Revisiting Waiting Times in DNA Evolution

Transcription factors are short stretches of DNA (or k-mers) mainly located in promoters sequences that enhance or repress gene expression. With respect to an initial distribution of letters on the DNA alphabet, Behrens and Vingron [3] consider a random sequence of length n that does not contain a given k-mer or word of size k. Under an evolution model of the DNA, they compute the probability pn that this k-mer appears after a unit time of 20 years. They prove that the waiting time for the first apparition of the k-mer is well approximated by Tn = 1/pn. Their work relies on the simplifying assumption that the kmer is not self-overlapping. They observe in particular that the waiting time is mostly driven by the initial distribution of letters. Behrens et al. [2] use an approach by automata that relaxes the assumption related to words overlaps. Their numerical evaluations confirms the validity of Behrens and Vingron approach for non self-overlapping words, but provides up to 44% corrections for highly self-overlapping words such as AAAAA. We devised an approach of the problem by clump analysis and generating functions; this approach leads to prove a quasi-linear behaviour of pn for a large range of values of n, an important result for DNA evolution. We present here this clump analysis, first by language decomposition, and next by an automaton construction; finally, we describe an equivalent approach by construction of Markov automata.