Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis.

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. 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 permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that 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 α-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.