Discovering Almost Any Hidden Motif from Multiple Sequences in Polynomial Time with Low Sample Complexity and High Success Probability

Abstract

We study a natural probabilistic model for motif discovery that has been used to experimentally test the effectiveness 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 = g ₁ g ₂...g _m is a string of m characters. Each background sequence is implanted a probabilistically generated approximate copy of G. For a probabilistically generated approximate copy b ₁ b ₂...b _m of G, every character is probabilistically generated such that the probability for b _i  ≠ g _i is at most α. It has been conjectured that multiple background sequences can help with finding faint motifs G.

In this paper, we develop an efficient algorithm that can discover a hidden motif from a set of sequences for any alphabet Σ with |Σ| ≥ 2 and is applicable to DNA motif discovery. We prove that for $\alpha<{1\over 4}(1-{1\over |\Sigma|})$ and any constant x ≥ 8, there exist positive constants c ₀, ε, δ ₁ and δ ₂ such that if the length ρ of motif G is at least δ ₁ logn, and there are k ≥ c ₀ logn input sequences, then in O(n ² + kn) time this algorithm finds the motif with probability at least $1-{1\over 2^x}$ for every $G\in \Sigma^{\rho}-\Psi_{\rho, h,\epsilon}(\Sigma)$, where ρ is the length of the motif, h is a parameter with ρ ≥ 4h ≥ δ ₂logn, and Ψ _{ρ, h,ε} (Σ) is a small subset of at most $2^{-\Theta(\epsilon^2 h)}$ fraction of the sequences in Σ ^ρ. The constants c ₀, ε, δ ₁ and δ ₂ do not depend on x when x is a parameter of order O(logn). Our algorithm can take any number k sequences as input.