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HEARING DISTANT ECHOES: USING EXTREMAL STATISTICS TO PROBE EVOLUTIONARY ORIGINS 112 while the standard deviation is estimated to be = 1.49. In contrast to Table 4.1, there is one tRNA, that for cystine, that scores exceptionally high. The tails of the extremal distributions in the logarithmic region probably behave like exp(âλt), where t is the test value as in the section "Local Sequence Comparisons" and λ is a constant, but this has not yet been proven rigorously. The usual intuition informed by the tail of a normal distribution has the probabilities behaving like exp(ât2)/2, which converges to 0 much more rapidly than the Poisson or exponential tails. Thus except for the cystine score, the remaining scores look very much like scores from random sequences. Simulations were performed, and the score 21.0 has an approximate p-value of 10â3, so that it is not possible to dismiss this matching for statistical reasons alone. As far as we know, no one has offered a biological explanation of this interesting match. As to the hypothesis of Bloch et al. (1983), while their work concluded that "matches are too frequent and extensive to be attributed to coincidence," it is not supported by the data but is instead the result of incorrect estimation of p-values. This data set received their most extensive analysis, and they concluded that over 30 percent of the matchings between E. coli 16S rRNA and tRNAs were significant at the level α = 0.10. Correct estimates show about 10 percent of the matching at the level α = 0.10. While the origin of life may be hiding in tRNA and 16S rRNA, it remains elusive. REFERENCES Aldous, D.J., 1989, Probability Approximations via the Poisson Clumping Heuristic, New York: Springer-Verlag. Arratia, R.A., L. Goldstein, and L. Gordon, 1989, "Two moments suffice for Poisson approximation: The Chen-Stein method," Annals of Probability 17, 9-25. Arratia, R.A., L. Goldstein, and L. Gordon, 1990, "Poisson approximation and the Chen-Stein method," Statistical Science 5, 403-434. Arratia, R.A., and M.S. Waterman, 1994, "A phase transition for the score in matching random sequences allowing deletions," Annals of Applied Probability 4, 200-225. Bloch, D.P., B. McArthur, R. Widdowson, D. Spector, R.C. Guimaraes, and J. Smith, 1983, "tRNA-rRNA sequence homologies: Evidence for a common evolutionary origin?," Journal of Molecular Evolution 19, 420-428. Chung, K.L., 1974, A Course in Probability Theory, 2nd ed, San Diego, CA: Academic Press.
HEARING DISTANT ECHOES: USING EXTREMAL STATISTICS TO PROBE EVOLUTIONARY ORIGINS 113 Chvátal, V, and D. Sankoff, 1975, "Longest common subsequences of two random sequences," Journal of Applied Probability 12, 306-315. Deken, J., 1979, "Some limit results for longest common subsequences," Discrete Mathematics 26, 17-31. Erdös, P., and A. Rényi, 1970, "On a new law of large numbers," Journal d' Analyse Mathematique 22, 103-111. Reprinted in 1976 in Selected Papers of Alfred Rényi, Vol. 3, 1962-1970s, Budapest: Akadémiai Kiadó. Feller, W., 1968, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., New York: Wiley and Sons. Goldstein, L., 1990, "Poisson approximation and DNA sequence matching," Communications in Statistics. Part A: Theory and Methods 19, 4167-4179. Kingman, J.F.C., 1973, "Subadditive ergodic theory," Annals of Probability 1, 883-909. Smith, T.F., and M.S. Waterman, 1981, "Identification of common molecular subsequences," Journal of Molecular Biology 147, 195-197. Steele, J.M., 1986, "An Efron-Stein inequality for nonsymmetric statistics," Annals of Statistics 14, 753-758. Waterman, M.S., and M. Eggert, 1987, "A new algorithm for best subsequence alignments with application to tRNA-rRNA comparisons," Journal of Molecular Biology 197, 723-728. Waterman, M.S., L. Gordon, and R. Arratia, 1987, "Phase transitions in sequence matches and nucleic acid structure," Proceedings of the National Academy of Sciences USA 84, 1239-1243. Waterman, M.S., and M. Vingron, 1994, "Rapid and accurate estimates of statistical significance for data base searches," Proceedings of the National Academy of Sciences USA 91, 4625-4628.
