By Grotendorst J., Marx D., Muramatsu A. (eds.)
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Similarly, for all P r [ C E l aijXi 5 (1 - X)z*] j , P r [ C E l ( l - aij)Xi 5 (1 - X)z*] I e - - x 2 z * / 2 . By the naive union bound, Pr[Z 5 (1 - ~ ) z * 5] 2ne-X2z*/2. Likewise, E x p [ C Z 1 X ~ ] = 5 s. Thus, by the Chernoff bound, P r [ C z l X i 2 (1+ E ) S ] I e--t2s/4, where 0 < E I2e - 1. Letting L = Czl Xi - s, we have Pr[L 2 I e ~ ~where ~ 0/ < ~S 5, (2e - 1)&. Since 2 2 2 - L, and using the above estimates, we get Pr[Z I (1 - X)z* - S&] 5 P r [ z 5 (1 - ~ ) z * ] Pr[L 2 6 4 1 5 2ne-X2z"/2 e-S2/4.
We show that this method works well for the more general Problem 1 as well. Recall that the signal (”subtle motifs”) is embedded in t random sequences. , an 1-mer without wild cards or dont-care characters), it is not necessarily contained in any of the t sequences. However, if we can obtain a correct alignment of the m sequences, then it is relatively easy to extract the consensus motif satisfying the (1, d ) constraint. In other words, one of the difficulties of the problem is that the sequences are unaligned.
Streit. Metagenomics: Advances in ecology and biotechnology. FEMS Microbiology Letters, 247(2):105-111,2005. 11. S. G. Tringe, C. von Mering, A. Kobayashi, A. A. Salamov, K. Chen, H. W. Chang, M. Podar, J. M. Short, E. J. Mathur, J. C. Detter, P. Bork, P. Hugenholtz, and E. M. Rubin. Comparative metagenomics of microbial communities. Science, 308:554557,2005. 12. G. W. Tyson, J. Chapman, P. Hugenholtz, E. E. Allen, R. J. Raml, P. M. Richardson, V. V. Solovyev, E. M. Rubin, D. S. Rokhsar, and J.