We present probabilistic arithmetic automata (PAAs), a general model to describe chains of operations whose operands depend on chance, along with two different algorithms to exactly calculate the distribution of the results obtained by such probabilistic calculations. PAAs provide a unifying framework to approach many problems arising in computational biology and elsewhere. Here, we present five different applications, namely (1) pattern matching statistics on random texts, including the computation of the distribution of occurrence counts, waiting time and clump size under HMM background models; (2) exact analysis of window-based pattern matching algorithms; (3) sensitivity of filtration seeds used to detect candidate sequence alignments; (4) length and mass statistics of peptide fragments resulting from enzymatic cleavage reactions; and (5) read length statistics of 454 sequencing reads. The diversity of these applications indicates the flexibility and unifying character of the presented framework. While the construction of a PAA depends on the particular application, we single out a frequently applicable construction method for pattern statistics: We introduce deterministic arithmetic automata (DAAs) to model deterministic calculations on sequences, and demonstrate how to construct a PAA from a given DAA and a finite-memory random text model. We show how to transform a finite automaton into a DAA and then into the corresponding PAA.
Since exact local alignment is too slow in practice, most heuristic homology search algorithms are based on a two-phase filtration technique @cite_27 @cite_9 @cite_23 @cite_28 . First, candidate sequences are selected that share a common pattern ( seed'') of matching characters with the query. These candidates (or hits'') are then further investigated by an exact method. Initially, (e.g., perfectly matching DNA 11-mers in the initial BLAST implementation) were used. PatternHunter (PH) by Ma et al. @cite_70 was the first tool to systematically advocate and investigate : PH looks for 18-mers with at least 11 matching positions distributed as 111*1**1*1**11*111 , where 1 denotes a necessary match and * denotes a don't care position (match or mismatch). Over time, various seed models have been proposed in the literature, including consecutive seeds @cite_27 @cite_9 , spaced seeds @cite_70 @cite_59 @cite_39 @cite_30 , subset seeds @cite_5 , vector seeds @cite_63 , and indel seeds @cite_38 .
@cite_30 @cite_38 @cite_28 @cite_70 @cite_9 @cite_39 @cite_27 @cite_23 @cite_59 @cite_63 @cite_5
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Motivation: Homology search finds similar segments between two biological sequences, such as DNA or protein sequences. The introduction of optimal spaced seeds in PatternHunter has increased both the sensitivity and the speed of homology search, and it has been adopted by many alignment programs such as BLAST. With the further improvement provided by multiple spaced seeds in PatternHunterII, Smith–Waterman sensitivity is approached at BLASTn speed. However, computing optimal multiple spaced seeds was proved to be NP-hard and current heuristic algorithms are all very slow (exponential). Results: We give a simple algorithm which computes good multiple seeds in polynomial time. Due to a completely different approach, the difference with respect to the previous methods is dramatic. The multiple spaced seed of PatternHunterII, with 16 weight 11 seeds, was computed in 12 days. It takes us 17 s to find a better one. Our approach changes the way of looking at multiple spaced seeds. Contact: [email protected] We are interested in detecting homologous genomic DNA sequences with the goal of locating approximate inverted, interspersed, and tandem repeats. Standard search techniques start by detecting small matching parts, called seeds, between a query sequence and database sequences. Contiguous seed models have existed for many years. Recently, spaced seeds were shown to be more sensitive than contiguous seeds without increasing the random hit rate. To determine the superiority of one seed model over another, a model of homologous sequence alignment must be chosen. Previous studies evaluating spaced and contiguous seeds have assumed that matches and mismatches occur within these alignments, but not insertions and deletions (indels). This is perhaps appropriate when searching for protein coding sequences (<5 of the human genome), but is inappropriate when looking for repeats in the majority of genomic sequence where indels are common. In this paper, we assume a model of homologous sequence alignment which includes indels and we describe a new seed model, called indel seeds, which explicitly allows indels. We present a waiting time formula for computing the sensitivity of an indel seed and show that indel seeds significantly outperform contiguous and spaced seeds when homologies include indels. We discuss the practical aspect of using indel seeds and finally we present results from a search for inverted repeats in the dog genome using both indel and spaced seeds. Contact: dyfmak@bu.edu Motivation: Filtration is an important technique used to speed up local alignment as exemplified in the BLAST programs. Recently, discovered that better filtering can be achieved by spacing out the matching positions according to a certain pattern, instead of contiguous positions to trigger a local alignment in their PatternHunter program. Such a match pattern is called a spaced seed. Results: Our numerical computation shows that the ranks of spaced seeds (based on sensitivity) change with the sequences similarity. Since homologous sequences may have diverse similarity, we assess the sensitivity of spaced seeds over a range of similarity levels and present a list of good spaced seeds for facilitating homology search in DNA genomic sequences. We validate that the listed spaced seeds are indeed more sensitive using three arbitrarily chosen pairs of DNA genomic sequences. Extending the single optimized spaced seed of PatternHunter20 to multiple ones, PatternHunter II simultaneously remedies the lack of sensitivity of Blastn and the lack of speed of Smith–Waterman, for homology search. At Blastn speed, PatternHunter II approaches Smith–Waterman sensitivity, bringing homology search methodology research back to a full circle.