The embedding problem for predistance matrices

A fundamental problem in molecular biology is the determination of the conformation of macromolecules from NMR data. Several successful distance geometry programs have been developed for this purpose, for example DISGEO. A particularly difficult facet of these programs is the embedding problem, that is the problem of determining those conformations whose distances between atoms are nearest those measured by the NMR techniques. The embedding problem is the distance geometry equivalent of the multiple minima problem, which arises in energy minimization approaches to conformation determination. We show that the distance geometry approach has some nice geometry not associated with other methods that allows one to prove detailed results with regard to the location of local minima. We exploit this geometry to develop some algorithms which are faster and find more minima than the algorithms presently used.

[1]  I. J. Schoenberg.  Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[2]  A. Householder,et al.  Discussion of a set of points in terms of their mutual distances , 1938 .

[3]  John von Neumann,et al.  The geometry of orthogonal spaces , 1950 .

[4]  Leonard M. Blumenthal,et al.  Theory and applications of distance geometry , 1954 .

[5]  Forrest W. Young,et al.  Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features , 1977 .

[6]  S. Schiffman.  Introduction to Multidimensional Scaling , 1981 .

[7]  Timothy F. Havel,et al.  The theory and practice of distance geometry , 1983, Bulletin of Mathematical Biology.

[8]  R. Dykstra.  An Algorithm for Restricted Least Squares Regression , 1983 .

[9]  Timothy F. Havel,et al.  A distance geometry program for determining the structures of small proteins and other macromolecules from nuclear magnetic resonance measurements of intramolecular1H−1H proximities in solution , 1984 .

[10]  J. Jungck,et al.  Mathematical tools for molecular genetics data: An annotated bibliography , 1984 .

[11]  J. Gower.  Properties of Euclidean and non-Euclidean distance matrices , 1985 .

[12]  N Go,et al.  Calculation of protein conformations by proton-proton distance constraints. A new efficient algorithm. , 1985, Journal of molecular biology.

[13]  R. Fletcher.  Semi-Definite Matrix Constraints in Optimization , 1985 .

[14]  Timothy F. Havel,et al.  An evaluation of the combined use of nuclear magnetic resonance and distance geometry for the determination of protein conformations in solution. , 1985, Journal of molecular biology.

[15]  J. Meulman.  A Distance Approach to Nonlinear Multivariate Analysis , 1986 .

[16]  M Karplus,et al.  The three‐dimensional structure of α1‐purothionin in solution: combined use of nuclear magnetic resonance, distance geometry and restrained molecular dynamics , 1986, The EMBO journal.

[17]  Michael W. Browne.  The Young-Householder algorithm and the least squares multidimensional scaling of squared distances , 1987 .

[18]  Kurt Wüthrich,et al.  The ellipsoid algorithm as a method for the determination of polypeptide conformations from experimental distance constraints and energy minimization , 1987 .

[19]  Shih-Ping Han,et al.  A successive projection method , 1988, Math. Program..

[20]  F. Critchley.  On certain linear mappings between inner-product and squared-distance matrices , 1988 .

[21]  Timothy F. Havel,et al.  Shortest-path problems and molecular conformation , 1988, Discret. Appl. Math..

[22]  Harold A. Scheraga.  Approaches to the Multiple-Minima Problem in Conformational Energy Calculations on Polypeptides and Proteins , 1988 .

[23]  Gordon M. Crippen,et al.  Distance Geometry and Molecular Conformation , 1988 .

[24]  Thomas L. Hayden,et al.  Approximation by matrices positive semidefinite on a subspace , 1988 .

[25]  K. Wüthrich.  Protein structure determination in solution by nuclear magnetic resonance spectroscopy. , 1989, Science.

[26]  R. Mathar,et al.  A cyclic projection algorithm via duality , 1989 .

[27]  Timothy F. Havel,et al.  Computational experience with an algorithm for tetrangle inequality bound smoothing. , 1989, Bulletin of mathematical biology.

[28]  W. Glunt,et al.  An alternating projection algorithm for computing the nearest euclidean distance matrix , 1990 .

[29]  Gordon M. Crippen,et al.  Global energy minimization by rotational energy embedding , 1990, J. Chem. Inf. Comput. Sci..

[30]  Timothy F. Havel,et al.  The sampling properties of some distance geometry algorithms applied to unconstrained polypeptide chains: A study of 1830 independently computed conformations , 1990, Biopolymers.

[31]  Timothy F. Havel,et al.  Bound Smoothing under Chirality Constraints , 1991, SIAM J. Discret. Math..

[32]  Wei-Min Liu,et al.  The cone of distance matrices , 1991 .