Taxon ordering in phylogenetic trees by means of evolutionary algorithms
- Francesco Cerutti^{1, 2},
- Luigi Bertolotti^{1, 2},
- Tony L Goldberg^{3} and
- Mario Giacobini^{1, 2}Email author
https://doi.org/10.1186/1756-0381-4-20
© Cerutti et al; licensee BioMed Central Ltd. 2011
Received: 11 June 2010
Accepted: 1 July 2011
Published: 1 July 2011
Abstract
Background
In in a typical "left-to-right" phylogenetic tree, the vertical order of taxa is meaningless, as only the branch path between them reflects their degree of similarity. To make unresolved trees more informative, here we propose an innovative Evolutionary Algorithm (EA) method to search the best graphical representation of unresolved trees, in order to give a biological meaning to the vertical order of taxa.
Methods
Starting from a West Nile virus phylogenetic tree, in a (1 + 1)-EA we evolved it by randomly rotating the internal nodes and selecting the tree with better fitness every generation. The fitness is a sum of genetic distances between the considered taxon and the r (radius) next taxa. After having set the radius to the best performance, we evolved the trees with (λ + μ)-EAs to study the influence of population on the algorithm.
Results
The (1 + 1)-EA consistently outperformed a random search, and better results were obtained setting the radius to 8. The (λ + μ)-EAs performed as well as the (1 + 1), except the larger population (1000 + 1000).
Conclusions
The trees after the evolution showed an improvement both of the fitness (based on a genetic distance matrix, then close taxa are actually genetically close), and of the biological interpretation. Samples collected in the same state or year moved close each other, making the tree easier to interpret. Biological relationships between samples are also easier to observe.
Background
A central goal of evolutionary biology is to describe the "Tree of Life", inferring relationships among all living organisms. First appeared in the XIX century, trees were often used to describe relationships among organisms, but only Charles Darwin, in his revolutionary Origin of the Species [1], was the first to define them as evolutionary trees. Instead of using phenotypic characters, as Darwin did first, nowadays such trees are commonly built on genetic information and models of molecular evolution.
A first approach to reorder the taxa according to a distance matrix was introduced by Moscato, Cotta and colleagues [7, 8]. In these works, the authors both build new phylogenies and improve existing ones generated by Neighbor Joining and hypercleaning methods. They approached the problem as a minimum Hamiltonian path problem, and used memetic algorithms to find the "solution that minimizes the length of a path of distances between species" [8].
Here we propose a different approach belonging to the Evolutionary Algorithms (EAs) family to search a better graphical representation of an unresolved tree [9, 10]. Given that each node can be freely rotated, one could group taxa with similar features, such as genetic similarity, geographic location or collection date, preserving the original topology. This approach could assist in drawing phylogenetic trees including more information, especially in highly unresolved trees.
Considering a tree with its own topology (either rooted or unrooted, and previously determined by means, for example, of neighbor joining, maximum likelihood, maximum parsimony, or Bayesian phylogenetics), the search space contains all the possible tree representations obtainable by node rotation. In this environment, the search problem would be to find the graphical representation that minimizes the distances between adjacent taxa. Such distances can be defined in separated matrices, obtained from different taxa features like genetic, temporal or geographic data.
where {d _{1}, d _{2}, ..., d _{ N } } are the degrees of the N internal nodes of the tree topology. Considering the previous case of the tree with 64 clades, this value would range from |neigh _{ s } | = 64 to |neigh _{ s } | = (64 × 63) = 2 = 2, 016.
In order to find the optimal solution to our problem, we chose to use (λ + μ)-EAs, where λ is the parent population from which the child population μ is generated. In general, a (λ + μ)-EA can be seen as a natural selection process (being inspired by the biological world), where the λ parents mate, generate μ children (different from them), and the fittest to the environment (i.e. the problem to be solved) survives and mate again. The λ parents for the next generation are given by the best (λ + μ) individuals. Generation after generation, the population itself is selected to fit the environment (i.e. to solve the problem).
As first approach, we decided to use a (1 + 1)-EA, that could be interpreted as first-ascent hill climber. In each generation a new tree is created by random swap between two branches connected to the same node and evaluated according to a fitness criterion. Defined the radius (r) as the number of taxa adjacent to a considered taxon, the fitness of each tree is evaluated as the sum of all the distances between each node and the closest r tips according to the genetic distance matrix. The fittest tree (i.e. that tree with a taxon order that minimizes the distances) survives the next generation. After having explored the parameter r in the algorithm and compared the results with random searches, we investigated the influence of population size with different combinations of λ and μ values [11]. In this work we only considered a genetic distance matrix applied to the phylogenetic tree of West Nile virus (WNV) presented by Bertolotti and colleagues and thus exhaustedly explained in [12].
Results and Discussion
The two experimental phases described at the end of the previous section are here described and discussed.
(1 + 1)-Evolutionary Algorithm
Using the tree shown in Figure 1 as starting point for our search, we performed 50 runs of the (1 + 1)-EA for each radius r = 1, 4, 8, 32 considered for the fitness evaluation. The radius is meant as the number of close taxa considered during the fitness evaluation of a single taxon. Thus, for each taxon of the tree, the sum of the distances between it and the r closest taxa is considered as partial fitness. The resulting sum is the fitness of the tree according to the given distances and to the r value. We arbitrarily chose 4 values to test the radius, as the evaluation with r = [1, 131] would be meaningless: increasing the value of r leads to a flattening of the fitness, as the distance between each taxon and the r next taxa would likely be similar if not equal.
Relative fitness improvement using (1 + 1)-EA
r= 1 | r= 4 | r= 8 | r= 32 | |||||
---|---|---|---|---|---|---|---|---|
hits | generation | hits | generation | hits | generation | hits | generation | |
Gold | 0 | N/A | 38 | 89, 349_{±7,906.3} | 50 | 39, 092_{±4,770.1} | 0 | N/A |
Silver | 26 | 93, 243_{±7,178.2} | 38 | 67, 230_{±9,409.3} | 50 | 2, 035_{±160.0} | 0 | N/A |
Bronze | 27 | 61, 810_{±9,053.5} | 39 | 41, 309_{±6,638.4} | 50 | 714_{±83.4} | 50 | 3, 239_{±195.7} |
With r = 1, no tree in Gold position was found, but more than half of the 50 runs reached both Silver and Bronze ranks. For r = 4 more than half of the runs reached the Gold position. Apparently, better results were obtained with r = 8, as all the 50 runs reached the Gold class. When r = 32 was used in the fitness evaluation, all the runs improved only until Bronze rank. Trees at r = 8 reached the Bronze class always in less than 1, 000 generations, while the other radii obtained this improvement only after 40, 000 generations in average.
To compare the relative fitness improvement using a different unit, we evaluated all the final trees with r = 75, a value that is well above the one used in the search algorithm and that still allows discrimination between the different curves in Figure 2.
Statistical analysis of relative fitness improvements of the 4 different groups evaluated at r = 75 showed a significant difference between them (Kruskal-Wallis rank sum test p < 0.001). Moreover, these data highlighted a significant ordering among the groups, having r = 1 worse than r = 4, r = 4 worse than r = 8, and r = 8 worse than r = 32 (all pairwise Wilcoxon rank sum test p < 0.001).
To justify the use of our fairly complex heuristic search method, we performed for comparison a random search (RS) algorithm with the same number of evaluations (n = 200000). Briefly, our (1 + 1)-EA found trees with relative fitness improvement of 0.8344 (for r = 1), 0: 7943 (for r = 4), 0.7478 (for r = 8), and 0: 8511 (for r = 32), whereas RS method found trees evaluated at 0.9065 (for r = 1), 0.8206 (for r = 4), 0.7699 (for r = 8), and 0.8878 (for r = 32). Wilcoxon rank sum test performed between EA and RS returned p ≪ 0.001 for each r values, showing that the evolutionary search methods consistently outperform random search with a strong statistical support.
(λ + μ)-Evolutionary Algorithms
As in the (1 + 1)-EA study, we performed 50 runs for each parameter of the population study. In details, we studied two different parameters for (1 + μ)-EA, where μ = (5, 10), and five combinations of parameters for (λ + μ)-EAs: (5 + 5), (5 + 10), (10 + 10), (50 + 50) and (1000 + 1000). As starting tree, the same WNV original tree was used and its fitness at r = 8 was considered as starting fitness for the computation of relative fitness improvement (final tree fitness/original tree fitness).
Hit rate and Convergence speed for (λ + μ)-EAs at r = 8
1+1 | 1+5 | 1+10 | 5+5 | 5+10 | 10+10 | 50+50 | 1000+1000 | |
---|---|---|---|---|---|---|---|---|
Hit Rate | 0.92 | 0.90 | 0.94 | 0.94 | 0.90 | 0.90 | 0.87 | 1.00 |
Conv. speed | 60917.24 | 51712.44 | 59373.83 | 53641.17 | 69464.67 | 71369.56 | 42705.77 | 4800.00 |
6810.97 | 6740.09 | 7168.53 | 6655.89 | 8098.49 | 8597.38 | 9149.91 | 304.75 |
Biological interpretation of the results
Conclusions
Since unresolved trees are hard to interpret, we studied the behavior of different parameters in (λ + μ)-EAs to explore the search space of all the solutions to our problem: making unresolved trees more meaningful. To do this we used an already published tree as starting point [12], and a matrix of genetic distances among samples, corrected with the best fit molecular substitution model, in the fitness computation. First, we investigated the influence of different radius values (r = 1, 4, 8, 32) in evaluating the fitness in the (1 + 1)-EA. The most informative results were obtained using r = 8, though all the considered radii performed better than the random search. Although, as expected, trees evolved using r = 32 showed improved fitness with respect to the other groups, the computational time needed to evolve these trees is long. Statistical analyses of the experimental data, together with the observations described above, suggest that a value for the radius in the fitness evaluation r = 8 might generally strike an acceptable balance between computational intensity and accuracy.
After the identification of an efficient radius, we investigated the influence of population sizes in (λ + μ)-EAs. All the combinations of λ and μ values returned solid results for the improvement in the graphical representation of the tree, adding important information to the vertical reading direction, although no differences in performance were observed for most of them. In fact, only the (1000 + 1000)-EA returned worse fitness of the final trees when compared to the other populations, but still better than the original tree. However, our experiments showed a significant gain in fitness values of one order of magnitude by this large population EA.
In a biological perspective, the best tree obtained in all the runs was improved by the (1 + 5)-EA. This outcome allows an easier interpretation of the tree, especially in those highly unresolved ones, with many siblings deriving from the same common ancestor. As previously discussed, in the improved tree, genetically more similar taxa are collected in a close spatiotemporal dimension and are now closer than in the original tree. This suggests that the use of genetic closeness between samples to improve the order of taxa in the tree somehow reflects the spatiotemporal relationship between them.
This work represents the first try to develop a new method to add a meaning to the order of taxa in a phylogenetic tree and it needs to be further investigated. So far we explored the dimension of the algorithm, studying the influence of the parameters on the performance during the process of selecting best trees. Next steps will be applying our method on different models analyzing different trees and different distances (e.g. geographic, temporal, serological) to test the algorithm in varying conditions.
Methods
The two algorithms applied in the article are here described and discussed, together with the phylogenetic tree used to validate the new technique, and the statistical methods employed to analyze the experimental results.
(1 + 1)-Evolutionary Algorithm
As starting tree for the algorithm, an original tree obtained by any available software for phylogenetic inference with any method (neighbor joining, maximum likelihood, maximum parsimony, Bayesian phylogenetics) is suitable. Starting from the original tree, in each generation a new tree is created by applying a random swap between two taxa connected to the same node in the tree. The fitness of the new tree is evaluated as sum of all the distances between each node and the closest r tips according to the genetic distances matrix (r being the radius in the fitness evaluation). If the fitness of the new tree is better than the one of the tree in the previous generation, the new tree replaces the old one, and the search procedure continues. If not, the old tree is retained. This process is iterated for 200000 generations, resulting in the creation and evaluation of 200000 new tentative solutions.
When evaluating the fitness of a tentative solution, the most straightforward method would be to sum up, for all taxa in the tree, the distance between the taxon under consideration and the two taxa next to it, i.e. taxa at radius r = 1. However, in order to have a wider overview to the taxa closeness, we tested different values of the radius in the fitness evaluation process. For each r value, 50 independent runs were performed. To justify the use of our heuristic search method, we performed for comparison a random search with the same number of evaluations (n = 200000) of trees generated by random swap of taxa, without memory of the previous generations.
(λ + μ)-Evolutionary Algorithm
A first exploration of simple population EAs was performed, where λ = 1 is the starting tree to improve, μ = 5, 10 and r = 8. Then, we evaluated the performances of the EAs when λ = μ ∈ {5, 10, 50, 1000}.
When λ > 1, λ - 1 trees generated by random swap are added to the original tree in the initial generation. In details, we tested the following populations: (1 + 5), (1 + 10), (5 + 5), (5 + 10), (10 + 10), (50 + 50) and (1000 + 1000).
The next generation of λ parents is selected by performing μ tournaments between couples chosen by random sampling with reintroduction among the λ + μ individuals, selecting the best μ ones. In this way the best individual can potentially be selected more than once, without excluding the other tentative solutions. To have a fair comparison between algorithms with different population sizes, we set a maximum generation limit so that the number of new tentative solutions that each EA evaluates is 200000. For μ = 5, 10, 50, 1000 this results in 40000, 20000, 4000, and 200 generations, respectively. For each parameter combination, 50 independent runs each were performed.
Experimental validation on WNV phylogenetic tree
WNV is a single stranded, positive-sense RNA virus member of the Japanese encephalitis serocomplex (Flaviviridae; Flavivirus) that is transmitted primarily through the bite of infected mosquitoes. Because of the recent introduction into North America, it has been possible to study the phylogenesis and the evolution of the virus. Such studies of the virus in North America reported highly unresolved trees [12, 14, 15]. In these cases, information on genetic, spatial or temporal clustering was not apparent, so that population substructure was investigated using different approaches.
In this study we used the tree presented by Bertolotti and colleagues in 2007, built using MrBayes software [16, 17] (Figure 1).
The original tree has a total of 132 taxa and 28 internal nodes. The root node has 76 branches, among which 62 were directly connected to terminal taxa (as magnified in Figure 1), pointing out that this part of tree is highly unresolved. Relating to the previous formulas, the resulting search space is |S| = 1.749 × 10^{137}, and every tree has 2, 975 neighbors, an area too large to be explored with a exhaustive search.
To calculate the fitness of each tentative solution, we used the matrix of genetic distances among samples, corrected with the best fit molecular substitution model (GTR+ Γ +I) [5, 18].
Statistical analysis
The data obtained by the runs were tested with the Shapiro-Wilk test to determine if the samples come from a normal distribution. As the test suggested a non-normal distribution, we applied the Kruskal-Wallis test for testing equality of population medians among groups. Finally we performed the Wilcoxon rank-sum test to study the variability of the results. All the tests were performed in R software [19].
Computational performance
The effective code of the algorithms was written in R language, using the package 'ape' [20]. The runs were performed on the cluster IBM-BCX available at the Supercomputing Group of the CINECA Systems & Tecnologies Department at the time of the work.
Declarations
Acknowledgements
The authors thank the Supercomputing Group of the CINECA Systems & Tecnologies Department for supporting in computations. MG acknowledges funding (60% grant) by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica. LB gratefully acknowledges financial support by Ricerca Sanitaria Finalizzata 2008 - Regione Piemonte. This work is supported by the National Science Foundation/National Institutes of Health Ecology of Infectious Diseases Program under award number EF-0840403.
Authors’ Affiliations
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