Nuclear magnetic resonance (NMR) spectroscopy allows scientists to study protein structure, dynamics and interactions in solution. A necessary first step for such applications is determining the resonance assignment, mapping spectral data to atoms and residues in the primary sequence. Automated resonance assignment algorithms rely on information regarding connectivity (e.g., through-bond atomic interactions) and amino acid type, typically using the former to determine strings of connected residues and the latter to map those strings to positions in the primary sequence. Significant ambiguity exists in both connectivity and amino acid type information. This paper focuses on the information content available in connectivity alone and develops a novel random-graph theoretic framework and algorithm for connectivity-driven NMR sequential assignment. Our random graph model captures the structure of chemical shift degeneracy, a key source of connectivity ambiguity. We then give a simple and natural randomized algorithm for finding optimal assignments as sets of connected fragments in NMR graphs. The algorithm naturally and efficiently reuses substrings while exploring connectivity choices; it overcomes local ambiguity by enforcing global consistency of all choices. By analyzing our algorithm under our random graph model, we show that it can provably tolerate relatively large ambiguity while still giving expected optimal performance in polynomial time. We present results from practical applications of the algorithm to experimental datasets from a variety of proteins and experimental set-ups. We demonstrate that our approach is able to overcome significant noise and local ambiguity in identifying significant fragments of sequential assignments. Key words: nuclear magnetic resonance (NMR) spectroscopy, automated sequential resonance assignment, random graph model, randomized algorithm, Hamiltonian path. 1.

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We develop an iterative relaxation algorithm, called RIBRA, for NMR protein backbone assignment. RIBRA applies nearest neighbor and weighted maximum independent set algorithms to solve the problem. To deal with noisy NMR spectral data, RIBRA is executed in an iterative fashion based on the quality of spectral peaks. We first produce spin system pairs using the spectral data without missing peaks, then the data group with one missing peak, and finally, the data group with two missing peaks. We test RIBRA on two real NMR datasets: hbSBD and hbLBD, and perfect BMRB data (with 902 proteins) and four synthetic BMRB data which simulate four kinds of errors. The accuracy of RIBRA on hbSBD and hbLBD are 91.4 % and 83.6%, respectively. The average accuracy of RIBRA on perfect BMRB datasets is 98.28%, and 98.28%, 95.61%, 98.16 % and 96.28 % on four kinds of synthetic datasets, respectively.

Consider a truck running along a road. It picks up a load Li at point βi and delivers it at αi, carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L1,..., Ln is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads Li with coordinates (αi, βi) and the distance from Li to Lj is given by � βj αi f(x)dx if βj ≥ αi and by � αi βj g(x)dx if βj < αi. This case of TSP is polynomially solvable with significant real-world applications. Gilmore and Gomory obtained a polynomial time solution for this TSP [5]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [9]. We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory’s polynomial time algorithm for the TSP. We achieve an approximation ratio of ∀x. Note that when f(x) = g(x), the approximation ratio is 3. (2 + γ) where γ ≥ f(x) g(x)

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