Handling long targets and errors in sequencing by hybridization.

Sequencing by hybridization (SBH) is a DNA sequencing technique, in which the sequence is reconstructed using its k-mer content. This content, which is called the spectrum of the sequence, is obtained by hybridization to a universal DNA array. Standard universal arrays contain all k-mers for some fixed k, typically 8 to 10. Currently, in spite of its promise and elegance, SBH is not competitive with standard gel-based sequencing methods. This is due to two main reasons: lack of tools to handle realistic levels of hybridization errors and an inherent limitation on the length of uniquely reconstructible sequence by standard universal arrays. In this paper, we deal with both problems. We introduce a simple polynomial reconstruction algorithm which can be applied to spectra from standard arrays and has provable performance in the presence of both false negative and false positive errors. We also propose a novel design of chips containing universal bases that differs from the one proposed by Preparata et al. (1999). We give a simple algorithm that uses spectra from such chips to reconstruct with high probability random sequences of length lower only by a squared log factor compared to the information theoretic bound. Our algorithm is very robust to errors and has a provable performance even if there are both false negative and false positive errors. Simulations indicate that its sensitivity to errors is also very small in practice.     

Optimal sequencing by hybridization in rounds.

Sequencing by hybridization (SBH) is a method for sequencing DNA. The Watson-Crick complementarity of DNA can be used to determine whether the DNA contains an oligonucleotide substring. A large number of oligonucleotides can be arranged on an array (SBH chip). A combinatorial method is used to construct the sequence from the collection of probes that occur in it. We develop an idea of Margaritis and Skiena and propose an algorithm that uses a series of small SBH chips to sequence long strings. The total number of probes used by our method matches the information theoretical lower bound up to a constant factor.     

Optimal reconstruction of a sequence from its probes.

An important combinatorial problem, motivated by DNA sequencing in molecular biology, is the reconstruction of a sequence over a small finite alphabet from the collection of its probes (the sequence spectrum), obtained by sliding a fixed sampling pattern over the sequence. Such construction is required for Sequencing-by-Hybridization (SBH), a novel DNA sequencing technique based on an array (SBH chip) of short nucleotide sequences (probes). Once the sequence spectrum is biochemically obtained, a combinatorial method is used to reconstruct the DNA sequence from its spectrum. Since technology limits the number of probes on the SBH chip, a challenging combinatorial question is the design of a smallest set of probes that can sequence an arbitrary DNA string of a given length. We present in this work a novel probe design, crucially based on the use of universal bases [bases that bind to any nucleotide (Loakes and Brown, 1994)] that drastically improves the performance of the SBH process and asymptotically approaches the information-theoretic bound up to a constant factor. Furthermore, the sequencing algorithm we propose is substantially simpler than the Eulerian path method used in previous solutions of this problem.     

On realizing shapes in the theory of RNA neutral networks

It is known (Reidys et al., 1997b. Bull. Math. Biol. 59(2), 339-397) that for any two secondary structures S,S' there exists an RNA sequence compatible with both, and that this result does not extend to more than two secondary structures. Indeed, a simple formula for the number of RNA sequences compatible with secondary structures S,S' plays a role in the algorithms of Flamm et al. (2001. RNA 7, 254-265) and of Abfalter et al. (2003. Proceedings of the German Conference on Bioinformatics, ) to design an RNA switch. Here we show that a natural extension of this problem is NP-complete. Unless P=NP, there is no polynomial time algorithm, which when given secondary structures S1,...,S(k), for k4, determines the least number of positions, such that after removal of all base pairs incident to these positions there exists an RNA nucleotide sequence compatible with the given secondary structures. We also consider a restricted version of this problem with a "fixed maximum" number of possible stars and show that it has a simple polynomial time solution.     

Deconvolving sequence variation in mixed DNA populations.

We present an original approach to identifying sequence variants in a mixed DNA population from sequence trace data. The heart of the method is based on parsimony: given a wildtype DNA sequence, a set of observed variations at each position collected from sequencing data, and a complete catalog of all possible mutations, determine the smallest set of mutations from the catalog that could fully explain the observed variations. The algorithmic complexity of the problem is analyzed for several classes of mutations, including block substitutions, single-range deletions, and single-range insertions. The reconstruction problem is shown to be NP-complete for single-range insertions and deletions, while for block substitutions, single character insertion, and single character deletion mutations, polynomial time algorithms are provided. Once a minimum set of mutations compatible with the observed sequence is found, the relative frequency of those mutations is recovered by solving a system of linear equations. Simulation results show the algorithm successfully deconvolving mutations in p53 known to cause cancer. An extension of the algorithm is proposed as a new method of high throughput screening for single nucleotide polymorphisms by multiplexing DNA.     

Is optimal solution of every NP-complete or NP-hard problem determined from its characteristic for DNA-based computing

Cook's Theorem [Cormen, T.H., Leiserson, C.E., Rivest, R.L., 2001. Introduction to Algorithms, second ed., The MIT Press; Garey, M.R., Johnson, D.S., 1979. Computer and Intractability, Freeman, San Fransico, CA] is that if one algorithm for an NP-complete or an NP-hard problem will be developed, then other problems will be solved by means of reduction to that problem. Cook's Theorem has been demonstrated to be correct in a general digital electronic computer. In this paper, we first propose a DNA algorithm for solving the vertex-cover problem. Then, we demonstrate that if the size of a reduced NP-complete or NP-hard problem is equal to or less than that of the vertex-cover problem, then the proposed algorithm can be directly used for solving the reduced NP-complete or NP-hard problem and Cook's Theorem is correct on DNA-based computing. Otherwise, a new DNA algorithm for optimal solution of a reduced NP-complete problem or a reduced NP-hard problem should be developed from the characteristic of NP-complete problems or NP-hard problems.     

An algorithm for approximate tandem repeats.

A perfect single tandem repeat is defined as a nonempty string that can be divided into two identical substrings, e.g., abcabc. An approximate single tandem repeat is one in which the substrings are similar, but not identical, e.g., abcdaacd. In this paper we consider two criterions of similarity: the Hamming distance (k mismatches) and the edit distance (k differences). For a string S of length n and an integer k our algorithm reports all locally optimal approximate repeats, r = umacro ?, for which the Hamming distance of umacro and ? is at most k, in O(nk log (n/k)) time, or all those for which the edit distance of umacro and ? is at most k, in O(nk log k log (n/k)) time. This paper concentrates on a more general type of repeat called multiple tandem repeats. A multiple tandem repeat in a sequence S is a (periodic) substring r of S of the form r = u(a)u', where u is a prefix of r and u' is a prefix of u. An approximate multiple tandem repeat is a multiple repeat with errors; the repeated subsequences are similar but not identical. We precisely define approximate multiple repeats, and present an algorithm that finds all repeats that concur with our definition. The time complexity of the algorithm, when searching for repeats with up to k errors in a string S of length n, is O(nka log (n/k)) where a is the maximum number of periods in any reported repeat. We present some experimental results concerning the performance and sensitivity of our algorithm. The problem of finding repeats within a string is a computational problem with important applications in the field of molecular biology. Both exact and inexact repeats occur frequently in the genome, and certain repeats occurring in the genome are known to be related to diseases in the human.     

An algorithm for the DNA sequence generation from k-tuple word contents of the minimal number of random fragments

An algorithm is described for generation of the long sequence written in a four letter alphabet from the constituent k-tuple words in the minimal number of separate, randomly defined fragments of the starting sequence. It is primarily intended for use in sequencing by hybridization (SBH) process- a potential method for sequencing human genome DNA (Drmanac et al., Genomics 4, pp. 114-128, 1989). The algorithm is based on the formerly defined rules and informative entities of the linear sequence. The algorithm requires neither knowledge on the number of appearances of a given k-tuple in sequence fragments, nor the information on which k-tuple words are on the ends of a fragment. It operates with the mixed content of k-tuples of the various lengths. The concept of the algorithm enables operations with the k-tuple sets containing false positive and false negative k-tuples. The content of the false k-tuples primarily affects the completeness of the generated sequence, and its correctness in the specific cases only. The algorithm can be used for the optimization of SBH parameters in the simulation experiments, as well as for the sequence generation in the real SBH experiments on the genomic DNA.     

Sequencing-by-hybridization at the information-theory bound: an optimal algorithm.

In a recent paper (Preparata et aL, 1999) we introduced a novel probing scheme for DNA sequencing by hybridization (SBH). The new gapped-probe scheme combines natural and universal bases in a well-defined periodic pattern. It has been shown (Preparata et al, 1999) that the performance of the gapped-probe scheme (in terms of the length of a sequence that can be uniquely reconstructed using a given size library of probes) is significantly better than the standard scheme based on oligomer probes. In this paper we present and analyze a new, more powerful, sequencing algorithm for the gapped-probe scheme. We prove that the new algorithm exploits the full potential of the SBH technology with high-confidence performance that comes within a small constant factor (about 2) of the information-theory bound. Moreover, this performance is achieved while maintaining running time linear in the target sequence length.     

An Algorithm for the DNA Sequence Generation from k-Tuple Word Contents of the Minimal Number of Random Fragments

Abstract

An algorithm is described for generation of the long sequence written in a four letter alphabet from the constituent k-tuple words in the minimal number of separate, randomly defined fragments of the starting sequence. It is primarily intended for use in sequencing by hybridization (SBH) process- a potential method for sequencing human genome DNA (Drmanac et al., Genomics 4, pp. 114–128, 1989). The algorithm is based on the formerly defined rules and informative entities of the linear sequence.

The algorithm requires neither knowledge on the number of appearances of a given k-tuple in sequence fragments, nor the information on which k-tuple words are on the ends of a fragment. It operates with the mixed content of k-tuples of the various lengths. The concept of the algorithm enables operations with the k-tuple sets containing false positive and false negative k-tuples. The content of the false k-tuples primarily affects the completeness of the generated sequence, and its correctness in the specific cases only. The algorithm can be used for the optimization of SBH parameters in the simulation experiments, as well as for the sequence generation in the real SBH experiments on the genomic DNA.     

Reconstruction of DNA sequencing by hybridization

MOTIVATION: It is widely recognized that the hybridization process is prone to errors and that the future of DNA sequencing by hybridization is predicated on the ability to successfully cope with such errors. However, the occurrence of hybridization errors results in the computational difficulty of the reconstruction of DNA sequencing by hybridization. The reconstruction problem of DNA sequencing by hybridization with errors is a strongly NP-hard problem. So far the problem has not been solved well. RESULTS: In this paper, a new approach is presented to solve the reconstruction problem of DNA sequencing by hybridization, which realizes the computational part of the SBH experiment. The proposed algorithm accepts both the negative and positive errors. The computational experiments show that the algorithm behaves satisfactorily, especially for the case with k-tuple repetitions and positive errors.     

Predicting the accuracy of multiple sequence alignment algorithms by using computational intelligent techniques

Multiple sequence alignments (MSAs) have become one of the most studied approaches in bioinformatics to perform other outstanding tasks such as structure prediction, biological function analysis or next-generation sequencing. However, current MSA algorithms do not always provide consistent solutions, since alignments become increasingly difficult when dealing with low similarity sequences. As widely known, these algorithms directly depend on specific features of the sequences, causing relevant influence on the alignment accuracy. Many MSA tools have been recently designed but it is not possible to know in advance which one is the most suitable for a particular set of sequences. In this work, we analyze some of the most used algorithms presented in the bibliography and their dependences on several features. A novel intelligent algorithm based on least square support vector machine is then developed to predict how accurate each alignment could be, depending on its analyzed features. This algorithm is performed with a dataset of 2180 MSAs. The proposed system first estimates the accuracy of possible alignments. The most promising methodologies are then selected in order to align each set of sequences. Since only one selected algorithm is run, the computational time is not excessively increased.     

Fast exact algorithms for the closest string and substring problems with application to the planted (L, d)-motif model

We present two parameterized algorithms for the closest string problem. The first runs in O(nL + nd · 17.97d) time for DNA strings and in O(nL + nd · 61.86d) time for protein strings, where n is the number of input strings, L is the length of each input string, and d is the given upper bound on the number of mismatches between the center string and each input string. The second runs in O(nL + nd · 13.92d) time for DNA strings and in O(nL + nd · 47.21d) time for protein strings. We then extend the first algorithm to a new parameterized algorithm for the closest substring problem that runs in O((n - 1)m2(L + d · 17.97d · m[log2(d+1)])) time for DNA strings and in O((n - 1)m2(L + d · 61.86d · m[log2(d+1)])) time for protein strings, where n is the number of input strings, L is the length of the center substring, L - 1 + m is the maximum length of a single input string, and d is the given upper bound on the number of mismatches between the center substring and at least one substring of each input string. All the algorithms significantly improve the previous bests. To verify experimentally the theoretical improvements in the time complexity, we implement our algorithm in C and apply the resulting program to the planted (L, d)-motif problem proposed by Pevzner and Sze in 2000. We compare our program with the previously best exact program for the problem, namely PMSPrune (designed by Davila et al. in 2007). Our experimental data show that our program runs faster for practical cases and also for several challenging cases. Our algorithm uses less memory too.     

A stochastic online algorithm for unloading boxes from a conveyor line

This article discusses the problem of unloading a sequence of boxes from a single conveyor line with a minimum number of moves. The problem under study is efficiently solvable with dynamic programming if the complete sequence of boxes is known in advance. In practice, however, the problem typically occurs in a real-time setting where the boxes are simultaneously placed on and picked from the conveyor line. Moreover, a large part of the sequence is often not visible. As a result, only a part of the sequence is known when deciding which boxes to move next. We develop an online algorithm that evaluates the quality of each possible move with a scenario-based stochastic method. Two versions of the algorithm are analyzed: in one version, the quality of each scenario is measured with an exact method, while a heuristic technique is applied in the second version. We evaluate the performance of the proposed algorithms using extensive computational experiments and establish a simple policy for determining which version to choose for specific problems. Numerical results show that the proposed approach consistently provides high-quality results, and compares favorably with the best known deterministic online algorithms. Indeed, the new approach typically provides results with relative gaps of 1–5% to the optimum, which is about 20–80% lower than those obtained with the best deterministic approach.     

An O(N2) algorithm for discovering optimal Boolean pattern pairs

We consider the problem of finding the optimal combination of string patterns, which characterizes a given set of strings that have a numeric attribute value assigned to each string. Pattern combinations are scored based on the correlation between their occurrences in the strings and the numeric attribute values. The aim is to find the combination of patterns which is best with respect to an appropriate scoring function. We present an O(N²) time algorithm for finding the optimal pair of substring patterns combined with Boolean functions, where N is the total length of the sequences. The algorithm looks for all possible Boolean combinations of the patterns, e.g., patterns of the form p nland notq, which indicates that the pattern pair is considered to occur in a given string s, if p occurs in s, and q does not occur in s. An efficient implementation using suffix arrays is presented, and we further show that the algorithm can be adapted to find the best k-pattern Boolean combination in O(N^k) time. The algorithm is applied to mRNA sequence data sets of moderate size combined with their turnover rates for the purpose of finding regulatory elements that cooperate, complement, or compete with each other in enhancing and/or silencing mRNA decay     

The complexity of the gapped consecutive-ones property problem for matrices of bounded maximum degree

The Gapped Consecutive-Ones Property (C1P) Problem, or the (k, δ)-C1P Problem is: given a binary matrix M and integers k and δ, decide if the columns of M can be ordered such that each row contains at most k blocks of 1's, and no two neighboring blocks of 1's are separated by a gap of more than δ 0's. This problem was introduced by Chauve et al. ( 2009b ). The classical polynomial-time solvable C1P Problem is equivalent to the (1, 0)-C1P problem. It has been shown that, for every unbounded or bounded k ≥ 2 and unbounded or bounded δ ≥ 1, except when (k, δ) = (2, 1), the (k, δ)-C1P Problem is NP-complete (Maňuch et al., 2011 ; Goldberg et al., 1995 ). In this article, we study the Gapped C1P Problem with a third parameter d, namely the bound on the maximum number of 1's in any row of M, or the bound on the maximum degree of M. This is motivated by the reconstruction of ancestral genomes (Ma et al., 2006 ; Chauve and Tannier, 2008 ), where, in binary matrices obtained from the experiments of Chauve and Tannier ( 2008 ), we have observed that the majority of the rows have low degree, while each high degree row contains many rows of low degree. The (d, k, δ)-C1P Problem has been shown to be polynomial-time solvable when all three parameters are fixed (Chauve et al., 2009b ). Since fixing d also fixes k (k ≤ d), the only case left to consider is the case when δ is unbounded, or the (d, k, ∞)-C1P Problem. Here we show that for every d > k ≥ 2, the (d, k, ∞)-C1P Problem is NP-complete.     

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways. In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present if and only if it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to “efficiently” solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter . Nevertheless, here it is proved that the probability of requiring a value of to obtain a solution for a random graph decreases exponentially: , making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.     

Efficient enumeration of phylogenetically informative substrings.

We study the problem of enumerating substrings that are common amongst genomes that share evolutionary descent. For example, one might want to enumerate all identical (therefore conserved) substrings that are shared between all mammals and not found in non-mammals. Such collection of substrings may be used to identify conserved subsequences or to construct sets of identifying substrings for branches of a phylogenetic tree. For two disjoint sets of genomes on a phylogenetic tree, a substring is called a tag if it is found in all of the genomes of one set and none of the genomes of the other set. We present a near-linear time algorithm that finds all tags in a given phylogeny; and a sublinear space algorithm (at the expense of running time) that is more suited for very large data sets. Under a stochastic model of evolution, we show that a simple process of tag-generation essentially captures all possible ways of generating tags. We use this insight to develop a faster tag discovery algorithm with a small chance of error. However, since tags are not guaranteed to exist in a given data set, we generalize the notion of a tag from a single substring to a set of substrings. We present a linear programming-based approach for finding approximate generalized tag sets. Finally, we use our tag enumeration algorithm to analyze a phylogeny containing 57 whole microbial genomes. We find tags for all nodes in the phylogeny except the root for which we find generalized tag sets.     

The position end-set tree: A small automaton for word recognition in biological sequences

We consider the basic function which locates a specific stringof symbols within a longer sequence. When one is expecting todo many substring searches it is worthwhile to build an auxiliaryindex to the sequence to aid in the search. We propose a methodto generate a compact index that can be viewed as a small (partial)deterministic finite automaton recognizing the subword structureof a sequence. We present an algorithm for its constructionon-line in linear time. Such a data structure permits the efficientlocalization of subwords in a sequence and can be used in thedevelopment of interactive sequence analysis software.     

SST: an algorithm for finding near-exact sequence matches in time proportional to the logarithm of the database size

MOTIVATION: Searches for near exact sequence matches are performed frequently in large-scale sequencing projects and in comparative genomics. The time and cost of performing these large-scale sequence-similarity searches is prohibitive using even the fastest of the extant algorithms. Faster algorithms are desired. RESULTS: We have developed an algorithm, called SST (Sequence Search Tree), that searches a database of DNA sequences for near-exact matches, in time proportional to the logarithm of the database size n. In SST, we partition each sequence into fragments of fixed length called 'windows' using multiple offsets. Each window is mapped into a vector of dimension 4(k) which contains the frequency of occurrence of its component k-tuples, with k a parameter typically in the range 4-6. Then we create a tree-structured index of the windows in vector space, with tree-structured vector quantization (TSVQ). We identify the nearest neighbors of a query sequence by partitioning the query into windows and searching the tree-structured index for nearest-neighbor windows in the database. When the tree is balanced this yields an O(logn) complexity for the search. This complexity was observed in our computations. SST is most effective for applications in which the target sequences show a high degree of similarity to the query sequence, such as assembling shotgun sequences or matching ESTs to genomic sequence. The algorithm is also an effective filtration method. Specifically, it can be used as a preprocessing step for other search methods to reduce the complexity of searching one large database against another. For the problem of identifying overlapping fragments in the assembly of 120 000 fragments from a 1.5 megabase genomic sequence, SST is 15 times faster than BLAST when we consider both building and searching the tree. For searching alone (i.e. after building the tree index), SST 27 times faster than BLAST. AVAILABILITY: Request from the authors.