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  • 1
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    Journal of chemical information and modeling 30 (1990), S. 222-227 
    ISSN: 1520-5142
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1520-4995
    Source: ACS Legacy Archives
    Topics: Biology , Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Bulletin of mathematical biology 45 (1983), S. 665-720 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The mathematics of distance geometry constitutes the basis of a group of algorithms for revealing the structural consequences of diverse forms of information about a macromolecule's conformation. These algorithms are of proven utility in the analysis of experimental conformational data. This paper presents the basic theorems of distance geometry in Euclidean space and gives formal proofs of the correctness and, where possible, of the complexity of these algorithms. The implications of distance geometry for the energy minimization of macromolecules are also discussed.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Bulletin of mathematical biology 51 (1989), S. 173-194 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract An important component of computer programs for determining the solution conformation of proteins and other flexible molecules from nuclear magnetic resonance data are the so-called “bound smoothing algorithms”, which compute lower and upper limits on the values of all the interatomic distances from the relatively sparse set which can usually be measured experimentally. To date, the only methods efficient enough for use in large problems take account of only the triangle inequality, but an appreciable improvement in the precision of the limits is possible if the algebraic relations between the distances among each quadruple of atoms are also considered. The goal of this paper is to use a recently improved algorithm for computing these “tetrangle inequality limits” to determine just how much improvement really is possible, given the types of experimental data that are usually available.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Foundations of physics 23 (1993), S. 1357-1374 
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1573-5001
    Keywords: 2D NMR ; Simulation of spectra ; COSY ; Correlated spectroscopy ; Proton-proton coupling constants ; Scalar coupling ; Parameter estimation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Chemistry and Pharmacology
    Notes: Summary A computational method is described that takes an initial estimate of the chemical shifts, line widths and scalar coupling constants for the protons in a molecule, and refines this estimate so as to improve the least-squares fit between an experimental COSY spectrum and the spectrum simulated from these parameters in the weak-coupling approximation. In order to evaluate the potential of such refinements for estimating these parameters from COSY experiments, the method has been applied to a large number of sample problems which were themselves simulated from standard conformations of the amino acids, along with 25 near-native conformations of the protein bovine pancreatic trypsin inhibitor. The results of this evaluation show that: (i) if the chemical shifts are known to within ca. 0.01 ppm and no noise or artifacts are present in the data, the method is capable of recovering the correct coupling constants, starting from essentially arbitrary values, to within 0.1 Hz in almost all cases. (ii) Although the precision of these estimates of the coupling constants is degraded by the limited resolution, noise and artifacts present in most experimental spectra, the large majority of coupling constants can still be recovered to within 1.0 Hz; the local minimum problem is not made significantly worse by such defects in the data. (iii) The method assigns an ‘effective’ line width to all the resonances, and in the process can resolve overlapping cross peaks. (iv) The method is not capable of determining the chemical shifts a priori, due to the presence of numerous local minima in the least-squares residual as a function of these parameters.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 1573-5001
    Keywords: DQF-COSY ; Correlated spectroscopy ; Proton-proton coupling constants ; Parameter estimation ; Peptide and protein conformation ; Decorsin ; VDA
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Chemistry and Pharmacology
    Notes: Summary A new computational method for simultaneously estimating all the proton-proton coupling constants in a molecule from COSY spectra [Yang, J.-X. and Havel, T.F. (1994) J. Biomol. NMR, 4, 807–826] is applied to experimental data from two polypeptides. The first of these is a cyclic hexapeptide denoted as VDA (-d-Ala1-Phe2-Trp3-Lys(Z)4-Val5-Phe6-), in deuterated DMSO, while the second is a 39-residue protein, called decorsin, in aqueous solution. The effect of different data processing strategies and different initial parameter values on the accuracy of the coupling constants was explored. In the case of VDA, most of the coupling constants did not depend strongly on the initial values chosen for the optimization or on how the data were processed. This, together with our previous experience using simulated data, implies strongly that these values are accurate estimates of the coupling constants. They also differ by an average of only 0.36 Hz from the values of the 14 coupling constants that could be measured independently by established methods. In the case of decorsin, many of the coupling constants exhibited a moderate dependence on their initial values and a strong dependence on how the data were processed. With the most successful data processing strategy, the amide-α coupling constants differed by an average of 1.11 Hz from the 21 values that could be measured by established methods, while two thirds of the three-bond coupling constants fell within 1.0 Hz of the ranges obtained by applying the Karplus relation to an independently computed ensemble of distance geometry structures. The averages of the coupling constants over multiple optimizations using random initial values were computed in order to obtain the best possible estimates of the coupling constants. Most clearly incorrect averages can be identified by large standard deviations in the coupling constants or the associated line widths and chemical shifts, and can be explained by strong coupling and/or overlap with the water signal, the diagonal peaks or other cross peaks.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical chemistry 21 (1997), S. 223-260 
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract This paper presents a new distance geometry algorithm for calculating atomic coordinates from estimates of the interatomic distances, which maintains the positions of the atoms in a known rigid substructure. Given an M × 3 matrix of coordinates X for the rigid substructure, this problem consists of finding the N × 3 matrix Y which yields the global minimum of the so-called STRAIN, i.e., $$\mathop {\min }\limits_Y \left\| {\left[ {\begin{array}{*{20}c} {XX^T } \\ {YX^T } \\ \end{array} \begin{array}{*{20}c} {XY^T } \\ {YY^T } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} A \\ {B^T } \\ \end{array} \begin{array}{*{20}c} B \\ C \\ \end{array} } \right]} \right\|_F^2 ,$$ where $$A = XX^{\text{{\rm T}}} $$ , and B, C are matrices of inner products calculated from the estimated distances. The vanishing of the gradient of the STRAIN is shown to be equivalent to a system of only six nonlinear equations in six unknowns for the inertial tensor associated with the solution Y. The entire solution space is characterized in terms of the geometry of the intersection curves between the unit sphere and certain variable ellipsoids. Upon deriving tight bilateral bounds on the moments of inertia of any possible solution, we construct a search procedure that reliably locates the global minimum. The effectiveness of this method is demonstrated on realistic simulated and chemical test problems.
    Type of Medium: Electronic Resource
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  • 9
    ISSN: 0006-3525
    Keywords: Chemistry ; Polymer and Materials Science
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: By generating classes of random structures for trypsin inhibitor and carp myogen, each consistent with a given set of experimental or theoretical information, we have assessed the relative utility of various experiments and theories in deducing the conformation of macromolecules. We compare the calculated structures with known x-ray coordinates and compute for each class an average error. Small errors mean that the experimental or theoretical constraints limit the structures to the vicinity of the crystal structure, whereas large errors show that the constraints permit a wide variety of tertiary conformations. We find the following points to hold true: (1) Qualitative information on all the distances, as might be obtained from the correct prediction of interresidue contacts, effectively determines the structure (error approximately 1 Å). (2) Quantitative information on a limited number of distances, as might be obtained from nmr or crosslinking experiments, significantly restricts the range of possible structures only when the number of distances given is comparable to the number of residues (error approximately 3 Å). (3) Quantitative information on the distances of each residue to the center of mass of the molecule, as might in part be obtained from solvent accessibility and solution x-ray studies, is not particularly restrictive by itself (error approximately 5 Å). (4) Complete qualitative local distance information, as might be obtained from secondary prediction and CD/ORD studies, is clearly consistent with a wide variety of tertiary structures (error approximately 7 Å).
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 0006-3525
    Keywords: Chemistry ; Polymer and Materials Science
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: In this paper we study the statistical geometry of ensembles of poly(L-alanine) conformations computed by several different distance geometry algorithms. Since basic theory only permits us to predict the statistical properties of such ensembles a priori when the distance constraints have a very simple form, the only constraints used for these calculations are those necessary to obtain reasonable bond lengths and angles, together with a lack of short- and long-range atomic overlaps. The geometric properties studied include the squared end-to-end distance and radius of gyration of the computed conformations, in addition to the usual rms coordinate and φ / ψ angle deviations among these conformations. The distance geometry algorithms evaluated include several variations of the well-known embed algorithm, together with optimizations of the torsion angles using the ellipsoid and variable target function algorithms.The conclusions may be summarized as follows: First, the distribution with which the trail distances are chosen in most implementations of the embed algorithm is not appropriate when no long-range upper bounds on the distances are present, because it leads to unjustifiable expanded conformations. Second, chosing the trail distances independently of one another leads to a lack of variation in the degree of expansion, which in turn produces a relatively low rms square coordinate difference among the members of the ensemble. Third, when short-range steric constraints are present, torsion angle optimizations that start from conformations obtained by choosing their φ / ψ angles randomly with a uniform distribution between - 180° and + 180° do not converge to conformations whose angles are uniformly distributed over the sterically allowed regions of the φ / ψ plane.Finally, in an appendix we show how the sampling obtained with the embed algorithm can be substantially improved upon by the proper application of existing methodology.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
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