You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: 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.

[2]: 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

[3]: Analyzing vertebrate genomes requires rapid mRNA DNA and cross-species protein alignments. A new tool, BLAT, is more accurate and 500 times faster than popular existing tools for mRNA DNA alignments and 50 times faster for protein alignments at sensitivity settings typically used when comparing vertebrate sequences. BLAT's speed stems from an index of all nonoverlapping K-mers in the genome. This index fits inside the RAM of inexpensive computers, and need only be computed once for each genome assembly. BLAT has several major stages. It uses the index to find regions in the genome likely to be homologous to the query sequence. It performs an alignment between homologous regions. It stitches together these aligned regions (often exons) into larger alignments (typically genes). Finally, BLAT revisits small internal exons possibly missed at the first stage and adjusts large gap boundaries that have canonical splice sites where feasible. This paper describes how BLAT was optimized. Effects on speed and sensitivity are explored for various K-mer sizes, mismatch schemes, and number of required index matches. BLAT is compared with other alignment programs on various test sets and then used in several genome-wide applications. http: genome.ucsc.edu hosts a web-based BLAT server for the human genome.

[4]: Motivation: Genomics and proteomics studies routinely depend on homology searches based on the strategy of finding short seed matches which are then extended. The exploding genomic data growth presents a dilemma for DNA homology search techniques: increasing seed size decreases sensitivity whereas decreasing seed size slows down computation. Results: We present a new homology search algorithm ‘PatternHunter’ that uses a novel seed model for increased sensitivity and new hit-processing techniques for significantly increased speed. At Blast levels of sensitivity, PatternHunter is able to find homologies between sequences as large as human chromosomes, in mere hours on a desktop. Availability: PatternHunter is available at http: www. bioinformaticssolutions.com, as a commercial package. It runs on all platforms that support Java. PatternHunter technology is being patented; commercial use requires a license from BSI, while non-commercial use will be free.

[5]: A new approach to rapid sequence comparison, basic local alignment search tool (BLAST), directly approximates alignments that optimize a measure of local similarity, the maximal segment pair (MSP) score. Recent mathematical results on the stochastic properties of MSP scores allow an analysis of the performance of this method as well as the statistical significance of alignments it generates. The basic algorithm is simple and robust; it can be implemented in a number of ways and applied in a variety of contexts including straight-forward DNA and protein sequence database searches, motif searches, gene identification searches, and in the analysis of multiple regions of similarity in long DNA sequences. In addition to its flexibility and tractability to mathematical analysis, BLAST is an order of magnitude faster than existing sequence comparison tools of comparable sensitivity.

[6]: 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.

[7]: We have developed three computer programs for comparisons of protein and DNA sequences. They can be used to search sequence data bases, evaluate similarity scores, and identify periodic structures based on local sequence similarity. The FASTA program is a more sensitive derivative of the FASTP program, which can be used to search protein or DNA sequence data bases and can compare a protein sequence to a DNA sequence data base by translating the DNA data base as it is searched. FASTA includes an additional step in the calculation of the initial pairwise similarity score that allows multiple regions of similarity to be joined to increase the score of related sequences. The RDF2 program can be used to evaluate the significance of similarity scores using a shuffling method that preserves local sequence composition. The LFASTA program can display all the regions of local similarity between two sequences with scores greater than a threshold, using the same scoring parameters and a similar alignment algorithm; these local similarities can be displayed as a "graphic matrix" plot or as individual alignments. In addition, these programs have been generalized to allow comparison of DNA or protein sequences based on a variety of alternative scoring matrices.

[8]: The BLAST programs are widely used tools for searching protein and DNA databases for sequence similarities. For protein comparisons, a variety of definitional, algorithmic and statistical refinements described here permits the execution time of the BLAST programs to be decreased substantially while enhancing their sensitivity to weak similarities. A new criterion for triggering the extension of word hits, combined with a new heuristic for generating gapped alignments, yields a gapped BLAST program that runs at approximately three times the speed of the original. In addition, a method is introduced for automatically combining statistically significant alignments produced by BLAST into a position-specific score matrix, and searching the database using this matrix. The resulting Position-Specific Iterated BLAST (PSIBLAST) program runs at approximately the same speed per iteration as gapped BLAST, but in many cases is much more sensitive to weak but biologically relevant sequence similarities. PSI-BLAST is used to uncover several new and interesting members of the BRCT superfamily.

[9]: 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.

[10]: We present improved techniques for finding homologous regions in DNA and protein sequences. Our approach focuses on the core regions of a local pairwise alignment; we suggest new ways to characterize these regions that allow marked improvements in both specificity and sensitivity over existing techniques for sequence alignment. For any such characterization, which we call a vector seed, we give an efficient algorithm that estimates the specificity and sensitivity of that seed under reasonable probabilistic models of sequence. We also characterize the probability of a match when an alignment is required to have multiple hits before it is detected. Our extensions fit well with existing approaches to sequence alignment, while still offering substantial improvement in runtime and sensitivity, particularly for the important problem of identifying matches between homologous coding DNA sequences.

[11]: We propose a general approach to compute the seed sensitivity, that can be applied to different definitions of seeds. It treats separately three components of the seed sensitivity problem--a set of target alignments, an associated probability distribution, and a seed model--that are specified by distinct finite automata. The approach is then applied to a new concept of subset seeds for which we propose an efficient automaton construction. Experimental results confirm that sensitive subset seeds can be efficiently designed using our approach, and can then be used in similarity search producing better results than ordinary spaced seeds.
Since exact local alignment is too slow in practice, most heuristic homology search algorithms are based on a two-phase filtration technique [7] [5] [8] [3] . 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. [4] 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 [7] [5] , spaced seeds [4] [9] [6] [1] , subset seeds [11] , vector seeds [10] , and indel seeds [2] .