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The study of labeling graphs exposed to various distance constraints is motivated by the problem of minimizing the span of non-interfering frequencies assigned to radio transmitters. However, fuzzy labeling models yield more precision, flexibility and compatibility to the system compared to the classical models. In this paper we show that whether any simple graph is fuzzy magic labelizing, by considering the concept of fuzzy magic labeling of graphs. In fact, we prove that every connected graph is a fuzzy magic labelizing graph. Finally, we give some applications for fuzzy magic labeling graphs.

We further develop our phi-function technique for solving Cutting and Packing problems. Here we introduce quasi-phi-functions for an analytical description of non-overlapping and containment constraints for 2D- and 3D-objects which can be continuously rotated and translated. These new functions can work well for various types of objects, such as ellipses, for which ordinary phi-functions are too complicated or have not been constructed yet. We also define normalized quasi-phi-functions and pseudonormalized quasi-phi-functions for modeling distance constraints. To show the advantages of our new quasi-phi-functions we apply them to the problem of placing a given collection of ellipses into a rectangular container of minimal area. We use radical free quasi-phi-functions to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. We present computational results that compare favourably with those published elsewhere recently.

This paper presents a collision avoidance (CA) algorithm for cooperative unmanned aerial vehicles (UAVs) sharing three-dimensional airspace. The method based on geometric optimization model aims to provide feasible optimal trajectory for the selected UAV, with a local optimization scope at operational level. It generates optimal flight trajectory by the objective function (the integration equation of distance, time and track adjustment costs) in response to a set of restrictions (performance, state and distance constraints) reducing the solution space. The CA maneuver has been validated with various simulations, owning such advantages as optimization spending a minimal cost, robustness considering the global traffic situation, scalability possessing explicit coordinates of waypoints and efficiency in implementing different tests of tuning parameters.

Multiprotein complexes, rather than individual proteins, make up a large part of the biological macromolecular machinery of a cell. Understanding the structure and organization of these complexes is critical to understanding cellular function. Chemical cross-linking coupled with mass spectrometry is emerging as a complementary technique to traditional structural biology methods and can provide low-resolution structural information for a multitude of purposes, such as distance constraints in computational modeling of protein complexes. In this review, we discuss the experimental considerations for successful application of chemical cross-linking-mass spectrometry in biological studies and highlight three examples of such studies from the recent literature. These examples (as well as many others) illustrate the utility of a chemical cross-linking-mass spectrometry approach in facilitating structural analysis of large and challenging complexes.

This paper introduces the concept of radical-free pseudonormalized Φ-functions, which allows one to describe constraints on minimum and maximum allowable distances between two-dimensional φ-objects. Translations and rotations of φ-objects in a two-dimensional Euclidean space are allowable. The theorem on the existence of a radical-free pseudonormalized Φ-function for a pair of arbitrary-shaped φ-objects whose frontiers are formed by the union of line segments and circular arcs is formulated. An efficient algorithm is proposed to derive pseudonormalized Φ-functions.

Three-dimensional (3D) structure determination of proteins is benefitted by long-range distance constraints comprising the methyl groups, which constitute the hydrophobic core of proteins. However, in methyl groups (of Ala, Ile, Leu, Met, Thr and Val) there is a significant overlap of C-13 and H-1 chemical shifts. Such overlap can be resolved using the recently proposed (3,2)D HCCH-COSY, a G-matrix Fourier transform (GFT) NMR based experiment, which facilitates editing of methyl groups into distinct spectral regions by combining their C-13 chemical shifts with that of the neighboring, directly attached, C-13 nucleus. Using this principle, we present three GFT experiments: (a) (4,3)D NOESY-HCCH, (b) (4,3)D H-1-TOCSY-HCCH and (c) (4,3)D C-13-TOCSY-HCCH. These experiments provide unique 4D spectral information rapidly with high sensitivity and resolution for side-chain resonance assignments and NOE analysis of methyl groups. This is exemplified by (4,3)D NOESY-HCCH data acquired for 17.9 kDa non-deuterated cytosolic human J-protein co-chaperone, which provided crucial long-range distance constraints for its 3D structure determination.

The paper investigates a capacitated vehicle routing problem with two objectives: (1) minimization of total travel cost and (2) minimization of the length of the longest route. We present algorithmic variants for the exact determination of the Pareto-optimal solutions of this bi-objective problem. Our approach is based on the adaptive e-constraint method. For solving the resulting single-objective subproblems, we apply a branch-and-cut technique, using (among others) a novel implementation of Held-Karp-type bounds. Incumbent solutions are generated by means of a single-objective genetic algorithm and, alternatively, by the multi-objective NSGA-II algorithm. Experimental results for a benchmark of 54 test instances from the TSPLIB are reported.

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in \( \mathbb{E}^d \) , which are centers of balls Bd(pi, ri) and Bd(qi, ri) of radius ri, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way.

We introduce a geometric buildup approach to the distance geometry problem in protein modeling, and discuss the necessary and sufficient conditions on the distances for rigid or unique determination of a protein structure. We describe a new buildup algorithm for determining protein structures rigidly instead of uniquely. The algorithm requires even fewer distance constraints than the general buildup algorithm. We present the test results from applying the algorithm to determining the protein structures with varying degrees of availability of the distances, and show that the new development increases the modeling ability of the geometric buildup method even more while retaining much of the computational feasibility of the method.

The information required for chemical shift assignments in large deuterated proteins was investigated using a Monte Carlo approach (Hitchens et al., 2002). In particular, the consequences of missing amide resonances on the reliability of assignments derived from C alpha and CO or from C alpha and C beta chemical shifts was investigated. Missing amide resonances reduce both the number of correct assignments as well as the confidence in these assignments. More significantly, a number of undetectable errors can arise when as few as 9% of the amide resonances are missing from the spectra. However, the use of information from residue specific labeling as well as local and long-range distance constraints improves the reliability and extent of assignment. It is also shown that missing residues have only a minor effect on the assignment of protein-ligand complexes using C alpha and CO chemical shifts and C alpha inter-residue connectivity, provided that the known chemical shifts of the unliganded protein are utilized in the assignment process.

In this paper we propose PSICO (Processing Structural Information with Constraint programming and Optimisation) as a constraint-based approach to determining protein structures compatible with distance constraints obtained from Nuclear Magnetic Resonance (NMR) data. We compare the performance of our proposed algorithm with DYANA (“Dynamics algorithm for NMR applications”) an existing commercial application based on simulated annealing. On a test case with experimental data on the dimeric protein Desulforedoxin, the method proposed here supplied similar results in less than 10 minutes compared to approximately 10 hours of computation time for DYANA. Although the quality of results can still be improved, this shows that CP technology can greatly reduce computation time, a major advantage because structural NMR technique generally demands multiple runs of structural computation.

High-resolution structural information on RNA and its functionally important complexes with proteins is dramatically underrepresented compared with proteins but is urgently needed for understanding cellular processes at the molecular and atomic level. Here we present an EPR-based protocol to help solving large RNA and protein-RNA complex structures in solution by providing long-range distance constraints between rigid fragments. Using enzymatic ligation of smaller RNA fragments, large doubly spin-labelled RNAs can be obtained permitting the acquisition of long distance distributions (>80 A) within a large protein-RNA complex. Using a simple and fast calculation in torsion angle space of the spin-label distributions with the program CYANA, we can derive simple distance constraints between the spin labels and use them together with short-range distance restraints derived from NMR to determine the structure of a 70 kDa protein-RNA complex composed of three subcomplexes.

Protein docking algorithms can be used to study the driving forces and reaction mechanisms of docking processes. They are also able to speed up the lengthy process of experimental structure elucidation of protein complexes by proposing potential structures. In this paper, we are discussing a variant of the protein-protein docking problem, where the input consists of the tertiary structures of proteins A and B plus an unassigned one-dimensional 1H-NMR spectrum of the complex AB. We present a new scoring function for evaluating and ranking potential complex structures produced by a docking algorithm. The scoring function computes a `theoretical' 1H-NMR spectrum for each tentative complex structure and subtracts the calculated spectrum from the experimental one. The absolute areas of the difference spectra are then used to rank the potential complex structures. In contrast to formerly published approaches (e.g. [Morelli et al. (2000) Biochemistry, 39, 2530–2537]) we do not use distance constraints (intermolecular NOE constraints). We have tested the approach with four protein complexes whose three-dimensional structures are stored in the PDB data bank [Bernstein et al. (1977)] and whose 1H-NMR shift assignments are available from the BMRB database. The best result was obtained for an example, where all standard scoring functions failed completely. Here, our new scoring function achieved an almost perfect separation between good approximations of the true complex structure and false positives.

We study the performance of the {\dgs} code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The {\dgs} code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds. Our computational results show that protein structures can be determined by solving a distance geometry problem with {\dgs} and that the approach based on {\dgs} is significantly more reliable and efficient than multi-starts with an optimization code.

LetV = {v 1,⋯, v n } be a set ofn points on the real line (existing facilities). The problem considered is to locatep new point facilities,F 1,⋯, F p , inV while satisfying distance constraints between pairs of existing and new facilities and between pairs of new facilities. Fori = 1, ⋯, p, j = 1, ⋯, n, the cost of locatingF i at pointv j isc ij . The objective is to minimize the total cost of setting up the new facilities. We present anO(p 3 n 2 logn) algorithm to solve the model.

Three different approaches to improve tertiary fold prediction using the genetic algorithm are discussed: (i) Refinement of the search strategy, (ii) combination of prediction and experiment and (iii) inclusion of experimental data as selection criteria into the genetic algorithm. Examples from our current work are presented for refined strategies against crowding in solution space, definition of domain boundaries and secondary structure in combination with experiment, and direct incorporation of experimentally known distance constraints into the fitness function.

A new program, ASNO (‘ASsign NOes’), for computer-supported NOE cross-peak assignments is described. ASNO is used for structure refinement in several rounds of NOESY cross-peak assignments and 3D structure calculations, where the preliminary structures are used as a reference to resolve ambiguities in NOE assignments which are otherwise based on the chemical shifts available from the sequence-specific resonance assignments. The practical use of ASNO for proteins is illustrated with the structure determination of Dendrotoxin K from Dendroaspis polylepis polylepis.