| Abstract |
Three-dimensional molecular structure is fundamental in drug design and discovery, docking, and chemical function identification. The enumeration of the feasible conformations provides a rigorous way to find an optimal or a few acceptable geometric configurations with respect to energy minimization. The input to our algorithm consists of a set of geometric constraints, typically obtained by the chemical covalent structure, and the output is a set, often complete, of valid conformations in specified neighborhoods of the conformation space. For molecules or molecular substructures with a few tens of degrees of freedom, our methods are able to fully enumerate all realizable conformations. Numerical linear algebra methods are employed to solve the problem of Euclidean embeddability, which is NP-complete. The main tools include, besides traditional iterative optimization, distance geometry and matrix perturbations for minimizing singular (or eigen-) values of real symmetric matrices. The computations are mostly numeric, for reasons of speed. Nonetheless, in our concern to guarantee the results, we are able to bound from below the number of degrees of freedom on the conformation manifold. Our implementation in the MATLAB (or SCILAB) environment is illustrated on small cyclic molecules as well as more general molecules, or substructures of larger molecules, with up to 20 degrees of freedom. The data may also come from NMR experiments of varying accuracy, in which case we hope that our program helps in detecting outliers. Corresponding problems, based on pairwise distances, appear in robotics and in computer aided design (CAD).
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Αλγεβρικές και αριθμητικές μέθοδοι με εφαρμογές στη Μοριακή Βιολογία
