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.     

Robustness of Topological Supertree Methods for Reconciling Dense Incompatible Data

Given a collection of rooted phylogenetic trees with overlapping sets of leaves, a compatible supertree $S$ is a single tree whose set of leaves is the union of the input sets of leaves and such that $S$ agrees with each input tree when restricted to the leaves of the input tree. Typically with trees from real data, no compatible supertree exists, and various methods may be utilized to reconcile the incompatibilities in the input trees. This paper focuses on a measure of robustness of a supertree method called its ``radius" $R$. The larger the value of $R$ is, the further the data set can be from a natural correct tree $T$ and yet the method will still output $T$. It is shown that the maximal possible radius for a method is $R = 1/2$. Many familiar methods, both for supertrees and consensus trees, are shown to have $R = 0$, indicating that they need not output a tree $T$ that would seem to be the natural correct answer. A polynomial-time method Normalized Triplet Supertree (NTS) with the maximal possible $R = 1/2$ is defined. A geometric interpretion is given, and NTS is shown to solve an optimization problem. Additional properties of NTS are described.     

Robinson-Foulds Supertrees

Background  

Supertree methods synthesize collections of small phylogenetic trees with incomplete taxon overlap into comprehensive trees, or supertrees, that include all taxa found in the input trees. Supertree methods based on the well established Robinson-Foulds (RF) distance have the potential to build supertrees that retain much information from the input trees. Specifically, the RF supertree problem seeks a binary supertree that minimizes the sum of the RF distances from the supertree to the input trees. Thus, an RF supertree is a supertree that is consistent with the largest number of clusters (or clades) from the input trees.     

Fast algorithm to reconstruct a species supertree from a set of protein trees

The problem of reconstructing a species supertree from a given set of protein, gene, and regulatorysite trees is the subject of this study. Under the traditional formulation, this problem is proven to be NP-hard. We propose a reformulation: to seek for a supertree, most of the clades of which contribute to the original protein trees. In such a variant, the problem seems to be biologically natural and a fast algorithm can be developed for its solution. The algorithm was tested on artificial and biological sets of protein trees, and it proved to be efficient even under the assumption of horizontal gene transfer. When horizontal transfer is not allowed, the algorithm correctness is proved mathematically; the time necessary for repeating the algorithm is assessed, and, in the worst case scenario, it is of the order n ³ · |V ₀|³, where n is the number of gene trees and |V ₀| is the number of tree species. Our software for supertree construction, examples of computations, and instructions can be freely accessed at . Events associated with horizontal gene transfer are not included either in this study or in any variant of the software. A general case is described in the authors’ report (journal Problems of Information Transmission, 2011).     

Unique Reconstruction of Tree-Like Phylogenetic Networks from Distances Between Leaves

In this paper, a class of rooted acyclic directed graphs (called TOM-networks) is defined that generalizes rooted trees and allows for models including hybridization events. It is argued that the defining properties are biologically plausible. Each TOM-network has a distance defined between each pair of vertices. For a TOM-network N, suppose that the set X consisting of the leaves and the root is known, together with the distances between members of X. It is proved that N is uniquely determined from this information and can be reconstructed in polynomial time. Thus, given exact distance information on the leaves and root, the phylogenetic network can be uniquely recovered, provided that it is a TOM-network. An outgroup can be used instead of a true root.     

Efficient FPT Algorithms for (Strict) Compatibility of Unrooted Phylogenetic Trees

In phylogenetics, a central problem is to infer the evolutionary relationships between a set of species X; these relationships are often depicted via a phylogenetic tree—a tree having its leaves labeled bijectively by elements of X and without degree-2 nodes—called the “species tree.” One common approach for reconstructing a species tree consists in first constructing several phylogenetic trees from primary data (e.g., DNA sequences originating from some species in X), and then constructing a single phylogenetic tree maximizing the “concordance” with the input trees. The obtained tree is our estimation of the species tree and, when the input trees are defined on overlapping—but not identical—sets of labels, is called “supertree.” In this paper, we focus on two problems that are central when combining phylogenetic trees into a supertree: the compatibility and the strict compatibility problems for unrooted phylogenetic trees. These problems are strongly related, respectively, to the notions of “containing as a minor” and “containing as a topological minor” in the graph community. Both problems are known to be fixed parameter tractable in the number of input trees k, by using their expressibility in monadic second-order logic and a reduction to graphs of bounded treewidth. Motivated by the fact that the dependency on k of these algorithms is prohibitively large, we give the first explicit dynamic programming algorithms for solving these problems, both running in time \(2^{O(k^2)} \cdot n\), where n is the total size of the input.     

Genus-level supertree of Cyprinidae (Actinopterygii: Cypriniformes), partitioned qualitative clade support and test of macro-evolutionary scenarios

We used the supertree approach of matrix representation with parsimony to reconstruct to date the most exhaustive (genus‐level) phylogeny of Cyprinidae. The supertree of Cyprinidae, representing 397 taxa (237 nominal genera) and 990 pseudocharacters, was well resolved (96%) through extended consensus majority rule, although 36 nodes (9.4%) were unsupported. The proportion of shared taxa among source trees was very low after calculation of the taxonomic coverage index (TCI = 0.059), which is proposed here as a more accurate alternative to the usual ratios calculated from the number of pseudo‐characters or source trees per taxon. We define a new index for the calculation of partitioned qualitative clade support, the partitioned rQS (_prQS), which offers a straightforward visualization of the relative supports of source tree partitions at supertree nodes.The use of _prQS showed that the molecular source tree partition contributed to most node supports within the supertree of Cyprinidae (73%, contra 21% for the morphological partition) and evidenced a fair proportion of conflict at nodes between the two partitions (21%), notably reflecting (i) the greater number and resolution of molecular source trees, and (ii) potential morphological convergences. Most of the higher‐level relationships within Cyprinidae were supported by both morphological and molecular source tree partitions. Our supertree showed a well‐supported dichotomy between a clade consisting of a ‘barbine’ + ‘rasborine’ lineage, sister group to (Barbinae [paraphyletic], (Cyprininae, Labeoninae)), and a clade consisting of other rasborines (large polytomy) and the two monophyletic groups ((Tincinae, Tanichthys), (Ecocarpia, (Acheilognathinae, (Gobioninae, Leuciscinae)))) and (Squaliobarbinae, (Xenocyprinae, Cultrinae)). Through the non‐monophyly of almost all the traditional subfamilies of Cyprinidae and 34 genera, our supertree exemplified the taxonomic chaos that reigns in the classification of the family. It also highlighted that further efforts should aim at increasing taxonomic sampling and generating alternative phylogenetic signals, notably for the still poorly apprehended Tincinae, Squaliobarbinae, Acheilognathinae, Gobioninae, and Rasborinae, the latter representing a key taxon for the understanding of early cyprinid evolution. Our supertree also proved useful for testing macro‐evolutionary scenarios at a wide taxonomic scale. Ancestral reconstructions using linear parsimony confirmed that the Oriental tropical region was the centre of origin of Cyprinidae, and identified three Oriental‐to‐Palaearctic, two Palaearctic‐to‐Nearctic, and one Oriental‐to‐Afrotropical major migration events. On the other hand, we almost completely rejected the hypothesis of presence of barbels as a plesiomorphic condition within Cyprinidae (although ambiguous for maxillary barbels of the Barbinae‐Cyprininae type). The supertree of Cyprinidae serves as a basis to discuss the applications and bias of the newly proposed _prQS, to provide future guidelines for a better achievement of cyprinid phylogeny, and to elaborate further on inter‐continental migrations and the adaptive value of barbels.     

Using max cut to enhance rooted trees consistency

Supertree methods are used to construct a large tree over a large set of taxa from a set of small trees over overlapping subsets of the complete taxa set. Since accurate reconstruction methods are currently limited to a maximum of a few dozen taxa, the use of a supertree method in order to construct the tree of life is inevitable. Supertree methods are broadly divided according to the input trees: When the input trees are unrooted, the basic reconstruction unit is a quartet tree. In this case, the basic decision problem of whether there exists a tree that agrees with all quartets is NP-complete. On the other hand, when the input trees are rooted, the basic reconstruction unit is a rooted triplet and the above decision problem has a polynomial time algorithm. However, when there is no tree which agrees with all triplets, it would be desirable to find the tree that agrees with the maximum number of triplets. However, this optimization problem was shown to be NP-hard. Current heuristic approaches perform min cut on a graph representing the triplets inconsistency and return a tree that is guaranteed to satisfy some required properties. In this work, we present a different heuristic approach that guarantees the properties provided by the current methods and give experimental evidence that it significantly outperforms currently used methods. This method is based on a divide and conquer approach, where the min cut in the divide step is replaced by a max cut in a variant of the same graph. The latter is achieved by a lightweight semidefinite programming-like heuristic that leads to very fast running times     

Minimum evolution using ordinary least-squares is less robust than neighbor-joining

The method of minimum evolution reconstructs a phylogenetic tree T for n taxa given dissimilarity data d. In principle, for every tree W with these n leaves an estimate for the total length of W is made, and T is selected as the W that yields the minimum total length. Suppose that the ordinary least-squares formula S _W (d) is used to estimate the total length of W. A theorem of Rzhetsky and Nei shows that when d is positively additive on a completely resolved tree T, then for all W ≠ T it will be true that S _W (d) > S _T (d). The same will be true if d is merely sufficiently close to an additive dissimilarity function. This paper proves that as n grows large, even if the shortest branch length in the true tree T remains constant and d is additive on T, then the difference S _W (d)-S _T (d) can go to zero. It is also proved that, as n grows large, there is a tree T with n leaves, an additive distance function d _T on T with shortest edge ε, a distance function d, and a tree W with the same n leaves such that d differs from d _T by only approximately ε/4, yet minimum evolution incorrectly selects the tree W over the tree T. This result contrasts with the method of neighbor-joining, for which Atteson showed that incorrect selection of W required a deviation at least ε/2. It follows that, for large n, minimum evolution with ordinary least-squares can be only half as robust as neighbor-joining.     

Minimum-flip supertrees: complexity and algorithms

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.     

On the relationship between leaf anatomy and CO2 diffusion through the mesophyll of hypostomatous leaves

Internal conductances to CO₂ transfer from the stomatal cavity to sites of carboxylation (g_i) in hypostomatous sun-and shade-grown leaves of citrus, peach and Macadamia trees (Lloyd et al. 1992) were related to anatomical characteristics of mesophyll tissues. There was a consistent relationship between absorptance of photosynthetically active radiation and chlorophyll concentration (mmol m^?2) for all leaves, including sclerophyllous Macadamia, whose transmittance was high despite its relatively thick leaves. In thin peach leaves, which had high g_i, the chloro-plast volume and mesophyll surface area exposed to intercellular air spaces (ias) per unit leaf area were similar to those in the thicker leaves of the evergreen species. Peach leaves, however, had the lowest leaf dry weight per area (D/a), the lowest tissue density (T_d) and the highest chloro-plast surface area (S_c) exposed to ias. There were negative correlations between g_i and leaf thickness or D/a, but positive correlations between g_i and S_c or S_c/T_d. We developed a one-dimensional diffusion model which partitioned g_i into a gaseous diffusion conductance through the ias (g_ias) plus a liquid-phase conductance through mesophyll cell walls (g_cw). The model accounted for a significant amount of variation (r₂=0.80) in measured g_i by incorporating both components. The g_ias component was related to the one-dimensional path-length for diffusion across the mesophyll and so was greater in thinner peach leaves than in leaves of evergreen species. The g_cw component was related to tissue density and to the degree of chloroplast exposure to the ias. Thus the negative correlations between g_i and leaf thickness or D/a related to g_ias whereas positive correlations between g_i and S_c or S_c/T_d, related to g_cw. The g_cw was consistently lower than g_ias, and thus represented a greater constraint on CO₂ diffusion in the mesophylls of these hypostomatous species.     

PhySIC_IST: cleaning source trees to infer more informative supertrees

Background  

Supertree methods combine phylogenies with overlapping sets of taxa into a larger one. Topological conflicts frequently arise among source trees for methodological or biological reasons, such as long branch attraction, lateral gene transfers, gene duplication/loss or deep gene coalescence. When topological conflicts occur among source trees, liberal methods infer supertrees containing the most frequent alternative, while veto methods infer supertrees not contradicting any source tree, i.e. discard all conflicting resolutions. When the source trees host a significant number of topological conflicts or have a small taxon overlap, supertree methods of both kinds can propose poorly resolved, hence uninformative, supertrees.     

F-tests for Hypotheses with Block Matrices and Under Conditions of Orthogonality in the General Multivariate Gauss-Markoff Model

The multivariate general Gauss-Markoff (MGM) model (U, XB, ∑?σ²V) when the matrices V ≥ 0 and ∑ > 0 are known and the scalar σ² > 0 is unknown, is considered. The present paper is a continuation of two earlier works (Oktaba, 1988a, b). If XB = X₁Σ + X₂Δ, then the F-test for verification the hypothesis WΣA = 0 is presented. Moreover, under conditions of orthogonality the decomposition of the matrix S_A (?BCA)′L^?(?BCA) into the sum of s = r(L) matrices is given, where ?BCA is the estimator of the parametric estimable functions ?BCA, Cov (?BCA) = A′ ∑?σ²L = ?C₄?′, B? = (X′T^?X)^?X′T^?U, C₄ = (X′T^?X)^?M, where M = M′ is any arbitrary matrix such that R(X) ? R(T), T=V+XMX′; T^? is any c-inverse. R(A) is the linear space generated by the colums of A. Then under additional assumption on normality of U the statistics F for testing ?BA = 0 is deduced. Under conditions of normality of U and decomposition of S_A, the statistics F₁, …, F_s for the hypotheses jⁱ BA = 0 (i = 1,…, s) are established.     

PhySIC: a veto supertree method with desirable properties

This paper focuses on veto supertree methods; i.e., methods that aim at producing a conservative synthesis of the relationships agreed upon by all source trees. We propose desirable properties that a supertree should satisfy in this framework, namely the non-contradiction property (PC) and the induction property (PI). The former requires that the supertree does not contain relationships that contradict one or a combination of the source topologies, whereas the latter requires that all topological information contained in the supertree is present in a source tree or collectively induced by several source trees. We provide simple examples to illustrate their relevance and that allow a comparison with previously advocated properties. We show that these properties can be checked in polynomial time for any given rooted supertree. Moreover, we introduce the PhySIC method (PHYlogenetic Signal with Induction and non-Contradiction). For k input trees spanning a set of n taxa, this method produces a supertree that satisfies the above-mentioned properties in O(kn(3) + n(4)) computing time. The polytomies of the produced supertree are also tagged by labels indicating areas of conflict as well as those with insufficient overlap. As a whole, PhySIC enables the user to quickly summarize consensual information of a set of trees and localize groups of taxa for which the data require consolidation. Lastly, we illustrate the behaviour of PhySIC on primate data sets of various sizes, and propose a supertree covering 95% of all primate extant genera. The PhySIC algorithm is available at http://atgc.lirmm.fr/cgi-bin/PhySIC.     

Stem hydraulic capacitance decreases with drought stress: implications for modelling tree hydraulics in the Mediterranean oak Quercus ilex

Hydraulic modelling is a primary tool to predict plant performance in future drier scenarios. However, as most tree models are validated under non‐stress conditions, they may fail when water becomes limiting. To simulate tree hydraulic functioning under moist and dry conditions, the current version of a water flow and storage mechanistic model was further developed by implementing equations that describe variation in xylem hydraulic resistance (R_X) and stem hydraulic capacitance (C_S) with predawn water potential (Ψ_PD). The model was applied in a Mediterranean forest experiencing intense summer drought, where six Quercus ilex trees were instrumented to monitor stem diameter variations and sap flow, concurrently with measurements of predawn and midday leaf water potential. Best model performance was observed when C_S was allowed to decrease with decreasing Ψ_PD. Hydraulic capacitance decreased from 62 to 25 kg m^?3 MPa^?1 across the growing season. In parallel, tree transpiration decreased to a greater extent than the capacitive water release and the contribution of stored water to transpiration increased from 2.0 to 5.1%. Our results demonstrate the importance of stored water and seasonality in C_S for tree hydraulic functioning, and they suggest that C_S should be considered to predict the drought response of trees with models.     

Evolution of glutamate dehydrogenase genes: Evidence for two paralogous protein families and unusual branching patterns of the archaebacteria in the universal tree of life

Summary The existence of two families of genes coding for hexameric glutamate dehydrogenases has been deduced from the alignment of 21 primary sequences and the determination of the percentages of similarity between each pair of proteins. Each family could also be characterized by specific motifs. One family (Family 1) was composed of gdh genes from six eubacteria and six lower eukaryotes (the primitive protozoan Giardia lamblia, the green alga Chlorella sorokiniana, and several fungi and yeasts). The other one (Family 11) was composed of gdh genes from two eubacteria, two archaebacteria, and five higher eukaryotes (vertebrates). Reconstruction of phylogenetic trees using several parsimony and distance methods confirmed the existence of these two families. Therefore, these results reinforced our previously proposed hypothesis that two close but already different gdh genes were present in the last common ancestor to the three Ur-kingdoms (eubacteria, archaebacteria, and eukaryotes). The branching order of the different species of Family I was found to be the same whatever the method of tree reconstruction although it varied slightly according the region analyzed. Similarly, the topological positions of eubacteria and eukaryotes of Family II were independent of the method used. However, the branching of the two archaebacteria in Family II appeared to be unexpected: (1) the thermoacidophilic Sulfolobus solfataricus was found clustered with the two eubacteria of this family both in parsimony and distance trees, a situation not predicted by either one of the contradictory trees recently proposed; and (2) the branching of the halophilic Halobacterium salinarium varied according to the method of tree construction: it was closer to the eubacteria in the maximum parsimony tree and to eukaryotesin distance trees. Therefore, whatever the actual position of the halophilic species, archaebacteria did not appear to be monophyletic in these gdh gene trees. This result questions the firmness of the presently accepted interpretation of previous protein trees which were supposed to root unambiguously the universal tree of life and place the archaebacteria in this tree. Offprint requests to: B. Labedan     

Shoots grafted into the upper crowns of tall Japanese cedar (Cryptomeria japonica D. Don) show foliar gas exchange characteristics similar to those of intact shoots

The lower foliar photosynthetic rates seen in shoots in the upper crowns of tall trees than those in seedlings could be caused by extrinsic factors, such as hydraulic conductance, for shoots or by irreversible intrinsic change in the meristems during tree development. To clarify which is most significant, we compared foliar gas exchange characteristics and water relations among scions of Japanese cedar (Cryptomeria japonica D. Don) grafted into the upper crowns of tall trees, rooted cuttings developed from scions of the same clones, and intact shoots in the upper crowns of the tall trees. Grafted shoots had the same water regime as intact shoots, as confirmed by their similar water potentials at the turgor loss point, which were more negative than those of the rooted cuttings. No significant difference was observed between the grafted and intact shoots in their light-saturated photosynthetic rate (P_max), stomatal conductance (g_s), photosynthetic capacity, carboxylation efficiency, ratio of intercellular to ambient CO₂ concentration (C_i/C_a), and carbon isotope composition (¹³C). Compared with the rooted cuttings, the grafted shoots showed significantly lower P_max, g_s, photosynthetic capacity, and carboxylation efficiency (to 49%, 31%, 68%, and 65%, respectively). The C_i/C_a and ¹³C indicated significantly stronger instantaneous and long-term stomatal limitation in the grafted shoots than in the rooted cuttings. These indicate that changes in extrinsic factors can reduce foliar photosynthetic rates in shoots in the upper crowns of tall trees as a result of stronger stomatal limitation and reduced photosynthetic activity, without irreversible intrinsic changes in the meristems.     

Direct Calculation of a Tree Length Using a Distance Matrix

Comparative studies of tree-building methods have shown minimum evolution to be in general an accurate criterion for selecting a true tree. To improve the use of this criterion, this paper proposes a method for rapidly and directly calculating a length of a dichotomous tree without having to resort to branch length calculations. This direct calculation (DC) method applies to the complete final topology, giving equal importance to each branch after a dichotomy. According to this method, the tree length S _DC is S _DC =∑ _i ∑ _j (D _ij /2 ^Bij ) = (∑ _i<j ∑D _ij 2 ^Bmax−Bij )/2 ^{Bmax −1} where D _ij is the observed distance between taxa i and j, B _ij is the number of branches connecting i and j, Bmax is the greatest B _ij in the tree, and the powers of two are due to the dichotomy of the tree. This tree length expression may be used as a rapid method for selecting the shortest tree from a set of hypothetical or subobtimal trees. Received: 2 March 2000 / Accepted: 24 March 2000     

Hydrocarbon fermentation: Kinetics of microbial cell growth

Modeling of microbial growth using nonmiscible substrate is studied when kinetics of substrate dissolution is rate limiting. When the substrate concentration is low, the growth rate is described by an analytical relation that can be identified as a Contois relationship. If the substrate concentration is greater than a critical value S_crit, the potentially useful hydrocarbon S* concentration is described by S* = S_crit/(1 + S_crit/S). A relationship was found between S_crit and the biomass concentration X. When X increased, S_crit decreased. The cell growth rate is related to a relation μ = μ_m[A(X/S_crit)(1 + S_crit/S) + 1]^?1. This model describes the evolution of the growth rate when exponential or linear growth occurs, which is related to physico-chemical properties and hydrodynamic fermentation conditions. Experimental data to support the model are presented.     

Tree Holes Used for Resting by Gray Mouse Lemurs (Microcebus murinus) in Madagascar: Insulation Capacities and Energetic Consequences

I studied the insulation capacity of tree holes used by gray mouse lemurs (Microcebus murinus) in a primary dry deciduous forest in western Madagascar during the cool dry season. Tree holes had an insulating effect, and fluctuations of air temperatures were less extreme inside the holes than outside them. The insulation capacity of the tree holes peaked between 0800 and 1100 hr, when ambient temperatures ranged between 25 and 30°C. To compare tree holes, I calculated the mean difference between the internal temperature )(T_i ) and the external temperature (T_e ) for each tree hole. Thus large differences indicate good insulation capacities. The mean difference of tree holes in living trees was significantly larger than that of tree holes in dead trees, which shows that insulation in living trees is more effective. During the dry season, the insulation capacity of tree holes in living trees decreased and was lowest in July, whereas the insulation capacity of holes in dead trees remained approximately constant. Physiological studies under natural temperature and light condition in Microcebus murinus reveal that daily torpor saves around 40% of the daily energy expenditure compared to normothermia. However, torpor can be maintained only up to the threshold body and ambient temperature of 28°C, whereat Microcebus murinus have to terminate torpor actively. By occupying insulating tree holes, mouse lemurs may stay longer in torpor, which increases their daily energy savings by an extra 5%.