Exact Distribution of a Spaced Seed Statistic for DNA Homology Detection

Let a seed, S, be a string from the alphabet {1, ∗}, of arbitrary length k, which starts and ends with a 1. For example, S = 11 ∗ 1. S occurs in a binary string T at position h if the length k substring of T ending at position h contains a 1 in every position where there is a 1 in S. We say that the 1s at the corresponding positions in T are covered. We are interested in calculating the probability distribution for the number of 1s covered by a seed S in an iid Bernoulli string of length n with probability of 1 equal to p. We refer to this new probability distribution as CnSp, for covered, with S being the seed. We present an efficient method to calculate this distribution exactly. Covered 1s represent matching positions detected in DNA sequences when using multiple hits of a spaced seed. Knowledge of the distribution provides a statistical threshold for distinguishing true homologies from randomly matching sequences.