Properties of phylogenetic trees generated by Yule-type speciation models. (English) Zbl 0977.92017
Summary: We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. We describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth (number of edges from the root) of the most recent common ancestor of a random subset of \(k\) species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs in rapid burst.
MSC:
| 92D15 | Problems related to evolution |
| 05C05 | Trees |
| 05C90 | Applications of graph theory |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Software:
TREECONReferences:
| [1] | D. Aldous, Probability distributions on cladograms, in: D. Aldous, R. Permantle (Eds.), Random Structures, vol. 76, Springer, Berlin, 1996, p. 1; D. Aldous, Probability distributions on cladograms, in: D. Aldous, R. Permantle (Eds.), Random Structures, vol. 76, Springer, Berlin, 1996, p. 1 · Zbl 0841.92015 |
| [2] | Brown, J., Probabilities of evolutionary trees, Syst. Biol., 43, 1, 78 (1994) |
| [3] | Harding, E., The probabilities of rooted tree-shapes generated by random bifurcation, Adv. Appl. Prob., 3, 44 (1971) · Zbl 0241.92012 |
| [4] | Heard, S., Patterns in tree balance among cladistic, phenetic, and randomly generated phylogenetic trees, Evolution, 46, 6, 1818 (1992) |
| [5] | Heard, S., Patterns in phylogenetic tree balance with variable and evolving speciation rates, Evolution, 50, 6, 2141 (1996) |
| [6] | Kubo, T.; Iwasa, Y., Inferring the rates of branching and extinction from molecular phylogenies, Evolution, 49, 694 (1995) |
| [7] | Mooers, A.; Heard, S., Inferring evolutionary process from phylogenetic tree shape, Quart. Rev. Biol., 72, 1, 31 (1997) |
| [8] | McKenzie, A.; Steel, M., Distributions of cherries for two models of trees, Math. Biosci., 164, 1, 81 (2000) · Zbl 0947.92021 |
| [9] | D. Aldous, Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today [online], Available: http://www.stat.berkeley.edu/users/aldous/ bibliog.html; D. Aldous, Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today [online], Available: http://www.stat.berkeley.edu/users/aldous/ bibliog.html · Zbl 1127.60313 |
| [10] | Rannala, B.; Yang, Z., Probability distribution of molecular evolutionary trees: a new method of phylogenetic inference, Molecular Evolution, 43, 304 (1996) |
| [11] | Mau, B.; Newton, M.; Larget, B., Bayesian phylogenetic inference via Markov chain Monte Carlo methods, Biometrics, 55, 1 (1999) · Zbl 1059.62675 |
| [12] | Li, S.; Pearl, D. K.; Doss, H., Phylogenetic tree construction using Markov chain Monte Carlo, J. Am. Statist. Assoc., 95, 450, 493 (2000) |
| [13] | Slowinski, J., Probabilities of \(n\)-trees under two models: a demonstration that asymmetrical interior nodes are not improbable, Syst. Zool., 39, 1, 89 (1990) |
| [14] | Page, R.; Holmes, E., (Molecular Evolution: A Phylogenetic Approach (1998), Blackwell Science: Blackwell Science Oxford), 11, (Chapter 2) |
| [15] | Slowinski, J.; Guyer, C., Testing the stochasticity of patterns of organismal diversity: an improved null model, Am. Nat., 134, 6, 907 (1989) |
| [16] | Nee, S.; May, R.; Harvey, P., The reconstructed evolutionary process, Philos. Trans. R. Soc. London B, 344, 305 (1994) |
| [17] | Losos, J.; Adler, F., Stumped by trees? A generalized null model for patterns of organismal diversity, Am. Nat., 145, 3, 329 (1995) |
| [18] | Kingman, J., On the genealogy of large populations, J. Appl. Prob., 19A, 27 (1982) · Zbl 0516.92011 |
| [19] | Tajima, F., Evolutionary relationships of DNA sequences in finite populations, Genetics, 105, 437 (1983) |
| [20] | D. Knuth, The Art of Computer Programming, vol. 3, Addison-Wesley, Reading, MA, 1997, p. 67 (Chapter 5, Problem 20); D. Knuth, The Art of Computer Programming, vol. 3, Addison-Wesley, Reading, MA, 1997, p. 67 (Chapter 5, Problem 20) · Zbl 0895.68055 |
| [21] | R. Stanley, Enumerative Combinatorics: Cambridge Studies in Advanced Mathematics, vol. 1, no. 49, Cambridge University, Cambridge, 1997, p. 312; R. Stanley, Enumerative Combinatorics: Cambridge Studies in Advanced Mathematics, vol. 1, no. 49, Cambridge University, Cambridge, 1997, p. 312 · Zbl 0889.05001 |
| [22] | Edwards, A., Estimation of the branch points of a branching diffusion process, J. R. Stat. Soc. Ser. B, 32, 155 (1970) · Zbl 0203.50801 |
| [23] | Sanderson, M., How many taxa must be sampled to identify the the root node of a large clade?, Syst. Biol., 45, 2, 168 (1996) |
| [24] | Lynch, W., More combinatorial properties of certain trees, Comput. J., 71, 299 (1965) · Zbl 0136.38801 |
| [25] | Mahmoud, H., (Evolution of Random Search Trees (1992), Wiley: Wiley New York), 72 · Zbl 0762.68033 |
| [26] | R. Stanley, Enumerative Combinatorics, in: Cambridge Studies in Advanced Mathematics, vol. 1, no. 49, Cambridge University, Cambridge, 1997, p. 18; R. Stanley, Enumerative Combinatorics, in: Cambridge Studies in Advanced Mathematics, vol. 1, no. 49, Cambridge University, Cambridge, 1997, p. 18 · Zbl 0889.05001 |
| [27] | Ridley, M.; Evolution, (Evolution (1993), Blackwell Science: Blackwell Science Cambridge, MA, USA), 460, Chapter 17 |
| [28] | Smith, A., Rooting molecular trees: problems and strategies, Biol. J. Linnean Soc., 51, 279 (1994) |
| [29] | Farris, J., Estimating phylogenetic trees from distance matrices, Am. Nat., 106, 646 (1972) |
| [30] | D.L. Swofford, G.J. Olsen, P.J. Waddell, D.M. Hillis, Phylogenetic inference, in: D.M. Hillis, C. Moritz, B.K. Mable (Eds.), Molecular Systmatics, Sinauer Associates Inc., Sunderland, MA, USA, 1996, p. 488; D.L. Swofford, G.J. Olsen, P.J. Waddell, D.M. Hillis, Phylogenetic inference, in: D.M. Hillis, C. Moritz, B.K. Mable (Eds.), Molecular Systmatics, Sinauer Associates Inc., Sunderland, MA, USA, 1996, p. 488 |
| [31] | de Peer, Y. V.; Wacher, R. D., Treecon: a software package for the construction and drawing of evolutionary trees, Comput. Applic. Biosci., 9, 177 (1993) |
| [32] | Haigh, J., The recovery of the root of a tree, J. Appl. Prob., 7, 79 (1970) · Zbl 0191.47702 |
| [33] | Zwet, W.v., A proof of Kakutani’s conjecture on random subdivision of longest intervals, Annal. Prob., 6, 1, 133 (1978) · Zbl 0374.60036 |
| [34] | Medhi, J., (Stochastic Processes (1982), Wiley: Wiley New York), 119, Chapter 4 · Zbl 0485.60001 |
| [35] | Cavalli-Sforza, L.; Edwards, A., Phylogenetic analysis, Evolution, 21, 550 (1967) |
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