Probabilistic Analysis of a Motif Discovery Algorithm for Multiple Sequences

We study a natural probabilistic model for motif discovery that has been used to experimentally test the quality of motif discovery programs. In this model, there are k background sequences, and each character in a background sequence is a random character from an alphabet Σ. A motif G = g1g2 · · · gm is a string of m characters. Each background sequence is implanted into a probabilistically generated approximate copy of G. For an approximate copy b1b2 · · · bm of G, every character bi is probabilistically generated such that the probability for bi = gi is at most α. In this paper, we give the first analytical proof that multiple background sequences do help with finding subtle and faint motifs. This work is a theoretical approach with a rigorous probabilistic analysis. We develop an algorithm that under the probabilistic model can find the implanted motif with high probability when the number of background sequences is reasonably large. Specifically, we prove that for α < 0.1771 and any constant x ≥ 8, there exist constants t0, δ0, δ1 > 0 such that if the length of the motif is at least δ0 logn, the alphabet has at least t0 characters, and there are at least δ1 logn0 input sequences, then in O(n3) time our algorithm finds the motif with probability at least 1− 1 2x , where n is the longest length of any input sequence and n0 ≤ n is an upper bound for the length of the motif.