A well-known problem in protein modeling is the determination of the structureof a protein with a given set of inter-atomic or inter-residue distances obtainedfrom either physical experiments or theoretical estimates. A general form of theproblem is known as the distance geometry problem in mathematics, the graphembedding problem in computer science, and the multidimensional scaling problemin statistics. The problem has applications in many other scientific and engineeringfields as well such as sensor network localization, image recognition, and proteinclassification. We describe the formulations and complexities of the problem in itsvarious forms, and introduce a geometric buildup approach to the problem. Centralto this approach is the idea that the coordinates of the atoms in a protein can bedetermined one atom at a time, with the distances from the determined atoms tothe undetermined ones. It can determine a structure more efficiently than otherconventional approaches, yet without requiring more distance constraints thannecessary. We present the general algorithm and its theory and review the recentdevelopment of the algorithm for controlling the propagation of the numericalerrors in the buildup process, for determining rigid vs. unique structures, and forhandling problems with inexact distances (distances with errors). We show theresults from applying the algorithm to some of the model problems and justify thepotential use of the algorithm in protein modeling.
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