Skip to main content

Advertisement

SpringerLink
Log in

Characterizing Local Optima for Maximum Parsimony

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Finding the best phylogenetic tree under the maximum parsimony optimality criterion is computationally difficult. We quantify the occurrence of such optima for well-behaved sets of data. When nearest neighbor interchange operations are used, multiple local optima can occur even for “perfect” sequence data, which results in hill-climbing searches that never reach a global optimum. In contrast, we show that when neighbors are defined via the subtree prune and regraft metric, there is a single local optimum for perfect sequence data, and thus, every such search finds a global optimum quickly. We further characterize conditions for which sequences simulated under the Cavender–Farris–Neyman and Jukes–Cantor models of evolution yield well-behaved search spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Allen BL, Steel M (2001) Subtree transfer operations and their induced metrics on evolutionary trees. Ann Comb 5(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Atteson K (1999) The performance of the neighbor-joining methods of phylogenetic reconstruction. Algorithmica 25:251–278

    Article  MathSciNet  MATH  Google Scholar 

  • Bastert O, Rockmore D, Stadler PF, Tinhofer G (2002) Landscapes on spaces of trees. Appl Math Comput 131(2–3):439–459

    MathSciNet  MATH  Google Scholar 

  • Bordewich M, Gascuel O, Huber KT, Moulton V (2009) Consistency of topological moves based on the balanced minimum evolution principle of phylogenetic inference. IEEE/ACM Trans Comput Biol Bioinform (TCBB) 6(1):110–117

    Article  Google Scholar 

  • Caceres AJJ, Castillo J, Jinnie J, St John K (2013) Walks on SPR neighborhoods. IEEE/ACM Trans Comput Biol Bioinform 10(1):236–239

    Article  Google Scholar 

  • Cartwright RA (2005) DNA assembly with gaps (Dawg): simulating sequence evolution. Bioinformatics 21(Suppl 3):iii31–iii38

    Article  MathSciNet  Google Scholar 

  • Cavender JA (1978) Taxonomy with confidence. Math Biosci 40(3):271–280

    Article  MathSciNet  MATH  Google Scholar 

  • Charleston MA (1995) Toward a characterization of landscapes of combinatorial optimization problems, with special attention to the phylogeny problem. J Comput Biol 2(3):439–450

    Article  Google Scholar 

  • Chen S, Montgomery J, Bolufé-Röhler A, Gonzalez-Fernandez Y (2015) Invited paper: a review of thresheld convergence. GECONTEC Revista Internacional de Gestión del Conocimiento y la Tecnología 3(1):1–13

    Google Scholar 

  • Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 2nd edn. MIT Press, Cambridge

    MATH  Google Scholar 

  • Dress A, Krüger M (1987) Parsimonious phylogenetic trees in metric spaces and simulated annealing. Adv Appl Math 8(1):8–37

    Article  MathSciNet  MATH  Google Scholar 

  • Farach M, Kannan S (1999) Efficient algorithms for inverting evolution. JACM 46(4):437–449

    Article  MathSciNet  MATH  Google Scholar 

  • Farris JS (1970) A method for computing wagner trees. Syst Zool 19:83–92

    Article  Google Scholar 

  • Farris JS (1973) A probability model for inferring evolutionary trees. Syst Biol 22(3):250–256

    Article  Google Scholar 

  • Fitch WM (1971) Towards defining the course of evolution: minimum change for a specific tree topology. Syst Zool 20(4):406–416

    Article  Google Scholar 

  • Fitch WM, Margoliash E et al (1967) Construction of phylogenetic trees. Science 155(760):279–284

    Article  Google Scholar 

  • Foulds LR, Graham RL (1982) The Steiner problem in phylogeny is NP-complete. Adv Appl Math 3(1):43–49

    Article  MathSciNet  MATH  Google Scholar 

  • Garnier J, Kallel L (2001) Efficiency of local search with multiple local optima. SIAM J Discrete Math 15(1):122–141

    Article  MathSciNet  MATH  Google Scholar 

  • Goloboff PA, Farris JS, Nixon KC (2008) TNT, a free program for phylogenetic analysis. Cladistics 24(5):774–786

    Article  Google Scholar 

  • Hein J (1990) Reconstructing evolution of sequences subject to recombination using parsimony. Math Biosci 98(2):185–200

    Article  MathSciNet  MATH  Google Scholar 

  • Hendy MD, Foulds LR, Penny D (1980) Proving phylogenetic trees minimal with l-clustering and set partitioning. Math Biosci 51(1):71–88

    Article  MathSciNet  MATH  Google Scholar 

  • Hendy MD, Penny D (1982) Branch and bound algorithms to determine minimal evolutionary trees. Math Biosci 59(2):277–290

    Article  MathSciNet  MATH  Google Scholar 

  • Hillis DM, Mable BK, Moritz C (1996) Molecular systematics. Sinauer Associates, Sunderland

    Google Scholar 

  • Jukes TH, Cantor CR, Munro HN (1969) Mammalian protein metabolism. Evol Protein Mol 3:21–132

    Google Scholar 

  • Kirkup B, Kim J (2003) From rolling hills to jagged mountains: scaling of heuristic searches for phylogenetic estimation. Mol Biol Evol (in revision)

  • Li M, Tromp J, Zhang L (1996) Some notes on the nearest neighbour interchange distance. In: COCOON ’96: proceedings of the second annual international conference on computing and combinatorics. Springer, London, pp 343–351

  • Maddison DR (1991) The discovery and importance of multiple islands of most-parsimonious trees. Syst Zool 40(3):315–328

    Article  Google Scholar 

  • Money D, Whelan S (2012) Characterizing the phylogenetic tree-search problem. Syst Biol 61(2):228–239

    Article  Google Scholar 

  • Moulton V, Steel M (1999) Retractions of finite distance functions onto tree metrics. Discrete Appl Math 91(1):215–233

    Article  MathSciNet  MATH  Google Scholar 

  • Neyman J (1971) Molecular studies of evolution: a source of novel statistical problems. In: Gupta SS, Yackeleds J (eds) Statistical decision theory and related topics. Academic Press, New York, pp 1–27

    Google Scholar 

  • Nixon KC (1999) The parsimony ratchet, a new method for rapid parsimony analysis. Cladistics 15(4):407–414

    Article  Google Scholar 

  • Robinson DF (1971) Comparison of labeled trees with valency three. J Comb Theory Ser B 11(2):105–119

    Article  MathSciNet  Google Scholar 

  • Rokas A, Williams BL, King N, Carroll SB (2003) Genome-scale approaches to resolving incongruence in molecular phylogenies. Nature 425:798–804

    Article  Google Scholar 

  • Saitou N, Nei M (1987) The neighbor-joining method: a new method for reconstructing phylogenetic trees. Mol Biol Evol 4:406–425

    Google Scholar 

  • Semple C, Steel M (2003) Phylogenetics. Oxford lecture series in mathematics and its applications, vol 24. Oxford University Press, Oxford

    Google Scholar 

  • Stamatakis A, Blagojevic F, Antonopoulos CD, Nikolopoulos DS (2007) Exploring new search algorithms and hardware for phylogenetics: RAxML meets the IBM cell. J VLSI Signal Process Syst 48(3):271–286

    Article  Google Scholar 

  • Steel MA (1994) The maximum likelihood point for a phylogenetic tree is not unique. Syst Biol 43(4):560–564

    Article  Google Scholar 

  • Swofford DL (2002) PAUP*. Phylogenetic analysis using parsimony (*and other methods). Version 4. Sinauer Associates, Sunderland

    Google Scholar 

  • Varón A, Vinh LS, Wheeler WC (2010) POY version 4: phylogenetic analysis using dynamic homologies. Cladistics 26(1):72–85

    Article  Google Scholar 

  • Wheeler WC (2012) Systematics: a course of lectures. Wiley-Blackwell, New York

    Book  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Mike Charleston, Kevaughn Gordon, Barbara Holland, and Ward Wheeler for insightful and helpful discussion and the anonymous referees and Associate Editor Mike Steel for their comments which greatly improved this paper. We would also like to thank the American Museum of Natural History for hosting the first author as a visiting summer student. Partial funding was provided by the US National Science Foundation (#0920920 to KS) and the Simons Foundation.

Author information

Authors and Affiliations

  1. Departments of Mathematics and Chemistry, Johns Hopkins University, Baltimore, MD, USA

    Ellen Urheim

  2. Department of Computer Science, Graduate Center, City University of New York, New York, NY, USA

    Eric Ford & Katherine St. John

  3. Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY, USA

    Eric Ford & Katherine St. John

  4. Invertebrate Zoology, American Museum of Natural History, New York, NY, USA

    Eric Ford & Katherine St. John

Authors
  1. Ellen Urheim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric Ford
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Katherine St. John
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katherine St. John.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Urheim, E., Ford, E. & St. John, K. Characterizing Local Optima for Maximum Parsimony. Bull Math Biol 78, 1058–1075 (2016). https://doi.org/10.1007/s11538-016-0174-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-016-0174-0

Keywords

Search

Navigation