Harmonic analysis of large systems. I. Methodology

Methods have been developed for the determination of vibrational frequencies and normal modes of large systems in the full conformational space (including all degrees of freedom) and in a reduced conformational space (reducing the number of degrees of freedom). The computational method, which includes Hessian generation and storage, full and iterative diagonalization techniques, and the refinement of the results, is presented. A method is given for the quasiharmonic analysis and the reduced basis quasiharmonic analysis. The underlying principle is that from the atomic fluctuations, an effective harmonic force field can be determined relative to the dynamic average structure. Normal mode analysis tools can be used to characterize quasiharmonic modes of vibration. These correspond to conventional normal modes except that anharmonic effects are included. Numerous techniques for the analyses of vibrational frequencies and normal modes are described. Criteria for the analysis of the similarity of low‐frequency normal modes is presented. The approach to determining the natural frequencies and normal modes of vibration described here is general and applicable to any large system. © 1995 John Wiley & Sons, Inc. This article is a U.S. Government work and, as such, is in the public domain in the United States of America.

[1]  James M. Ortega,et al.  On Sturm Sequences for Tridiagonal Matrices , 1960, JACM.

[2]  H. Berendsen,et al.  ALGORITHMS FOR MACROMOLECULAR DYNAMICS AND CONSTRAINT DYNAMICS , 1977 .

[3]  U. Singh,et al.  A combined 2D-NMR and molecular dynamics analysis of the structure of the actinomycin D: d(ATGCAT)2 complex. , 1989, Journal of biomolecular structure & dynamics.

[4]  Bernard R. Brooks,et al.  Inelastic neutron scattering analysis of low frequency motion in proteins: A normal mode study of the bovine pancreatic trypsin inhibitor , 1986 .

[5]  Y. Sanejouand,et al.  A new approach for determining low‐frequency normal modes in macromolecules , 1994 .

[6]  G. Ciccotti,et al.  Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes , 1977 .

[7]  M. Karplus,et al.  Normal modes for specific motions of macromolecules: application to the hinge-bending mode of lysozyme. , 1985, Proceedings of the National Academy of Sciences of the United States of America.

[8]  M Karplus,et al.  Dynamics of myoglobin: comparison of simulation results with neutron scattering spectra. , 1990, Proceedings of the National Academy of Sciences of the United States of America.

[9]  M. Karplus,et al.  CHARMM: A program for macromolecular energy, minimization, and dynamics calculations , 1983 .

[10]  N Go,et al.  A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis. , 1990, Biophysical chemistry.

[11]  C. Bender,et al.  The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvectors of very large symmetric matrices , 1973 .

[12]  M. Karplus,et al.  Protein dynamics in solution and in a crystalline environment: a molecular dynamics study. , 1982, Biochemistry.

[13]  N. Go,et al.  Dynamics of a small globular protein in terms of low-frequency vibrational modes. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[14]  D Perahia,et al.  Normal modes of symmetric protein assemblies. Application to the tobacco mosaic virus protein disk. , 1992, Biophysical journal.

[15]  James Hardy Wilkinson,et al.  Householder's Method for the Solution of the Algebraic Eigenproblem , 1960, Comput. J..

[16]  D. Case,et al.  Harmonic dynamics of a DNA hexamer in the absence and presence of the intercalator ethidium , 1989, Biopolymers.

[17]  M. Karplus,et al.  Harmonic dynamics of proteins: normal modes and fluctuations in bovine pancreatic trypsin inhibitor. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[18]  D. ben-Avraham,et al.  Normal mode analysis of G-actin. , 1993, Journal of molecular biology.

[19]  M Karplus,et al.  Dynamics of proteins: elements and function. , 1983, Annual review of biochemistry.

[20]  Ronald M. Levy,et al.  Vibrational approach to the dynamics of an α‐helix , 1979 .

[21]  Isaiah Shavitt,et al.  The Method of Configuration Interaction , 1977 .

[22]  D. Case,et al.  Langevin modes of macromolecules: Applications to crambin and DNA hexamers , 1990, Biopolymers.

[23]  M Karplus,et al.  The normal modes of the gramicidin-A dimer channel. , 1988, Biophysical journal.

[24]  Yonezo Morino,et al.  A Note on the Classification of Normal Vibrations of Molecules , 1952 .

[25]  Bernard R. Brooks,et al.  Protein simulation below the glass-transition temperature. Dependence on cooling protocol , 1994 .

[26]  M. Levitt,et al.  Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. , 1985, Journal of molecular biology.

[27]  James Hardy Wilkinson,et al.  The Calculation of the Eigenvectors of Codiagonal Matrices , 1958, Comput. J..

[28]  C. Lanczos An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[29]  M Tasumi,et al.  Normal vibrations of proteins: glucagon. , 1982, Biopolymers.

[30]  R. Sharon,et al.  Accurate simulation of protein dynamics in solution. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[31]  M. Karplus,et al.  Evaluation of the configurational entropy for proteins: application to molecular dynamics simulations of an α-helix , 1984 .

[32]  Bruce Tidor,et al.  Inelastic neutron scattering analysis of low-frequency motions in proteins : harmonic and damped harmonic models of bovine pancreatic tryspin inhibitor , 1990 .

[33]  M. Karplus,et al.  Method for estimating the configurational entropy of macromolecules , 1981 .

[34]  M. Karplus,et al.  Multiple conformational states of proteins: a molecular dynamics analysis of myoglobin. , 1987, Science.

[35]  S. Krimm,et al.  Vibrational analysis of conformation in peptides, polypeptides, and proteins , 1983, Biopolymers.

[36]  A. Szabó,et al.  Langevin modes of macromolecules , 1986 .

[37]  M Karplus,et al.  Transition from B to Z DNA: contribution of internal fluctuations to the configurational entropy difference. , 1985, Science.