Large Eigenvalue Problems in Coarse-Grained Dynamic Analyses of Supramolecular Systems

Computational methods ranging from all-atom molecular dynamics simulations to coarse-grained normal-mode analyses based on simplified elastic networks provide a general framework to studying molecular dynamics. Despite recent successes in analyzing very large systems with up to 100 million atoms, those methods are currently limited to studying small- to medium-size molecular systems when used on standard desktop computers, because of computational limitations. The hope to circumvent those limitations rests on the development of improved algorithms with novel implementations that mitigate their computationally challenging parts. In this paper, we have addressed the computational challenges associated with computing coarse-grained normal modes of very large molecular systems, focusing on the calculation of the eigenpairs of the Hessian of the potential energy function from which the normal modes are computed. We have described and implemented a new method for handling this Hessian based on tensor products. This new formulation is shown to reduce space requirements and to improve the parallelization of its implementation. We have implemented and tested four different methods for computing some eigenpairs of the Hessian, namely, the standard, robust Lanczos method, a simple modification of this method based on polynomial filtering, a functional-based method recently proposed for normal-mode analyses of viruses, and a block Chebyshev-Davidson method with inner-outer restart. We have shown that the latter provides the most efficient implementation when computing eigenpairs of extremely large Hessian matrices corresponding to large viral capsids. We have also shown that, for those viral capsids, a large number of eigenpairs is actually needed, on the order of thousands, noticing however that this large number is still a small fraction of the total number of possible eigenpairs (a few percent).