B.R. Brooks

Normal mode analysis of macromolecular systems with the mobile block Hessian method

A. Ghysels, V. Van Speybroeck, D. Van Neck, B.R. Brooks, M. Waroquier
AIP Conference Proceedings
1642 (2015), 559
2015
P1

Abstract 

Until recently, normal mode analysis (NMA) was limited to small proteins, not only because the required energy minimization is a computationally exhausting task, but also because NMA requires the expensive diagonalization of a 3N(a) x 3N(a) matrix with N-a the number of atoms. A series of simplified models has been proposed, in particular the Rotation-Translation Blocks (RTB) method by Tama et al. for the simulation of proteins. It makes use of the concept that a peptide chain or protein can be seen as a subsequent set of rigid components, i.e. the peptide units. A peptide chain is thus divided into rigid blocks with six degrees of freedom each.

Recently we developed the Mobile Block Hessian (MBH) method, which in a sense has similar features as the RTB method. The main difference is that MBH was developed to deal with partially optimized systems. The position/orientation of each block is optimized while the internal geometry is kept fixed at a plausible - but not necessarily optimized - geometry. This reduces the computational cost of the energy minimization. Applying the standard NMA on a partially optimized structure however results in spurious imaginary frequencies and unwanted coordinate dependence. The MBH avoids these unphysical effects by taking into account energy gradient corrections. Moreover the number of variables is reduced, which facilitates the diagonalization of the Hessian.

In the original implementation of MBH, atoms could only be part of one rigid block. The MBH is now extended to the case where atoms can be part of two or more blocks. Two basic linkages can be realized: (1) blocks connected by one link atom, or (2) by two link atoms, where the latter is referred to as the hinge type connection. In this work we present the MBH concept and illustrate its performance with the crambin protein as an example.

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
2012
A1

Abstract 

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.

Normal modes for large molecules with arbitrary link constraints in the mobile block Hessian approach

A. Ghysels, D. Van Neck, B.R. Brooks, V. Van Speybroeck, M. Waroquier
Journal of Chemical Physics
130 (8), 084107
2009
A1

Abstract 

In a previous paper [ Ghysels et al., J. Chem. Phys. 126, 224102 (2007) ] the mobile block Hessian (MBH) approach was presented. The method was designed to accurately compute vibrational modes of partially optimized molecular structures. The key concept was the introduction of several blocks of atoms, which can move as rigid bodies with respect to a local, fully optimized subsystem. The choice of the blocks was restricted in the sense that none of them could be connected, and also linear blocks were not taken into consideration. In this paper an extended version of the MBH method is presented that is generally applicable and allows blocks to be adjoined by one or two common atoms. This extension to all possible block partitions of the molecule provides a structural flexibility varying from very rigid to extremely relaxed. The general MBH method is very well suited to study selected normal modes of large macromolecules (such as proteins and polymers) because the number of degrees of freedom can be greatly reduced while still keeping the essential motions of the molecular system. The reduction in the number of degrees of freedom due to the block linkages is imposed here directly using a constraint method, in contrast to restraint methods where stiff harmonic couplings are introduced to restrain the relative motion of the blocks. The computational cost of this constraint method is less than that of an implementation using a restraint method. This is illustrated for the α-helix conformation of an alanine-20-polypeptide. © 2009 American Institute of Physics

Mobile Block Hessian Approach with Adjoined Blocks: An Efficient Approach for the Calculation of Frequencies in Macromolecules

A. Ghysels, V. Van Speybroeck, E. Pauwels, D. Van Neck, B.R. Brooks, M. Waroquier
Journal of Chemical Theory and Computation (JCTC)
5 (5), 1203-1215
2009
A1

Abstract 

In an earlier work, the authors developed a new method, the mobile block Hessian (MBH) approach, to accurately calculate vibrational modes for partially optimized molecular structures [ J. Chem. Phys. 2007, 126 (22), 224102.]. It is based on the introduction of blocks, consisting of groups of atoms, that can move as rigid bodies. The internal geometry of the blocks need not correspond to an overall optimization state of the total molecular structure. The standard MBH approach considers free blocks with six degrees of freedom. In the extended MBH approach introduced herein, the blocks can be connected by one or two adjoining atoms, which further reduces the number of degrees of freedom. The new approach paves the way for the normal-mode analysis of biomolecules such as proteins. It rests on the hypothesis that low-frequency modes of proteins can be described as pure rigid-body motions of blocks of consecutive amino acid residues. The method is validated for a series of small molecules and further applied to alanine dipeptide as a prototype to describe vibrational interactions between two peptide units; to crambin, a small protein with 46 amino acid residues; and to ICE/caspase-1, which contains 518 amino acid residues.

Vibrational subsystem analysis: A method for probing free energies and correlations in the harmonic limit

H. Lee Woodcock III, W. Zheng, A. Ghysels, Y. Shao, J. Kong, B.R. Brooks
Journal of Chemical Physics
129 (21), 214109
2008
A1

Abstract 

A new vibrational subsystem analysis (VSA) method is presented for coupling global motion to a local subsystem while including the inertial effects of the environment. The premise of the VSA method is a partitioning of a system into a smaller region of interest and a usually larger part referred to as environment. This method allows the investigation of local-global coupling, a more accurate estimation of vibrational free energy contribution for parts of a large system, and the elimination of the “tip effect” in elastic network model calculations. Additionally, the VSA method can be used as a probe of specific degrees of freedom that may contribute to free energy differences. The VSA approach can be employed in many ways, but it will likely be most useful for estimating activation free energies in QM/MM reaction path calculations. Four examples are presented to demonstrate the utility of this method.

Comparative study of various normal mode analysis techniques based on partial Hessians

A. Ghysels, V. Van Speybroeck, E. Pauwels, S. Catak, B.R. Brooks, D. Van Neck, M. Waroquier
Journal of Computational Chemistry
31 (5), 994-1007
2010
A1

Abstract 

Standard normal mode analysis becomes problematic for complex molecular systems, as a result of both the high computational cost and the excessive amount of information when the full Hessian matrix is used. Several partial Hessian methods have been proposed in the literature, yielding approximate normal modes. These methods aim at reducing the computational load and/or calculating only the relevant normal modes of interest in a specific application. Each method has its own (dis)advantages and application field but guidelines for the most suitable choice are lacking. We have investigated several partial Hessian methods, including the Partial Hessian Vibrational Analysis (PHVA), the Mobile Block Hessian (MBH), and the Vibrational Subsystem Analysis (VSA). In this article, we focus on the benefits and drawbacks of these methods, in terms of the reproduction of localized modes, collective modes, and the performance in partially optimized structures. We find that the PHVA is suitable for describing localized modes, that the MBH not only reproduces localized and global modes but also serves as an analysis tool of the spectrum, and that the VSA is mostly useful for the reproduction of the low frequency spectrum. These guidelines are illustrated with the reproduction of the localized amine-stretch, the spectrum of quinine and a bis-cinchona derivative, and the low frequency modes of the LAO binding protein. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010

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
2011
A1

Abstract 

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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