1471-2105-9-166 1471-2105 Research article <p>Fast NJ-like algorithms to deal with incomplete distance matrices</p> Criscuolo Alexis criscuol@lirmm.fr Gascuel Olivier gascuel@lirmm.fr

Equipe Méthodes et Algorithmes pour la Bioinformatique, LIRMM, CNRS – Université Montpellier 2, 161 rue Ada, 34392 Montpellier Cedex 05, France

Groupe Phylogénie Moléculaire, ISEM, CNRS – Université Montpellier 2, C.C. 064, 34095 Montpellier Cedex 05, France

Equipe Bioinformatique Théorique, LSIIT, Université Louis Pasteur, Strasbourg 1, Pôle API, Boulevard Sébastien Brant, BP 10413, 67412 Illkirch Cedex, France

 BMC Bioinformatics 1471-2105 2008 9 1 166 http://www.biomedcentral.com/1471-2105/9/166 18366787 10.1186/1471-2105-9-166 25 10 2007 26 3 2008 26 3 2008 2008 Criscuolo and Gascuel; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Distance-based phylogeny inference methods first estimate evolutionary distances between every pair of taxa, then build a tree from the so-obtained distance matrix. These methods are fast and fairly accurate. However, they hardly deal with incomplete distance matrices. Such matrices are frequent with recent multi-gene studies, when two species do not share any gene in analyzed data. The few existing algorithms to infer trees with satisfying accuracy from incomplete distance matrices have time complexity in O(n4) or more, where n is the number of taxa, which precludes large scale studies. Agglomerative distance algorithms (e.g. NJ 12) are much faster, with time complexity in O(n3) which allows huge datasets and heavy bootstrap analyses to be dealt with. These algorithms proceed in three steps: (a) search for the taxon pair to be agglomerated, (b) estimate the lengths of the two so-created branches, (c) reduce the distance matrix and return to (a) until the tree is fully resolved. But available agglomerative algorithms cannot deal with incomplete matrices.

Results

We propose an adaptation to incomplete matrices of three agglomerative algorithms, namely NJ, BIONJ 3 and MVR 4. Our adaptation generalizes to incomplete matrices the taxon pair selection criterion of NJ (also used by BIONJ and MVR), and combines this generalized criterion with that of ADDTREE 5. Steps (b) and (c) are also modified, but O(n3) time complexity is kept. The performance of these new algorithms is studied with large scale simulations, which mimic multi-gene phylogenomic datasets. Our new algorithms – named NJ*, BIONJ* and MVR* – infer phylogenetic trees that are as least as accurate as those inferred by other available methods, but with much faster running times. MVR* presents the best overall performance. This algorithm accounts for the variance of the pairwise evolutionary distance estimates, and is well suited for multi-gene studies where some distances are accurately estimated using numerous genes, whereas others are poorly estimated (or not estimated) due to the low number (absence) of sequenced genes being shared by both species.

Conclusion

Our distance-based agglomerative algorithms NJ*, BIONJ* and MVR* are fast and accurate, and should be quite useful for large scale phylogenomic studies. When combined with the SDM method 6 to estimate a distance matrix from multiple genes, they offer a relevant alternative to usual supertree techniques 7. Binaries and all simulated data are downloadable from 8.

Background

Phylogeny inference methods can be classified into two main categories: character-based (e.g. maximum-parsimony or maximum-likelihood) and distance-based approaches. The latter have low running times which are quite useful (mandatory in some cases) to perform large-scale studies and bootstrap analyses. A number of computer simulations 91011121314151617 have shown that distance methods are fairly accurate, though not as accurate as likelihood-based methods that are much more time consuming. Using any distance-based method first requires to estimate the pairwise evolutionary distances between every taxon pair. These distances are usually estimated from DNA, RNA or protein sequences, but can also be obtained from DNA-DNA hybridization experiments or, e.g., computed from morphological data (see 18 for a review on distance estimation from various data types).

In the last few years, phylogenomic studies (i.e. phylogeny reconstruction from large gene collections 7) have instigated to the development of fast tree-building techniques being able to infer trees from datasets comprising hundreds of genes and taxa. The low-level gene combination involves concatenating the different genes into a unique supermatrix of characters, and then analyzing this matrix with a standard tree building method. This approach was shown to perform poorly when combined with distance methods, due to inaccurate distance estimations from such large heterogeneous character matrix 6. Better distance-based trees are obtained by extracting the phylogenetic information from each gene separately, and then combining resulting information sources into a unique distance supermatrix. The Average Consensus Supertree (ACS 19) and Super Distance Matrix (SDM 6) techniques input a collection of distance matrices being estimated from each gene separately (the so-called medium-level combination), or being equivalent to the gene trees (the high-level combination). These distance matrices are deformed, without modifying their topological message, and then averaged to obtain the distance supermatrix, which is finally analyzed using a distance-based tree building algorithm.

Estimating the distance supermatrix is fast. However, missing entries may occur in distance supermatrices depending on the extent of taxon overlap within the source matrices. For example, with the two large data sets of Driskell et al. 20, which were collected from Swiss-Prot and Gen-Bank thanks to a computer program, the ratio of missing distances is ~19% and ~1.2%, respectively. These distances are missing because only a few genes are sequenced within each species, meaning that a number of species pairs do not share any sequenced gene in common and cannot be compared using available data. However, Driskell et al. showed that, despite the sparseness of data and the fact that only a small subset of these data is potentially phylogenetically informative, a topological signal still emerges, which provides useful insights into the tree of life (see 20 and below for details). Analogous findings were reported by a number of authors in various contexts 212223, and tree building from sparse data has become topical, as can be seen from the flourishing literature on supertrees.

However, tree building from incomplete distance matrices is NP-hard 24, and thus practical algorithms are heuristics. The indirect approach involves first estimating missing distances by applying an ultrametric 25, additive 26, decomposition-based 27, or quartet-based 28 completion algorithm. The TREX package 29 provides several implementations of such algorithms to be used before tree building using any standard method with the completed matrix. The direct approach involves using a weighted least-squares (WLS) algorithm and associating missing distances with null weight (i.e. infinite variance), which means that missing distances are simply discarded from WLS computations (18, pp. 449). The FITCH algorithm 30 from the PHYLIP package 31 and the MWMODIF algorithm 32 from TREX implement this technique. A combination of both direct and indirect methods is provided by MW* 33 (also available in TREX); this algorithm first applies an ultrametric or additive completion algorithm (depending on the density of missing distances) and then infers a tree using MWMODIF, where weights are set to 1.0 for known distances, 0.5 for estimated distances, and 0.0 for missing distances (if any remain). All these (direct or indirect) algorithms have O(n4) time complexity or more, where n is the number of taxa. This limits their application to medium-sized datasets (say 200 taxa without bootstrapping, see below).

Agglomerative algorithms are much faster and allow dealing with thousands of taxa, as soon as the distance matrix is complete. The most popular of them is the Neighbor-Joining (NJ) algorithm 12. Starting from a star tree, agglomerative algorithms iteratively perform the three following steps, until the tree is completely resolved:

(a) select a taxon pair xy that is agglomerated into a new node u;

(b) estimate the length of the two so-created external branches ux and uy;

(c) replace x and y by u in the distance matrix, and estimate the new distances between u and the not-yet-agglomerated taxa.

Step (a) is more time consuming than the two other steps, because of testing all the O(n2) taxon pairs to select the optimal one. To this purpose, NJ optimizes a numerical criterion that is denoted as Qxy. This criterion admits several interpretations related to the Minimum Evolution principle 134, but also to the acentrality of the considered pair 3536. In this last interpretation (used here), Qxy measures how much the path joining x to y is far from the other taxa i ≠ x, y. The xy pair maximizing Qxy corresponds to the two taxa which are most distant from the other ones and is the best candidate for agglomeration. Another criterion, denoted as Nxy, is used by ADDTREE 5; this second criterion is based on the four point condition 3738 and counts the number of taxon quartets xyij where x and y are neighbors. When the distance matrix exactly corresponds to a tree (it is then said to be additive), Nxy indicates all pairs of sibling taxa in the tree, whereas Qxy indicates just one such taxon pair. We shall see that this property of Nxy is a great advantage when dealing with incomplete distance matrices. Indeed, Qxy is sometimes unusable whereas Nxy is still informative.

Steps (b) and (c) essentially correspond to distance averaging, which requires O(n) run time. These three steps being repeated n - 2 times, agglomerative algorithms require O(n3) time when using the Qxy pair selection criterion, and O(n4) with Nxy 39.

Several refinements of the NJ algorithm have been proposed. BIONJ 3 minimizes the variances associated to the new distances being estimated during each reduction step (c). This way, BIONJ makes use at each iteration of reliable distance estimates to select the new taxon pairs to be agglomerated. To this aim, BIONJ uses a simple Poisson model of the variances and covariances of the distances being contained in the initial distance matrix. BIONJ was generalized into the Minimum Variance Reduction algorithm (MVR 4), a WLS variant of which can deal with any distance variance model, but which does not account for the distance covariances. It has been shown using computer simulations that this variant (named WLS-MVR in 4 but referred here as MVR for simplicity) has similar accuracy as NJ when applied to distance matrices estimated from one-gene alignments 4. WEIGHBOR 40 further refines BIONJ approach and uses an agglomeration criterion which accounts for the variances of evolutionary distances. All these algorithms require O(n3) time. Faster, sophisticated distance-based algorithms have been proposed in the last few years 414243444546, some of them being clearly more accurate than NJ and BIONJ (e.g. FASTME 42 and STC 44, in O(n2 log(n)) and O(n2), respectively).

In this paper, we propose an adaptation of the agglomerative scheme to quickly infer phylogenetic trees from incomplete distance matrices. We show that the Qxy criterion may be rewritten to express the mean acentrality of the xy taxon pair. In the same way, the Nxy criterion may be rewritten to express the mean number of taxon quartets where x and y are neighbors. By estimating these two means using all available (non-missing) distances, we define the two criteria Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ and Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ which allow for the selection of taxon pairs in step (a), even when the distance matrix is incomplete. Using these two new criteria in the agglomerative scheme requires O(n3) and O(n4) run time, respectively. A limitation of Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ and Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ is that they cannot be computed when the distance corresponding to the xy pair is missing (see Methods for more). However, this difficulty is inherent to the problem of building trees from incomplete distance matrices and is encountered (in various forms) by all methods to deal with this problem. Moreover, Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ partly circumvents this difficulty thanks to its ability to indicate several relevant pairs, rather than a single one with Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ (see Methods for more). As running Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ requires O(n4) time, we use a filtering technique: at each step (a) we use Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ to select the s most promising pairs for agglomeration, and then use Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ to select the best of these s pairs. This computational trick (and other refinements, see Methods) greatly improves the accuracy compared to using Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ only, while requiring O(sn3) time, where s is a small constant (s = 15 in our experiments). Finally, the original NJ, BIONJ and MVR formulae corresponding to steps (b) and (c) essentially are distance averaging and are easily adapted to incomplete matrices. The three new algorithms are named NJ*, BIONJ* and MVR*, respectively.

Results and Discussion

Several computer simulations are presented in this section to assess the performance of NJ*, BIONJ* and MVR*. We first compare the agglomeration criteria Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@, Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ and their combination with distance matrices that are additive, but contain missing entries. Then, using more realistic datasets, we compare NJ*, BIONJ*, MVR* to FITCH 30 and MW* 33, in terms of both topological accuracy and run times.

Comparison of agglomeration criteria

Our approach is similar to Makarenkov and Lapointe's 33. We analyze with various algorithms and criteria a distance matrix with randomly deleted entries. The distance matrix we use is additive, i.e. is obtained from a tree by computing the path length distance between every taxon pair. Let T denote this tree and (Tij) be the corresponding distance matrix, where Tij is the path-length (or patristic) distance between taxa i and j in T. When no entry is missing, such an additive matrix uniquely defines T, which is recovered by any consistent algorithms (as are all algorithms being tested here). When entries are missing in (Tij), recovering T becomes a difficult task (see above), and we measure how well the algorithms perform when given an increasing number of missing distances. Such data thus are not realistic from a biological stand point, as evolutionary distances estimated from sequences are not additive, but this is a simple and standard approach to compare algorithms and agglomeration criteria.

We use for the correct tree T the phylogeny of 75 placental mammals from 6. The percentage of missing entries is Pmiss = 1%, 5%, 10%, 20%, 30%. For each Pmiss value, 500 replicates are randomly generated. From each of these 5 × 500 incomplete additive distance matrices, a tree T^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmivaqLbaKaaaaa@2D16@ is inferred by FITCH, MW* and BIONJ*. Various values of the s parameter are tested for BIONJ*, in order to compare the topological accuracy of Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@, Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@, and of the combination of these two agglomeration criteria. With s = 1, BIONJ* uses Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ only. With, s > 1, the taxon pairs corresponding to the s highest values of Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ are reanalyzed with Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ (and with other criteria when ties occur; see Methods). When s becomes large (which is denoted as s = max) BIONJ* uses Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ only, as all taxon pairs are retained in the first selection step.

Each inferred tree T^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmivaqLbaKaaaaa@2D16@ is compared to the correct tree T by using the quartet distance dq 47. This topological distance measures the number of resolved 4-taxon subtrees which are induced by one tree but not the other, and thus is more precise than the widely used bipartition distance 48 which counts the number of internal branches present in one tree but not in the other. Moreover, the quartet distance is less affected than the bipartition distance by small topological errors, e.g. wrong position of a single taxon 49. This distance is normalized: dq = 0 indicates that T and T^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmivaqLbaKaaaaa@2D16@ are identical, whereas dq = 1 means that both trees do not share any resolved 4-taxon subtrees. Averages of the 500 dq measures for each Pmiss value are displayed in Figure 1, for FITCH, MW*, and BIONJ* with various s values.

<p>Figure 1</p>

Topological accuracy depending on the rate of missing entries

Topological accuracy depending on the rate of missing entries. Horizontal axis: percentage of missing distances (Pmiss. Vertical axis: topological accuracy measured by the mean (over 500 trials) quartet distance (dq) between the correct and inferred trees. s: number of taxon pairs that BIONJ* first selects using NJ-like Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ criterion (6), and then analyzes using score-based Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ criterion (9) (and criteria (8), (10), (11) in case of ties). The distance matrix is additive, and thus all methods recover the correct tree when Pmiss = 0.

All curves in Figure 1 are decreasing; as expected, the correct tree T is better recovered (i.e. the mean dq value between T^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmivaqLbaKaaaaa@2D16@ and T decreases) as the proportion of missing distance Pmiss becomes closer to 0. Using Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ in BIONJ* greatly improves the agglomeration step; e.g. with Pmiss = 10%, mean dq values of BIONJ* are ~0.0015 and ~0.0008, with s = 1 and s = 15, respectively. However, there is no significant difference between s = 15 and s = max (as assessed by a sign-test 50 based on the 500 replicates, all p-values are much larger than 0.05), meaning that a small value of s (e.g. s = 15) seems to be enough to focus on the most relevant pairs, while avoiding the computational burden of using Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ only. Further experiments (see below) confirm this finding. FITCH and BIONJ* (with s = 15 and s = max) have similar accuracy, while MW* tends to perform better than the other algorithms with these data. However, we shall see that algorithm ordering is different with more realistic simulations. These experiments thus confirm the advantage of combining Qxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyuae1aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@3110@ and Nxy∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta40aa0baaSqaaiabdIha4jabdMha5bqaaiabgEHiQaaaaaa@310A@ within BIONJ*, and similar results (not shown) are obtained with NJ* and MVR*.

Comparison of reconstruction algorithms with distance supermatrices

We re-use a simulation protocol that we have used previously to compare a number of tree-reconstruction methods in a phylogenomic context 6. This protocol involves generating sequences and evolving them along trees, and is more realistic than the comparison described above. We first summarize this protocol, and then report the results that are obtained with the simulated datasets by FITCH, MW*, NJ*, BIONJ* and MVR*. To estimate the distance supermatrix that is the input of these algorithms, we use the SDM method (6, see also Methods) which computes a supermatrix that summarizes the topological signal being contained in a collection {(Δij1),(Δij2),...,(Δijk)} MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaiWaaeaadaqadaqaaiabfs5aenaaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIXaqmaaaakiaawIcacaGLPaaacqGGSaaldaqadaqaaiabfs5aenaaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIYaGmaaaakiaawIcacaGLPaaacqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaaldaqadaqaaiabfs5aenaaDaaaleaacqWGPbqAcqWGQbGAaeaacqWGRbWAaaaakiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@482D@ of k distance matrices. Simulations 6 have shown the high-quality of this distance supermatrix in both medium- and high-level gene combinations.

Simulations are as follows (see 6 for more details). Starting from a randomly generated tree T with n = 48 taxa, evolution of k genes is simulated, with k = 2, 4, ..., 20. For each of the k genes, some taxa are randomly deleted. Two deletion probabilities are used: 25% to preserve high overlap between the different taxon sets, and 75% to induce low overlap. From these k partially deleted gene alignments, k distance matrices are estimated to compose the collection CΔ of source matrices. The SDM method is then run with CΔ to obtain a distance supermatrix corresponding to a medium-level combination of the k partially deleted genes. To study the high-level combination, a phylogenetic tree is inferred by PhyML 17 from each of the k partially deleted genes; then, the path length distance between each taxon pair for each of the k phylogenies is computed, to form the collection CT of k additive distance matrices that are equivalent to the k PhyML trees. Finally, SDM is applied to CT to obtain a distance supermatrix corresponding to a high-level gene combination.

This simulation protocol is repeated 500 times for each value of k and each deletion proportion. We obtain this way (10 gene collection sizes × 500 collections × 2 overlap conditions × 2 gene combination levels) = 20,000 distance supermatrices, which are denoted as (ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@) and are frequently incomplete. Indeed, if taxon i is missing for gene p, then Δijp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdchaWbaaaaa@3189@ is missing – which is denoted as Δijp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdchaWbaaaaa@3189@ = ∅—, and if Δijp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdchaWbaaaaa@3189@ = ∅ for all p = 1,2, ..., k, then ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@ = ∅. With 25% deletion rate, almost all distance supermatrices are complete when k ≥ 14. With 75% deletion rate, all distance supermatrices are incomplete, but the number of missing distances decreases as k increases (missing distance proportions range from 42% to 11%).

FITCH and MW* are run with default options. In accordance with Figure 1, s is set to 15 for NJ*, BIONJ* and MVR*. With BIONJ*, Vij variances (associated with ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@ distance estimates) are naturally defined by Vij ∝ ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@ if ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@ ≠ ∅, else Vij = ∅. Variances used by MVR* comply with the same rule, but account for other parameters such as the length and the number of sequences being used to estimate each ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@ distance (see Methods). Accuracy of the five algorithms is measured with the topological distance dq, as above, and averaged for the 500 replicates corresponding to each of the conditions. Results are reported in Table 1 for the medium-level gene combination, and in Table 2 for the high-level gene combination. For each value of k, the first- and second-best mean dq values are indicated in bold&underlined and bold, respectively, and a sign-test 50 based on the 500 replicates is used to assess the significance of the difference between these two best values.

<p>Table 1</p>

Topological accuracy with medium-level distance supermatrices

(a): 25% taxon deletion rate

k =

FITCH

MW*

NJ*

BIONJ*

MVR*

p-value

2

0.0841

0.0906

0.0926

    0.0841

0.0857

0.286

4

0.0504

0.0546

0.0595

    0.0494

0.0524

0.466

6

0.0400

0.0445

0.0454

    0.0370

0.0410

0.015

8

0.0330

0.0356

0.0386

    0.0318

0.0320

0.958

10

0.0271

0.0300

0.0317

    0.0265

0.0286

0.364

12

0.0294

0.0317

0.0354

    0.0284

0.0314

0.030

14

0.0245

0.0266

0.0286

    0.0235

0.0251

0.816

16

0.0290

0.0318

0.0327

    0.0282

0.0303

0.028

18

0.0252

0.0278

0.0280

    0.0234

0.0265

0.020

20

0.0242

0.0259

0.0281

    0.0230

0.0247

0.955

(b): 75% taxon deletion rate

k =

FITCH

MW*

NJ*

BIONJ*

MVR*

p-value

2

0.2154

0.2174

0.2187

    0.2131

0.2163

0.920

4

    0.1683

0.1778

0.1818

0.1713

0.1713

0.060

6

    0.1347

0.1443

0.1534

0.1418

0.1400

≈ 0.0

8

    0.1089

0.1253

0.1302

0.1137

0.1114

0.176

10

    0.0878

0.1039

0.1117

0.0959

0.0901

0.033

12

    0.0825

0.0968

0.1021

0.0875

0.0842

0.470

14

    0.0652

0.0749

0.0850

0.0710

0.0676

0.464

16

    0.0583

0.0731

0.0802

0.0658

0.0625

0.335

18

    0.0516

0.0617

0.0687

0.0552

0.0555

0.074

20

    0.0503

0.0600

0.0682

0.0560

0.0509

0.189

In the medium-level combination, distance matrices are directly estimated from each of the genes and then combined (using SDM) into the distance supermatrix. Topolological accuracy is mesured by the mean (over 500 trials) quartet distance (dq) between the correct and inferred trees. k: number of genes. p-value: sign-test significance when comparing the 500 dq values of the two best methods that are indicated in bold and underlined (1st method) and bold (2nd one)

 <p>Table 2</p>

Topological accuracy with high-level distance supermatrices

(a): 25% taxon deletion rate

k =

FITCH

MW*

NJ*

BIONJ*

MVR*

p-value

2

0.0558

0.0561

0.0586

0.0566

    0.0522

≈ 0.0

4

0.0337

0.0345

0.0361

0.0351

    0.0319

≈ 0.0

6

0.0253

0.0265

0.0272

0.0261

    0.0235

≈ 0.0

8

0.0227

0.0228

0.0213

0.0217

    0.0212

0.094

10

0.0187

0.0188

0.0194

0.0192

    0.0171

0.047

12

0.0197

0.0207

0.0215

0.0199

    0.0191

0.949

14

    0.0160

0.0164

0.0164

0.0165

0.0162

0.882

16

0.0208

    0.0204

0.0210

0.0213

0.0206

≈ 0.0

18

    0.0170

0.0177

0.0177

0.0173

0.0174

0.271

20

0.0162

0.0168

0.0171

0.0160

    0.0158

0.648

(b): 75% taxon deletion rate

k =

FITCH

MW*

NJ*

BIONJ*

MVR*

p-value

2

0.1876

0.1877

0.1824

0.1822

    0.1817

0.282

4

0.1396

0.1397

0.1390

0.1381

    0.1345

0.018

6

0.1095

0.1125

0.1134

0.1119

    0.1065

0.166

8

0.0865

0.0892

0.0926

0.0870

    0.0823

0.005

10

0.0690

0.0739

0.0766

0.0723

    0.0671

0.023

12

0.0641

0.0670

0.0705

0.0677

    0.0616

0.015

14

0.0508

0.0538

0.0567

0.0534

    0.0493

≈ 0.0

16

0.0504

0.0518

0.0554

0.0512

    0.0457

≈ 0.0

18

0.0409

0.0416

0.0485

0.0424

    0.0402

0.922

20

0.0403

0.0435

0.0453

0.0431

    0.0371

≈ 0.0

In the high-level combination, ML trees are first inferred separately for every genes, and then these trees are turned into path-length distance matrices which are combined (using SDM) into the distance supermatrix. For symbols and notation, see note to Table 1.

In the medium-level gene combination, NJ* and MW* are outperformed by other algorithms. With a 25% deletion rate, BIONJ* has best topological accuracy, followed by FITCH. However, the sign-test indicates that the difference between these two algorithms is moderately significant as the p-value is lower than 0.05 for only five k values (= 6, 8, 12, 16, and 18). With a 75% deletion rate, FITCH is best, but again the sign-test shows that FITCH, BIONJ* and MVR* are broadly equivalent.

With high-level combination distance supermatrices, NJ* and MW* still tend to be outperformed by other algorithms. BIONJ* is in between, and the best mean dq values are observed with MVR* which is followed by FITCH. The sign-test broadly confirms the significance of this observation, though the accuracy difference between MVR* and FITCH is relatively low.

Altogether, these experiments show that MVR* is at least as accurate as FITCH, that BIONJ* has similar performance, while NJ* and MW* are behind these three algorithms. Comparing these findings with the results from (see Figure 2 in 6), we see that (in the high-level framework, Table 2) MVR* is more accurate than the standard Matrix Representation with Parsimony method (MRP, 5152), in most cases; e.g. with k = 10, MVR* has mean dq values of 0.0171 and 0.0663, for 25% and 75% deletion rate, respectively, while mean dq values of MRP equal 0.0175 and 0.1152. MVR* (combined with SDM) outperforms MRP with sparse information (75% deletion rate and/or low number of genes), while both approaches are nearly equivalent when the information is abundant (25% deletion rate). An explanation 53 of this finding could be that the distance approach not only uses the topology of the source trees (as MRP) but also their branch lengths. Distance-based supertrees thus contain more information than MRP supertrees, which makes a noticeable difference when the information is sparse, but does not impact much the results with abundant information (see also following simulation results).

Results with simulations based on Driskell et al. 20 dataset

This section aims to measure the accuracy of the different tree building algorithms when applied to simulated datasets being more realistic than those commonly used in a phylogenomic perspective. Most notably, uniformly random gene deletion (used in previous section, following 54) is not fully realistic because some genes (e.g. cytochrome b) are sequenced for most species, while some other genes are rarely sequenced (or rare among living species). It follows that the gene presence/absence pattern is different with real datasets to this being induced by uniformly random gene deletion (see 20555657 for illustrative examples). To this purpose, we use the character supermatrix from Driskell et al. 20, which comprises 69 green plant species and 254 genes, and was built via an automated exploration process of GenBank. This matrix contains a total number of 2777 sequences and has 87% missing characters, which are unequally distributed among taxa. Only 3 taxa have more than 50% genes, whereas 42 have 10% genes or less. In the same way, a few genes are present in most taxa (e.g., the 2 most sequenced genes belong to 59 taxa), whereas other genes are rare (e.g. 121 genes are present in at most 5 taxa). However, these k = 254 genes are complementary and the SDM distance supermatrix only contains ~1.2% missing entries. This low proportion of missing entries is favorable to tree reconstruction, but still requires an algorithm able to deal with incomplete matrices.

We use a simulation protocol analogous to that described above 6. The only difference is the deletion procedure, with random deletion replaced by the gene presence/absence pattern of (see Figure 2B in 20). We generate 100 datasets this way with n = 69 taxa and k = 254 genes. From these 100 datasets, we infer 100 distance matrix collections CΔ and 100 tree collections CT. Each of these 2 × 100 collections is dealt with by SDM, to obtain a distance supermatrix (ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@) that contains the same missing entries as those induced by the original dataset 20. We use these matrices to compare FITCH, MW*, NJ*, BIONJ* and MVR*, based on dq quartet distance between the correct and inferred trees (see above). Our three algorithms are run with both s = 15 and s = max. Results of MRP are also computed, using TNT 58 to infer the most parsimonious trees. TNT is run with 25 random addition sequences, TBR branch swapping and ratchet. The MRP supertree is defined in the standard way 59 as the strict consensus of the most parsimonious trees. Results are displayed in Table 3, which is similar to Tables 1 and 2; the first- and second-best mean dq values are indicated in bold&underlined and bold, respectively, and sign-tests are used to assess the significance of the differences between MVR* (our best algorithm), FITCH and MRP.

<p>Table 3</p>

Topological accuracy with datasets generated from Driskell et al. [20]

(a): medium level

FITCH

MW*

NJ*

BIONJ*

MVR*

p-value

MVR* – FITCH

MVR* – BIONJ*

0.0234

0.0268

0.0289

0.0227

    0.0171

≈ 0.0

≈ 0.0

(b): high level

FITCH

MW*

NJ*

BIONJ*

MVR*

MRP

p-value

MVR* – FITCH

MVR* – MRP

0.0161

0.0165

0.0182

0.0172

    0.0101

0.0119

0.001

0.193

(a): Medium-level combination of the distance matrices being directly estimated from the gene sequences. (b): High-level combination; ML trees are first inferred separately for every genes; MRP turns the gene trees into a matrix of partial binary characters, which is analyzed with parsimony; with the other (distance) methods, the gene trees are turned into path-length distance matrices which are combined into the distance supermatrix. All combinations of source distance matrices are achieved using SDM. p-value: sign-test significance when comparing the 100 dq values of MVR* (our best algorithm) to those of FITCH and MRP. For other symbols and notation, see note to Table 1.

NJ*, BIONJ* and MVR* do not show any significant difference when used with s = 15 and s = max (as assessed by the sign-test, all p-values are much larger than 0.05, results not shown). This confirms the results of the previous experiments to compare our various agglomeration criteria. NJ* has the worst accuracy, especially in the high-level combination framework. MW*, FITCH and BIONJ* show similar performance, while MVR* is best among distance approaches in the two gene combination levels. Moreover, the difference between MVR* and FITCH is highly significant (sign-test p-value ≈ 0.0). In the high-level framework, MVR* tends to be better than MRP, although the information is quite abundant (254 genes, ~1.2% of missing distances); however, the difference is not significant with 100 replicates (sign-test p-value ≈ 0.2). The results among distance methods are explained by the fact that MVR* uses fairly accurate estimates (VijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@334C@) of the variances of the distances in (ΔijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@337D@). Indeed, dataset 20 induces a highly heterogeneous distribution of missing sequences, meaning that some distances are well estimated thanks to a large number of sequences, while some others are poorly estimated via a few sequences. This is accounted for by MVR* in (VijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@334C@) calculations (see Methods), while MW*, FITCH and BIONJ* lack this information and use inaccurate estimations of (VijSDM MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aa0baaSqaaiabdMgaPjabdQgaQbqaaiabbofatjabbseaejabb2eanbaaaaa@334C@). The difference between these two approaches (i.e. MVR* on the one hand, and MW*, FITCH and BIONJ* on the other hand) is somewhat hidden when using uniformly random sequence deletion, because with the latter all distances are broadly estimated with the same number of genes. With biologically realistic pattern of gene presence/absence, the difference becomes important, especially for the high-level combination. Thus, this last set of simulations confirms the findings of the previous ones and supports the capacity of MVR* for dealing with phylogenomic data.

Run time comparison

Run times with various dataset sizes have been measured on a PC Pentium IV 1.8 GHz (1 Gb RAM) and are displayed in Table 4. We do not report the run times of NJ* and BIONJ*, as they are nearly the same as those of MVR*. In fact, NJ* and BIONJ* are ~2% faster than MVR*, because they are simpler, but these simplifications does not concern the heavy O(n3) parts of the algorithms (see Methods). We also report the run times of SDM 6, which are in the same range as the fastest tree building algorithms, except with Driskell et al. 20-like datasets, where SDM has to summarize a large number (254) of source matrices, but where the number of taxa (69) is relatively low. In this case, the run time of SDM is analogous to that of FITCH and MW* and remains quite handy (~5 minutes per dataset).

<p>Table 4</p>

Run times

(a): 25% taxon deletion rate

SDM

FITCH

MW*

MVR*

K =

k =

k =

k =

10

2

10

20

2

10

20

2

10

20

n = 48

< 1

11

23

23

21

39

41

< 1

<