All-Atom Calculation of the Normal Modes of Bacteriorhodopsin Using a Sliding Block Iterative Diagonalization Method

Conventional normal-mode analysis of molecular vibrations requires computation and storage of the Hessian matrix. For a typical biological system such storage can reach several gigabytes posing difficulties for straightforward implementation. In this work we discuss an iterative block method to carry out full diagonalization of the Hessian while only storing a few vectors in memory. The iterative approach is based on the conjugate gradient formulation of the Davidson algorithm for simultaneous optimization of L roots, where in our case 10 < L < 300. The procedure is modified further by automatically adding a new vector into the search space for each locked (converged) root and keeping the new vector orthogonal to the eigenvectors previously determined. The higher excited states are then converged with the orthonormality constraint to the locked roots by applying a projector which is carried out using a read-rewind step done once per iteration. This allows for convergence of as many roots as desired without increasing the computer memory. The required Hessian-vector products are calculated on the fly as follows, Kp = dgp/dt, where K is the mass weighted Hessian, and gp is the gradient along p. The method has been implemented into the TINKER suite of molecular design codes. Preliminary results are presented for the normal modes of bacteriorhodopsin (bR) up to 300 cm(-)(1) and for the high frequency range between 2840 and 3680 cm(-)(1). There is evidence of a highly localized, noncollective mode at ∼1.4 cm(-)(1), caused by long-range interactions acting between the cytoplasmic and extracellular domains of bR.