Abstract
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical sets of matches between identical pairs of trees. This ambiguity is uninformative, and detrimental to any probabilistic analysis. In this work, we consider tree alignments up to equivalence. Our first result is a precise asymptotic enumeration of tree alignments, obtained from a context-free grammar by means of basic analytic combinatorics. Our second result focuses on alignments between two given ordered trees. By refining our grammar to align specific trees, we obtain a decomposition scheme for the space of alignments, and use it to design an efficient dynamic programming algorithm for sampling alignments under the Gibbs-Boltzmann probability distribution. This generalizes existing tree alignment algorithms, and opens the door for a probabilistic analysis of the space of suboptimal RNA secondary structures alignments.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
In this work, unless explicitly specified, all trees will be rooted and ordered.
- 2.
The present results can be trivially extended to any edit scoring system that is a positive linear combination of the numbers of insertions, deletions and matches.
References
Andrade, H., Area, I., Nieto, J.J., Torres, A.: The number of reduced alignments between two dna sequences. BMC Bioinformatics 15, 94 (2014). http://dx.doi.org/10.1186/1471-2105-15-94
Blin, G., Denise, A., Dulucq, S., Herrbach, C., Touzet, H.: Alignments of RNA structures. IEEE/ACM Trans. Comput. Biol. Bioinform. 7(2), 309–322 (2010). http://doi.acm.org/10.1145/1791396.1791409
Chauve, C., Courtiel, J., Ponty, Y.: Counting, generating and sampling tree alignments. In: ALCOB - 3rd International Conference on Algorithms for Computational Biology - 2016. Trujillo, Spain, Jun 2016. https://hal.inria.fr/hal-01154030
Do, C.B., Gross, S.S., Batzoglou, S.: CONTRAlign: discriminative training for protein sequence alignment. In: Apostolico, A., Guerra, C., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2006. LNCS (LNBI), vol. 3909, pp. 160–174. Springer, Heidelberg (2006)
Dress, A., Morgenstern, B., Stoye, J.: The number of standard and of effective multiple alignments. Appl. Math. Lett. 11(4), 43–49 (1998). http://www.sciencedirect.com/science/article/pii/S0893965998000548
Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press, Cambridge (2009)
Herrbach, C., Denise, A., Dulucq, S.: Average complexity of the Jiang-Wang-Zhang pairwise tree alignment algorithm and of a RNA secondary structure alignment algorithm. Theor. Comput. Sci. 411(26–28), 2423–2432 (2010). http://dx.doi.org/10.1016/j.tcs.2010.01.014
Höchsmann, M., Töller, T., Giegerich, R., Kurtz, S.: Local similarity in RNA secondary structures. Proc. Ieee Comput. Soc. Bioinform Conf. 2, 159–168 (2003)
Höchsmann, M., Voss, B., Giegerich, R.: Pure multiple rna secondary structure alignments: a progressive profile approach. IEEE/ACM Trans. Comput. Biol. Bioinformatics 1(1), 53–62 (2004). http://dx.doi.org/10.1109/TCBB.2004.11
Jiang, T., Wang, L., Zhang, K.: Alignment of trees - an alternative to tree edit. Theor. Comput. Sci. 143(1), 137–148 (1995). http://dx.doi.org/10.1016/0304-3975(95)80029-9
Ponty, Y., Saule, C.: A combinatorial framework for designing (pseudoknotted) RNA algorithms. In: Przytycka, T.M., Sagot, M.-F. (eds.) WABI 2011. LNCS, vol. 6833, pp. 250–269. Springer, Heidelberg (2011). http://dx.doi.org/10.1007/978-3-642-23038-7_22
Schirmer, S., Giegerich, R.: Forest alignment with affine gaps and anchors, applied in RNA structure comparison. Theor. Comput. Sci. 483, 51–67 (2013). http://dx.doi.org/10.1016/j.tcs.2012.07.040
Torres, A., Cabada, A., Nieto, J.J.: An exact formula for the number of alignments between two DNA sequences. DNA Seq. 14(6), 427–430 (2003)
Vingron, M., Argos, P.: Determination of reliable regions in protein sequence alignments. Protein Eng. 3(7), 565–569 (1990). http://peds.oxfordjournals.org/content/3/7/565.abstract
Waterman, M.S.: Introduction to Computational Biology: Maps, Sequences, and Genomes. CRC Press, Pevzner (1995)
Wilf, H.S.: A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects. Adv. Math. 24, 281–291 (1977)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Chauve, C., Courtiel, J., Ponty, Y. (2016). Counting, Generating and Sampling Tree Alignments. In: Botón-Fernández, M., Martín-Vide, C., Santander-Jiménez, S., Vega-Rodríguez, M.A. (eds) Algorithms for Computational Biology. AlCoB 2016. Lecture Notes in Computer Science(), vol 9702. Springer, Cham. https://doi.org/10.1007/978-3-319-38827-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-38827-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-38826-7
Online ISBN: 978-3-319-38827-4
eBook Packages: Computer ScienceComputer Science (R0)