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Likelihoodmapping: A simple method to visualize phylogenetic content of a sequence alignment
Abstract
We introduce a graphical method, likelihoodmapping, to visualize the phylogenetic content of a set of aligned sequences. The method is based on an analysis of the maximum likelihoods for the three fully resolved tree topologies that can be computed for four sequences. The three likelihoods are represented as one point inside an equilateral triangle. The triangle is partitioned in different regions. One region represents starlike evolution, three regions represent a wellresolved phylogeny, and three regions reflect the situation where it is difficult to distinguish between two of the three trees. The location of the likelihoods in the triangle defines the mode of sequence evolution. If n sequences are analyzed, then the likelihoods for each subset of four sequences are mapped onto the triangle. The resulting distribution of points shows whether the data are suitable for a phylogenetic reconstruction or not.
The sequencebased study of phylogenetic relationships among different organisms has become routine. Parallel to the increasing amount of sequence information available a variety of methods have been suggested to reconstruct a phylogenetic tree (1) or a phylogenetic network (2–4). So far, few approaches have been proposed to elucidate the phylogenetic content in a set of aligned sequence a priori (5, 6). The socalled statistical geometry in sequence space analyzes the distribution of numerical invariants for all possible subsets of four sequences. The resulting distributions make it possible to distinguish between tree, star, and netlike geometry of the data. Moreover, based on the averages of the invariants, the method allows one to draw a graph that illustrates the mode of evolution. While the description of this diagram is straightforward if sequences consist only of purines and pyrimidines, it gets difficult if more complex alphabets (nucleic acids, amino acids) are used (7). Statistical geometry in sequence space has been successfully applied to study the evolution of tRNAs (8) or HIV (9).
In this paper, we present an alternative approach, likelihoodmapping, to display phylogenetic information contained in a sequence alignment. The method is applicable to nucleic acid sequences, amino acid sequences, or any other alphabet provided a model of sequence evolution (1, 10, 11) that can be implemented in a maximum likelihood tree reconstruction program (12, 13). Our approach allows one to visualize the treelikeness of all quartets in a single graph and therefore renders a quick interpretation of the phylogenetic content. We will exemplify the method by applying it to simulated sequences that evolved on a startree or on a completely resolved tree. The analysis of two biological data sets (14, 15) will conclude the paper.
METHOD
Four Sequences.
Let us consider a set of four sequences, a socalled quartet. For this quartet the maximum likelihoods (not loglikelihoods) belonging to the three possible fully resolved tree topologies (Fig. 1) are computed, using any model of sequence evolution (1, 10, 11). Let L_{i} be the maximum likelihood of tree T_{i} where i = 1, 2, 3. We can compute, according to Bayes’ theorem, the posterior probabilities 1 for each tree. Note the p_{i} are true probabilities satisfying p_{1}+ p_{2} + p_{3} = 1 and 0 ≤ p_{i} ≤ 1 in contrast to the maximum likelihoods L_{i} that are only conditional probabilities with L_{1}+ L_{2}+ L_{3} ≠ 1. The probabilities (p_{1}, p_{2}, p_{3}) can be viewed as the barycentric coordinates of the point P belonging to the twodimensional simplex 2 where the e_{i} are real valued and independent. They point to the three corners of the simplex. As a special case S_{2} can be illustrated as an equilateral triangle. This construction allows an easy geometric interpretation of the p_{i} values. For a given point P ∈ S_{2} the p_{i} are simply the lengths of the perpendiculars from the point P to the three sides of the triangle (Fig. 2).
If P is close to one corner of the triangle, the likelihoods (p_{1}, p_{2}, p_{3}) are clearly favoring one tree over the other two. Thus, every corner of the triangle corresponds to one of the three quartet topologies T_{1}, T_{2}, or T_{3}. In a typical maximumlikelihood analysis one chooses the tree T_{i} with 3 It is easy to compute the corresponding basins of attraction for each tree topology (Fig. 3A). The location of a point P in the simplex gives an immediate impression which tree is preferred.
Unfortunately, this picture is too optimistic. For real data it is not always possible to resolve the phylogenetic relationships of four sequences. This is either a consequence of limiting resolution due to short sequences (“noise”) or the true evolutionary tree was a star phylogeny. To account for this case, we introduce a region in the triangle S_{2} representing the star phylogeny. The center c of the simplex is the point where all probabilities take on the value p_{i}= that is the three trees are equally likely. Thus, if P is near the center the phylogenetic relationship cannot be resolved and is better displayed by a star phylogeny. On the other hand, it also might be possible that one can exclude one of the three trees but cannot choose from among the two remaining alternatives. This is the case, if T_{1} and T_{2} show probabilities p_{1}= p_{2}= ½ and if p_{3}= 0, for example. Near point x_{12} (see Fig. 3A) the phylogenetic relationship is best displayed by a netlike geometry that excludes tree T_{3}. Similarly, near points x_{13} and x_{23} it is impossible to unambiguously favor one tree. Based on these seven attractors in the triangle (marked with dots in Fig. 3B) the corresponding basins of attraction are easily computed. Each point in one of the seven regions has smallest Euclidean distance to its attractor. By A_{∗} we denote the region where the star tree is the optimal tree. Its area equals the sum of the areas of A_{1}, A_{2}, and A_{3}, the regions where one tree is clearly better then the remaining ones. The regions A_{ij} represent the situation where we cannot distinguish trees T_{i} and T_{j}. The area of A_{ij} equals the sum of the area of A_{i} and A_{j}.
There is yet another way to describe the basins of attraction. If one considers the threedimensional simplex S_{3} where the fourth corner represents the star phylogeny, the basins of attraction can be viewed as projections of their corresponding volumes of the tetrahedron S_{3} onto the twodimensional plane.
The General Case.
For a set of n aligned sequences there are exactly () different possible quartets of sequences. To get an overall impression of the phylogenetic signal present in the data we compute the probabilityvectors P for the quartets and draw the corresponding points in the simplex. If only few sequences are analyzed, P vectors of all () quartets are considered, otherwise a random sample of, e.g., 1,000 quartets is sufficient to obtain a comprehensive picture of the phylogenetic quality of the data set. The resulting distribution of points in the triangle S_{2} forms a distinct pattern allowing us to predict a priori whether an ntaxon tree will show a good resolution or not. If most of the points P are found, e.g., in regions A_{12}, A_{13}, A_{23}, or in the startree region A_{∗}, it is clear that the overall tree will be highly multifurcating. That is, evolution was either starlike or not treelike at all. However, the opposite conclusion is not necessarily true: Even if all quartets are completely resolved, that is, almost all P vectors are in A_{1}, A_{2}, and A_{3}, it is possible that the overall ntaxon tree is not completely resolved (13, 16).
FourCluster LikelihoodMapping.
Instead of looking at all quartets, the analysis of treelikeness for four disjoint groups of sequences (clusters) is also possible. Let C_{1}, C_{2}, C_{3}, and C_{4} be a set of four clusters with c_{1}, c_{2}, c_{3}, and c_{4} sequences. Then, we compute the probability vectors P for the c_{1}·c_{2}·c_{3}·c_{4} possible quartets and plot the corresponding points on the triangle S_{2}. While the p_{i} values are randomly assigned to the trees T_{1}, T_{2}, and T_{3}, when all quartets are studied, the assignment of p_{i} to tree T_{i} is now fixed. Each tree represents one of the three possible phylogenetic relationships among the clusters. As an illustration, think of the S_{i} in Fig. 1 as a representative of cluster C_{i}. The distribution of the c_{1}·c_{2}·c_{3}·c_{4} probability vectors over the basins of attractions allows one not only to identify the correct phylogenetic relationship of the four clusters but also shows the support for this and alternative groupings. This type of likelihoodmapping analysis is a helpful tool to illustrate how well supported an internal branch of a given tree topology is.
RESULTS
Simulation Studies.
Fig. 4 displays the result of a typical likelihoodmapping analysis. A simulated set of 16 DNAsequences was used to show the distribution of probability vectors P as a function of sequence length and the evolutionary history.
If evolution was according to a star topology then the probability vectors are concentrated in the center of the simplex with rays emanating to the corners of the triangle. This picture does not change with increasing sequence length. However, the proportion of quartets found in area A_{∗} increases (Table 1). If sequence evolution followed a completely resolved tree then the proportion of points P located inside A_{1}+ A_{2}+ A_{3} increases with longer sequences, as an indication that noise due to sampling artifacts is diminished. Correspondingly, the number of quartets in the remaining regions decreases. For sequences of length 500 bp the nontreelike regions of the triangle are empty (Table 1). Thus, Fig. 4 illustrates that likelihoodmapping enables an easy distinction between starlike or treelike evolution. The influence of sequence length (“noise”) on treelikeness of the data is easily recognized.
Data Analysis.
We illustrate the power of likelihoodmapping using two data sets published recently (14, 15). The first set (14) comprises eight partial cytochromeb sequences (135 bp) and nine putative dinosaur sequences (17). The second alignment (1,850 bp) consists of ribosomal DNA from major arthropod classes (three myriapods, two chelicerates, two crustaceans, three hexapods) and six other sequences (human, Xenopus, Tubifex, Caenorhabditis, mouse, and rat). Likelihoodmapping suggests (Fig. 5) that the Zischler et al. (14) data show a fair amount of starlikeness with 17.5% of all quartet points in region A_{∗} in contrast to only 0.2% for the ribosomal DNA. This result is corroborated by the bootstrap analysis as shown in refs. 14 and 15. Because of the short sequence length the percentage of quartets mapped into regions A_{12}, A_{13}, and A_{23} is with 10.1% for the sequences from ref. 14, very high compared with 1.6% for the rDNA sequences. However, the cytochromeb data still contain a reasonable amount of treelikeness as 72.4% of all quartets are placed in the areas A_{1}, A_{2},, and A_{3}. The treelikeness of the ribosomal DNA is extremely high (A_{1}+ A_{2}+ A_{3}= 98.3%). The a posteriori analysis based on bootstrap values (15) shows that all groupings in the tree receive high support.
FourCluster Likelihood Mapping.
A further application of likelihoodmapping allows testing of an internal edge of a tree as given from any tree reconstruction method. As an example we consider the sister group status of myriapods and chelicerates as suggested by Friedrich and Tautz (15). Fig. 6 shows that 90.4% of all quartets between the four corresponding clusters support the branching pattern that groups chelicerates and myriapods versus crustaceans, hexapods, and the remaining sequences. We find only very low support (6.9%) for the topology that pairs myriapods with crustaceans plus hexapods rather than with chelicerates or with the rest. Based on likelihoodmapping we cannot reject the hypothesis of monophyly of myriapods and chelicerates. However, the outcome of statistical tests as suggested in ref. 18 remains to be seen. But this is outside the scope of this paper.
DISCUSSION
The evaluation of the phylogenetic contents in a data set is of prime importance if one wants to avoid false conclusions about evolutionary relationships among organisms. Methods abound that evaluate the reliability of a reconstructed tree a posteriori (1). Likelihoodmapping† can be viewed as a complementary approach to existing methods of a priori or a posteriori evaluations of treelikeness. Our method may be helpful when analyzing controversial phylogenies. Similar to statistical geometry in sequence space (5–7) likelihoodmapping is based on the analysis of quartets, the basic ingredients to reconstruct trees (16). Moreover, the description of seven basins of attraction (Fig. 3) that can be characterized as fully resolved (A_{1}, A_{2}, and A_{3}), starlike (A_{∗}), or intermediate between two trees (A_{12}, A_{13}, and A_{23}) is also of great importance in the quartetpuzzling tree search algorithm (13, 19). Using a variant of likelihoodmapping it is also possible to detect recombination (A.v.H., unpublished data).
Here, we have provided a simple, but versatile, approach to visualize the phylogenetic content of a data set. We have shown that the method has reasonable predictive power. While we have presented only a visual tool to analyze the phylogenetic signal of sequences it is certainly necessary to develop solid statistical tests, that provide evidence as to the significance of clusters (18) or to a deviation from treelikeness. For example, the assumption of equal prior probability for the trees may be debatable. It remains to be seen how approaches like Jeffrey’s prior (20) or the inclusion of the variance of likelihood estimates (21) will influence the analysis.
Finally, one should keep in mind that the interpretation of the result of a likelihoodmapping analysis strongly depends on sequence length. The alignment of human mitochondrial controlregion data (22) comprises 1,137 positions, and 82.5% of the quartets belong to the regions that represent fully resolved trees. Thus, the result suggests that the data are very well suited to reconstruct a well resolved tree. However, we observe 8.3% of all quartets in the starlike region A_{∗} of the triangle. This value is too high for a completely resolved phylogeny (see Table 1). Therefore, we expect a phylogeny that is well resolved in certain parts of the tree only.
Acknowledgments
We thank Roland Fleissner, Nick Goldman, Sonja Meyer, Svante Pääbo, and Gunter Weiss for fruitful and stimulating discussions. We also would like to thank Hans Zischler and Diethard Tautz for providing the sequence alignments. Walter Fitch made helpful comments on a late version of the manuscript. Finally, we would like to acknowledge financial support from the Deutsche Forschungsgemeinschaft.
Footnotes

↵* To whom reprint requests should be addressed. email: arndt{at}zi.biologie.unimuenchen.de.

Walter M. Fitch, University of California, Irvine, CA

↵† Likelihoodmapping analysis is available as part of the maximumlikelihood tree reconstruction program puzzle Version 3.0 (13, 19). It can be retrieved free of charge over the Internet from URLs ftp://ftp.ebi.ac.uk/pub/software and http://www.zi.biologie.unimuenchen.de/~strimmer/puzzle.html.
 Received September 28, 1996.
 Accepted April 23, 1997.
 Copyright © 1997, The National Academy of Sciences of the USA
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