Abstract
The phylogenetic tree (PT) problem has been studied by a number of researchers as an application of the Steiner tree problem, a well-known network optimisation problem. Of all the methods developed for phylogenies the maximum parsimony (MP) method is a simple and commonly used method because it relies on directly observable changes in the input nucleotide or amino acid sequences. In this paper we show that the non-uniqueness of the evolutionary pathways in the MP method leads us to consider a new model of PTs. In this so-called probability representation model, for each site a node in a PT is modelled by a probability distribution of nucleotide or amino acid states, and hence the PT at a given site is a probability Steiner tree, i.e. a Steiner tree in a high-dimensional vector space. In spite of the generality of the probability representation model, in this paper we restrict our study to constructing probability phylogenetic trees (PPT) using the parsimony criterion, as well as discussing and comparing our approach with the classical MP method. We show that for a given input set although the optimal topology as well as the total tree length of the PPT is the same as the PT constructed by the classical MP method, the inferred ancestral states and branch lengths are different and the results given by our method provide a plausible alternative to the classical ones.
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References
Althaus E, Naujoks R (2006) Computing steiner minimum trees in hamming metric. SIAM ISBN 0-89871-605-5/06/01 (SODA 06)
Brazil M, Thomas DA, Nielsen BK, Winter P, Wulff-Nilsen C, Zachariasen M (2009) A novel approach to phylogenetic trees: d-dimensional geometric Steiner trees. Networks 53: 104–111
Ewens WJ, Grant GR (2005) Statistical methods in bioinformatics: an introduction. Springer, New York
Felsenstein J (1981) Evolutional trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17: 368–376
Felsenstein J (2004) Inferring phylogenies. Sinauer Associates Inc, Sunderland
Felsenstein J (2011) http://evolution.genetics.washington.edu/phylip.html
Fitch WM (1971) Toward defining the course of evolution: minimum changes for a specific tree topology. Syst Zool 20: 406–416
Fomin F, Grandoni F, Kratsch D (2008) Faster Steiner tree computation in polynomial-space. LNCS 5193: 430–441
Foulds LR, Graham RL (1982) The Steiner problem in phylogeny is NP-complete. Adv Appl Math 3: 43–49
Fuchs B, Kern W, Mölle D, Richter S, Rossmanith P, Wang X (2007) Dynamic programming for minimum Steiner trees. Theory Comput Syst 41(3): 493–500
Grant M, Boyd S (2011) CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/~boyd/cvx
Hwang FK, Richards DS, Winter P (1992) The Steiner tree problem. Elsevier Science Publishers B.V., The Netherlands
Liberti L, Maculan N (2006) Global optimization: from theory to implementation. Springer, New York
Liébecq C (1992) Biochemical nomenclature and related documents, 2nd edn. Portland Press, London, pp 109–114
Nei M, Kumar S (2000) Molecular evolution and phylogenetics. Oxford University Press Inc, USA
Sankoff D (1975) Minimal mutation trees of sequences. SIAM J Appl Math 28: 35–42
Stanhope MJ, Smith MR, Waddell VG, Porter CA, Shivji MS, Goodman M (1996) Mammalian evolution and the interphotoreceptor Retinoid Binding Protein (IRBP) gene: convincing evidence for several superordinal clades. J Mol Evol 43: 83–92
Stewart C-B, Schilling JW, Wilson AC (1987) Adaptive evolution in the stomach lysozymes of foregut fermenters. Nature 330: 401–404
Swofford DL, Olsen GJ, Waddell PJ, Hillis DM (1996) Phylogenetic inference. In: Hillis DM, Moritz C, Mable BK (eds) Molecular systematics. 2nd edn. Sinauer Associates, Sunderland, pp 407–514
Tamura K, Dudley J, Nei M, Kumar S (2011) http://www.megasoftware.net/
Tuimala J (2006) A primer to phylogenetic analysis using the PHYLIP package (fifth edition), http://www.life.umd.edu/labs/delwiche/bsci348s/lab/phylip2.pdf
Weng J, Mareels I, Thomas D (2009) Computing Steiner points and probability Steiner points in ℓ1 and ℓ2 metric spaces, discrete mathematics. Algorithm Appl 1: 541–554
Weng J, Thomas DA, Mareels I (2011) Maximum parsimony, substitution model and probability phylogenetic trees. J Comput Biol 18: 67–80 (online in June 2010)
Xia X (2011) http://dambe.bio.uottawa.ca/dambe.asp
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Weng, J.F., Mareels, I. & Thomas, D.A. Probability Steiner trees and maximum parsimony in phylogenetic analysis. J. Math. Biol. 64, 1225–1251 (2012). https://doi.org/10.1007/s00285-011-0442-4
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DOI: https://doi.org/10.1007/s00285-011-0442-4