Faster exact Markovian probability functions for motif occurrences: a   DFA-only approach
  Background: The computation of the statistical properties of motif occurrences has an obviously relevant practical application: for example, patterns that are significantly over- or under-represented in the genome are interesting candidates for biological roles. However, the problem is computationally hard; as a result, virtually all the existing pipelines use fast but approximate scoring functions, in spite of the fact that they have been shown to systematically produce incorrect results. A few interesting exact approaches are known, but they are very slow and hence not practical in the case of realistic sequences. Results: We give an exact solution, solely based on deterministic finite-state automata (DFAs), to the problem of finding not only the p-value, but the whole relevant part of the Markovian probability distribution function of a motif in a biological sequence. In particular, the time complexity of the algorithm in the most interesting regimes is far better than that of Nuel (2006), which was the fastest similar exact algorithm known to date; in many cases, even approximate methods are outperformed. Conclusions: DFAs are a standard tool of computer science for the study of patterns, but so far they have been sparingly used in the study of biological motifs. Previous works do propose algorithms involving automata, but there they are used respectively as a first step to build a Finite Markov Chain Imbedding (FMCI), or to write a generating function: whereas we only rely on the concept of DFA to perform the calculations. This innovative approach can realistically be used for exact statistical studies of very long genomes and protein sequences, as we illustrate with some examples on the scale of the human genome. 
Paolo Ribeca and Emanuele Raineri