The subspace iteration method in protein normal mode analysis

Abstract

Normal mode analysis plays an important role in relating the conformational dynamics of proteins to their biological function. The subspace iteration method is a numerical procedure for normal mode analysis that has enjoyed widespread success in the structural mechanics community due to its numerical stability and computational efficiency in calculating the lowest normal modes of large systems. Here, we apply the subspace iteration method to proteins to demonstrate its advantageous properties in this area of computational protein science. An effective algorithm for choosing the number of iteration vectors in the method is established, offering a considerable improvement over the original implementation. In the present application, computational time scales linearly with the number of normal modes computed. Additionally, the method lends itself naturally to normal mode analyses of multiple neighboring macromolecular conformations, as demonstrated in a conformational change pathway analysis of adenylate kinase. These properties, together with its computational robustness and intrinsic scalability to multiple processors, render the subspace iteration method an effective and reliable computational approach to protein normal mode analysis.

The subspace iteration method – Revisited

“…(18), in appropriately modified form, could be used to obtain a better value of q than given by Eq. (19). In fact, this approach was followed in Ref.…”

“…In fact, this approach was followed in Ref. [19] and the resulting solution times scaled linearly. It is interesting to note that in the original development of the subspace iteration procedure the same expression in the brackets was used but its minimum [1,16].…”

“…In the analysis given next, we follow the derivation used in Ref. [19] where it was assumed that the eigenvalues vary linearly in magnitude (as it is approximately the case for proteins).…”

“…The eigenvectors just computed can be used, for example, in optimization problems of structures when the frequencies are calculated as the structure changes [16], in solving random eigenvalue problems when using Monte Carlo simulations [13], or in computational biology when evaluating the frequencies and mode shapes of proteins on conformational pathways [19]. In these cases, the use of the calculated eigenvectors of the previous solution as the starting vectors of the next solution can be very effective.…”

Conformational dynamics of supramolecular protein assemblies

“…Proteins are modeled as homogeneous linear isotropic materials characterized by three independent effective material parameters: the Young's modulus ( E ), the mass density (  ), and Poisson's ratio ( ), where proteins are assumed to have mass density 1.35 g/cm 3 and Poisson's ratio 0.3 [43], which is typical of crystalline solids. While the effective Young's modulus is generally unknown for proteins, it can be obtained by fitting thermal fluctuations of α-carbon atoms in the FE model to those obtained using either the all-atom normal mode analysis or the RTB procedure when atomic coordinates are available, which generally ranges from two to five GPa [18,44]. Because most structures in the EMDB lack atomic coordinates, normal mode amplitudes and dependent properties are computed using a Young's modulus of 2 GPa, representing an approximate lower bound on protein stiffness, and correspondingly an upper bound on molecular RMSFs [18].…”