  The flexibility-rigidity index (FRI) is a newly proposed method for the
construction of atomic rigidity functions. The FRI method analyzes protein
rigidity and flexibility and is capable of predicting protein B-factors without
resorting to matrix diagonalization. A fundamental assumption used in the FRI
is that protein structures are uniquely determined by various internal and
external interactions, while the protein functions, such as stability and
flexibility, are solely determined by the structure. As such, one can predict
protein flexibility without resorting to the protein interaction Hamiltonian.
Consequently, bypassing the matrix diagonalization, the original FRI has a
computational complexity of O(N^2). This work introduces a fast FRI (fFRI)
algorithm for the flexibility analysis of large macromolecules. The proposed
fFRI further reduces the computational complexity to O(N). Additionally, we
propose anisotropic FRI (aFRI) algorithms for the analysis of protein
collective dynamics. The aFRI algorithms admit adaptive Hessian matrices, from
a completely global 3N*3N matrix to completely local 3*3 matrices. However,
these local 3*3 matrices have built in much non-local correlation. Furthermore,
we compare the accuracy and efficiency of FRI with some {established}
approaches to flexibility analysis, namely, normal mode analysis (NMA) and
Gaussian network model (GNM). The accuracy of the FRI method is tested. The
FRI, particularly the fFRI, is orders of magnitude more efficient and about 10%
more accurate overall than some of the most popular methods in the field. The
proposed fFRI is able to predict B-factors for alpha-carbons of the HIV virus
capsid (313,236 residues) in less than 30 seconds on a single processor using
only one core. Finally, we demonstrate the application of FRI and aFRI to
protein domain analysis.
