Alignment of biological networks by integer linear programming: virus-host protein-protein interaction networks.

BACKGROUND: The alignment of protein-protein interaction networks was recently formulated as an integer quadratic programming problem, along with a linearization that can be solved by integer linear programming software tools. However, the resulting integer linear program has a huge number of variables and constraints, rendering it of no practical use. RESULTS: We present a compact integer linear programming reformulation of the protein-protein interaction network alignment problem, which can be solved using state-of-the-art mathematical modeling and integer linear programming software tools, along with empirical results showing that small biological networks, such as virus-host protein-protein interaction networks, can be aligned in a reasonable amount of time on a personal computer and the resulting alignments are structurally coherent and biologically meaningful. CONCLUSIONS: The implementation of the integer linear programming reformulation using current mathematical modeling and integer linear programming software tools provided biologically meaningful alignments of virus-host protein-protein interaction networks.

AligNet: alignment of protein-protein interaction networks.

BACKGROUND: All molecular functions and biological processes are carried out by groups of proteins that interact with each other. Metaproteomic data continuously generates new proteins whose molecular functions and relations must be discovered. A widely accepted structure to model functional relations between proteins are protein-protein interaction networks (PPIN), and their analysis and alignment has become a key ingredient in the study and prediction of protein-protein interactions, protein function, and evolutionary conserved assembly pathways of protein complexes. Several PPIN aligners have been proposed, but attaining the right balance between network topology and biological information is one of the most difficult and key points in the design of any PPIN alignment algorithm. RESULTS: Motivated by the challenge of well-balanced and efficient algorithms, we have designed and implemented AligNet, a parameter-free pairwise PPIN alignment algorithm aimed at bridging the gap between topologically efficient and biologically meaningful matchings. A comparison of the results obtained with AligNet and with the best aligners shows that AligNet achieves indeed a good balance between topological and biological matching. CONCLUSION: In this paper we present AligNet, a new pairwise global PPIN aligner that produces biologically meaningful alignments, by achieving a good balance between structural matching and protein function conservation, and more efficient computations than state-of-the-art tools.

On Sackin's original proposal: the variance of the leaves' depths as a phylogenetic balance index.

BACKGROUND: The Sackin indexS of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's paper (Syst Zool 21:225-6, 1972) is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in (Evolution 47:1171-81, 1993), where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. RESULTS: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space [Formula: see text] of bifurcating rooted phylogenetic trees with n≤183 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n≥184. We provide then an algorithm that finds the trees in [Formula: see text] with minimum V value in time O(n log(n)). Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index (Mir et al., Math Biosci 241:125-36, 2013) of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature. CONCLUSION: The phylogenetics community has been wise in preferring the sum S(T) of the leaves' depths of a phylogenetic tree T over their variance V(T) as a balance index, because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But it is still a valid and useful shape index.

On the minimum value of the Colless index and the bifurcating trees that achieve it.

Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (Syst Zool 31:100-104, 1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with n leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the tree shapes that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

A balance index for phylogenetic trees based on rooted quartets.

We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's [Formula: see text]-model and Aldous' [Formula: see text]-model and on arbitrary trees under the [Formula: see text]-[Formula: see text]-model.

The Probabilities of Trees and Cladograms under Ford's α-Model.

Ford's α-model is one of the most popular random parametric models of bifurcating phylogenetic tree growth, having as specific instances both the uniform and the Yule models. Its general properties have been used to study the behavior of phylogenetic tree shape indices under the probability distribution it defines. But the explicit formulas provided by Ford for the probabilities of unlabeled trees and phylogenetic trees fail in some cases. In this paper we give correct explicit formulas for these probabilities.

The comparison of tree-sibling time consistent phylogenetic networks is graph isomorphism-complete.

Several polynomial time computable metrics on the class of semibinary tree-sibling time consistent phylogenetic networks are available in the literature; in particular, the problem of deciding if two networks of this kind are isomorphic is in P. In this paper, we show that if we remove the semibinarity condition, then the problem becomes much harder. More precisely, we prove that the isomorphism problem for generic tree-sibling time consistent phylogenetic networks is polynomially equivalent to the graph isomorphism problem. Since the latter is believed not to belong to P, the chances are that it is impossible to define a metric on the class of all tree-sibling time consistent phylogenetic networks that can be computed in polynomial time.

Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf.

BACKGROUND: Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa. RESULTS: For every (rooted) phylogenetic tree T, let its cophenetic vectorφ(T) consist of all pairs of cophenetic values between pairs of taxa in T and all depths of taxa in T. It turns out that these cophenetic vectors single out weighted phylogenetic trees with nested taxa. We then define a family of cophenetic metrics dφ,p by comparing these cophenetic vectors by means of Lp norms, and we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics. CONCLUSIONS: The cophenetic metrics can be safely used on weighted phylogenetic trees with nested taxa and no restriction on degrees, and they can be computed in O(n2) time, where n stands for the number of taxa. The metrics dφ,1 and dφ,2 have positive skewed distributions, and they show a low rank correlation with the Robinson-Foulds metric and the nodal metrics, and a very high correlation with each other and with the splitted nodal metrics. The diameter of dφ,p, for p⩾1 , is in O(n(p+2)/p), and thus for low p they are more discriminative, having a wider range of values.

Diffusional conductances to CO2 as a target for increasing photosynthesis and photosynthetic water-use efficiency.

A key objective for sustainable agriculture and forestry is to breed plants with both high carbon gain and water-use efficiency (WUE). At the level of leaf physiology, this implies increasing net photosynthesis (A N) relative to stomatal conductance (g s). Here, we review evidence for CO2 diffusional constraints on photosynthesis and WUE. Analyzing past observations for an extensive pool of crop and wild plant species that vary widely in mesophyll conductance to CO2 (g m), g s, and foliage A N, it was shown that both g s and g m limit A N, although the relative importance of each of the two conductances depends on species and conditions. Based on Fick's law of diffusion, intrinsic WUE (the ratio A N/g s) should correlate on the ratio g m/g s, and not g m itself. Such a correlation is indeed often observed in the data. However, since besides diffusion A N also depends on photosynthetic capacity (i.e., V c,max), this relationship is not always sustained. It was shown that only in a very few cases, genotype selection has resulted in simultaneous increases of both A N and WUE. In fact, such a response has never been observed in genetically modified plants specifically engineered for either reduced g s or enhanced g m. Although increasing g m alone would result in increasing photosynthesis, and potentially increasing WUE, in practice, higher WUE seems to be only achieved when there are no parallel changes in g s. We conclude that for simultaneous improvement of A N and WUE, genetic manipulation of g m should avoid parallel changes in g s, and we suggest that the appropriate trait for selection for enhanced WUE is increased g m/g s.

Exact formulas for the variance of several balance indices under the Yule model.

One of the main applications of balance indices is in tests of nullmodels of evolutionary processes. The knowledge of an exact formula for a statistic of a balance index, holding for any number n of leaves, is necessary in order to use this statistic in tests of this kind involving trees of any size. In this paper we obtain exact formulas for the variance under the Yule model of the Sackin, the Colless and the total cophenetic indices of binary rooted phylogenetic trees with n leaves.

Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term.

Divide-and-conquer dividing by a half recurrences, of the form [Formula: see text] appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quartet index of the maximally balanced bifurcating phylogenetic trees with n leaves, and the sum of the bitwise AND operator applied to pairs of complementary numbers up to n.

The Generalized Robinson-Foulds Distance for Phylogenetic Trees.

The Robinson-Foulds (RF) distance, one of the most widely used metrics for comparing phylogenetic trees, has the advantage of being intuitive, with a natural interpretation in terms of common splits, and it can be computed in linear time, but it has a very low resolution, and it may become trivial for phylogenetic trees with overlapping taxa, that is, phylogenetic trees that share some but not all of their leaf labels. In this article, we study the properties of the Generalized Robinson-Foulds (GRF) distance, a recently proposed metric for comparing any structures that can be described by multisets of multisets of labels, when applied to rooted phylogenetic trees with overlapping taxa, which are described by sets of clusters, that is, by sets of sets of labels. We show that the GRF distance has a very high resolution, it can also be computed in linear time, and it is not (uniformly) equivalent to the RF distance.

Squaring within the Colless index yields a better balance index.

The Colless index for bifurcating phylogenetic trees, introduced by Colless (1982), is defined as the sum, over all internal nodes v of the tree, of the absolute value of the difference of the sizes of the clades defined by the children of v. It is one of the most popular phylogenetic balance indices, because, in addition to measuring the balance of a tree in a very simple and intuitive way, it turns out to be one of the most powerful and discriminating phylogenetic shape indices. But it has some drawbacks. On the one hand, although its minimum value is reached at the so-called maximally balanced trees, it is almost always reached also at trees that are not maximally balanced. On the other hand, its definition as a sum of absolute values of differences makes it difficult to study analytically its distribution under probabilistic models of bifurcating phylogenetic trees. In this paper we show that if we replace in its definition the absolute values of the differences of clade sizes by the squares of these differences, all these drawbacks are overcome and the resulting index is still more powerful and discriminating than the original Colless index.

Nodal distances for rooted phylogenetic trees.

Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. For rooted phylogenetic trees, however, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class of rooted phylogenetic trees. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class of rooted phylogenetic trees by comparing these matrices through metrics in spaces M(n)(R) of real-valued n x n matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using L(p) metrics on M(n)(R), with p [epsilon] R(>0).

A distance metric for a class of tree-sibling phylogenetic networks.

MOTIVATION: The presence of reticulate evolutionary events in phylogenies turn phylogenetic trees into phylogenetic networks. These events imply in particular that there may exist multiple evolutionary paths from a non-extant species to an extant one, and this multiplicity makes the comparison of phylogenetic networks much more difficult than the comparison of phylogenetic trees. In fact, all attempts to define a sound distance measure on the class of all phylogenetic networks have failed so far. Thus, the only practical solutions have been either the use of rough estimates of similarity (based on comparison of the trees embedded in the networks), or narrowing the class of phylogenetic networks to a certain class where such a distance is known and can be efficiently computed. The first approach has the problem that one may identify two networks as equivalent, when they are not; the second one has the drawback that there may not exist algorithms to reconstruct such networks from biological sequences. RESULTS: We present in this article a distance measure on the class of semi-binary tree-sibling time consistent phylogenetic networks, which generalize tree-child time consistent phylogenetic networks, and thus also galled-trees. The practical interest of this distance measure is 2-fold: it can be computed in polynomial time by means of simple algorithms, and there also exist polynomial-time algorithms for reconstructing networks of this class from DNA sequence data. AVAILABILITY: The Perl package Bio::PhyloNetwork, included in the BioPerl bundle, implements many algorithms on phylogenetic networks, including the computation of the distance presented in this article. SUPPLEMENTARY INFORMATION: Some counterexamples, proofs of the results not included in this article, and some computational experiments are available at Bioinformatics online.

Efficient reconstruction of metabolic pathways by bidirectional chemical search.

One of the main challenges in systems biology is the establishment of the metabolome: a catalogue of the metabolites and biochemical reactions present in a specific organism. Current knowledge of biochemical pathways as stored in public databases such as KEGG, is based on carefully curated genomic evidence for the presence of specific metabolites and enzymes that activate particular biochemical reactions. In this paper, we present an efficient method to build a substantial portion of the artificial chemistry defined by the metabolites and biochemical reactions in a given metabolic pathway, which is based on bidirectional chemical search. Computational results on the pathways stored in KEGG reveal novel biochemical pathways.

Extended Newick: it is time for a standard representation of phylogenetic networks.

BACKGROUND: Phylogenetic trees resulting from molecular phylogenetic analysis are available in Newick format from specialized databases but when it comes to phylogenetic networks, which provide an explicit representation of reticulate evolutionary events such as recombination, hybridization or lateral gene transfer, the lack of a standard format for their representation has hindered the publication of explicit phylogenetic networks in the specialized literature and their incorporation in specialized databases. Two different proposals to represent phylogenetic networks exist: as a single Newick string (where each hybrid node is splitted once for each parent) or as a set of Newick strings (one for each hybrid node plus another one for the phylogenetic network). RESULTS: The standard we advocate as extended Newick format describes a whole phylogenetic network with k hybrid nodes as a single Newick string with k repeated nodes, and this representation is unique once the phylogenetic network is drawn or the ordering among children nodes is fixed. The extended Newick format facilitates phylogenetic data sharing and exchange, and also allows for the practical use of phylogenetic networks in computer programs and scripts. This standard has been recently agreed upon by a number of computational biologists, is already supported by several phylogenetic tools, and avoids the different drawbacks of using an a priori unknown number of Newick strings without any additional mark-up to represent a phylogenetic network. CONCLUSION: The adoption of the extended Newick format as a standard for the representation of phylogenetic network is an important step towards the publication of explicit phylogenetic networks in peer-reviewed journals and their incorporation in a future database of published phylogenetic networks.

A perl package and an alignment tool for phylogenetic networks.

BACKGROUND: Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of evolutionary events acting at the population level, like recombination between genes, hybridization between lineages, and lateral gene transfer. While most phylogenetics tools implement a wide range of algorithms on phylogenetic trees, there exist only a few applications to work with phylogenetic networks, none of which are open-source libraries, and they do not allow for the comparative analysis of phylogenetic networks by computing distances between them or aligning them. RESULTS: In order to improve this situation, we have developed a Perl package that relies on the BioPerl bundle and implements many algorithms on phylogenetic networks. We have also developed a Java applet that makes use of the aforementioned Perl package and allows the user to make simple experiments with phylogenetic networks without having to develop a program or Perl script by him or herself. CONCLUSION: The Perl package is available as part of the BioPerl bundle, and can also be downloaded. A web-based application is also available (see availability and requirements). The Perl package includes full documentation of all its features.

Tripartitions do not always discriminate phylogenetic networks.

Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of non-treelike evolutionary events, like recombination, hybridization, or lateral gene transfer. In a recent series of papers devoted to the study of reconstructibility of phylogenetic networks, Moret, Nakhleh, Warnow and collaborators introduced the so-called tripartition metric for phylogenetic networks. In this paper we show that, in fact, this tripartition metric does not satisfy the separation axiom of distances (zero distance means isomorphism, or, in a more relaxed version, zero distance means indistinguishability in some specific sense) in any of the subclasses of phylogenetic networks where it is claimed to do so. We also present a subclass of phylogenetic networks whose members can be singled out by means of their sets of tripartitions (or even clusters), and hence where the latter can be used to define a meaningful metric.

Sound Colless-like balance indices for multifurcating trees.

The Colless index is one of the most popular and natural balance indices for bifurcating phylogenetic trees, but it makes no sense for multifurcating trees. In this paper we propose a family of Colless-like balance indices [Formula: see text] that generalize the Colless index to multifurcating phylogenetic trees. Each [Formula: see text] is determined by the choice of a dissimilarity D and a weight function [Formula: see text]. A balance index is sound when the most balanced phylogenetic trees according to it are exactly the fully symmetric ones. Unfortunately, not every Colless-like balance index is sound in this sense. We prove then that taking f(n) = ln(n + e) or f(n) = en as weight functions, the resulting index [Formula: see text] is sound for every dissimilarity D. Next, for each one of these two functions f and for three popular dissimilarities D (the variance, the standard deviation, and the mean deviation from the median), we find the most unbalanced phylogenetic trees according to [Formula: see text] with any given number n of leaves. The results show that the growth pace of the function f influences the notion of "balance" measured by the indices it defines. Finally, we introduce our R package "CollessLike," which, among other functionalities, allows the computation of Colless-like indices of trees and their comparison to their distribution under Chen-Ford-Winkel's α-γ-model for multifurcating phylogenetic trees. As an application, we show that the trees in TreeBASE do not seem to follow either the uniform model for multifurcating trees or the α-γ-model, for any values of α and γ.