Clades, clans, and reciprocal monophyly under neutral evolutionary models

The Yule model and the coalescent model are two neutral stochastic models for generating trees in phylogenetics and population genetics, respectively. Although these models are quite different, they lead to identical distributions concerning the probability that pre-specified groups of taxa form monophyletic groups (clades) in the tree. We extend earlier work to derive exact formulae for the probability of finding one or more groups of taxa as clades in a rooted tree, or as ‘clans’ in an unrooted tree. Our findings are relevant for calculating the statistical significance of observed monophyly and reciprocal monophyly in phylogenetics.     

Uprooted Phylogenetic Networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system \(\Sigma (N)\) induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.     

Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent

Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals—each with many genes—splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are four species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendant branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.     

Fast local search for unrooted Robinson-Foulds supertrees

A Robinson-Foulds (RF) supertree for a collection of input trees is a tree containing all the species in the input trees that is at minimum total RF distance to the input trees. Thus, an RF supertree is consistent with the maximum number of splits in the input trees. Constructing RF supertrees for rooted and unrooted data is NP-hard. Nevertheless, effective local search heuristics have been developed for the restricted case where the input trees and the supertree are rooted. We describe new heuristics, based on the Edge Contract and Refine (ECR) operation, that remove this restriction, thereby expanding the utility of RF supertrees. Our experimental results on simulated and empirical data sets show that our unrooted local search algorithms yield better supertrees than those obtained from MRP and rooted RF heuristics in terms of total RF distance to the input trees and, for simulated data, in terms of RF distance to the true tree.     

Algorithms for MDC-based multi-locus phylogeny inference: beyond rooted binary gene trees on single alleles

One of the criteria for inferring a species tree from a collection of gene trees, when gene tree incongruence is assumed to be due to incomplete lineage sorting (ILS), is Minimize Deep Coalescence (MDC). Exact algorithms for inferring the species tree from rooted, binary trees under MDC were recently introduced. Nevertheless, in phylogenetic analyses of biological data sets, estimated gene trees may differ from true gene trees, be incompletely resolved, and not necessarily rooted. In this article, we propose new MDC formulations for the cases where the gene trees are unrooted/binary, rooted/non-binary, and unrooted/non-binary. Further, we prove structural theorems that allow us to extend the algorithms for the rooted/binary gene tree case to these cases in a straightforward manner. In addition, we devise MDC-based algorithms for cases when multiple alleles per species may be sampled. We study the performance of these methods in coalescent-based computer simulations.     

Closure operations in phylogenetics

Closure operations are a useful device in both the theory and practice of tree reconstruction in biology and other areas of classification. These operations take a collection of trees (rooted or unrooted) that classify overlapping sets of objects at their leaves, and infer further tree-like relationships. In this paper we investigate closure operations on phylogenetic trees; both rooted and unrooted; as well as on X-splits, and in a general abstract setting. We derive a number of new results, particularly concerning the completeness (and incompleteness) and complexity of various types of closure rules.     

Inferring phylogeny from whole genomes

MOTIVATION: Inferring species phylogenies with a history of gene losses and duplications is a challenging and an important task in computational biology. This problem can be solved by duplication-loss models in which the primary step is to reconcile a rooted gene tree with a rooted species tree. Most modern methods of phylogenetic reconstruction (from sequences) produce unrooted gene trees. This limitation leads to the problem of transforming unrooted gene tree into a rooted tree, and then reconciling rooted trees. The main questions are 'What about biological interpretation of choosing rooting?', 'Can we find efficiently the optimal rootings?', 'Is the optimal rooting unique?'. RESULTS: In this paper we present a model of reconciling unrooted gene tree with a rooted species tree, which is based on a concept of choosing rooting which has minimal reconciliation cost. Our analysis leads to the surprising property that all the minimal rootings have identical distributions of gene duplications and gene losses in the species tree. It implies, in our opinion, that the concept of an optimal rooting is very robust, and thus biologically meaningful. Also, it has nice computational properties. We present a linear time and space algorithm for computing optimal rooting(s). This algorithm was used in two different ways to reconstruct the optimal species phylogeny of five known yeast genomes from approximately 4700 gene trees. Moreover, we determined locations (history) of all gene duplications and gene losses in the final species tree. It is interesting to notice that the top five species trees are the same for both methods. AVAILABILITY: Software and documentation are freely available from http://bioputer.mimuw.edu.pl/~gorecki/urec     

Genealogies: Pedigrees and Phylogenies are Reticulating Networks Not Just Divergent Trees

Pedigrees illustrate the genealogical relationships among individuals, and phylogenies do the same for groups of organisms (such as species, genera, etc.). Here, I provide a brief survey of current concepts and methods for calculating and displaying genealogical relationships. These relationships have long been recognized to be reticulating, rather than strictly divergent, and so both pedigrees and phylogenies are correctly treated as networks rather than trees. However, currently most pedigrees are instead presented as “family trees”, and most phylogenies are presented as phylogenetic trees. Nevertheless, the historical development of concepts shows that networks pre-dated trees in most fields of biology, including the study of pedigrees, biology theory, and biology practice, as well as in historical linguistics in the social sciences. Trees were actually introduced in order to provide a simpler conceptual model for historical relationships, since trees are a specific type of simple network. Computationally, trees and networks are a part of graph theory, consisting of nodes connected by edges. In this mathematical context they differ solely in the absence or presence of reticulation nodes, respectively. There are two types of graphs that can be called phylogenetic networks: (1) rooted evolutionary networks, and (2) unrooted affinity networks. There are quite a few computational methods for unrooted networks, which have two main roles in phylogenetics: (a) they act as a generic form of multivariate data display; and (b) they are used specifically to represent haplotype networks. Evolutionary networks are more difficult to infer and analyse, as there is no mathematical algorithm for reconstructing unique historical events. There is thus currently no coherent analytical framework for computing such networks.     

URec: a system for unrooted reconciliation

URec is a software based on a concept of unrooted reconciliation. It can be used to reconcile a set of unrooted gene trees with a rooted species tree or a set of rooted species trees. Moreover, it computes detailed distribution of gene duplications and gene losses in a species tree. It can be used to infer optimal species phylogenies for a given set of gene trees. URec is implemented in C++ and can be easily compiled under Unix and Windows systems. Availability: Software is freely available for download from our website at http://bioputer.mimuw.edu.pl/~gorecki/urec. This webpage also contains Windows executables and a number of advanced examples with explanations.     

Property and efficiency of the maximum likelihood method for molecular phylogeny

Summary The maximum likelihood (ML) method for constructing phylogenetic trees (both rooted and unrooted trees) from DNA sequence data was studied. Although there is some theoretical problem in the comparison of ML values conditional for each topology, it is possible to make a heuristic argument to justify the method. Based on this argument, a new algorithm for estimating the ML tree is presented. It is shown that under the assumption of a constant rate of evolution, the ML method and UPGMA always give the same rooted tree for the case of three operational taxonomic units (OTUs). This also seems to hold approximately for the case with four OTUs. When we consider unrooted trees with the assumption of a varying rate of nucleotide substitution, the efficiency of the ML method in obtaining the correct tree is similar to those of the maximum parsimony method and distance methods. The ML method was applied to Brown et al.'s data, and the tree topology obtained was the same as that found by the maximum parsimony method, but it was different from those obtained by distance methods.     

Improved parameterized complexity of the maximum agreement subtree and maximum compatible tree problems

Given a set of evolutionary trees on a same set of taxa, the maximum agreement subtree problem (MAST), respectively, maximum compatible tree problem (MCT), consists of finding a largest subset of taxa such that all input trees restricted to these taxa are isomorphic, respectively compatible. These problems have several applications in phylogenetics such as the computation of a consensus of phylogenies obtained from different data sets, the identification of species subjected to horizontal gene transfers and, more recently, the inference of supertrees, e.g., Trees Of Life. We provide two linear time algorithms to check the isomorphism, respectively, compatibility, of a set of trees or otherwise identify a conflict between the trees with respect to the relative location of a small subset of taxa. Then, we use these algorithms as subroutines to solve MAST and MCT on rooted or unrooted trees of unbounded degree. More precisely, we give exact fixed-parameter tractable algorithms, whose running time is uniformly polynomial when the number of taxa on which the trees disagree is bounded. The improves on a known result for MAST and proves fixed-parameter tractability for MCT.     

Root location in random trees: A polarity property of all sampling consistent phylogenetic models except one

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we show that among models that treat all taxa equally, and are sampling consistent (i.e. the distribution on trees is not affected by taxa yet to be included), all such models, except one (the so-called PDA model), convey some information as to the location of the ancestral root in an unrooted tree.     

The combinatorics of tandem duplication trees

We developed a recurrence relation that counts the number of tandem duplication trees (either rooted or unrooted) that are consistent with a set of n tandemly repeated sequences generated under the standard unequal recombination (or crossover) model of tandem duplications. The number of rooted duplication trees is exactly twice the number of unrooted trees, which means that on average only two positions for a root on a duplication tree are possible. Using the recurrence, we tabulated these numbers for small values of n. We also developed an asymptotic formula that for large n provides estimates for these numbers. These numbers give a priori probabilities for phylogenies of the repeated sequences to be duplication trees. This work extends earlier studies where exhaustive counts of the numbers for small n were obtained. One application showed the significance of finding that most maximum-parsimony trees constructed from repeat sequences from human immunoglobins and T-cell receptors were tandem duplication trees. Those findings provided strong support to the proposed mechanisms of tandem gene duplication. The recurrence relation also suggests efficient algorithms to recognize duplication trees and to generate random duplication trees for simulation. We present a linear-time recognition algorithm.     

Graph theory, tionary trees and classification

Many methods have been used for analysing information about organisms in order to understand tionary relationships and/or to determine classifications. The reationship between some of these methods is illustrated for the character state matrix, incompatibility and similarity matrices, minimal unrooted and rooted trees, and tionary classifications. Existing methods of determining the shortest possible tree are described. In addition a new method of building a minimal tree is introduced which starts with the largest possible subset (clique) of characters that is compatible for all pairs of characters. The remaining characters are ranked in order of their increasing number of incompatibilities. These characters are added singly, a tree constructed and then tested for minimality by previously described methods for partitioning characters into subsets. The procedure is repeated at least until the tree can no longer be proved minimal. The relationship between trees and tionary and phylogenetic classifications has been neglected but three methods are metioned and a new criterion suggested. It is suggested that graph theory, rather than statistics, is better suited for the primary analysis of comparative data.     

Accuracy of phylogenetic trees estimated from DNA sequence data

The relative merits of four different tree-making methods in obtaining the correct topology were studied by using computer simulation. The methods studied were the unweighted pair-group method with arithmetic mean (UPGMA), Fitch and Margoliash's (FM) method, thd distance Wagner (DW) method, and Tateno et al.'s modified Farris (MF) method. An ancestral DNA sequence was assumed to evolve into eight sequences following a given model tree. Both constant and varying rates of nucleotide substitution were considered. Once the DNA sequences for the eight extant species were obtained, phylogenetic trees were constructed by using corrected (d) and uncorrected (p) nucleotide substitutions per site. The topologies of the trees obtained were then compared with that of the model tree. The results obtained can be summarized as follows: (1) The probability of obtaining the correct rooted or unrooted tree is low unless a large number of nucleotide differences exists between different sequences. (2) When the number of nucleotide substitutions per sequence is small or moderately large, the FM, DW, and MF methods show a better performance than UPGMA in recovering the correct topology. The former group of methods is particularly good for obtaining the correct unrooted tree. (3) When the number of substitutions per sequence is large, UPGMA is at least as good as the other methods, particularly for obtaining the correct rooted tree. (4) When the rate of nucleotide substitution varies with evolutionary lineage, the FM, DW, and MF methods show a better performance in obtaining the correct topology than UPGMA, except when a rooted tree is to be produced from data with a large number of nucleotide substitutions per sequence.(ABSTRACT TRUNCATED AT 250 WORDS)      

Split Probabilities and Species Tree Inference Under the Multispecies Coalescent Model

Using topological summaries of gene trees as a basis for species tree inference is a promising approach to obtain acceptable speed on genomic-scale datasets, and to avoid some undesirable modeling assumptions. Here we study the probabilities of splits on gene trees under the multispecies coalescent model, and how their features might inform species tree inference. After investigating the behavior of split consensus methods, we investigate split invariants—that is, polynomial relationships between split probabilities. These invariants are then used to show that, even though a split is an unrooted notion, split probabilities retain enough information to identify the rooted species tree topology for trees of 5 or more taxa, with one possible 6-taxon exception.     

All galls are divided into three or more parts: recursive enumeration of labeled histories for galled trees

Objective

In mathematical phylogenetics, a labeled rooted binary tree topology can possess any of a number of labeled histories, each of which represents a possible temporal ordering of its coalescences. Labeled histories appear frequently in calculations that describe the combinatorics of phylogenetic trees. Here, we generalize the concept of labeled histories from rooted phylogenetic trees to rooted phylogenetic networks, specifically for the class of rooted phylogenetic networks known as rooted galled trees.

Results

Extending a recursive algorithm for enumerating the labeled histories of a labeled tree topology, we present a method to enumerate the labeled histories associated with a labeled rooted galled tree. The method relies on a recursive decomposition by which each gall in a galled tree possesses three or more descendant subtrees. We exhaustively provide the numbers of labeled histories for all small galled trees, finding that each gall reduces the number of labeled histories relative to a specified galled tree that does not contain it.

Conclusion

The results expand the set of structures for which labeled histories can be enumerated, extending a well-known calculation for phylogenetic trees to a class of phylogenetic networks.

     

Genomic duplication problems for unrooted gene trees

Background

Discovering the location of gene duplications and multiple gene duplication episodes is a fundamental issue in evolutionary molecular biology. The problem introduced by Guigó et al. in 1996 is to map gene duplication events from a collection of rooted, binary gene family trees onto theirs corresponding rooted binary species tree in such a way that the total number of multiple gene duplication episodes is minimized. There are several models in the literature that specify how gene duplications from gene families can be interpreted as one duplication episode. However, in all duplication episode problems gene trees are rooted. This restriction limits the applicability, since unrooted gene family trees are frequently inferred by phylogenetic methods.

Results

In this article we show the first solution to the open problem of episode clustering where the input gene family trees are unrooted. In particular, by using theoretical properties of unrooted reconciliation, we show an efficient algorithm that reduces this problem into the episode clustering problems defined for rooted trees. We show theoretical properties of the reduction algorithm and evaluation of empirical datasets.

Conclusions

We provided algorithms and tools that were successfully applied to several empirical datasets. In particular, our comparative study shows that we can improve known results on genomic duplication inference from real datasets.

     

Properties of phylogenetic trees generated by Yule-type speciation models

We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. In this paper we describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth (number of edges from the root) of the most recent common ancestor of a random subset of k species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs in rapid bursts.     

Phylogenetic identification of lateral genetic transfer events

Background  

Lateral genetic transfer can lead to disagreements among phylogenetic trees comprising sequences from the same set of taxa. Where topological discordance is thought to have arisen through genetic transfer events, tree comparisons can be used to identify the lineages that may have shared genetic information. An 'edit path' of one or more transfer events can be represented with a series of subtree prune and regraft (SPR) operations, but finding the optimal such set of operations is NP-hard for comparisons between rooted trees, and may be so for unrooted trees as well.