks to integrate SCS data with matching bulk sequencing data by minimizing a linear combination of potential false negatives (due to allele dropout or variance in sequence coverage), false positives (due to read errors) among mutation calls, and the number of mutations that violate ISA (real or because of incorrect copy number estimation). We then describe a combinatorial formulation to solve this problem which ensures that several lineage constraints imposed by the use of variant allele frequencies (VAFs, derived from bulk sequence data) are satisfied. We express our formulation both in the form of an integer linear program (ILP) and—as a first in tumor phylogeny reconstruction—a Boolean constraint satisfaction problem (CSP) and solve them by leveraging state-of-the-art ILP/CSP solvers. The resulting method, which we name PhISCS, is the first to integrate SCS and bulk sequencing data while accounting for ISA violating mutations. In contrast to the alternative methods, typically based on probabilistic approaches, PhISCS provides a guarantee of optimality in reported solutions. Using simulated and real data sets, we demonstrate that PhISCS is more general and accurate than all available approaches.

Implications of non-uniqueness in phylogenetic deconvolution of bulk DNA samples of tumors

BackgroundTumors exhibit extensive intra-tumor heterogeneity, the presence of groups of cellular populations with distinct sets of somatic mutations. This heterogeneity is the result of an evolutionary process, described by a phylogenetic tree. In addition to enabling clinicians to devise patient-specific treatment plans, phylogenetic trees of tumors enable researchers to decipher the mechanisms of tumorigenesis and metastasis. However, the problem of reconstructing a phylogenetic tree T given bulk sequencing data from a tumor is more complicated than the classic phylogeny inference problem. Rather than observing the leaves of T directly, we are given mutation frequencies that are the result of mixtures of the leaves of T. The majority of current tumor phylogeny inference methods employ the perfect phylogeny evolutionary model. The underlying Perfect Phylogeny Mixture (PPM) combinatorial problem typically has multiple solutions.ResultsWe prove that determining the exact number of solutions to the PPM problem is #P-complete and hard to approximate within a constant factor. Moreover, we show that sampling solutions uniformly at random is hard as well. On the positive side, we provide a polynomial-time computable upper bound on the number of solutions and introduce a simple rejection-sampling based scheme that works well for small instances. Using simulated and real data, we identify factors that contribute to and counteract non-uniqueness of solutions. In addition, we study the sampling performance of current methods, identifying significant biases.ConclusionsAwareness of non-uniqueness of solutions to the PPM problem is key to drawing accurate conclusions in downstream analyses based on tumor phylogenies. This work provides the theoretical foundations for non-uniqueness of solutions in tumor phylogeny inference from bulk DNA samples.

Does Relaxing the Infinite Sites Assumption Give Better Tumor Phylogenies? An ILP-Based Comparative Approach.

Most of the evolutionary history reconstruction approaches are based on the infinite sites assumption, which states that mutations appear once in the evolutionary history. The Perfect Phylogeny model is the result of the infinite sites assumption and has been widely used to infer cancer evolution. Nonetheless, recent results show that recurrent and back mutations are present in the evolutionary history of tumors, hence the Perfect Phylogeny model might be too restrictive. We propose an approach that allows losing previously acquired mutations and multiple acquisitions of a character. Moreover, we provide an ILP formulation for the evolutionary tree reconstruction problem. Our formulation allows us to tackle both the Incomplete Directed Phylogeny problem and the Clonal Reconstruction problem when general evolutionary models are considered. The latter problem is fundamental in cancer genomics, the goal is to study the evolutionary history of a tumor considering as input data the fraction of cells having a certain mutation in a set of cancer samples. For the Clonal Reconstruction problem, an experimental analysis shows the advantage of allowing mutation losses. Namely, by analyzing real and simulated datasets, our ILP approach provides a better interpretation of the evolutionary history than a Perfect Phylogeny. The software is at https://github.com/AlgoLab/gppf.

Accurate and efficient cell lineage tree inference from noisy single cell data: the maximum likelihood perfect phylogeny approach

Cells in an organism share a common evolutionary history, called cell lineage tree. Cell lineage tree can be inferred from single cell genotypes at genomic variation sites. Cell lineage tree inference from noisy single cell data is a challenging computational problem. Most existing methods for cell lineage tree inference assume uniform uncertainty in genotypes. A key missing aspect is that real single cell data usually has non-uniform uncertainty in individual genotypes. Moreover, existing methods are often sampling based and can be very slow for large data. In this article, we propose a new method called ScisTree, which infers cell lineage tree and calls genotypes from noisy single cell genotype data. Different from most existing approaches, ScisTree works with genotype probabilities of individual genotypes (which can be computed by existing single cell genotype callers). ScisTree assumes the infinite sites model. Given uncertain genotypes with individualized probabilities, ScisTree implements a fast heuristic for inferring cell lineage tree and calling the genotypes that allow the so-called perfect phylogeny and maximize the likelihood of the genotypes. Through simulation, we show that ScisTree performs well on the accuracy of inferred trees, and is much more efficient than existing methods. The efficiency of ScisTree enables new applications including imputation of the so-called doublets. The program ScisTree is available for download at: https://github.com/yufengwudcs/ScisTree. Supplementary data are available at Bioinformatics online.