Dekel Tsur

Identification of post-translational modifications by blind search of mass spectra

Most tandem mass spectrometry (MS/MS) database search algorithms perform a restrictive search that takes into account only a few types of post-translational modifications (PTMs) and ignores all others. We describe an unrestrictive PTM search algorithm, MS-Alignment, that searches for all types of PTMs at once in a blind mode, that is, without knowing which PTMs exist in nature. Blind PTM identification makes it possible to study the extent and frequency of different types of PTMs, still an open problem in proteomics. Application of this approach to lens proteins resulted in the largest set of PTMs reported in human crystallins so far. Our analysis of various MS/MS data sets implies that the biological phenomenon of modification is much more widespread than previously thought. We also argue that MS-Alignment reveals some uncharacterized modifications that warrant further experimental validation.

Cluster graph modification problems

In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making the fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p>=2; Cluster Deletion is polynomial for p=2 but NP-complete for p>2; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for a variant of Cluster Editing when p=2.

Protein identification by spectral networks analysis

Advances in tandem mass spectrometry (MS/MS) steadily increase the rate of generation of MS/MS spectra. As a result, the existing approaches that compare spectra against databases are already facing a bottleneck, particularly when interpreting spectra of modified peptides. Here we explore a concept that allows one to perform an MS/MS database search without ever comparing a spectrum against a database. We propose to take advantage of spectral pairs, which are pairs of spectra obtained from overlapping (often nontryptic) peptides or from unmodified and modified versions of the same peptide. Having a spectrum of a modified peptide paired with a spectrum of an unmodified peptide allows one to separate the prefix and suffix ladders, to greatly reduce the number of noise peaks, and to generate a small number of peptide reconstructions that are likely to contain the correct one. The MS/MS database search is thus reduced to extremely fast pattern-matching (rather than time-consuming matching of spectra against databases). In addition to speed, our approach provides a unique paradigm for identifying posttranslational modifications by means of spectral networks analysis.

Cluster Graph Modification Problems

In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters.We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p ? 2; Cluster Deletion is polynomial for p = 2 but NP-complete for p > 2; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for Cluster Editing when p = 2.

Sparse RNA folding: Time and space efficient algorithms

The currently fastest algorithm for RNA Single Strand Folding requires O(nZ) time and @Q(n^2) space, where n denotes the length of the input string and Z is a sparsity parameter satisfying n=

Tradeoffs in Worst-Case Equilibria

We investigate the problem of routing traffic through a congested network in an environment of non-cooperative users. We use the worst-case coordination ratio suggested by Koutsoupias and Papadimitriou to measure the performance degradation due to the lack of a centralized traffic regulating authority. We provide a full characterization of the worst-case coordination ratio in the restricted assignment and unrelated parallel links models. In particular, we quantify the tradeoff between the ”negligibility” of the traffic controlled by each user and the coordination ratio. We analyze both pure and mixed strategies systems and identify the range where their performance is similar.

Approximate labelled subtree homeomorphism

Given two undirected trees T and P, the Subtree Homeomorphism Problem is to find whether T has a subtree t that can be transformed into P by removing entire subtrees, as well as repeatedly removing a degree-2 node and adding the edge joining its two neighbors. In this paper we extend the Subtree Homeomorphism Problem to a new optimization problem by enriching the subtree-comparison with node-to-node similarity scores. The new problem, called Approximate Labelled Subtree Homeomorphism (ALSH), is to compute the homeomorphic subtree of T which also maximizes the overall node-to-node resemblance. We describe an O(m^2n/logm+mnlogn) algorithm for solving ALSH on unordered, unrooted trees, where m and n are the number of vertices in P and T, respectively. We also give an O(mn) algorithm for rooted ordered trees and O(mnlogm) and O(mn) algorithms for unrooted cyclically ordered and unrooted linearly ordered trees, respectively.

Large scale sequencing by hybridization

Sequencing by hybridization is a method for reconstructing a DNA sequence based on its k-mer content. This content, called the spectrum of the sequence, can be obtained from hybridization with a universal DNA chip. However, even with a sequencing chip containing all 49 9-mers and assuming no hybridization errors, only about 400 bases-long sequences can be reconstructed unambiguously. Drmanac et al. suggested sequencing long DNA targets by obtaining spectra of many short overlapping fragments of the target, inferring their relative positions along the target and computing spectra of subfragments that are short enough to be uniquely recoverable. Drmanac et al. do not treat the realistic case of errors in the hybridization process. In this paper we study the effect of such errors. We show that the probability of ambiguous reconstruction in the presence of (false negative) errors is close to the probability in the errorless case. More precisely, the ratio between these probabilities is 1 + &Ogr;(p/(1 - p)4 · 1/d) where d is the average distance between neighboring subfragments, and p is the probability of a false negative. We also obtain lower and upper bounds for the probability of unambiguous reconstruction based on errorless spectrum. For realistic chip sizes, these bounds are tighter than those given by Arratia et al. Finally, we report results on simulations with real DNA sequences, showing that even in the presence of 50% false negative errors, a target of cosmid length can be recovered with less than 0.1% miscalled bases.

Faster subtree isomorphism

We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O(~k/sup 1.8//log k\ n) time algorithm for this problem, where k and n are the number of vertices in H and G respectively. This improves over the O(k/sup 1.5/n) algorithms of Chung (1987) and Matula (1978). We also give a randomized (Las Vegas) O(min(k/sup 1.45/n, kn/sup 1.43/))-time algorithm for the decision problem.

Generalized LCS

The Longest Common Subsequence (LCS) is a well studied problem, having a wide range of implementations. Its motivation is in comparing strings. It has long been of interest to devise a similar measure for comparing higher dimensional objects, and more complex structures. In this paper we study the Longest Common Substructure of two matrices and show that this problem is NP-hard. We also study the Longest Common Subforest problem for multiple trees including a constrained version, as well. We show NP-hardness for k>2 unordered trees in the constrained LCS. We also give polynomial time algorithms for ordered trees and prove a lower bound for any decomposition strategy for k trees.