Techniques from graph theory are applied to analyze the bond networks in proteins and identify the flexible and rigid regions. The bond network consists of distance constraints defined by thecovalentandhydrogenbondsandsaltbridgesin the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degreesoffreedomwithinthisconstraintnetworkand that identifies all the rigid and flexible substructures in the protein, including overconstrained regions(withmorecrosslinkingbondsthanareneeded to rigidify the region) and underconstrained or flexible regions, in which dihedral bond rotations can occur. The number of extra constraints or remaining degrees of bond-rotational freedom within asubstructurequantifiesitsrelativerigidity/flexibility and provides aflexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximatelyamilliontimesfasterthanmolecular dynamics simulations and captures the essential conformational flexibility of the protein main and side-chains from analysis of asingle, static three-dimensional structure. This approach is demonstrated by comparison with experimental measures of flexibility for three proteins in which hinge and loop motion are essential for biological function: HIV protease, adenylate kinase, and dihydrofolate reductase
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