Exact Distribution of a Spaced Seed Statistic for Applications in DNA Repeat Detection

Let a seed, S, be a string from the alphabet {1, ∗} which starts and ends with a 1. For example S = 11 ∗ 1. S occurs in a binary string B at position k if S can be positioned so that the last letter in S aligns with the kth letter in B, and each 1 in S aligns with a 1 in B. A 1 in B is covered by S if there exists some occurrence of S in B such that the 1 in B aligns with a 1 in the occurrence of S. We show how to compute the exact probability distribution for the number of 1s covered by a seed S in an i.i.d Bernoulli string of length n with probability of 1 equal to p. We refer to the new probability distribution as CnSp, for covered, with S being the seed. When S consists entirely of 1s, for example S = 111, this reduces to the familiar Rnkp which is the probability distribution for the number of 1s which occur in runs of length k or longer (k is the number of 1s in S). Importantly, our method is probability independent in that the calculation yields a formula in terms of the probability parameter p, and does not require fixing the value of p in advance. The CnSp distribution has applications in the detection of approximate DNA repeats using spaced seeds.