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.
An extension to single seed models is the design of a multiple seed. This is a set of spaced seeds to be used simultaneously, such that a similarity is detected when it is found by (at least) one of the seeds. The idea to use a family of spaced seeds for BLAST-type DNA local alignment has been suggested by Ma et al. @cite_70 and was implemented in PatternHunter II @cite_31 . It has also been applied to local protein alignment in @cite_32 . Recent approaches @cite_34 @cite_3 approximate the sensitivity of multiple spaced seeds by means of correlation functions. Since finding optimal multiple seeds is challenging, most authors concentrate on the design of efficient sets of seeds, leading to higher sensitivity than optimal single seeds @cite_31 @cite_65 @cite_25 .
@cite_70 @cite_65 @cite_32 @cite_3 @cite_31 @cite_34 @cite_25
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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] Optimal spaced seeds were developed as a method to increase sensitivity of local alignment programs similar to BLASTN. Such seeds have been used before in the program PatternHunter, and have given improved sensitivity and running time relative to BLASTN in genome–genome comparison. We study the problem of computing optimal spaced seeds for detecting homologous coding regions in unannotated genomic sequences. By using well-chosen seeds, we are able to improve the sensitivity of coding sequence alignment over that of TBLASTX, while keeping runtime comparable to BLASTN. We identify good seeds by first giving effective hidden Markov models of conservation in alignments of homologous coding regions. We give an efficient algorithm to compute the optimal spaced seed when conservation patterns are generated by these models. Our results offer the hope of improved gene finding due to fewer missed exons in DNA DNA comparison, and more effective homology search in general, and may have applications outside of bioinformatics. Large-scale comparison of genomic DNA is of fundamental importance in annotating functional elements of genomes. To perform large comparisons efficiently, BLAST (Methods: Companion Methods Enzymol 266 (1996) 460, J. Mol. Biol. 215 (1990) 403, Nucleic Acids Res. 25(17) (1997) 3389) and other widely used tools use seeded alignment, which compares only sequences that can be shown to share a common pattern or ''seed'' of matching bases. The literature suggests that the choice of seed substantially affects the sensitivity of seeded alignment, but designing and evaluating seeds is computationally challenging. This work addresses the problem of designing a seed to optimize performance of seeded alignment. We give a fast, simple algorithm based on finite automata for evaluating the sensitivity of a seed in a Markov model of ungapped alignments, along with extensions to mixtures and inhomogeneous Markov models. We give intuition and theoretical results on which seeds are good choices. Finally, we describe Mandala, a software tool for seed design, and show that it can be used to improve the sensitivity of alignment in practice. 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