| A well-known problem in protein modeling is the determination of the structure of a protein with a given set of interatomic distances obtained from either physical experiments or theoretical estimates. A more general form of this problem is known as the distance geometry problem in mathematics, which can be solved in polynomial time if a complete set of exact distances is given, but is generally intractable for a general sparse set of distance data. We investigate the solution of the problem within a geometric buildup framework. We propose a new geometric buildup algorithm for solving the problem using special least-squares approximation techniques, which not only prevents the accumulation of the rounding errors in the buildup calculations successfully, but also tolerates small errors in given distances. In NMR spectroscopy, however, distances can only be obtained with their rough ranges, and hence an ensemble of solutions satisfying the given constraints becomes critical to find. We propose a new approach to the problem of determining an ensemble of protein structures with a set of interatomic distance bounds. Similar to X-ray crystallography, we assume that the protein has an equilibrium structure and the atoms fluctuate around their equilibrium positions. Then, the problem can be formulated as a generalized distance geometry problem to find the equilibrium positions and maximal possible fluctuation radii for the atoms in the protein, subject to the condition that the fluctuations should be within the given distance bounds. We describe the scientific background of the work, the motivation of the new approach and the formulation of the problem. We develop a geometric buildup algorithm for an approximate solution to the problem and present some preliminary test results. We also discuss related theoretical and computational issues and potential impacts of this work in NMR protein modeling. |
