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 PesonenEmail author
  • Krister O. E. Henriksson
  • Jose Ramon López-Blanco
  • Pablo Chacón
Original Paper


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.


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


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Janne Pesonen
    • 1
    Email author
  • 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

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