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Molecular distance geometry methods: From continuous to discrete
, 2009
"... Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context ..."
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Cited by 19 (17 self)
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Distance geometry problems arise from the need to position entities in the Euclidean Kspace given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties and Nuclear Magnetic Resonance experiments; sensor networks can estimate their relative distance by recording the power loss during a twoway exchange; finally, when drawing graphs in 2D or 3D, the graph to be drawn is given, and therefore distances between vertices can be computed. Distance geometry problems involve a search in a continuous Euclidean space, but sometimes the problem structure helps reduce the search to a discrete set of points. In this paper we survey some continuous and discrete methods for solving some problems of molecular distance geometry. 1
On the Computation of Protein Backbones by using Artificial Backbones of Hydrogens
"... NMR experiments provide information from which some of the distances between pairs of hydrogen atoms of a protein molecule can be estimated. Such distances can be exploited in order to identify the threedimensional conformation of the molecule: this problem is known in the literature as the Molecu ..."
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Cited by 11 (8 self)
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NMR experiments provide information from which some of the distances between pairs of hydrogen atoms of a protein molecule can be estimated. Such distances can be exploited in order to identify the threedimensional conformation of the molecule: this problem is known in the literature as the Molecular Distance Geometry Problem (MDGP). In this paper, we show how an artificial backbone of hydrogens can be defined which allows the reformulation of the MDGP as a combinatorial problem. This is done with the aim of solving the problem by the Branch and Prune (BP) algorithm, which is able to solve it efficiently. Moreover, we show how the real backbone of a protein conformation can be computed by using the coordinates of the hydrogens found by the BP algorithm. Formal proofs of the presented results are provided, as well as computational experiences on a set of instances whose size ranges from 60 to 6000 atoms.
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Cited by 1 (0 self)
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
Distance Geometry Problem Using LeastSquares Approximation ∗
, 2008
"... Abstract. We propose a new geometric buildup algorithm for the solution of the distance geometry problem in protein modeling, which can prevent the accumulation of the rounding errors in the buildup calculations successfully and also tolerate errors in given distances. In this algorithm, we use all ..."
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Abstract. We propose a new geometric buildup algorithm for the solution of the distance geometry problem in protein modeling, which can prevent the accumulation of the rounding errors in the buildup calculations successfully and also tolerate errors in given distances. In this algorithm, we use all instead of a subset of available distances for the determination of each unknown atom and obtain the position of the atom by using a leastsquares approximation instead of an exact solution to the system of distance equations. We show that the leastsquares approximation can be obtained by using a special singular value decomposition method, which not only tolerates and minimizes the distance errors, but also prevents the rounding errors from propagation effectively. We describe the leastsquares formulations and their solution methods, and present the test results from applying the new algorithm for the determination of a set of protein structures with varying degrees of availability and accuracy of the distances. We show that the new development of the algorithm increases the modeling ability of the geometric buildup approach significantly from both theoretical and practical points of view. Key words Biomolecular modeling, protein structure determination, distance geometry, linear and nonlinear systems of equations, linear and nonlinear optimization
THE SOLUTION OF THE DISTANCE GEOMETRY PROBLEM IN PROTEIN MODELING VIA GEOMETRIC BUILDUP ∗
, 2007
"... Abstract. A wellknown problem in protein modeling is the determination of the structure of a protein with a given set of interatomic or interresidue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry proble ..."
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Abstract. A wellknown problem in protein modeling is the determination of the structure of a protein with a given set of interatomic or interresidue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a geometric buildup approach to the problem. Central to this approach is the idea that the coordinates of the atoms in a protein can be determined one atom at a time, with the distances from the determined atoms to the undetermined ones. It can determine a structure more efficiently than other conventional approaches, yet without requiring more distance constraints than necessary. We present the general algorithm and its theory and review the recent development of the algorithm for controlling the propagation of the numerical errors in the buildup process, for determining rigid vs. unique structures, and for handling problems with inexact distances (distances with
Evolutionary Computation on the Connex Architecture
"... We discuss massively parallel implementation issues of the following heuristic optimization methods: Evolution Strategy, Genetic Algorithms, Harmony Search, and Simulated Annealing. For the first time, we implement these algorithms on the Connex architecture, a recently designed array of 1024 proces ..."
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We discuss massively parallel implementation issues of the following heuristic optimization methods: Evolution Strategy, Genetic Algorithms, Harmony Search, and Simulated Annealing. For the first time, we implement these algorithms on the Connex architecture, a recently designed array of 1024 processing elements. We use the VectorC programming environment, an extension of the C language adapted for Connex.