Moduli Spaces of Phylogenetic Trees Describing Tumor Evolutionary Patterns

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8609)


Cancers follow a clonal Darwinian evolution, with fitter subclones replacing more quiescent cells, ultimately giving rise to macroscopic disease. High-throughput genomics provides the opportunity to investigate these processes and determine specific genetic alterations driving disease progression. Genomic sampling of a patient’s cancer provides a molecular history, represented by a phylogenetic tree. Cohorts of patients represent a forest of related phylogenetic structures. To extract clinically relevant information, one must represent and statistically compare these collections of trees. We propose a framework based on an application of the work by Billera, Holmes and Vogtmann on phylogenetic tree spaces to the case of unrooted trees of intra-individual cancer tissue samples. We observe that these tree spaces are globally nonpositively curved, allowing for statistical inference on populations of patient histories. A projective tree space is introduced, permitting visualizations of evolutionary patterns. Published data from four types of human malignancies are explored within our framework.


phylogenetic tree moduli space tumor evolution genomics 


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  1. 1.
    Aitchison, J.: The Statistical Analysis of Compositional Data. Journal of the Royal Statistical Society 44, 139–177 (1982)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Alexandrov, A.: Über eine verallgemeinerung der riemannschen geometrie. Schr. Forschungsinst. Math. Berlin 1, 33–84 (1957)Google Scholar
  3. 3.
    Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the Space of Phylogenetic Trees. Advances in Applied Mathematics 27(4), 733–767 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Campbell, P.J., Yachida, S., Mudie, L.J., Stephens, P.J., Pleasance, E.D., Stebbings, L.A., Morsberger, L.A., Latimer, C., McLaren, S., Lin, M.L., McBride, D.J., Varela, I., Nik-Zainal, S.A., Leroy, C., Jia, M., Menzies, A., Butler, A.P., Teague, J.W., Griffin, C.A., Burton, J., Swerdlow, H., Quail, M.A., Stratton, M.R., Iacobuzio-Donahue, C., Futreal, P.A.: The patterns and dynamics of genomic instability in metastatic pancreatic cancer. Nature 467(7319), 1109–1113 (2010)CrossRefGoogle Scholar
  5. 5.
    Carlotti, E., Wrench, D., Matthews, J., Iqbal, S., Davies, A., Norton, A., Hart, J., Lai, R., Montoto, S., Gribben, J.G., Lister, T.A., Fitzgibbon, J.: Transformation of follicular lymphoma to diffuse large B-cell lymphoma may occur by divergent evolution from a common progenitor cell or by direct evolution from the follicular lymphoma clone. Blood 113(15), 3553–3557 (2009)Google Scholar
  6. 6.
    Ding, L., Ley, T.J., Larson, D.E., Miller, C.A., Koboldt, D.C., Welch, J.S., Ritchey, J.K., Young, M.A., Lamprecht, T., McLellan, M.D., McMichael, J.F., Wallis, J.W., Lu, C., Shen, D., Harris, C.C., Dooling, D.J., Fulton, R.S., Fulton, L.L., Chen, K., Schmidt, H., Kalicki-Veizer, J., Magrini, V.J., Cook, L., McGrath, S.D., Vickery, T.L., Wendl, M.C., Heath, S., Watson, M.A., Link, D.C., Tomasson, M.H., Shannon, W.D., Payton, J.E., Kulkarni, S., Westervelt, P., Walter, M.J., Graubert, T.A., Mardis, E.R., Wilson, R.K., DiPersio, J.F.: Clonal evolution in relapsed acute myeloid leukaemia revealed by whole-genome sequencing. Nature 481(7382), 506–510 (2012)CrossRefGoogle Scholar
  7. 7.
    Li, S., Hricik, T., Chung, S.S., Bar, H., Brown, A.L., Patel, J.P., Rapoport, F., Liu, L., Sheridan, C., Ishii, J., Zumbo, P., Gandara, J., Lewis, I.D., To, L.B., Becker, M.W., Guzman, M.L., D’Andrea, R.J., Michor, F., Park, C.Y., Carroll, M., Levine, R.L., Mason, C.E., Melnick, A.M.: Epigenetic deregulation in relapsed acute myeloid leukemia. Blood 122(21), 2499 (2013)Google Scholar
  8. 8.
    Miller, E., Owen, M., Provan, J.S.: Polyhedral computational geometry for averaging metric phylogenetic trees (2014)Google Scholar
  9. 9.
    Ohtake, S., Miyawaki, S., Fujita, H., Kiyoi, H., Shinagawa, K., Usui, N., Okumura, H., Miyamura, K., Nakaseko, C., Miyazaki, Y., Fujieda, A., Nagai, T., Yamane, T., Taniwaki, M., Takahashi, M., Yagasaki, F., Kimura, Y., Asou, N., Sakamaki, H., Handa, H., Honda, S., Ohnishi, K., Naoe, T., Ohno, R.: Randomized study of induction therapy comparing standard-dose idarubicin with high-dose daunorubicin in adult patients with previously untreated acute myeloid leukemia: the JALSG AML201 Study. Blood 117(8), 2358–2365 (2011)CrossRefGoogle Scholar
  10. 10.
    Okosun, J., Bödör, C., Wang, J., Araf, S., Yang, C.Y., Pan, C., Boller, S., Cittaro, D., Bozek, M., Iqbal, S., Matthews, J., Wrench, D., Marzec, J., Tawana, K., Popov, N., O’Riain, C., O’Shea, D., Carlotti, E., Davies, A., Lawrie, C.H., Matolcsy, A., Calaminici, M., Norton, A., Byers, R.J., Mein, C., Stupka, E., Lister, T.A., Lenz, G., Montoto, S., Gribben, J.G., Fan, Y., Grosschedl, R., Chelala, C., Fitzgibbon, J.: Integrated genomic analysis identifies recurrent mutations and evolution patterns driving the initiation and progression of follicular lymphoma. Nature Genetics 46(2), 176–181 (2014)CrossRefGoogle Scholar
  11. 11.
    Owen, M., Provan, J.: A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Transactions on Computational Biology, 1–18 (2011)Google Scholar
  12. 12.
    Pasqualucci, L., Khiabanian, H., Fangazio, M., Vasishtha, M., Messina, M., Holmes, A.B., Ouillette, P., Trifonov, V., Rossi, D., Tabbò, F., Ponzoni, M., Chadburn, A., Murty, V.V., Bhagat, G., Gaidano, G., Inghirami, G., Malek, S.N., Rabadan, R., Dalla-Favera, R.: Genetics of follicular lymphoma transformation. Cell Reports 6(1), 130–140 (2014)CrossRefGoogle Scholar
  13. 13.
    Sturm, K.: Probability measures on metric spaces of nonpositive curvature. Contemporary Mathematics, 1–34 (2003)Google Scholar
  14. 14.
    Tzoneva, G., Perez-Garcia, A., Carpenter, Z., Khiabanian, H., Tosello, V., Allegretta, M., Paietta, E., Racevskis, J., Rowe, J.M., Tallman, M.S., Paganin, M., Basso, G., Hof, J., Kirschner-Schwabe, R., Palomero, T., Rabadan, R., Ferrando, A.: Activating mutations in the NT5C2 nucleotidase gene drive chemotherapy resistance in relapsed ALL. Nature medicine 19(3), 368–371 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Systems BiologyColumbia UniversityNew YorkUSA
  2. 2.Department of Biomedical InformaticsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA

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