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to evaluate the PAUP tree building method using maximum likelihood criteria, but a single tree-building time was over30minutes for a dataset that contains50sequences, so we could not finish our calculations.

The log likelihood values of the trees were estimated using Felsenstein-Churchill model [45], applying the DNAML program from the PHYLIP program package. This model assumes that the transition and transversion ratios vary from site to site corresponding to a Hidden Markov Model [73].

Table 5.1: The average of the RF differences for the model trees. The number of leaves was chosen to ben = 50. Theλ parameter of the standard exponential distribution for which the lengths of the edges obeys, was set to 0.10.51.0. In these experiments we used the PAUP package as we described in Section 5.3.2.

Number of Length of ancestor =900

input trees ORIG MCC GREEDY STRICT MAJ.

K2P λ= 0.1

10 25 50

8.63

7.33 7.50 8.46

8.57 8.57 8.48

12.14 18.71 28.62

8.57 9.70 8.47 F84

λ= 0.1

10 25 50

8.75

7.73 8.29 8.53

8.69 8.57 8.50

12.15 18.96 28.52

8.67 8.59 8.50 K2P

λ= 0.5

10 25 50

15.61

9.75 9.78 10.92

13.54 13.30 13.23

19.63 25.07 29.95

13.46 13.32 13.20 F84

λ= 0.5

10 25 50

15.86

8.87 11.00 11.11

13.43 13.14 12.86

19.57 24.82 29.69

13.41 13.15 12.83 K2P

λ= 1.0

10 25 50

39.23

37.59 36.09 37.33

38.06 38.04 37.89

32.71 33.39 34.70

37.92 38.08 37.91 F84

λ= 1.0

10 25 50

40.47

38.26 38.63 39.70

38.94 38.85 39.29

32.62 33.58 34.85

39.00 38.91 39.29

5.3.3 Evaluation of performance

We evaluated the consensus tree building methods in terms of RF differences described in Section 2.1. We tried to cover all types of datasets which may occur in real life. The size of the databases was limited by the computational burden of the tree reconstruction methods. Therefore, the sizes of the model datasets were set to 50100 200.

The branch length of the trees was generated according to an standard exponential distribution. The λparameter was set to one of three different values: 0.10.51.0.

With a 1.0 settings the model dataset had a rather high divergence. In addition, we tested the methods using two different evolutionary models: the Kimura-2-parameter (K2P) and Felsenstein84 (F84) models.

5.3 Experiments 45

Figure 5.1: The MCC consensus tree of the hydrogenase group. The taxonomic groups themselves can be seen separately on the right side of the picture.

Table 5.1 lists the results for n = 50 leaves. The results reveal a few general trends. First, in each case it is worth applying a consensus method; on the other hand among the consensus methods the MCC approach using the maximum likelihood based weighting (described in Section 5.2.3) outperforms all the other algorithms in terms of symmetric difference, except when the dataset displays a high divergence (i.e. λ= 1.0).

The differences between the performances are only marginally influenced by the choice of the number of input trees. The choice of theλparameter, however, hardly influences the performance. We should mention here that if the task is very complicated (e.g.

with K2P and F84 when λ = 0.1), i.e. the tree reconstruction is not so accurate, the consensus tree methods will not achieve any significant improvement on the results. In this case only the strict consensus method produces a noticeable improvement, because it generates trees which are not so resolved (i.e. their cluster set contain only a few clusters). It indicates that the information content of the strict consensus trees are very low (i.e. highly unresolved), this means that the input profile contains very different trees.

The strict consensus method achieves a higher symmetric difference, however, when the task is not so complicated. Table 5.2 shows the results for datasets of100sequences (the results for the dataset of 200 sequences are not shown). The general trends are quite similar to those mentioned above.

In Table 5.3 the results on amino acid sequences are presented. The general trend is very similar to the results of previous two experiments, but in this case the MCC outperforms the other consensus tree methods even in the case of λ= 1.

Figure 5.2: The Majority consensus tree of the hydrogenase group.

Table 5.2: The average of the RF differences for the model trees. Here the number of leaves is n= 100. The λ parameter of the standard exponential distribution for which the lengths of the edges obey, is set to0.10.51.0. In this experiment we used the PAUP package described in Section 5.3.2.

Number of Length of ancestor =900

input trees ORIG MCC GREEDY STRICT MAJ.

K2P λ= 0.1

10 25 50

12.55

9.51 11.39 11.77

11.94 11.97 11.85

17.47 24.56 32.62

11.95 12.00 11.84 F84

λ= 0.1

10 25 50

12.16

9.78 11.14 11.17

11.38 11.50 11.26

16.84 24.13 32.32

11.44 11.52 11.28 K2P

λ= 0.5

10 25 50

24.05

15.16 17.02 18.88

21.57 21.48 21.48

28.30 33.96 39.35

21.51 21.43 21.49 F84

λ= 0.5

10 25 50

23.65

14.33 14.32 17.94

21.33 21.29 21.21

27.96 34.40 39.86

21.27 21.31 21.19 K2P

λ= 1.0

10 25 50

87.80

86.24 89.98 87.87

87.12 87.23 87.21

72.01 69.49 68.37

87.14 87.19 87.24 F84

λ= 1.0

10 25 50

87.05

82.75 86.60 74.77

86.57 86.43 86.41

71.41 68.65 68.02

86.45 86.36 86.39

5.3 Experiments 47

Table 5.3: The average of the RF differences for the model trees. The ancestral sequence was an amino acid sequence with length of 500 in this test. The number of leaves was chosen n= 100. Theλ parameter of the standard exponential distribution for which the lengths of the edges obey, was set to 0.10.51.0. In this experiment we used the PAUP package described in Section 5.3.2.

Number of Length of ancestor = 500

input trees ORIG MCC GREEDY STRICT MAJ.

BLOSUM λ= 0.1

10 25 50

9.74

8.56 8.88 9.40

9.78 9.70 9.70

12.74 18.84 26.74

9.82 9.68 9.68 BLOSUM

λ= 0.5

10 25 50

12.27

10.08 10.72 10.84

11.86 11.69 11.42

15.75 21.88 29.55

11.71 11.69 11.36 BLOSUM

λ= 1.0

10 25 50

18.87

10.95 13.60 15.01

16.36 16.11 16.04

21.64 26.15 29.94

16.34 16.13 16.09 In our experiments the computational time requirements for all four evaluated consensus methods were no more than few seconds in each test case, which is negligible when we consider the time requirements for the tree building phase.

5.3.4 A real-life dataset

The real-life dataset we used is the group of71[NiFe]hydrogenases. Hydrogenases are metallo-enzymes that catalyze the reaction H2 ­2H++ 2e. They can be found in bacteria, archae and cyanobacteria. The [Ni-Fe] hydrogenases are usually placed into 4 different taxonomic groups [60].

The multiple alignment of the sequences was performed by the ClustalW program [30] using the Neighbor-Joining tree as the guide tree and the BLOSUM80 matrix [28]

as the scoring matrix. The aligned sequences are available at the supplementary web site. In our experiments we employed the following tree building methods:

1. Single Linkage 2. Neighbor-Joining [9]

3. PAUP using Maximum Likelihood criteria [20]

4. PAUP using Maximum Parsimony criteria [20]

5. Multi-Stack [11]

6. Tree Puzzle [74]

7. MrBayes [75]

Here trees for both small and large subunits of hydrogenase were used for consensus determination. Altogether the profile contained14trees. The strict consensus algorithm resulted in a fully unresolved tree. The majority consensus and the greedy consensus tree were the same in this experiment and it is presented in Figure 9.3. The taxonomic group ID is shown after the GROUP keyword in the name of the proteins. The grouping of hydrogenases resulted from these consensus methods is confused. The previously determined taxonomic groups were badly done. In contrast the MCC approach resulted in a tree which classifies the taxonomic groups perfectly. Every single enzyme is correctly placed within its correct taxonomic group justifying the classification of the hydrogenases which was based not just on phylogenetic but also on structural and functional similarities of different enzymes [60].