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 $\Sigma$. A motif $G=g_1g_2\cdots g_m$ is a string of $m$ characters. Each background sequence is implanted into a probabilistically generated approximate copy of $G$. For an approximate copy $b_1b_2\cdots b_m$ of $G$, every character $b_i$ is probabilistically generated such that the probability for $b_i\neq g_i$ is at most $\alpha$. 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 $\alpha0$ such that if the length of the motif is at least $\delta_0\log n$, the alphabet has at least $t_0$ characters, and there are at least $\delta_1\log n_0$ input sequences, then in $O(n^3)$ time our algorithm finds the motif with probability at least $1-\frac{1}{2^x}$, where $n$ is the longest length of any input sequence and $n_0\leq n$ is an upper bound for the length of the motif.