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Concept: Delaunay triangulation


Comparison of the binding sites of proteins is an effective means for predicting protein functions based on their structure information. Despite the importance of this problem and much research in the past, it is still very challenging to predict the binding ligands from the atomic structures of protein binding sites. Here, we designed a new algorithm, TIPSA (Triangulation-based Iterative-closest-point for Protein Surface Alignment), based on the iterative closest point (ICP) algorithm. TIPSA aims to find the maximum number of atoms that can be superposed between two protein binding sites, where any pair of superposed atoms has a distance smaller than a given threshold. The search starts from similar tetrahedra between two binding sites obtained from 3D Delaunay triangulation and uses the Hungarian algorithm to find additional matched atoms. We found that, due to the plasticity of protein binding sites, matching the rigid body of point clouds of protein binding sites is not adequate for satisfactory binding ligand prediction. We further incorporated global geometric information, the radius of gyration of binding site atoms, and used nearest neighbor classification for binding site prediction. Tested on benchmark data, our method achieved a performance comparable to the best methods in the literature, while simultaneously providing the common atom set and atom correspondences.

Concepts: DNA, Proteins, Protein, Prediction, Futurology, Atom, Binding site, Delaunay triangulation


Concentration addition (CA) was proposed as a reasonable default approach for the ecological risk assessment of chemical mixtures. However, CA cannot predict the toxicity of mixture at some effect zones if not all components have definite effective concentrations at the given effect, such as some compounds induce hormesis. In this paper, we developed a new method for the toxicity prediction of various types of binary mixtures, an interpolation method based on the Delaunay triangulation (DT) and Voronoi tessellation (VT) as well as the training set of direct equipartition ray design (EquRay) mixtures, simply IDVequ. At first, the EquRay was employed to design the basic concentration compositions of five binary mixture rays. The toxic effects of single components and mixture rays at different times and various concentrations were determined by the time-dependent microplate toxicity analysis. Secondly, the concentration-toxicity data of the pure components and various mixture rays were acted as a training set. The DT triangles and VT polygons were constructed by various vertices of concentrations in the training set. The toxicities of unknown mixtures were predicted by the linear interpolation and natural neighbor interpolation of vertices. The IDVequ successfully predicted the toxicities of various types of binary mixtures.

Concepts: Chemical substance, Mixture, Chemical compound, Toxicity, Computational geometry, Delaunay triangulation, Voronoi diagram, CGAL


The problem of finding the number and optimal positions of relay nodes for restoring the network connectivity in partitioned Wireless Sensor Networks (WSNs) is Non-deterministic Polynomial-time hard (NP-hard) and thus heuristic methods are preferred to solve it. This paper proposes a novel polynomial time heuristic algorithm, namely, Relay Placement using Space Network Coding (RPSNC), to solve this problem, where Space Network Coding, also called Space Information Flow (SIF), is a new research paradigm that studies network coding in Euclidean space, in which extra relay nodes can be introduced to reduce the cost of communication. Unlike contemporary schemes that are often based on Minimum Spanning Tree (MST), Euclidean Steiner Minimal Tree (ESMT) or a combination of MST with ESMT, RPSNC is a new min-cost multicast space network coding approach that combines Delaunay triangulation and non-uniform partitioning techniques for generating a number of candidate relay nodes, and then linear programming is applied for choosing the optimal relay nodes and computing their connection links with terminals. Subsequently, an equilibrium method is used to refine the locations of the optimal relay nodes, by moving them to balanced positions. RPSNC can adapt to any density distribution of relay nodes and terminals, as well as any density distribution of terminals. The performance and complexity of RPSNC are analyzed and its performance is validated through simulation experiments.

Concepts: Algorithm, Vector space, Problem solving, Computational complexity theory, Delaunay triangulation, Spanning tree protocol, Heuristic, Wireless sensor network


We propose a new analytical method for detecting and computing contacts between atoms in biomolecules. It is based on the alpha shape theory and proceeds in three steps. First, we compute the weighted Delaunay triangulation of the union of spheres representing the molecule. In the second step, the Delaunay complex is filtered to derive the dual complex. Finally, contacts between spheres are collected. In this approach, two atoms i and j are defined to be in contact if their centers are connected by an edge in the dual complex. The contact areas between atom i and its neighbors are computed based on the caps formed by these neighbors on the surface of i; the total area of all these caps is partitioned according to their spherical Laguerre Voronoi diagram on the surface of i. This method is analytical and its implementation in a new program BallContact is fast and robust. We have used BallContact to study contacts in a database of 1551 high resolution protein structures. We show that with this new definition of atomic contacts, we generate realistic representations of the environments of atoms and residues within a protein. In particular, we establish the importance of nonpolar contact areas that complement the information represented by the accessible surface areas. This new method bears similarity to the tessellation methods used to quantify atomic volumes and contacts, with the advantage that it does not require the presence of explicit solvent molecules if the surface of the protein is to be considered. © 2012 Wiley Periodicals, Inc.

Concepts: Molecule, Chemistry, Atom, Computer, Computational geometry, Delaunay triangulation, Voronoi diagram, CGAL


In recent years, more 3D protein structures have become available, which has made the analysis of large molecular structures much easier. There is a strong demand for geometric models for the study of protein-related interactions. Alpha shape and Delaunay triangulation are powerful tools to represent protein structures and have advantages in characterizing the surface curvature and atom contacts. This review presents state-of-the-art applications of alpha shape and Delaunay triangulation in the studies on protein-DNA, protein-protein, protein-ligand interactions and protein structure analysis.

Concepts: Protein structure, Structure, Molecule, Sociology, Differential geometry, Computational geometry, Delaunay triangulation


We propose the first GPU solution to compute the 2D constrained Delaunay triangulation (CDT) of a planar straight line graph (PSLG) consisting of points and edges. There are many existing CPU algorithms to solve the CDT problem in computational geometry, yet there has been no prior approach to solve this problem efficiently using the parallel computing power of the GPU. For the special case of the CDT problem where the PSLG consists of just points, which is simply the normal Delaunay triangulation problem, a hybrid approach using the GPU together with the CPU to partially speed up the computation has already been presented in the literature. Our work, on the other hand, accelerates the entire computation on the GPU. Our implementation using the CUDA programming model on NVIDIA GPUs is numerically robust, and runs up to an order of magnitude faster than the best sequential implementations on the CPU. This result is reflected in our experiment with both randomly generated PSLGs and real-world GIS data having millions of points and edges.

Concepts: Algorithm, Parallel computing, Computer science, Computational geometry, Delaunay triangulation, CUDA, Graphics processing unit, GPGPU


Voronoia4RNA ( is a structural database storing precalculated atomic volumes, atomic packing densities (PDs) and coordinates of internal cavities for currently 1869 RNAs and RNA-protein complexes. Atomic PDs are a measure for van der Waals interactions. Regions of low PD, containing water-sized internal cavities, refer to local structure flexibility or compressibility. RNA molecules build up the skeleton of large molecular machineries such as ribosomes or form smaller flexible structures such as riboswitches. The wealth of structural data on RNAs and their complexes allows setting up representative data sets and analysis of their structural features. We calculated atomic PDs from atomic volumes determined by the Voronoi cell method and internal cavities analytically by Delaunay triangulation. Reference internal PD values were derived from a non-redundant sub-data set of buried atoms. Comparison of internal PD values shows that RNA is more tightly packed than proteins. Finally, the relation between structure size, resolution and internal PD of the Voronoia4RNA entries is discussed. RNA, protein structures and their complexes can be visualized by the Jmol-based viewer Provi. Variations in PD are depicted by a color code. Internal cavities are represented by their molecular boundaries or schematically as balls.

Concepts: Protein, Gene, RNA, Ribosome, Messenger RNA, Atom, Delaunay triangulation, Voronoi diagram


For Wireless Sensor Networks (WSNs), the Voronoi partition of a region is a challenging problem owing to the limited sensing ability of each sensor and the distributed organization of the network. In this paper, an algorithm is proposed for each sensor having a limited sensing range to compute its limited Voronoi cell autonomously, so that the limited Voronoi partition of the entire WSN is generated in a distributed manner. Inspired by Graham’s Scan (GS) algorithm used to compute the convex hull of a point set, the limited Voronoi cell of each sensor is obtained by sequentially scanning two consecutive bisectors between the sensor and its neighbors. The proposed algorithm called the Boundary Scan (BS) algorithm has a lower computational complexity than the existing Range-Constrained Voronoi Cell (RCVC) algorithm and reaches the lower bound of the computational complexity of the algorithms used to solve the problem of this kind. Moreover, it also improves the time efficiency of a key step in the Adjust-Sensing-Radius (ASR) algorithm used to compute the exact Voronoi cell. Extensive numerical simulations are performed to demonstrate the correctness and effectiveness of the BS algorithm. The distributed realization of the BS combined with a localization algorithm in WSNs is used to justify the WSN nature of the proposed algorithm.

Concepts: Algorithm, Computational complexity theory, Computational geometry, Delaunay triangulation, Voronoi diagram, Convex hull, Wireless sensor network, Graham scan


Breathing parameters can be measured by motion capture systems by placing photo-reflective markers on the chest wall. A computational model is mandatory to compute the breathing volume and to calculate temporal and kinematical features by the gathered markers trajectories. Despite different methods based on different geometrical approaches can be adopted to compute volumes, no information about their differences in the respiratory evaluation are available. This study investigated the performances of four methods (conventional, prism-based, convex hull with boundary condition, based on Delaunay triangulation) using an optoelectronic motion capture system, on twelve healthy participants during 30 s of breathing. Temporal trends of volume traces, tidal volume values, and breathing durations were compared between methods and spirometry (used as reference instrument). Additionally, thoraco-abdominal motion patterns were compared between methods by analysing the compartmental contributions and their variability. Results shows comparable similarities between the volume traces obtained using spirometry, prism-based and conventional methods. Prism-based and convex hull with boundary condition methods show lower bias in tidal volumes estimation up to 0.06 L, compared to the conventional and Delaunay triangulation methods. Prism-based method shows maximum differences of 30 mL in the comparison of compartmental contributions to the total volume, by resulting in a maximum deviation of 1.6% in the percentage contribution analysis. In conclusion, our finding demonstrated the accuracy of the non-invasive MoCap-based breathing analysis with the prism-based method tested. Data provided in this study will lead researchers and clinicians in the computational method choice for temporal and volumetric breathing analysis.

Concepts: Volume, Thermodynamics, Sphere, Motion capture, Computational geometry, Delaunay triangulation, Simplex, Convex hull


A general and direct computational scheme to locate the surface separating arbitrarily shaped domains made up of molecules (or any other particles) has been developed and is described and illustrated for several, both artificial and physical examples. The proposed scheme consists of two modules: (i) triangulation and (ii) assignment of simplices to domains. Three different triangulation methods are employed, viz., the Delaunay triangulation, regular triangulation, and quasi-triangulation. In the triangulated system, the assignment step is carried out in two different ways, one based on the characteristic metric of a particular triangulation procedure and the other on the concept of a touching sphere. Some of the combinations of the triangulation and assignment steps lead to methods already used by others to find interfacial or surface molecules, namely the alpha-shape-based method of Usabiaga nad Duque [Phys. Rev. E 79 (2009) 046709] and GITIM of Sega et al. [J. Chem. Phys. 138 (2013) 044110]. The resulting surface is defined not only as a discrete set of particles, but it is build up of facets of the triangulation forming a broken line in two dimensions or a polyhedral surface in three dimensions. Individual molecular layers are identified in a very straightforward manner, starting with the interfacial layer itself and proceeding into the interior of the phase. The proposed scheme is illustrated first by identifying border molecules of pre-sampled domains of several shapes in a plane and then applied to five physically meaningful examples: thin films, near critical water, liquid water slab in an electric field, liquid water at a solid wall, and water at condition of electric-field-induced jetting. Performance of the considered methods is critically assessed. Treatment of domains forming percolating clusters through periodic boundary conditions is also described along with the determination of their periodicity and dimensionality.

Concepts: Physics, Water, Molecule, Chemistry, Atom, Dimension, Manifold, Delaunay triangulation