Likelihood-mapping: A simple method to visualize phylogenetic content of a sequence alignment

Four Sequences.

Let us consider a set of four sequences, a so-called quartet. For this quartet the maximum likelihoods (not log-likelihoods) 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 p₁+ p₂ + p₃ = 1 and 0 ≤ p_i ≤ 1 in contrast to the maximum likelihoods L_i that are only conditional probabilities with L₁+ L₂+ L₃ ≠ 1. The probabilities (p₁, p₂, p₃) can be viewed as the barycentric coordinates of the point P belonging to the two-dimensional simplex 2 where the e_i are real valued and independent. They point to the three corners of the simplex. As a special case S₂ can be illustrated as an equilateral triangle. This construction allows an easy geometric interpretation of the p_i values. For a given point P ∈ S₂ the p_i are simply the lengths of the perpendiculars from the point P to the three sides of the triangle (Fig. 2).

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Figure 1

The fully resolved tree topologies T₁, T₂, and T₃ connecting four sequences S₁, S₂, S₃, and S₄.

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Figure 2

Map of the probability vector P = (p₁, p₂, p₃) onto an equilateral triangle. Barycentric coordinates are used, i.e. the lengths of the perpendiculars from point P to the triangle sides are equal to the probabilities p_i. The corners T₁, T₂, and T₃ represent three quartet topologies with corresponding coordinates (probabilities) (1, 0, 0), (0, 1, 0), and (0, 0, 1).

If P is close to one corner of the triangle, the likelihoods (p₁, p₂, p₃) are clearly favoring one tree over the other two. Thus, every corner of the triangle corresponds to one of the three quartet topologies T₁, T₂, or T₃. In a typical maximum-likelihood 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.

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Figure 3

(A) Basins of attraction for the three topologies T₁, T₂, and T₃. The gray area shows the region where the probability for tree T₁ is largest. In the center c = (1/3, 1/3, 1/3) all trees are equally likely, at the points x₁₂ = (1/2, 1/2, 0), x₁₃ = (1/2, 0, 1/2), and x₂₃ = (0, 1/2, 1/2) two trees have the same likelihood whereas the remaining one has probability zero. (B) The seven basins of attraction allowing not only fully resolved trees but also the star phylogeny and three regions where it is not possible to decide between two topologies. The dots indicate the corresponding seven attractors. A₁, A₂, A₃ show the tree-like regions. A₁₂, A₁₃, A₂₃represent the net-like regions and A_∗ displays the star-like area.

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 S₂ 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 T₁ and T₂ show probabilities p₁= p₂= ½ and if p₃= 0, for example. Near point x₁₂ (see Fig. 3A) the phylogenetic relationship is best displayed by a net-like geometry that excludes tree T₃. Similarly, near points x₁₃ and x₂₃ 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 A₁, A₂, and A₃, 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 three-dimensional simplex S₃ where the fourth corner represents the star phylogeny, the basins of attraction can be viewed as projections of their corresponding volumes of the tetrahedron S₃ onto the two-dimensional 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 probability-vectors 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 S₂ forms a distinct pattern allowing us to predict a priori whether an n-taxon tree will show a good resolution or not. If most of the points P are found, e.g., in regions A₁₂, A₁₃, A₂₃, or in the star-tree region A_∗, it is clear that the overall tree will be highly multifurcating. That is, evolution was either star-like or not tree-like 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 A₁, A₂, and A₃, it is possible that the overall n-taxon tree is not completely resolved (13, 16).

Four-Cluster Likelihood-Mapping.

Instead of looking at all quartets, the analysis of tree-likeness for four disjoint groups of sequences (clusters) is also possible. Let C₁, C₂, C₃, and C₄ be a set of four clusters with c₁, c₂, c₃, and c₄ sequences. Then, we compute the probability vectors P for the c₁·c₂·c₃·c₄ possible quartets and plot the corresponding points on the triangle S₂. While the p_i values are randomly assigned to the trees T₁, T₂, and T₃, 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 c₁·c₂·c₃·c₄ 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 likelihood-mapping analysis is a helpful tool to illustrate how well supported an internal branch of a given tree topology is.