Abstract
Recently there has been renewed interest in phylogenetic inference methods based on phylogenetic invariants, alongside the related Markov invariants. Broadly speaking, both these approaches give rise to polynomial functions of sequence site patterns that, in expectation value, either vanish for particular evolutionary trees (in the case of phylogenetic invariants) or have well understood transformation properties (in the case of Markov invariants). While both approaches have been valued for their intrinsic mathematical interest, it is not clear how they relate to each other, and to what extent they can be used as practical tools for inference of phylogenetic trees. In this paper, by focusing on the special case of binary sequence data and quartets of taxa, we are able to view these two different polynomial-based approaches within a common framework. To motivate the discussion, we present three desirable statistical properties that we argue any invariant-based phylogenetic method should satisfy: (1) sensible behaviour under reordering of input sequences; (2) stability as the taxa evolve independently according to a Markov process; and (3) explicit dependence on the assumption of a continuous-time process. Motivated by these statistical properties, we develop and explore several new phylogenetic inference methods. In particular, we develop a statistically bias-corrected version of the Markov invariants approach which satisfies all three properties. We also extend previous work by showing that the phylogenetic invariants can be implemented in such a way as to satisfy property (3). A simulation study shows that, in comparison to other methods, our new proposed approach based on bias-corrected Markov invariants is extremely powerful for phylogenetic inference. The binary case is of particular theoretical interest as—in this case only—the Markov invariants can be expressed as linear combinations of the phylogenetic invariants. A wider implication of this is that, for models with more than two states—for example DNA sequence alignments with four-state models—we find that methods which rely on phylogenetic invariants are incapable of satisfying all three of the stated statistical properties. This is because in these cases the relevant Markov invariants belong to a class of polynomials independent from the phylogenetic invariants.
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Notes
It is sometimes important to distinguish between a method (as theoretically conceived) and its implementation in software (for example, ambiguities often arise in quartet methods in regard to random tie breaking). Throughout this article we will assume the two match up perfectly without further comment.
To avoid unimportant technicalities, we will assume “generic” parameter settings throughout this article. In particular, we assume that all Markov matrices are non-singular and \(\pi _i\ne 0\) for \(i=1,2\).
At a formal level (not strictly required here), the reader should note that since we are working over the complex field, the flattening should be defined so in place of the matrix transpose in (2.2) we have the conjugate transpose operation.
This observation admits a significant generalization—developed by Sumner (2017)—to any number of taxa and any number of states k.
References
Allman ES, Rhodes JA (2003) Phylogenetic invariants of the general Markov model of sequence mutation. Math Biosci 186:113–144
Allman ES, Rhodes JA (2008) Phylogenetic ideals and varieties for the general Markov model. Adv Appl Math 40:127–148
Allman ES, Rhodes JA, Taylor A (2014) A semialgebraic description of the general markov model on phylogenetic trees. SIAM J Discrete Math 28(2):736–755
Bates DJ, Oeding L (2011) Toward a salmon conjecture. Exp Math 20(3):358–370
Casanellas M, Fernández-Sánchez J (2010) Relevant phylogenetic invariants of evolutionary models. J Math Pures Appl 96:207–229
Cavender JA, Felsenstein J (1987) Invariants of phylogenies in a simple case with discrete states. J Classif 4:57–71
Chor B, Hendy MD, Holland BR, Penny D (2000) Multiple maxima of likelihood in phylogenetic trees: an analytic approach. Mol Biol Evol 17:1529–1541
Davey JW, Hohenlohe PA, Etter PD, Boone JQ, Catchen JM, Blaxter ML (2011) Genome-wide genetic marker discovery and genotyping using next-generation sequencing. Nat Rev Genet 12(7):499–510
Draisma J, Kuttler J (2008) On the ideals of equivariant tree models. Math Ann 344:619–644
Eriksson N (2008) Using invariants for phylogenetic tree construction. In: Putinar M, Sullivant S (eds) Emerging applications of algebraic geometry. Springer, Berlin
Felsenstein J (1978) Cases in which parsimony or compatibility methods will be positively misleading. Syst Zool 27:401–410
Felsenstein J (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17:368–376
Felsenstein J (2004) Inferring phylogenies. Sinauer Associates, Sunderland
Fernández-Sánchez J, Casanellas M (2015) Invariant versus classical quartet inference when evolution is heterogeneous across sites and lineages. Syst. Biol. 65(2):280–291. http://sysbio.oxfordjournals.org/content/early/2015/11/11/sysbio.syv086.abstract
Friedland S (2013) On tensors of border rank l in \(\mathbb{C}^{m\times n\times l}\). Linear Algebra Appl 438(2):713–737
Friedland S, Gross E (2012) A proof of the set-theoretic version of the salmon conjecture. J Algebra 356(1):374–379
Hillis D, Huelsenbeck J, Swofford D (1994) Hobgoblin of phylogenetics? Nature 369:363–364
Holland BR, Sumner JG, Jarvis PD (2013) Low-parameter phylogenetic inference under the general Markov model. Syst Biol 62:78–92
Huelsenbeck JP, Hillis DM (1993) Success of phylogenetic methods in the four-taxon case. Syst Biol 42(3):247–264
Jarvis PD, Sumner JG (2014) Adventures in invariant theory. ANZIAM J 56(02):105–115
Lake JA (1987) A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony. Mol Biol Evol 4:167–191
Lemmon EM, Lemmon AR (2013) High-throughput genomic data in systematics and phylogenetics. Annu Rev Ecol Evol Syst 44:99–121
Olver PJ (2003) Classical invariant theory. Cambridge University Press, Cambridge
Rusinko JP, Hipp B (2012) Invariant based quartet puzzling. Algorithms Mol Biol 7(1):1
Saitou N, Nei M (1987) The neighbor-joining method: a new method for reconstructing phylogenetic trees. Mol Biol Evol 4(4):406–425
Sumner JG (2017) Dimensional reduction for the general Markov Model on phylogenetic trees. Bull Math Biol 79(3):619–634. doi:10.1007/s11538-017-0249-6
Sumner JG, Charleston MA, Jermiin LS, Jarvis PD (2008) Markov invariants, plethysms, and phylogenetics. J Theor Biol 253:601–615
Sumner JG, Fernández-Sánchez J, Jarvis PD (2012) Lie Markov models. J Theor Biol 298:16–31
Sumner JG, Jarvis PD (2009) Markov invariants and the isotropy subgroup of a quartet tree. J Theor Biol 258:302–310
Swofford DL, Waddell PJ, Huelsenbeck JP, Foster PG, Lewis PO, Rogers JS (2001) Bias in phylogenetic estimation and its relevance to the choice between parsimony and likelihood methods. Syst Biol 50(4):525–539
Wolfram Research Inc (2010) Mathematica 8. Wolfram Research Inc, Champaign, IL
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We would like to thank the two anonymous reviewers whose thoughtful and careful reading of our manuscript led to a greatly improved final version.
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This work was supported by the Australian Research Council Discovery Early Career Fellowship DE130100423 (JGS) and the University of Tasmania Visiting Scholars Program (AT).
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Sumner, J.G., Taylor, A., Holland, B.R. et al. Developing a statistically powerful measure for quartet tree inference using phylogenetic identities and Markov invariants. J. Math. Biol. 75, 1619–1654 (2017). https://doi.org/10.1007/s00285-017-1129-2
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DOI: https://doi.org/10.1007/s00285-017-1129-2