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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 the covalent and hydrogen bonds and salt bridges in the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degrees of freedom within this constraint network and that identifies all the rigid and flexible substructures in the protein, including overconstrained regions (with more crosslinking bonds than are needed 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 a substructure quantifies its relative rigidity/flexibility and provides a flexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximately a million times faster than molecular dynamics simulations and captures the essential conformational flexibility of the protein main and side-chains from analysis of a single, 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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A recurrent problem in materials science is the prediction of the percolation threshold of suspensions and composites containing complex-shaped constituents. We consider an idealized material built up from freely overlapping objects randomly placed in a matrix, and numerically compute the geometrical percolation threshold p, where the objects first form a continuous phase. Ellipsoids of revolution, ranging from the extreme oblate limit of platelike particles to the extreme prolate limit of needlelike particles, are used to study the inhuence of object shape on the value of p, . The reciprocal threshold 1/p, (p, equals the critical volume fraction occupied by the overlapping ellipsoids) is found to scale linearly with the ratio of the larger ellipsoid dimension to the smaller dimension in both the needle and plate limits. Ratios of the estimates of p, are taken with other important functionals of object shape (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic conductivity) in an attempt to obtain a universal description of p, . Unfortunately, none of the possibilities considered proves to be invariant over the entire shape range, so that p, appears to be a rather unique functional of object shape. It is conjectured, based on the numerical evidence, that 1/p, is minimal for a sphere of all objects having a finite volume.

It is believed that covalent glasses can be divided into two classes: those with high average coordination (amorphous solids) and those with low average coordination (polymeric glasses). We present the first conclusive evidence that this division is correct by calculating the elastic properties of random networks with different average coordination (r). The results show that the elastic constants depend predominantly on (r) and go to zero when (r) = 2.4 with an exponent /= 1.5 ±0.2.