BACKGROUND: The advent of next-generation and long-read sequencing technologies has provided an ever-increasing wealth of phylogenetic data that require specially designed algorithms to decipher the underlying evolutionary relationships. As large-scale data become increasingly accessible, there is a concomitant need for efficient computational libraries that facilitate the development and dissemination of specialized algorithms for phylogenetic comparative biology. RESULTS: We introduce Phylo-rs: a fast, extensible, general-purpose library for phylogenetic analysis and inference written in the Rust programming language. Phylo-rs leverages a combination of speed, memory-safety, and native WebAssembly support offered by Rust to provide a robust set of memory-efficient data structures and elementary phylogenetic algorithms. Phylo-rs focuses on the efficient and convenient deployment of software aimed at large-scale phylogenetic analysis and inference. Scalability analysis against popular libraries shows that Phylo-rs performs comparably or better on key algorithms. We utilized it to assess the phylogenetic diversity of influenza A virus in swine, identifying virus groups that are undergoing evolutionary expansion that could be targeted for control through multivalent vaccines. Additionally, we used Phylo-rs to enhance phylogenetic inference by visualizing tree space from Markov chain Monte Carlo (MCMC) Bayesian analysis, efficiently computing approximately five billion tree pair distances to evaluate convergence and select MCMC runs for genomic epidemiology. CONCLUSION: Phylo-rs enables the design and implementation of cutting-edge software for phylogenetic analysis, thereby facilitating the application and dissemination of theoretical advancements in biology. Phylo-rs is available under an open-source license on GitHub at https://github.com/sriram98v/phylo-rs , with documentation available at https://docs.rs/phylo/latest/phylo/ .

The cophenetic distance is a well-established metric in biology used to compare pairs of trees represented in a vector format. This distance was introduced by Cardona and his co-authors, building on the foundational work of Sokal and Rohlf, which dates back over 60 years. It is widely recognized for its versatility since it can analyze trees with edge weights using various vector norms. However, when comparing large-scale trees, the quadratic runtime of the current best-known (i.e., naïve) algorithm for computing the cophenetic distance can become prohibitive. Recently, a new algorithmic framework with near-linear time complexity has been developed to calculate the distances of a generalized class of cophenetic distances, which are derived from the work of Sokal and Rohlf. This improvement not only allows the cophenetic distance to be utilized in large-scale studies but also enhances the versatility of these studies by incorporating generalized variants of the cophenetic distance. However, the framework is limited to applying only the L1 and L2 vector norms, which significantly restricts the versatility of generalized cophenetic distances in large-scale applications. To address this limitation, we present a near-linear time algorithmic framework for computing the generalized cophenetic distances across all Lp vector norms. In our scalability study, we showcase the practical performance of our unrestricted algorithmic framework. Furthermore, we investigate the applicability of the generalized cophenetic distances by analyzing the distributions of key components of these distances under various vector norms.

The domain of software design is gaining a long overdue recognition as a vital discipline within software engineering, necessitating innovative approaches to its teaching in undergraduate education. Despite the growing importance of software design, academic programs often treat it as a secondary skill, overshadowed by the strong emphasis on coding. Only a few schools offer a design degree or dedicated design courses. This disparity between industry demands and educational practices underscores the need for novel pedagogical strategies. In our innovative work, we discuss a new way of effectively teaching software design-by-analogy for undergraduates to help them rapidly acquire the essential skills needed to design complex software without getting entangled in complex code generation and management. Software design does not necessarily follow the same clear delineation/separation between modules and components naturally apparent in tangible engineering domains. We employ “Conceptual Blending” to help students map their everyday experiences onto software design concepts. The process begins with students analyzing a simple two-arm watch to identify its user interface and create a finite state automaton for its interaction design. Success rates decline as the complexity of the watches increases, underscoring the software design challenges. By comparing these exercises to software interfaces, students learn to apply design techniques such as navigation modeling and prototyping, ensuring they can create intuitive, user-friendly software.

The classic quantitative measure of phylogenetic diversity (PD) has been used to address problems in conservation biology, microbial ecology, and evolutionary biology. PD is the minimum total length of the branches in a phylogeny required to cover a specified set of taxa on the phylogeny. A general goal in the application of PD has been identifying a set of taxa of size k that maximize PD on a given phylogeny; this has been mirrored in active research to develop efficient algorithms for the problem. Other descriptive statistics, such as the minimum PD, average PD, and standard deviation of PD, can provide invaluable insight into the distribution of PD across a phylogeny (relative to a fixed value of k). However, there has been limited or no research on computing these statistics, especially when required for each clade in a phylogeny, enabling direct comparisons of PD between clades. We introduce efficient algorithms for computing PD and the associated descriptive statistics for a given phylogeny and each of its clades. In simulation studies, we demonstrate the ability of our algorithms to analyze large-scale phylogenies with applications in ecology and evolutionary biology. The software is available at https://github.com/flu-crew/PD_stats.

Diverse Natural Language Processing tasks employ constituency parsing to understand the syntactic structure of a sentence according to a phrase structure grammar. Many state-of-the-art constituency parsers are proposed, but they may provide different results for the same sentences, especially for corpora outside their training domains. This paper adopts the truth discovery idea to aggregate constituency parse trees from different parsers by estimating their reliability in the absence of ground truth. Our goal is to consistently obtain high-quality aggregated constituency parse trees. We formulate the constituency parse tree aggregation problem in two steps, structure aggregation and constituent label aggregation. Specifically, we propose the first truth discovery solution for tree structures by minimizing the weighted sum of Robinson-Foulds (RF) distances, a classic symmetric distance metric between two trees. Extensive experiments are conducted on benchmark datasets in different languages and domains. The experimental results show that our method, CPTAM, outperforms the state-of-the-art aggregation baselines. We also demonstrate that the weights estimated by CPTAM can adequately evaluate constituency parsers in the absence of ground truth.

MOTIVATION: A phylogenetic network is a powerful model to represent entangled evolutionary histories with both divergent (speciation) and convergent (e.g. hybridization, reassortment, recombination) evolution. The standard approach to inference of hybridization networks is to (i) reconstruct rooted gene trees and (ii) leverage gene tree discordance for network inference. Recently, we introduced a method called RF-Net for accurate inference of virus reassortment and hybridization networks from input gene trees in the presence of errors commonly found in phylogenetic trees. While RF-Net demonstrated the ability to accurately infer networks with up to four reticulations from erroneous input gene trees, its application was limited by the number of reticulations it could handle in a reasonable amount of time. This limitation is particularly restrictive in the inference of the evolutionary history of segmented RNA viruses such as influenza A virus (IAV), where reassortment is one of the major mechanisms shaping the evolution of these pathogens. RESULTS: Here, we expand the functionality of RF-Net that makes it significantly more applicable in practice. Crucially, we introduce a fast extension to RF-Net, called Fast-RF-Net, that can handle large numbers of reticulations without sacrificing accuracy. In addition, we develop automatic stopping criteria to select the appropriate number of reticulations heuristically and implement a feature for RF-Net to output error-corrected input gene trees. We then conduct a comprehensive study of the original method and its novel extensions and confirm their efficacy in practice using extensive simulation and empirical IAV evolutionary analyses. AVAILABILITY AND IMPLEMENTATION: RF-Net 2 is available at https://github.com/flu-crew/rf-net-2. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: The classic multispecies coalescent (MSC) model provides the means for theoretical justification of incomplete lineage sorting-aware species tree inference methods. This has motivated an extensive body of work on phylogenetic methods that are statistically consistent under MSC. One such particularly popular method is ASTRAL, a quartet-based species tree inference method. Novel studies suggest that ASTRAL also performs well when given multi-locus gene trees in simulation studies. Further, Legried et al. recently demonstrated that ASTRAL is statistically consistent under the gene duplication and loss model (GDL). GDL is prevalent in evolutionary histories and is the first core process in the powerful duplication-loss-coalescence evolutionary model (DLCoal) by Rasmussen and Kellis. RESULTS: In this work, we prove that ASTRAL is statistically consistent under the general DLCoal model. Therefore, our result supports the empirical evidence from the simulation-based studies. More broadly, we prove that the quartet-based inference approach is statistically consistent under DLCoal. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Tree reconciliation costs are a popular choice to account for the discordance between the evolutionary history of a gene family (i.e., a gene tree), and the species tree through which this family has evolved. This discordance is accounted for by the minimum number of postulated evolutionary events necessary for reconciling the two trees. Such events include gene duplication, loss, and deep coalescence, and are used to define different types of tree reconciliation costs. For example, the duplication-loss cost for a gene tree and species tree accounts for the minimum number of gene duplications and losses necessary to reconcile these trees. Fundamental to the understanding of how gene trees and species trees relate to each other are the diameters of tree reconciliation costs. While such diameters have been well-researched, still absent from these studies are the unconstrained diameters for two of the classic tree reconciliation costs, namely the duplication-loss cost and the loss cost. Here, we show the essential mathematical properties of these diameters and provide efficient solutions for computing them. Finally, we analyze the distributions of these diameters using simulated datasets.

Solving median tree problems is a classic approach for inferring species trees from a collection of discordant gene trees. Median tree problems are typically NP-hard and dealt with by local search heuristics. Unfortunately, such heuristics generally lack provable correctness and precision. Algorithmic advances addressing this uncertainty have led to exact dynamic programming formulations suitable to solve a well-studied group of median tree problems for smaller phylogenetic analyses. However, these formulations allow computing only very few optimal species trees out of possibly many such trees, and phylogenetic studies often require the analysis of all optimal solutions through their consensus tree. Here, we describe a significant algorithmic modification of the dynamic programming formulations that compute the cluster counts of all optimal species trees from which various types of consensus trees can be efficiently computed. Through experimental studies, we demonstrate that our parallel implementation of the modified dynamic programming formulation is more efficient than a previous implementation of the original formulation. Finally, we show that the parallel implementation can rapidly identify novel reassorted influenza A viruses potentially facilitating pandemic preparedness efforts.

MOTIVATION: The evolution of complexity is one of the most fascinating and challenging problems in modern biology, and tracing the evolution of complex traits is an open problem. In bacteria, operons and gene blocks provide a model of tractable evolutionary complexity at the genomic level. Gene blocks are structures of co-located genes with related functions, and operons are gene blocks whose genes are co-transcribed on a single mRNA molecule. The genes in operons and gene blocks typically work together in the same system or molecular complex. Previously, we proposed a method that explains the evolution of orthologous gene blocks (orthoblocks) as a combination of a small set of events that take place in vertical evolution from common ancestors. A heuristic method was proposed to solve this problem. However, no study was done to identify the complexity of the problem. RESULTS: Here, we establish that finding the homologous gene block problem is NP-hard and APX-hard. We have developed a greedy algorithm that runs in polynomial time and guarantees an O(ln⁡n) approximation. In addition, we formalize our problem as an integer linear program problem and solve it using the PuLP package and the standard CPLEX algorithm. Our exploration of several candidate operons reveals that our new method provides more optimal results than the results from the heuristic approach, and is significantly faster. AVAILABILITY AND IMPLEMENTATION: The software and data accompanying this paper are available under the GPLv3 and CC0 license respectively on: https://github.com/nguyenngochuy91/Relevant-Operon.

Inference of phylogenetic networks - the evolutionary histories of species involving speciation as well as reticulation events - has proved to be an extremely challenging problem even for smaller datasets easily tackled by supertree inference methods. An effective way to boost the scalability of distance-based supertree methods originates from the Pareto (for clusters) property, which is a highly desirable property for phylogenetic consensus methods. In particular, one can employ strict consensus merger algorithms to boost the scalability and accuracy of supertree methods satisfying Pareto; cf. SuperFine. In this work, we establish a Pareto-like property for phylogenetic networks. Then we consider the recently introduced RF-Net method that heuristically solves the so-called RF-Network problem and which was demonstrated to be an efficient and effective tool for the inference of hybridization and reassortment networks. As our main result, we provide a constructive proof (entailing an explicit refinement algorithm) that the Pareto property applies to the RF-Network problem when the solution space is restricted to the popular class of tree-child networks. This result implies that strict consensus merger strategies, similar to SuperFine, can be directly applied to boost both accuracy and scalability of RF-Net significantly. Finally, we further investigate the optimum solutions to the RF-Network problem; in particular, we describe structural properties of all optimum (tree-child) RF-networks in relation to strict consensus clusters of the input trees.

MOTIVATION: Complexity is a fundamental attribute of life. Complex systems are made of parts that together perform functions that a single component, or subsets of components, cannot. Examples of complex molecular systems include protein structures such as the F1Fo-ATPase, the ribosome, or the flagellar motor: each one of these structures requires most or all of its components to function properly. Given the ubiquity of complex systems in the biosphere, understanding the evolution of complexity is central to biology. At the molecular level, operons are classic examples of a complex system. An operon's genes are co-transcribed under the control of a single promoter to a polycistronic mRNA molecule, and the operon's gene products often form molecular complexes or metabolic pathways. With the large number of complete bacterial genomes available, we now have the opportunity to explore the evolution of these complex entities, by identifying possible intermediate states of operons. RESULTS: In this work, we developed a maximum parsimony algorithm to reconstruct ancestral operon states, and show a simple vertical evolution model of how operons may evolve from the individual component genes. We describe several ancestral states that are plausible functional intermediate forms leading to the full operon. We also offer Reconstruction of Ancestral Gene blocks Using Events or ROAGUE as a software tool for those interested in exploring gene block and operon evolution. AVAILABILITY AND IMPLEMENTATION: The software accompanying this paper is available under GPLv3 license on: https://github.com/nguyenngochuy91/Ancestral-Blocks-Reconstruction. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Median tree problems are powerful tools for inferring large-scale phylogenetic trees that hold enormous promise for society at large. Such problems seek a median tree for a given collection of input trees under some problem-specific distance. Here, we introduce a median tree problem under the classic Manhattan path-difference distance. We show that this problem is NP-hard, devise an ILP formulation, and provide an effective local search heuristic that is based on solving a local search problem exactly. Our algorithm for the local search problem improves asymptotically by a factor of $n$n on the best-known (naïve) solution, where $n$n is the overall number of taxa in the input trees. Finally, comparative phylogenetic studies using considerably large empirical data and an accuracy analysis for smaller phylogenetic trees reveal the ability of our novel heuristic.

Synthesizing large-scale phylogenetic trees is a fundamental problem in evolutionary biology. Median tree problems have evolved as a powerful tool to reconstruct such trees. Such problems seek a median tree for a given collection of input trees under some problem-specific tree distance. There has been an increased interest in the median tree problem for the classical path-difference distance between trees. While this problem is NP-hard, standard local search heuristics have been described that are based on solving a local search problem exactly. For a more effective heuristic we devise a time efficient algorithm for the local search problem that improves on the best-known solution by a factor of $n$n, where $n$n is the size of the input trees. Furthermore, we introduce a novel hybrid version of the standard local search that is exploiting our new algorithm for a more refined heuristic search. Finally, we demonstrate the performance of our hybrid heuristic in a comparative study with other commonly used methods that synthesize species trees using published empirical data sets.

Median tree inference under path-difference metrics has shown great promise for large-scale phylogeny estimation. Similar to these metrics is the family of cophenetic metrics that originates from a classic dendrogram comparison method introduced more than 50 years ago. Despite the appeal of this family of metrics, the problem of computing median trees under cophenetic metrics has not been analyzed. Like other standard median tree problems relevant in practice, as we show here, this problem is also NP-hard. NP-hard median tree problems have been successfully addressed by local search heuristics that are solving thousands of instances of a corresponding (local neighborhood) search problem. For the local neighborhood search problem under a cophenetic metric, the best known (naïve) algorithm has a time complexity that is typically prohibitive for effective heuristic searches. Building on the pioneering work on path-difference median trees, we develop efficient algorithms for Manhattan and Euclidean cophenetic search problems that improve on the naïve solution by a linear and a quadratic factor, respectively. We demonstrate the performance and effectiveness of the resulting heuristic methods in a comparative study using benchmark empirical datasets.

Synthesizing median trees from a collection of gene trees under the biologically motivated gene tree parsimony (GTP) costs has provided credible species tree estimates. GTP costs are defined for each of the classic evolutionary processes. These costs count the minimum number of events necessary to reconcile the gene tree with the species tree where the leaf-genes are mapped to the leaf-species through a function called labeling. To better understand the synthesis of median trees under these costs, there is an increased interest in analyzing their diameters. The diameters of a GTP cost between a gene tree and a species tree are the maximum values of this cost of one or both topologies of the trees involved. We are concerned about the diameters of the GTP costs under bijective labelings. While these diameters are linear time computable for the gene duplication and deep coalescence costs, this has been unknown for the classic gene duplication and loss, and for the loss cost. For the first time, we show how to compute these diameters and proof that this can be achieved in linear time, and thus, completing the computational time analysis for all of the bijective diameters under the GTP costs.

The minimizing-deep-coalescence (MDC) approach infers a median (species) tree for a given set of gene trees under the deep coalescence cost. This cost accounts for the minimum number of deep coalescences needed to reconcile a gene tree with a species tree where the leaf-genes are mapped to the leaf-species through a function called leaf labeling. In order to better understand the MDC approach we investigate here the diameter of a gene tree, which is an important property of the deep coalescence cost. This diameter is the maximal deep coalescence costs for a given gene tree under all leaf labelings for each possible species tree topology. While we prove that this diameter is generally infinite, this result relies on the diameter's unrealistic assumption that species trees can be of infinite size. Providing a more practical definition, we introduce a natural extension of the gene tree diameter that constrains the species tree size by a given constant. For this new diameter, we describe an exact formula, present a complete classification of the trees yielding this diameter, derive formulas for its mean and variance, and demonstrate its ability using comparative studies.

The deep coalescence cost accounts for discord caused by deep coalescence between a gene tree and a species tree. It is a major concern that the diameter of a gene tree (the tree's maximum deep coalescence cost across all species trees) depends on its topology, which can largely obfuscate phylogenetic studies. While this bias can be compensated by normalizing the deep coalescence cost using diameters, obtaining them efficiently has been posed as an open problem by Than and Rosenberg. Here, we resolve this problem by describing a linear time algorithm to compute the diameter of a gene tree. In addition, we provide a complete classification of the species trees yielding this diameter to guide phylogenetic analyses.

was held at the University of North Carolina at Charlotte (Charlotte, NC, USA) on May 20-22, 2013.For 9 years, the ISBRA symposium has been a forum for exchange of diverse ideas and research results broadly organized under the umbrella disciplines of bioinformatics and computational biology.The five keynote addresses at ISBRA 2013 included talks on RNA structure (Dr.Steve Harvey), computational behavioral ecology (Dr.Tanya Berger-Wolf), peptide identification from mass spectrometry (Dr.Bin Ma), gene regulation (Dr.Martha Bulyk), and biological network analysis (Dr.Luonan Chen).The research presented by conference participants was equally diverse.Reflecting current trends in the field, a substantial number of presentations were focused on genomics research, so we have chosen to present two linked supplements in BMC Bioinformatics, and one in BMC Genomics.ISBRA 2013 was attended by 115 participants from the US, Canada, China, southeast Asia, and several European nations.

BACKGROUND: Evolutionary studies are complicated by discordance between gene trees and the species tree in which they evolved. Dealing with discordant trees often relies on comparison costs between gene and species trees, including the well-established Robinson-Foulds, gene duplication, and deep coalescence costs. While these costs have provided credible results for binary rooted gene trees, corresponding cost definitions for non-binary unrooted gene trees, which are frequently occurring in practice, are challenged by biological realism. RESULT: We propose a natural extension of the well-established costs for comparing unrooted and non-binary gene trees with rooted binary species trees using a binary refinement model. For the duplication cost we describe an efficient algorithm that is based on a linear time reduction and also computes an optimal rooted binary refinement of the given gene tree. Finally, we show that similar reductions lead to solutions for computing the deep coalescence and the Robinson-Foulds costs. CONCLUSION: Our binary refinement of Robinson-Foulds, gene duplication, and deep coalescence costs for unrooted and non-binary gene trees together with the linear time reductions provided here for computing these costs significantly extends the range of trees that can be incorporated into approaches dealing with discordance.

The articles in this special section were presented at the Ninth International Symposium on Bioinformatics Research and Applications (ISBRA 2013), which was held at the University of North Carolina at Charlotte, NC.

The minimizing deep coalescence (MDC) problem seeks a species tree that reconciles the given gene trees with the minimum number of deep coalescence events, called deep coalescence (DC) cost. To better assess MDC species trees we investigate into a basic mathematical property of the DC cost, called the diameter. Given a gene tree, a species tree, and a leaf labeling function that assigns leaf-genes of the gene tree to a leaf-species in the species tree from which they were sampled, the DC cost describes the discordance between the trees caused by deep coalescence events. The diameter of a gene tree and a species tree is the maximum DC cost across all leaf labelings for these trees. We prove fundamental mathematical properties describing precisely these diameters for bijective and general leaf labelings, and present efficient algorithms to compute the diameters and their corresponding leaf labelings. In particular, we describe an optimal, i.e., linear time, algorithm for the bijective case. Finally, in an experimental study we demonstrate that the average diameters between a gene tree and a species tree grow significantly slower than their naive upper bounds, suggesting that our exact bounds can significantly improve on assessing DC costs when using diameters.

The use of genomic data sets for phylogenetics is complicated by the fact that evolutionary processes such as gene duplication and loss, or incomplete lineage sorting (deep coalescence) cause incongruence among gene trees. One well-known approach that deals with this complication is gene tree parsimony, which, given a collection of gene trees, seeks a species tree that requires the smallest number of evolutionary events to explain the incongruence of the gene trees. However, a lack of efficient algorithms has limited the use of this approach. Here, we present efficient algorithms for SPR and TBR-based local search heuristics for gene tree parsimony under the 1) duplication, 2) loss, 3) duplication-loss, and 4) deep coalescence reconciliation costs. These novel algorithms improve upon the time complexities of previous algorithms for these problems by a factor of n, where n is the number of species in the collection of gene trees. Our algorithms provide a substantial improvement in runtime and scalability compared to previous implementations and enable large-scale gene tree parsimony analyses using any of the four reconciliation costs. Our algorithms have been implemented in the software packages DupTree and iGTP, and have already been used to perform several compelling phylogenetic studies.

Tree comparison functions are widely used in phylogenetics for comparing evolutionary trees. Unrooted trees can be compared with rooted trees by identifying all rootings of the unrooted tree that minimize some provided comparison function between two rooted trees. The plateau property is satisfied by the provided function, if all optimal rootings form a subtree, or plateau, in the unrooted tree, from which the rootings along every path toward a leaf have monotonically increasing costs. This property is sufficient for the linear-time identification of all optimal rootings and rooting costs. However, the plateau property has only been proven for a few rooted comparison functions, requiring individual proofs for each function without benefitting from inherent structural features of such functions. Here, we introduce the consistency condition that is sufficient for a general function to satisfy the plateau property. For consistent functions, we introduce general linear-time solutions that identify optimal rootings and all rooting costs. Further, we identify novel relationships between consistent functions in terms of plateaus, especially the plateau of the well-studied duplication-loss function is part of a plateau of every other consistent function. We introduce a novel approach for identifying consistent cost functions by defining a formal language of Boolean costs. Formulas in this language can be interpreted as cost functions. Finally, we demonstrate the performance of our general linear-time solutions in practice using empirical and simulation studies.

Phylogenetic inference is a computationally difficult problem, and constructing high-quality phylogenies that can build upon existing phylogenetic knowledge and synthesize insights from new data remains a major challenge. We introduce knowledge-enhanced phylogenetic problems for both supertree and supermatrix phylogenetic analyses. These problems seek an optimal phylogenetic tree that can only be assembled from a user-supplied set of, possibly incompatible, phylogenetic relationships. We describe exact polynomial time algorithms for the knowledge-enhanced versions of the NP-hard Robinson Foulds, gene duplication, duplication and loss, and deep coalescence supertree problems. Further, we demonstrate that our algorithms can rapidly improve upon results of local search heuristics for these problems. Finally, we introduce a knowledge-enhanced search heuristic that can be applied to any discrete character data set using the maximum parsimony (MP) phylogenetic problem. Although this approach is not guaranteed to find exact solutions, we show that it also can improve upon solutions from commonly used MP heuristics.

BACKGROUND: Biological networks provide fundamental insights into the functional characterization of genes and their products, the characterization of DNA-protein interactions, the identification of regulatory mechanisms, and other biological tasks. Due to the experimental and biological complexity, their computational exploitation faces many algorithmic challenges. RESULTS: We introduce novel weighted quasi-biclique problems to identify functional modules in biological networks when represented by bipartite graphs. In difference to previous quasi-biclique problems, we include biological interaction levels by using edge-weighted quasi-bicliques. While we prove that our problems are NP-hard, we also describe IP formulations to compute exact solutions for moderately sized networks. CONCLUSIONS: We verify the effectiveness of our IP solutions using both simulation and empirical data. The simulation shows high quasi-biclique recall rates, and the empirical data corroborate the abilities of our weighted quasi-bicliques in extracting features and recovering missing interactions from biological networks.

BACKGROUND: Gene tree - species tree reconciliation problems infer the patterns and processes of gene evolution within a species tree. Gene tree parsimony approaches seek the evolutionary scenario that implies the fewest gene duplications, duplications and losses, or deep coalescence (incomplete lineage sorting) events needed to reconcile a gene tree and a species tree. While a gene tree parsimony approach can be informative about genome evolution and phylogenetics, error in gene trees can profoundly bias the results. RESULTS: We introduce efficient algorithms that rapidly search local Subtree Prune and Regraft (SPR) or Tree Bisection and Reconnection (TBR) neighborhoods of a given gene tree to identify a topology that implies the fewest duplications, duplication and losses, or deep coalescence events. These algorithms improve on the current solutions by a factor of n for searching SPR neighborhoods and n2 for searching TBR neighborhoods, where n is the number of taxa in the given gene tree. They provide a fast error correction protocol for ameliorating the effects of gene tree error by allowing small rearrangements in the topology to improve the reconciliation cost. We also demonstrate a simple protocol to use the gene rearrangement algorithm to improve gene tree parsimony phylogenetic analyses. CONCLUSIONS: The new gene tree rearrangement algorithms provide a fast method to address gene tree error. They do not make assumptions about the underlying processes of genome evolution, and they are amenable to analyses of large-scale genomic data sets. These algorithms are also easily incorporated into gene tree parsimony phylogenetic analyses, potentially producing more credible estimates of reconciliation cost.

BACKGROUND: Evolutionary methods are increasingly challenged by the wealth of fast growing resources of genomic sequence information. Evolutionary events, like gene duplication, loss, and deep coalescence, account more then ever for incongruence between gene trees and the actual species tree. Gene tree reconciliation is addressing this fundamental problem by invoking the minimum number of gene duplication and losses that reconcile a rooted gene tree with a rooted species tree. However, the reconciliation process is highly sensitive to topological error or wrong rooting of the gene tree, a condition that is not met by most gene trees in practice. Thus, despite the promises of gene tree reconciliation, its applicability in practice is severely limited. RESULTS: We introduce the problem of reconciling unrooted and erroneous gene trees by simultaneously rooting and error-correcting them, and describe an efficient algorithm for this problem. Moreover, we introduce an error-corrected version of the gene duplication problem, a standard application of gene tree reconciliation. We introduce an effective heuristic for our error-corrected version of the gene duplication problem, given that the original version of this problem is NP-hard. Our experimental results suggest that our error-correcting approaches for unrooted input trees can significantly improve on the accuracy of gene tree reconciliation, and the species tree inference under the gene duplication problem. Furthermore, the efficiency of our algorithm for error-correcting reconciliation is capable of handling truly large-scale phylogenetic studies. CONCLUSIONS: Our presented error-correction approach is a crucial step towards making gene tree reconciliation more robust, and thus to improve on the accuracy of applications that fundamentally rely on gene tree reconciliation, like the inference of gene-duplication supertrees.

BACKGROUND: To infer a species phylogeny from unlinked genes, phylogenetic inference methods must confront the biological processes that create incongruence between gene trees and the species phylogeny. Intra-specific gene variation in ancestral species can result in deep coalescence, also known as incomplete lineage sorting, which creates incongruence between gene trees and the species tree. One approach to account for deep coalescence in phylogenetic analyses is the deep coalescence problem, which takes a collection of gene trees and seeks the species tree that implies the fewest deep coalescence events. Although this approach is promising for phylogenetics, the consensus properties of this problem are mostly unknown and analyses of large data sets may be computationally prohibitive. RESULTS: We prove that the deep coalescence consensus tree problem satisfies the highly desirable Pareto property for clusters (clades). That is, in all instances, each cluster that is present in all of the input gene trees, called a consensus cluster, will also be found in every optimal solution. Moreover, we introduce a new divide and conquer method for the deep coalescence problem based on the Pareto property. This method refines the strict consensus of the input gene trees, thereby, in practice, often greatly reducing the complexity of the tree search and guaranteeing that the estimated species tree will satisfy the Pareto property. CONCLUSIONS: Analyses of both simulated and empirical data sets demonstrate that the divide and conquer method can greatly improve upon the speed of heuristics that do not consider the Pareto consensus property, while also guaranteeing that the proposed solution fulfills the Pareto property. The divide and conquer method extends the utility of the deep coalescence problem to data sets with enormous numbers of taxa.

BACKGROUND: The gene duplication (GD) problem seeks a species tree that implies the fewest gene duplication events across a given collection of gene trees. Solving this problem makes it possible to use large gene families with complex histories of duplication and loss to infer phylogenetic trees. However, the GD problem is NP-hard, and therefore, most analyses use heuristics that lack any performance guarantee. RESULTS: We describe the first integer linear programming (ILP) formulation to solve instances of the gene duplication problem exactly. With simulations, we demonstrate that the ILP solution can solve problem instances with up to 14 taxa. Furthermore, we apply the new ILP solution to solve the gene duplication problem for the seed plant phylogeny using a 12-taxon, 6,084-gene data set. The unique, optimal solution, which places Gnetales sister to the conifers, represents a new, large-scale genomic perspective on one of the most puzzling questions in plant systematics. CONCLUSIONS: Although the GD problem is NP-hard, our novel ILP solution for it can solve instances with data sets consisting of as many as 14 taxa and 1,000 genes in a few hours. These are the largest instances that have been solved to optimally to date. Thus, this work can provide large-scale genomic perspectives on phylogenetic questions that previously could only be addressed by heuristic estimates.

BACKGROUND: The abundance of new genomic data provides the opportunity to map the location of gene duplication and loss events on a species phylogeny. The first methods for mapping gene duplications and losses were based on a parsimony criterion, finding the mapping that minimizes the number of duplication and loss events. Probabilistic modeling of gene duplication and loss is relatively new and has largely focused on birth-death processes. RESULTS: We introduce a new maximum likelihood model that estimates the speciation and gene duplication and loss events in a gene tree within a species tree with branch lengths. We also provide an, in practice, efficient algorithm that computes optimal evolutionary scenarios for this model. We implemented the algorithm in the program DrML and verified its performance with empirical and simulated data. CONCLUSIONS: In test data sets, DrML finds optimal gene duplication and loss scenarios within minutes, even when the gene trees contain sequences from several hundred species. In many cases, these optimal scenarios differ from the lca-mapping that results from a parsimony gene tree reconciliation. Thus, DrML provides a new, practical statistical framework on which to study gene duplication.

Interactions of protein domains control essential cellular processes. Thus, inferring the evolutionary histories of multidomain proteins in the context of their families can provide rewarding insights into protein function. However, methods to infer these histories are challenged by the complexity of macroevolutionary events. Here, we address this challenge by describing an algorithm that computes a novel network-like structure, called plexus, which represents the evolution of domains and their combinations. Finally, we demonstrate the performance of this algorithm with empirical data sets.

BACKGROUND: Genomic data provide a wealth of new information for phylogenetic analysis. Yet making use of this data requires phylogenetic methods that can efficiently analyze extremely large data sets and account for processes of gene evolution, such as gene duplication and loss, incomplete lineage sorting (deep coalescence), or horizontal gene transfer, that cause incongruence among gene trees. One such approach is gene tree parsimony, which, given a set of gene trees, seeks a species tree that requires the smallest number of evolutionary events to explain the incongruence of the gene trees. However, the only existing algorithms for gene tree parsimony under the duplication-loss or deep coalescence reconciliation cost are prohibitively slow for large datasets. RESULTS: We describe novel algorithms for SPR and TBR based local search heuristics under the duplication-loss cost, and we show how they can be adapted for the deep coalescence cost. These algorithms improve upon the best existing algorithms for these problems by a factor of n, where n is the number of species in the collection of gene trees. We implemented our new SPR based local search algorithm for the duplication-loss cost and demonstrate the tremendous improvement in runtime and scalability it provides compared to existing implementations. We also evaluate the performance of our algorithm on three large-scale genomic data sets. CONCLUSION: Our new algorithms enable, for the first time, gene tree parsimony analyses of thousands of genes from hundreds of taxa using the duplication-loss and deep coalescence reconciliation costs. Thus, this work expands both the size of data sets and the range of evolutionary models that can be incorporated into genome-scale phylogenetic analyses.

BACKGROUND: The ever-increasing wealth of genomic sequence information provides an unprecedented opportunity for large-scale phylogenetic analysis. However, species phylogeny inference is obfuscated by incongruence among gene trees due to evolutionary events such as gene duplication and loss, incomplete lineage sorting (deep coalescence), and horizontal gene transfer. Gene tree parsimony (GTP) addresses this issue by seeking a species tree that requires the minimum number of evolutionary events to reconcile a given set of incongruent gene trees. Despite its promise, the use of gene tree parsimony has been limited by the fact that existing software is either not fast enough to tackle large data sets or is restricted in the range of evolutionary events it can handle. RESULTS: We introduce iGTP, a platform-independent software program that implements state-of-the-art algorithms that greatly speed up species tree inference under the duplication, duplication-loss, and deep coalescence reconciliation costs. iGTP significantly extends and improves the functionality and performance of existing gene tree parsimony software and offers advanced features such as building effective initial trees using stepwise leaf addition and the ability to have unrooted gene trees in the input. Moreover, iGTP provides a user-friendly graphical interface with integrated tree visualization software to facilitate analysis of the results. CONCLUSIONS: iGTP enables, for the first time, gene tree parsimony analyses of thousands of genes from hundreds of taxa using the duplication, duplication-loss, and deep coalescence reconciliation costs, all from within a convenient graphical user interface.

BACKGROUND: There is much interest in developing fast and accurate supertree methods to infer the tree of life. Supertree methods combine smaller input trees with overlapping sets of taxa to make a comprehensive phylogenetic tree that contains all of the taxa in the input trees. The intrinsically hard triplet supertree problem takes a collection of input species trees and seeks a species tree (supertree) that maximizes the number of triplet subtrees that it shares with the input trees. However, the utility of this supertree problem has been limited by a lack of efficient and effective heuristics. RESULTS: We introduce fast hill-climbing heuristics for the triplet supertree problem that perform a step-wise search of the tree space, where each step is guided by an exact solution to an instance of a local search problem. To realize time efficient heuristics we designed the first nontrivial algorithms for two standard search problems, which greatly improve on the time complexity to the best known (naïve) solutions by a factor of n and n2 (the number of taxa in the supertree). These algorithms enable large-scale supertree analyses based on the triplet supertree problem that were previously not possible. We implemented hill-climbing heuristics that are based on our new algorithms, and in analyses of two published supertree data sets, we demonstrate that our new heuristics outperform other standard supertree methods in maximizing the number of triplets shared with the input trees. CONCLUSION: With our new heuristics, the triplet supertree problem is now computationally more tractable for large-scale supertree analyses, and it provides a potentially more accurate alternative to existing supertree methods.

The gene-duplication problem is to infer a species supertree from a collection of gene trees that are confounded by complex histories of gene-duplication events. This problem is NP-complete and thus requires efficient and effective heuristics. Existing heuristics perform a stepwise search of the tree space, where each step is guided by an exact solution to an instance of a local search problem. A classical local search problem is the {\tt NNI} search problem, which is based on the nearest neighbor interchange operation. In this work, we 1) provide a novel near-linear time algorithm for the {\tt NNI} search problem, 2) introduce extensions that significantly enlarge the search space of the {\tt NNI} search problem, and 3) present algorithms for these extended versions that are asymptotically just as efficient as our algorithm for the {\tt NNI} search problem. The exceptional speedup achieved in the extended {\tt NNI} search problems makes the gene-duplication problem more tractable for large-scale phylogenetic analyses. We verify the performance of our algorithms in a comparison study using sets of large randomly generated gene trees.

MOTIVATION: Deciphering the location of gene duplications and multiple gene duplication episodes on the Tree of Life is fundamental to understanding the way gene families and genomes evolve. The multiple gene duplication problem provides a framework for placing gene duplication events onto nodes of a given species tree, and detecting episodes of multiple gene duplication. One version of the multiple gene duplication problem was defined by Guigó et al. in 1996. Several heuristic solutions have since been proposed for this problem, but no exact algorithms were known. RESULTS: In this article we solve this longstanding open problem by providing the first exact and efficient solution. We also demonstrate the improvement offered by our algorithm over the best heuristic approaches, by applying it to several simulated as well as empirical datasets.

UNLABELLED: DupTree is a new software program for inferring rooted species trees from collections of gene trees using the gene tree parsimony approach. The program implements a novel algorithm that significantly improves upon the run time of standard search heuristics for gene tree parsimony, and enables the first truly genome-scale phylogenetic analyses. In addition, DupTree allows users to examine alternate rootings and to weight the reconciliation costs for gene trees. DupTree is an open source project written in C++. AVAILABILITY: DupTree for Mac OS X, Windows, and Linux along with a sample dataset and an on-line manual are available at http://genome.cs.iastate.edu/CBL/DupTree

The gene-duplication problem is to infer a species supertree from gene trees that are confounded by complex histories of gene duplications. This problem is NP-hard and thus requires efficient and effective heuristics. Existing heuristics perform a stepwise search of the tree space, where each step is guided by an exact solution to an instance of a local search problem. We improve on the time complexity of the local search problem by a factor of n2= log n, where n is the size of the resulting species supertree. Typically, several thousand instances of the local search problem are solved throughout a stepwise heuristic search. Hence, our improvement makes the gene-duplication problem much more tractable for large-scale phylogenetic analyses.

The input to a supertree problem is a collection of phylogenetic trees that intersect pairwise in their leaf sets; the goal is to construct a single tree that retains as much as possible of the information in the input. This task is complicated by inconsistencies due to errors. We consider the case where the input trees are rooted and are represented by the clusters they exhibit. The problem is to find the minimum number of flips needed to resolve all inconsistencies, where each flip moves a taxon into or out of a cluster. We prove that the minimum-flip problem is NP-complete, but show that it is fixed-parameter tractable and give approximation algorithms for special cases.

UNLABELLED: Rainbow is a program that provides a graphic user interface to construct supertrees using different methods. It also provides tools to analyze the quality of the supertrees produced. Rainbow is available for Mac OS X, Windows and Linux. AVAILABILITY: Rainbow is a free open-source software. Its binary files, source code, and manual can be downloaded from the Rainbow web page: http://genome.cs.iastate.edu/Rainbow/

MOTIVATION: Numerous homologous sequences from diverse species can be retrieved from databases using programs such as BLAST. However, due to multigene families, evolutionary relationship often cannot be easily determined and proper functional assignment becomes difficult. Thus, discrimination between orthologues and paralogues within BLAST output lists of homologous sequences becomes more and more important. RESULT: We therefore developed a method that attempts to construct a reconciled tree from a gene tree of selected sequences and its corresponding phylogenetic tree of the species involved (species tree). An interface on the Web is developed to enable users to analyse the BLAST result. BLAST outputs are parsed and, for the selected sequences, multiple alignments are constructed either globally or for local regions. Bootstrapped trees are returned and compared with the expected species tree. In cases of discrepancies, gene duplications are assumed and a reconciled tree is computed. The reconciled tree shows probable orthologues and paralogues as predicted.