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Multiple pattern matching: a Markov chain approach

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Abstract

RNA motifs typically consist of short, modular patterns that include base pairs formed within and between modules. Estimating the abundance of these patterns is of fundamental importance for assessing the statistical significance of matches in genomewide searches, and for predicting whether a given function has evolved many times in different species or arose from a single common ancestor. In this manuscript, we review in an integrated and self-contained manner some basic concepts of automata theory, generating functions and transfer matrix methods that are relevant to pattern analysis in biological sequences. We formalize, in a general framework, the concept of Markov chain embedding to analyze patterns in random strings produced by a memoryless source. This conceptualization, together with the capability of automata to recognize complicated patterns, allows a systematic analysis of problems related to the occurrence and frequency of patterns in random strings. The applications we present focus on the concept of synchronization of automata, as well as automata used to search for a finite number of keywords (including sets of patterns generated according to base pairing rules) in a general text.

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References

  1. Aho A.V. and Corasick M.J. (1975). Efficient string matching: an aid to bibliographic search. Commun. ACM 18(6): 333–340

    Article  MATH  MathSciNet  Google Scholar 

  2. Aston J.A.D. and Martin D.E.K. (2005). Waiting time distributions of competing patterns in higher-order Markovian sequences. J. Appl. Prob. 42(4): 977–988

    Article  MATH  MathSciNet  Google Scholar 

  3. Biggins J.D. and Cannings C. (1987). Markov renewal processes, counters and repeated sequences in Markov chains. Adv. Appl. Prob. 19: 521–545

    Article  MATH  MathSciNet  Google Scholar 

  4. Benson, G.: Tandem repeats finder: a program to analyze DNA sequences. Nucleic Acids Res. 573–580 (1999)

  5. Bourdeau V., Ferbeyre G., Pageau M., Paquin B. and Cedergren R. (1999). The distribution of RNA motifs in natural sequences. Nucleic Acids Res. 27(22): 4457–4467

    Article  Google Scholar 

  6. Bender E.A. and Kochman F. (1993). The distribution of subword counts is usually normal. Eur. J. Comb. 14(4): 265–275

    Article  MATH  MathSciNet  Google Scholar 

  7. Buhler, J., Keich, U., Sun, Y.: Designing seeds for similarity search in genomic DNA. In: RECOMB ’03: Proceedings of the seventh annual international conference on Research in computational molecular biology, pp. 67–75 (2003)

  8. Bussemaker H.J., Li H. and Siggia E.D. (2000). Building a dictionary for genomes: identification of presumptive regulatory sites by statistical analysis. Proc. Natl. Acad. Sci. USA 97(18): 10096–10100

    Article  MathSciNet  Google Scholar 

  9. Brémaud P. (1998). Markov Chains: Gibbs fields, Monte Carlo Simulation and Queues. Springer, Heidelberg

    Google Scholar 

  10. Bourdon, J., Vallée, B.: Generalized pattern matching statistics. In: Colloquium on Mathematics and Computer Science: Algorithms and Trees, Trends in Mathematics, pp. 249–265. Birkhauser, 2002

  11. Bourdon, J., Vallée, B.: Pattern matching statistics on correlated sources. In: Proceedings of the seventh Latin American Symposium on Theoretical Informatics (LATIN’06), pp. 224–237, Valdivia, Chile (2006)

  12. Breen S., Waterman M.S. and Zhang N. (1985). Renewal theory for several patterns. J. Appl. Prob. 22: 228–234

    Article  MATH  MathSciNet  Google Scholar 

  13. Clément J., Flajolet P. and Vallée B. (2001). Dynamical sources in information theory: a general analysis of trie structures. Algorithmica 29(1): 307–369

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, X.: Limit theorems for functional of ergodic Markov chains with general state space, vol. 139. Memoirs of the American Mathematical Society, 1999

  15. Crochemore M. and Rytter W. (2002). Jewels of Stringology. World Scientific, Singapore

    Google Scholar 

  16. Cech T.R., Zaug A.J. and Grabowski P.J. (1981). In vitro splicing of the ribosomal RNA precursor of Tetrahymena: involvement of a guanosine nucleotide in the excision of the intervening sequence. Cell 27(3 Pt 2): 487–496

    Article  Google Scholar 

  17. Durrett R. (1999). Essentials of Stochastic Processes. Springer, Heidelberg

    MATH  Google Scholar 

  18. Durrett R. (2004). Probability: Theory and Examples, third edition. Duxbury Press, North Scituate

    Google Scholar 

  19. Eddy S.R. and Durbin R. (1994). RNA sequence analysis using covariance models. Nucleic Acids Res. 22(11): 2079–2088

    Article  Google Scholar 

  20. Ferbeyre G., Bourdeau V., Pageau M., Miramontes P. and Cedergren R. (2000). Distribution of hammerhead and hammerhead-like RNA motifs through the GenBank. Genome Res. 10(7): 1011–1019

    Article  Google Scholar 

  21. Fu J.C. and Chang Y.M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. J. Appl. Prob. 39(1): 70–80

    Article  MATH  MathSciNet  Google Scholar 

  22. Fu J.C. and Chang Y.M. (2003). On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials. J. Appl. Prob. 40(3): 623–642

    Article  MATH  MathSciNet  Google Scholar 

  23. Feller W. (1968). An Introduction to Probability Theory and Its Applications third edition. Wiley, New York

    Google Scholar 

  24. Felsenstein J. (1981). Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol. 17(6): 368–376

    Article  Google Scholar 

  25. Fu J.C. and Koutras M.V. (1994). Distribution theory of runs: a Markov chain approach. J. Am. Statist. Assoc. 89(427): 1050–1058

    Article  MATH  MathSciNet  Google Scholar 

  26. Flajolet P., Kirschenhofer P. and Tichy R.F. (1988). Deviations from uniformity in random strings. Probab. Th. Rel. Fields 80(1): 139–150

    Article  MATH  MathSciNet  Google Scholar 

  27. Fu, J.C., Lou, W.Y.W.: Distribution theory of runs and patterns and its applications. A finite Markov chain imbedding approach. World Scientific, Singapor (2003)

  28. Flajolet, P., Sedgewick, R.: Analytic Combinatorics, 2006. Electronic version available online at http://algo.inria.fr/flajolet/Publications/book060418.pdf

  29. Flajolet P., Szpankowski W. and Vallée B. (2006). Hidden word statistics. J. ACM 53(1): 147–183

    Article  MathSciNet  Google Scholar 

  30. Gani J. and Irle A. (1999). On patterns in sequences of random events. Mh. Math. 127: 295–309

    Article  MATH  MathSciNet  Google Scholar 

  31. Goulden I.P. and Jackson D.M. (2004). Combinatorial Enumeration. Dover, New York

    MATH  Google Scholar 

  32. Griffiths-Jones S., Moxon S., Marshall M., Khanna A., Eddy S.R. and Bateman A. (2005). RFAM: annotating non-coding RNAs in complete genomes. Nucleic Acids Res 33(Database issue): 121–124

    Article  Google Scholar 

  33. Gerber H.U. and Li S.-Y.R. (1981). The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Process. Appl. 11(1): 101–108

    Article  MATH  MathSciNet  Google Scholar 

  34. Guibas L.J. and Odlyzko A.M. (1978). Maximal prefix-synchronized codes. SIAM J. Appl. Math. 35(2): 401–418

    Article  MATH  MathSciNet  Google Scholar 

  35. Guibas L.J. and Odlyzko A.M. (1981). Periods in strings. J. Combin. Theory Ser. A 30(1): 19–42

    Article  MATH  MathSciNet  Google Scholar 

  36. Guibas L.J. and Odlyzko A.M. (1981). String overlaps, pattern matching and nontransitive games. J. Comb. Theory Ser. A 30(2): 183–208

    Article  MATH  MathSciNet  Google Scholar 

  37. Guerrier-Takada C., Gardiner K., Marsh T., Pace N. and Altman S. (1983). The RNA moiety of ribonuclease P is the catalytic subunit of the enzyme. Cell 35(3 Pt 2): 849–857

    Article  Google Scholar 

  38. Hentze M.W., Caughman S.W., Rouault T.A., Barriocanal J.G., Dancis A., Harford J.B. and Klausner R.D. (1987). Identification of the iron-responsive element for the translational regulation of human ferritin mRNA. Science 238(4833): 1570–1573

    Article  Google Scholar 

  39. Han Q. and Hirano K. (2003). Sooner and later waiting time problems for patterns in Markov dependent trials. J. Appl. Prob. 40(1): 73–86

    Article  MATH  MathSciNet  Google Scholar 

  40. Hopcroft J.E. and Ullman J.D. (1979). Introduction to automata theory, languages and computation. Addison-Wesley, Reading

    MATH  Google Scholar 

  41. Jones G.L. (2004). On the Markov chain central limit theorem. Probab. Surv. 1: 299–320

    Article  MathSciNet  MATH  Google Scholar 

  42. Knight R., De Sterck H., Markel R., Smit S., Oshmyansky A. and Yarus M. (2005). Abundance of correctly folded RNA motifs in sequence space, calculated on computational grids. Nucleic Acids Res. 33(18): 5924–5935

    Article  Google Scholar 

  43. Klein R.J. and Eddy S.R. (2003). RSEARCH: finding homologs of single structured RNA sequences. BMC Bioinform. 4: 44

    Article  Google Scholar 

  44. Kimura M. (1981). Estimation of evolutionary distances between homologous nucleotide sequences. Proc. Natl. Acad. Sci. USA 78(1): 454–458

    Article  MATH  Google Scholar 

  45. Knuth D.E., Pratt V.R. and Morris J.H. (1977). Fast pattern matching in strings. SIAM J. Comput. 6(2): 323–350

    Article  MATH  MathSciNet  Google Scholar 

  46. Kucherov, G., Noe, L., Roytberg, M.: A unifying framework for seed sensitivity and its application to subset seeds (extended abstract), (2006)

  47. Knight R. and Yarus M. (2003). Finding specific RNA motifs: function in a zeptomole world?. RNA 9(2): 218–230

    Article  Google Scholar 

  48. Lewis, B.P., Burge, C.B., Bartel, D.P.: Conserved seed pairing, often flanked by adenosines, indicates that thousands of human genes are microRNA targets. Cell 120(1), 15–20, Jan 2005. Letter

  49. Li S.-Y.R. (1980). A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Probab. 8(6): 1171–1176

    MATH  MathSciNet  Google Scholar 

  50. Lladser, M.: Minimal markov chain embeddings of pattern problems. In: Proceedings of the 2007 Information Theory and Applications Workshop, University of California, San Diego (2007)

  51. Lothaire M., Rota G.-C., Doran B., Ismail M., Lam T.Y., Wutwak E., Flajolet P. and Lutwak E. (2005). Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge

    Google Scholar 

  52. Lu C., Tej S.S., Luo S., Haudenschild C.D., Meyers B.C. and Green P.J. (2005). Elucidation of the small RNA component of the transcriptome. Science 309(5740): 1567–1569

    Article  Google Scholar 

  53. Martin D. (2005). Distribution of the number of successes in success runs of length at least k in higher-order Markovian sequences. Methodol. Comput. Appl. Probab. 7(4): 543–554

    Article  MATH  MathSciNet  Google Scholar 

  54. Nicodème P. (2003). Regexpcount, a symbolic package for counting problems on regular expressions and words. Fundamenta Informaticae 56(1-2): 71–88

    MATH  MathSciNet  Google Scholar 

  55. Nicodème P., Salvy B. and Flajolet P. (2002). Motif statistics. Theoret. Comput. Sci. 287(2): 593–617

    Article  MATH  MathSciNet  Google Scholar 

  56. Pozdnyakov V.I. and Kulldorff M. (2006). Waiting times for patterns and a method of gambling teams. Am. Math. Month. 113(2): 134–143

    Article  MathSciNet  MATH  Google Scholar 

  57. Park Y. and Spouge J.L. (2004). Searching for multiple words in a Markov sequence. INFORMS J. Comput. 16(4): 341–347

    Article  MathSciNet  Google Scholar 

  58. Robin S.S. and Daudin J.J. (2001). Exact distribution of the distances between any occurrences of a set of words. Ann. Inst. Statist. Math. 53(4): 895–905

    Article  MATH  MathSciNet  Google Scholar 

  59. Régnier M. and Denise A. (2004). Rare events and conditional events on random strings. DMTCS 6(2): 191–214

    MATH  MathSciNet  Google Scholar 

  60. Rivas E. and Eddy S.R. (2000). The language of RNA: a formal grammar that includes pseudoknots. Bioinformatics 16(4): 334–340

    Article  Google Scholar 

  61. Régnier M. (2000). A unified approach to word occurrences probabilities. Discrete Appl. Math. 104(1): 259–280

    Article  MATH  MathSciNet  Google Scholar 

  62. Régnier, M., Lifanov, A., Makeev, V.: Three variations on word counting. In: Proceedings German Conference on Bioinformatics, GCB’00, Heidelberg, pp. 75–82. Logos-Verlag, 2000

  63. Robin S., Rodolphe F. and Schbath S. (2005). DNA, Words and Models. Cambridge University Press, New York

    MATH  Google Scholar 

  64. Régnier M. and Szpankowski W. (1998). On pattern frequency occurrences in a Markovian sequence. Algorithmica 22(4): 631–649

    Article  MATH  MathSciNet  Google Scholar 

  65. Salehi-Ashtiani K. and Szostak J.W. (2001). In vitro evolution suggests multiple origins for the hammerhead ribozyme. Nature 414(6859): 82–84

    Article  Google Scholar 

  66. Shao J. (2003). Mathematical Statistics, second edition. Springer, Heidelberg

    Google Scholar 

  67. Sipser, M.: Introduction to the Theory of Computation. International Thomson Publishing, (1996)

  68. Singh R., Robida M.D. and Karimpour S. (2006). Building biological complexity with limited genes. Curr. Genom. 7: 97–114

    Article  Google Scholar 

  69. Sabeti P.C., Unrau P.J. and Bartel D.P. (1997). Accessing rare activities from random RNA sequences: the importance of the length of molecules in the starting pool. Chem. Biol. 4(10): 767–774

    Article  Google Scholar 

  70. Tang J. and Breaker R.R. (2000). Structural diversity of self-cleaving ribozymes. Proc. Natl. Acad. Sci. USA 97(11): 5784–5789

    Article  Google Scholar 

  71. Vallée B. (2001). Dynamical sources in information theory: fundamental intervals and word prefixes. Algorithmica 29(1): 262–306

    Article  MATH  MathSciNet  Google Scholar 

  72. Waterman, M.S.: Introduction to computational biology: maps, sequences and genomes. Chapman & Hall, WAT m 95:1 1.Ex (1995)

  73. Wilf H.S. (1994). Generatingfunctiology, second edition. Academic, New York

    Google Scholar 

  74. Welch M., Majerfeld I. and Yarus M. (1997). 23S rRNA similarity from selection for peptidyl transferase mimicry. Biochemistry 36(22): 6614–6623

    Article  Google Scholar 

  75. Winkler W., Nahvi A. and Breaker R.R. (2002). Thiamine derivatives bind messenger RNAs directly to regulate bacterial gene expression. Nature 419(6910): 952–956

    Article  Google Scholar 

  76. Yarus M., Caporaso J.G. and Knight R. (2005). Origins of the genetic code: the escaped triplet theory. Annu. Rev. Biochem. 74: 179–198

    Article  Google Scholar 

  77. Yarus M. and Welch M. (2000). Peptidyl transferase: ancient and exiguous. Chem. Biol. 7(10): 187–190

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO, 80309-0526, USA

    Manuel E. Lladser

  2. Department of Physics, University of Colorado at Boulder, 390 UCB, Boulder, CO, 80309, USA

    M. D. Betterton

  3. Department of Chemistry and Biochemistry, University of Colorado at Boulder, 215 UCB, Boulder, CO, 80309, USA

    Rob Knight

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  1. Manuel E. Lladser
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  2. M. D. Betterton
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  3. Rob Knight
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Correspondence to Manuel E. Lladser.

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We are grateful to Robert S. Maier and several researchers in the Analysis of Algorithms (AofA) community for their helpful comments and suggestions while preparing this manuscript.

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Lladser, M.E., Betterton, M.D. & Knight, R. Multiple pattern matching: a Markov chain approach. J. Math. Biol. 56, 51–92 (2008). https://doi.org/10.1007/s00285-007-0109-3

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