Journal of Mathematical Chemistry

, Volume 50, Issue 6, pp 1521–1549 | Cite as

Normal mode analysis of molecular motions in curvilinear coordinates on a non-Eckart body-frame: an application to protein torsion dynamics

  • Janne Pesonen
  • Krister O. E. Henriksson
  • Jose Ramon López-Blanco
  • Pablo Chacón
Original Paper

Abstract

Normal mode analysis (NMA) was introduced in 1930s as a framework to understand the structure of the observed vibration-rotation spectrum of several small molecules. During the past three decades NMA has also become a popular alternative to figuring out the large-scale motion of proteins and other macromolecules. However, the “standard” NMA is based on approximations, which sometimes are unphysical. Especially problematic is the assumption that atoms move only “infinitesimally”, which, of course, is an oxymoron when large amplitude motions are concerned. The “infinitesimal” approximation has the further unfortunate side effect of masking the physical importance of the coupling between vibrational and rotational degrees of freedom. Here, we present a novel formulation of the NMA, which is applied for finite motions in non-Eckart body-frame. Contrary to standard normal mode theory, our approach starts by assuming a harmonic potential in generalized coordinates, and tries to avoid the linearization of the coordinates. It also takes explicitly into account the Coriolis terms, which couple vibrations and rotations, and the terms involving Christoffel symbols, which are ignored by default in the standard NMA. We also computationally explore the effect of various terms to the solutions of the NMA equation of motions.

Keywords

Normal modes Curvilinear coordinates Non-Eckart frame Coriolis coupling Vibration-rotation Lagrangian Non-Euclidean metric Finite displacements Christoffel symbol Equations ofmotion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wilson E.B., Decius J.C., Cross P.C.: Molecular Vibrations. (Dover, New York 1980)Google Scholar
  2. 2.
    G. Sørensen, A new approach to the hamiltonian of nonrigid molecules, in Large Amplitude Motion in Molecules II. Topics in Current Chemistry, vol. 82, ed. by F. L. Boschke (Springer, Berlin, 1979), pp. 97–175Google Scholar
  3. 3.
    Cui, Q., Bahar, I. (eds.),: Normal Mode Analysis—Theory and Practice to Biological and Chemical Systems. (Chapman & Hall, CRC, London 2006)Google Scholar
  4. 4.
    Skjaerven L., Hollup S.M., Reuter N.: Normal mode analysis for proteins. J. Mol. Struct. THEOCHEM. 898(1–3), 42–48 (2009)CrossRefGoogle Scholar
  5. 5.
    Dykeman E.C., Sankey O.F.: Normal mode analysis and applications in biological physics. J. Phys. Condens. Matter. 22(42), 423202 (2010)CrossRefGoogle Scholar
  6. 6.
    Bahar I. et al.: Normal mode analysis of biomolecular structures: functional mechanisms of membrane proteins. Chem. Rev. 110(3), 1463–1497 (2010)CrossRefGoogle Scholar
  7. 7.
    Lopez-Blanco J.R., Garzon J.I., Chacon P.: iMod: multipurpose normal mode analysis in internal coordinates. Bioinformatics 27(20), 2843–2850 (2011)CrossRefGoogle Scholar
  8. 8.
    Kitao A., Hayward S., Go N.: Comparison of normal mode analyses on a small globular protein in dihedral angle space and cartesian coordinate space. Biophys. Chem. 52(2), 107–114 (1994)CrossRefGoogle Scholar
  9. 9.
    Noguti T., Go N.: Dynamics of native globular proteins in terms of dihedral angles. J. Phys. Soc. Jpn. 52, 3283–3288 (1983)CrossRefGoogle Scholar
  10. 10.
    Go N., Noguti T., Nishikawa T.: Dynamics of a small globular protein in terms of low-frequency vibrational modes. PNAS 80, 3696–3700 (1983)CrossRefGoogle Scholar
  11. 11.
    Levitt M., Sander C., Stern P.S.: Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. J. Mol. Biol. 181, 423–447 (1985)CrossRefGoogle Scholar
  12. 12.
    Tirion M.: Large amplitude elastic motions in proteins from a single-parameter, atomic analysis. Phys. Rev. Lett. 77, 1905–1908 (1996)CrossRefGoogle Scholar
  13. 13.
    Tama F., Sanejouand Y.-H.: Conformational change of proteins arising from normal mode calculations. Protein Eng. 14(1), 1–6 (2001)CrossRefGoogle Scholar
  14. 14.
    Tama F., Wriggers W., Brooks C.L. III: Exploring global distortions of biological macromolecules and assemblies from low-resolution structural information and elastic network theory. J. Mol. Biol. 321, 297–305 (2002)CrossRefGoogle Scholar
  15. 15.
    Kovacs J.A., Chacon P., Abagyan R.: Predictions of protein flexibility: first-order measures. Proteins 56, 661–668 (2004)CrossRefGoogle Scholar
  16. 16.
    van Vlijmen H.W.T., Karplus M.: Normal mode calculations of icosahedral viruses with full dihedral flexibility by use of molecular symmetry. J. Mol. Biol. 350, 528–542 (2005)CrossRefGoogle Scholar
  17. 17.
    Rueda M., Chacon P., Orozco M.: Thorough validation of protein normal mode analysis: a comparative study with essential dynamcis. Structure 15, 565–575 (2007)CrossRefGoogle Scholar
  18. 18.
    Fuchigami S., Omori S., Ikeguchi M., Kidera A.: Normal mode analysis of protein dynamics in a non-eckart frame. J. Chem. Phys. 132(10), 104109 (2010)CrossRefGoogle Scholar
  19. 19.
    Eckart C.: Some studies concerning rotating axes and polyatomic molecules. Phys. Rev. 47, 552–558 (1935)CrossRefGoogle Scholar
  20. 20.
    Malhiot R.J., Ferigle S.M.: Eckart conditions in Wilson’s treatment of molecular vibrations. J. Chem. Phys. 22(4), 717–719 (1954)CrossRefGoogle Scholar
  21. 21.
    Louck J.D., Galbraith H.W.: Eckart vectors, Eckart frames, and polyatomic molecules. Rev. Mod. Phys. 48(1), 69 (1976)CrossRefGoogle Scholar
  22. 22.
    Littlejohn R.G., Reinsch M.: Gauge fields in the separation of rotations and internal motions in the n-body problem. Rev. Mod. Phys. 69(1), 213–276 (1997)CrossRefGoogle Scholar
  23. 23.
    Littlejohn R.G., Mitchell K.: Gauge Theory of Small Vibrations in Polyatomic Molecules. In: Newton, P, Holmes, P, Weinstein, A (eds) Geometry, mechanics, and dynamics, pp. 407–428. Springer, New York (2002)CrossRefGoogle Scholar
  24. 24.
    Meremianin A.V.: Body frames in the separation of collective angles in quantum n-body problems. J. Chem. Phys. 120(17), 7861–7876 (2004)CrossRefGoogle Scholar
  25. 25.
    Yanao T.M., Takatsuka K.: Kinematic effects associated with molecular frames in structural isomerization dynamics of clusters. J. Chem. Phys. 120(19), 8924–8936 (2004)CrossRefGoogle Scholar
  26. 26.
    Califano S.: Vibrational States. (Wiley, London 1976)Google Scholar
  27. 27.
    Goldstein H., Poole C., Safko J.: Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco 2002)Google Scholar
  28. 28.
    Kay D.C.: Tensor Calculus (Schaum’s outline series). (McGraw-Hill, New York 1988)Google Scholar
  29. 29.
    Noguti T., Go N.: A method of rapid calculation of a second derivative matrix of conformational energy for large molecules. J. Phys. Soc. Jpn. 52(10), 3685–3690 (1983)CrossRefGoogle Scholar
  30. 30.
    Braun W., Yoshioki S., Go N.: Formulation of static and dynamic conformational energy analysis of biopolymer systems consisting of two or more molecules. J. Phys. Soc. Jpn. 53(9), 3269 (1984)CrossRefGoogle Scholar
  31. 31.
    Hestenes D.: New Foundations for Classical Mechanics, 2nd edn. (Kluwer Academic Publishers, Dordrecht 1999)Google Scholar
  32. 32.
    Wei H.: Eckart frames for planar molecules. J. Chem. Phys. 118, 7202 (2003)CrossRefGoogle Scholar
  33. 33.
    Kudin K.N., Dymarsky A.Y.: Eckart axis conditions and the minimization of the root-mean-square deviation: Two closely related problems. J. Chem. Phys. 122, 224105 (2005)CrossRefGoogle Scholar
  34. 34.
    Pesonen J., Henriksson K.O.E.: Polymer conformations in internal (polyspherical) coordinates. J. Comput. Chem. 31(9), 1873–1881 (2010)Google Scholar
  35. 35.
    Henriksson K.O.E., Pesonen J.: Polymer dynamics in torsion space. J. Comput. Chem. 31(9), 1882–1888 (2010)Google Scholar
  36. 36.
    Pesonen J.: Kinetic energy operators in linearized internal coordinates. J. Chem. Phys. 128(4), 044319 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Janne Pesonen
    • 1
  • Krister O. E. Henriksson
    • 2
  • Jose Ramon López-Blanco
    • 3
  • Pablo Chacón
    • 3
  1. 1.Department of ChemistryUniversity of HelsinkiHelsinkiFinland
  2. 2.Department of PhysicsUniversity of HelsinkiHelsinkiFinland
  3. 3.Department of Biological Physical ChemistryRocasolano Physical Chemistry Institute, CSICMadridSpain

Personalised recommendations