Markov invariants, plethysms, and phylogenetics

Journal of Theoretical Biology

Volumn 253, Issue 3, 2008, Pages 601-615

  Sumner, J G ^a,b   Charleston, M A ^a,c   Jermiin, L S ^c   Jarvis, P D ^b  

^a UNIVERSITY OF SYDNEY   (Australia)

^b UNIVERSITY OF TASMANIA   (Australia)

^c UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Branching rules; Invariants; Phylogenetics; Plethysm; Schur functions

Indexed keywords

MARKOV CHAIN; MODELING; PHYLOGENY;

ARTICLE; MODEL; PHYLOGENETIC TREE; PHYLOGENY; PRIORITY JOURNAL; PROBABILITY;

COMPUTATIONAL BIOLOGY; EVOLUTION, MOLECULAR; MARKOV CHAINS; MODELS, GENETIC; PHYLOGENY;

EID: 46849115564     PISSN: 00225193     EISSN: 10958541     Source Type: Journal    
DOI: 10.1016/j.jtbi.2008.04.001     Document Type: Article

References (70)

1. Allman E.S., and Rhodes J.A. Phylogenetic invariants of the general Markov model of sequence mutation. Math. Biosci. 186 (2003) 113-144
2. Baker A. Matrix Groups: An Introduction to Lie Group Theory (2003), Springer, Berlin
3. Barry D., and Hartigan J.A. Asynchronous distance between homologous DNA sequences. Biometrics 43 (1987) 261-276
4. Bashford J.D., Jarvis P.D., Sumner J.G., and Steel M.A. U (1) × U (1) × U (1) symmetry of the Kimura 3ST model and phylogenetic branching processes. J. Phys. A 37 (2004) L1-L9
5. Biedenharn L.C., and Louck J.D. The Racah-Wigner Algebra in Quantum Theory (1981), Addison-Wesley, Reading, MA
6. Biedenharn L.C., and Louck J.D. Inhomogeneous basis set of symmetric polynomials defined by tableaux. Proc. Natl. Acad. Sci. USA 87 (1990) 1441-1445
7. Bryant, D., Galtier, N., Poursat, M.-A., 2005. Likelihood Calculation in Molecular Phylogenetics. Mathematics of Evolution and Phylogenetics. Oxford University Press, Oxford, pp. 33-62.
8. Burnham K.P., and Anderson D. Model Selection and Multi-Model Inference (2002), Springer, Berlin
9. Casanellas M., and Fernández-Sánchez J. Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees. Mol. Biol. Evol. 24 (2007) 288-293
10. Cavender J.A., and Felsenstein J. Invariants of phylogenies in a simple case with discrete states. J. Class. 4 (1987) 57-71
11. Coffman V., Kundu J., and Wootters W.K. Distributed entanglement. Phys. Rev. A 61 5 (2000) 052306
12. Drummond A., Forsberg R., and Rodrigo A.G. The inference of stepwise changes in substitution rates using serial sequence samples. Mol. Biol. Evol. 18 (2001) 1365-1371
13. Dur W., Vidal G., and Cirac J.I. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62 (2000) 062314
14. Eriksson, N., Using invariants for phylogenetic tree construction, to appear. eprint arXiv:0709.2890.
15. Eriksson, N., Yao, Y., 2008. Metric learning for phylogenetic invariants. eprint arXiv:q-bio/0703034.
16. Evans S.N., and Speed T.P. Invariants of some probability models used in phylogenetic inference. Ann. Statist. 21 1 (1993) 355-377
17. Fauser B., Jarvis P.D., King R.C., and Wybourne B.G. New branching rules induced by plethysm. J. Phys. A 39 (2006) 2611-2655
18. Felsenstein, J., 2005. PHYLIP (Phylogeny Inference Package) version 3.6. Distributed by the author. Department of Genome Sciences, University of Washington, Seattle.
19. Fulton W., and Harris J. Representation Theory. Graduate Text in Mathematics (1991), Springer, Berlin
20. Goodman G.S. An intrinsic time for non-stationary finite Markov chains. Probab. Theor. Relat. Field 16 (1970) 165-180
21. Goodman R., and Wallach N.R. Representations and Invariants of the Classical Groups (1998), Cambridge University Press, Cambridge
22. Hillis D., Huelsenbeck J., and Swofford D. Hobgoblin of phylogenetics?. Nature 369 (1994) 363-364
23. Huelsenbeck J.P. Performance of phylogenetic methods in simulation. Syst. Biol. 44 (1995) 17-48
24. Huelsenbeck J.P., Larget B., and Alfaro M.E. Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo. Mol. Biol. Evol. 21 (2004) 1123-1133
25. Huson D.H., Nettles S.M., and Warnow T.J. Disk-covering, a fast-converging method for phylogenetic tree reconstruction. J. Comput. Biol. 6 (1999) 369-386
26. Iosifescu M. Finite Markov Processes and Their Applications (1980), Wiley, Chichester
27. Itzykson C., and Zuber J.-B. Quantum Field Theory (1980), McGraw-Hill, New York
28. Jarvis P.D., Bashford J.D., and Sumner J.G. Path integral formulation and Feynman rules for phylogenetic branching models. J. Phys. A 38 (2005) 9621-9647
29. Jayaswal V., Jermiin L.S., and Robinson J. Estimation of phylogeny using a general Markov model. Evol. Bioinformatics Online 1 (2005) 62-80
30. Jermiin L.S., Ho S.Y.W., Ababneh F., Robinson J., and Larkum A.W.D. Hetero: a program to simulate the evolution of DNA on four-taxon trees. Appl. Bioinformatics 2 (2003) 159-163
31. Jermiin L.S., Ho S.Y.W., Ababneh F., Robinson J., and Larkum A.W.D. The biasing effect of compositional heterogeneity on phylogenetic estimates may be underestimated. Syst. Biol. 53 (2004) 638-643
32. Jermiin, L.S., Jayaswal, V., Ababneh, F., Robinson, J., 2008. Phylogenetic model evaluation. Bioinformatics, Data, Sequences Analysis and Evolution, vol. I. Humana Press, Totowa, NJ, pp. 331-363.
33. Johnson J.E. Markov-type Lie groups in GL (n, R). J. Math. Phys. 26 (1985) 252-257
34. Kelarev A. Ring Constructions and Applications (2002), World Scientific, Singapore
35. King R.C. Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975) 429-449
36. Lake J.A. A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony. Mol. Biol. Evol. 4 (1987) 167-191
37. Lake J.A. Reconstructing evolutionary trees from DNA and protein sequences: paralinear distances. Proc. Natl. Acad. Sci. 91 (1994) 1455-1459
38. Landsberg, J.M., Manivel, L., 2008. Generalizations of Strassen's equations for secant varieties of Segre varieties. Commun. Algebra, to appear. eprint arXiv:math/0601097.
39. Littlewood D.E. The Theory of Group Characters (1940), Clarendon Press, Oxford
40. Littlewood D.E. Plethysm and the inner product of S-functions. J. Lond. Math. Soc. s1-32 (1955) 18-22
41. Lockhart P.J., Steel M.A., Hendy M.D., and Penny D. Recovering evolutionary trees under a more realistic model of sequence evolution. Mol. Biol. Evol. 11 (1994) 605-612
42. Lockhart P.J., Steel M.A., Barbrook A.C., Huson D.H., and Howe C.J. A covariotide model describes the evolution of oxygenic photosynthesis. Mol. Biol. Evol. 15 (1998) 1183-1188
43. Lockhart, P.J., Novis, P., Milligan, B.G., Riden, J., Rambaut, A., Larkum, A.W.D., 2006. Heterotachy and tree building: a case study with plastids and eubacteria. Mol. Biol. Evol. 40-45.
44. MacDonald I.G. Symmetric Functions and Hall Polynomials (1979), Clarendon Press, Oxford
45. Massingham, T., Goldman, N., 2007. Statistics of the log-det estimator. MBE Advance Access published August 16, 2007.
46. McCullagh P. Tensor Methods in Statistics (1987), Chapman and Hall, NY
47. Molev, A., 2007. On the fusion procedure for the symmetric group. Rep. Math. Physics, to appear. eprint arXiv:math/0612207.
48. Mourad B. On a Lie-theoretic approach to generalised doubly stochastic matrices and applications. Linear and Multilinear Algebra 52 (2004) 99-113
49. Olver P.J. Classical Invariant Theory (2003), Cambridge University Press, Cambridge
50. Pagel M., and Meade A. A phylogenetic mixture model for detecting pattern-heterogeneity in gene sequence or character-state data. Syst. Biol. 53 (2004) 571-581
51. Paradis E., Claude J., and Strimmer K. APE: analyses of phylogenetics and evolution in R language. Bioinformatics 20 (2004) 289-290
52. R Development Core Team, 2006. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
53. Semple C., and Steel M. Phylogenetics (2003), Oxford Press
54. Seshadri C.S. On a theorem of Weitzenböck in invariant theory. J. Math. Kyoto. Univ. 1 (1962) 403-409
55. Steel M. Should phylogenetic models be trying to 'fit an elephant'?. Genetics 21 (2005) 307-309
56. Steel M.A. Recovering a tree from the leaf colourations it generates under a Markov model. Appl. Math. Lett. 7 (1994) 19-24
57. Steel M.A., Szekely L., Erdos P.L., and Waddell P. A complete family of phylogenetic invariants for any number of taxa under Kimura's 3ST model. N.Z. J. Bot. 31 (1993) 289-296
58. Steel M.A., Szekely L., and Hendy M. Reconstructing trees when sequence sites evolve at variable rates. J. Comput. Biol. 1 (1994) 153-163
59. Sturmfels, B., Open problems in algebraic statistics. In: Putinar, M., Sullivant, S. (Eds.), Emerging Applications of Algebraic Geometry, I.M.A. Volumes in Mathematics and its Applications, to appear. eprint arXiv:math/0707.4558.
60. Sumner, J.G., 2006. Entanglement, Invariants, and Phylogenetics. Ph.D. Thesis, University of Tasmania 〈http://eprints.utas.edu.au〉.
61. Sumner, J.G., 2008. Phylogenetic quartet inference using the squangles. University of Sydney 〈http://www.it.usyd.edu.au/∼mcharles/software〉.
62. Sumner J.G., and Jarvis P.D. Entanglement invariants and phylogenetic branching. J. Math. Biol. 51 (2005) 18-36
63. Sumner J.G., and Jarvis P.D. Using the tangle: a consistent construction of phylogenetic distance matrices. Math. Biosci. 204 (2006) 49-67
64. Thorne J.L., Kishino H., and Painter I.S. Estimating the rate of evolution of the rate of molecular evolution. Mol. Biol. Evol. 15 (1998) 1647-1657
65. Weitzenböck R. Über die Invarianten von linearen Gruppen. Acta Math. 58 (1931) 231-293
66. Weyl H. The Theory of Groups and Quantum Mechanics (1950), Dover Publications, NY
67. Whippman M.L. Branching rules for simple Lie groups. J. Math. Phys. 6 (1965) 1534-1539
68. Wolfram Research Inc., 2005. Mathematica 5.2.
69. Wybourne, B.G., 2004. Schur: an interactive programme for calculating properties of Lie groups. version 6.03. http://sourceforge.net/projects/schur.
70. Zharkikh A. Estimation of evolutionary distance between nucleotide sequences. J. Mol. Evol. 39 (1994) 315-329