B.T. Miller

Comparing normal modes across different models and scales: Hessian reduction versus coarse-graining

A. Ghysels, B.T. Miller, F.C. Pickard III, B.R. Brooks
Journal of Computational Chemistry
33(28), 2250–2275


Dimension reduction is often necessary when attempting to reach longer length and time scales in molecular simulations. It is realized by constraining degrees of freedom or by coarse-graining the system. When evaluating the accuracy of a dimensional reduction, there is a practical challenge: the models yield vectors with different lengths, making a comparison by calculating their dot product impossible. This article investigates mapping procedures for normal mode analysis. We first review a horizontal mapping procedure for the reduced Hessian techniques, which projects out degrees of freedom. We then design a vertical mapping procedure for the “implosion” of the all-atom (AA) Hessian to a coarse-grained scale that is based upon vibrational subsystem analysis. This latter method derives both effective force constants and an effective kinetic tensor. Next, a series of metrics is presented for comparison across different scales, where special attention is given to proper mass-weighting. The dimension-dependent metrics, which require prior mapping for proper evaluation, are frequencies, overlap of normal mode vectors, probability similarity, Hessian similarity, collectivity of modes, and thermal fluctuations. The dimension-independent metrics are shape derivatives, elastic modulus, vibrational free energy differences, heat capacity, and projection on a predefined basis set. The power of these metrics to distinguish between reasonable and unreasonable models is tested on a toy alpha helix system and a globular protein; both are represented at several scales: the AA scale, a Gō-like model, a canonical elastic network model, and a network model with intentionally unphysical force constants. Published 2012 Wiley Periodicals, Inc.

Efficient Calculation of QM/MM Frequencies with the Mobile Block Hessian

A. Ghysels, H. Lee Woodcock III, J.D. Larkin, B.T. Miller, Y. Shao, J. Kong, D. Van Neck, V. Van Speybroeck, M. Waroquier, B.R. Brooks
Journal of Chemical Theory and Computation (JCTC)
7 (2), 496–514


The calculation of the analytical second derivative matrix (Hessian) is the bottleneck for vibrational analysis in QM/MM systems when an electrostatic embedding scheme is employed. Even with a small number of QM atoms in the system, the presence of MM atoms increases the computational cost dramatically: the long-range Coulomb interactions require that additional coupled perturbed self-consistent field (CPSCF) equations need to be solved for each MM atom displacement. This paper presents an extension to the Mobile Block Hessian (MBH) formalism for QM/MM calculations with blocks in the MM region and its implementation in a parallel version of the Q-Chem/CHARMM interface. MBH reduces both the CPU time and the memory requirements compared to the standard full Hessian QM/MM analysis, without the need to use a cutoff distance for the electrostatic interactions. Special attention is given to the treatment of link atoms which are usually present when the QM/MM border cuts through a covalent bond. Computational efficiency improvements are highlighted using a reduced chorismate mutase enzyme system, consisting of 24 QM atoms and 306 MM atoms, as a test example. In addition, the drug bortezomib, used for cancer treatment of myeloma, has been studied as a test case with multiple MBH block choices and both a QM and QM/MM description. The accuracy of the calculated Hessians is quantified by imposing Eckart constraints, which allows for the assessment of numerical errors in second derivative procedures. The results show that MBH within the QM/MM description not only is a computationally attractive method but also produces accurate results.

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