Microsecond dynamics in proteins by two-dimensional ESR. II. Addressing computational challenges.

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In previous work, we provided simulations for the case of domain motions within proteins that are themselves slowly tumbling in a solution. In order to perform these simulations, it was found that the standard approach of solving the relevant stochastic Liouville equation using the efficient Lanczos algorithm for this case breaks down, so algorithms were employed that rely on the Arnoldi iteration. While they lead to accurate simulations, they are very time-consuming. In this work, we focus on a variant known as the rational Arnoldi algorithm. We show that this can achieve a significant reduction in computation time. The stochastic Liouville matrix, which is of very large dimension, N, is first reduced to a much smaller dimension, m, e.g., from N ∼ O(10 4 ) to m ∼ 60, that spans the relevant Krylov subspace from which the spectrum is predicted. This requires the selection of the m frequency shifts to be utilized. A method of adaptive shift choice is introduced to optimize this selection. We also find that these procedures help in optimizing the pruning procedure that greatly reduces the dimension of the initial N dimensional stochastic Liouville matrix in such subsequent computations.