Normal Mode Analysis Techniques in Structural Biology

Abstract

The dynamic simulation of macromolecular systems with biologically relevant sizes and time scales is critical for understanding macromolecular function. In this context, normal mode analysis (NMA) approximates the complex dynamical behaviour of a macromolecule as a simple set of harmonic oscillators vibrating around a given equilibrium conformation. This technique, originated from classical mechanics, was first applied to investigate the dynamical properties of small biological systems more than 30 years ago. During this time, a wealth of evidence has accumulated to support NMA as a successful tool for simulating macromolecular motions at extended length scales. Today, NMA combined with coarse‐grained representations has become an efficient alternative to molecular dynamics simulations for studying the slow and large‐amplitude motions of macromolecular machines. Interesting insights into these systems can be obtained very quickly with NMA to characterise their flexibility, to predict the directions of their collective conformational changes, or to help in the interpretation of experimental structural data. The recently developed methods and applications of NMA together with an introduction of the underlying theory will be briefly reviewed here.

Key Concepts:

  • NMA computes all motions around an equilibrium conformation (normal modes).

  • Normal modes are orthogonal displacement vectors with an associated frequency.

  • The collective functional motions of the macromolecules are well described by lowest frequency modes.

  • NMA inexpensive and accurately simulates the slow and large‐amplitude motions of biomolecules, but local reorganisations and the absolute time scale or amplitude of the motions are poorly predicted.

  • NMA can be used to characterise macromolecular flexibility, to predict the directions of collective conformational changes, and to interpret structural experimental data.

Keywords: normal mode analysis; dynamic simulation; conformational changes; X‐ray/NMR structure; coarse-grained

Figure 1.

Dynamics of biological molecules and computational approaches that can be used to investigate dynamical properties.

Figure 2.

Normal mode theory. (a) Description of a simple harmonic oscillator: A particle m is attached to a spring with a force constant k and its displacement is x. (b) Normal mode analysis: harmonic approximation of the potential energy surface. For any biological system, the real energy surface is rugged (dotted line) but for the normal mode analysis, the surface is approximated as a harmonic surface (plain line). (c) Normal mode coordinates are independent (uncoupled) but not Cartesian coordinates (coupled). Although the contour lines represent the equipotential points of a parabolic force field in a two‐dimensional space, the blue and red axis correspond to Cartesian (X) and normal mode (q) systems of coordinates, respectively. When some particle is released at any of the normal mode axes, its trajectory stays on this axis. In contrast, when the particle is released at some other point, its motion needs to be described by both Cartesian axes. (d) Simple normal mode vectors for the water molecule. Each arrow represents the direction of motion that each atom will undergo as obtained from normal mode theory. The three distinct motions predicted by NMA for the water molecule, i.e. symmetric and asymmetric stretching modes plus a bending mode, are in agreement with experimental observations.

Figure 3.

Normal mode analysis applied to a biological macromolecule. Representative atomic displacements corresponding to collective and local motions are shown for the adenylate kinase protein (chain A from PDB ID 4ake), the enzyme that catalyses the reaction: 2ADP↔ATP+AMP. Numbers are the mode indices sorted from low to high frequencies. Whereas the arrows represent the direction and relative amplitude of the motions, the different colours indicate the regions that are moving together. In the lowest frequency normal modes (collective motions), large groups of atoms experience a concerted motion, whereas in higher‐frequency modes (local modes), only some small groups are moving together (local motions). It is worth noting that the arrow lengths have been normalised for visualisation purposes, otherwise local motions should be smaller. These images were generated with the iMODS server (López‐Blanco et al., ) (http://imods.chaconlab.org).

Figure 4.

Conformational open‐to‐closed transition pathway of the GroEL protein based on NMA. Starting from the open monomer (chain A from PDB ID 1sx4) (coloured structure at top left corner), a combination of low‐frequency modes were used iteratively to generate the intermediate structures (middle and bottom rows). Only those modes that reduced the differences with the closed conformation (chain H from PDB ID 1sx4) (grey) were employed. This transition was generated with the morphing tool of iMODS server (http://imods.chaconlab.org).

Figure 5.

Flexible fitting of an atomic structure into a low‐resolution density map based on NMA. The high‐resolution structure of the thermosome (PDB ID 1a6d) (rainbow colours) has been flexibly fitted into a low‐resolution cryo‐Electron Microscopy density map (EMDB 1396) (grey transparency) with iMODFIT (Lopéz‐Blanco and Chacón, ) by using the low‐frequency modes to maximise the overlap between the map and the structure.

close

References

Atilgan AR, Durell SR, Jernigan RL et al. (2001) Anisotropy of fluctuation dynamics of proteins with an elastic network model. Biophysical Journal 80: 505–515.

Bahar I, Atilgan AR and Erman B (1997) Direct evaluation of thermal fluctuations in proteins using a single‐parameter harmonic potential. Folding and Design 2: 173–181.

Bahar I, Lezon TR, Bakan A and Shrivastava IH (2010a) Normal mode analysis of biomolecular structures: functional mechanisms of membrane proteins. Chemical Reviews 110: 1463–1497.

Bahar I, Lezon TR, Yang LW and Eyal E (2010b) Global dynamics of proteins: bridging between structure and function. Annual Review of Biophysics 39: 23–42.

Bahar I and Rader AJ (2005) Coarse‐grained normal mode analysis in structural biology. Current Opinion in Structural Biology 15: 586–592.

Bastolla U (2014) Computing protein dynamics from protein structure with elastic network models. Wiley Interdisciplinary Reviews: Computational Molecular Science 4(5): 488–503.

Brooks B and Karplus M (1983) Harmonic dynamics of proteins: normal modes and fluctuations in bovine pancreatic trypsin inhibitor. Proceedings of the National Academy of Sciences of the USA 80: 6571–6575.

Chacón P, Tama F and Wriggers W (2003) Mega‐Dalton biomolecular motion captured from electron microscopy reconstructions. Journal of Molecular Biology 326: 485–492.

Cui Q and Bahar I (2010) Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems. Boca Raton, FL: Chapman & Hall/CRC.

Eyal E, Dutta A and Bahar I (2011) Cooperative dynamics of proteins unraveled by network models. Wiley Interdisciplinary Reviews: Computational Molecular Science 1: 426–439.

Flores SC, Keating KS, Painter J et al. (2008) HingeMaster: normal mode hinge prediction approach and integration of complementary predictors. Proteins: Structure, Function and Genetics 73: 299–319.

Go N, Noguti T and Nishikawa T (1983) Dynamics of a small globular protein in terms of low‐frequency vibrational modes. Proceedings of the National Academy of Sciences of the USA 80: 3696–3700.

Goldstein H, Poole CP and Safko JL (2002) Classical Mechanics, 3rd edn. San Francisco, CA: Addison-Wesley.

Hinsen K, Petrescu AJ, Dellerue S, Bellissent‐Funel MC and Kneller GR (2000) Harmonicity in slow protein dynamics. Chemical Physics 261: 25–37.

Ingólfsson HI, Lopez CA, Uusitalo JJ et al. (2014) The power of coarse graining in biomolecular simulations. Wiley Interdisciplinary Reviews: Computational Molecular Science 4: 225–248.

Kovacs JA, Chacón P and Abagyan R (2004) Predictions of protein flexibility: first‐order measures. Proteins: Structure, Function and Genetics 56: 661–668.

Krebs WG, Alexandrov V, Wilson CA et al. (2002) Normal mode analysis of macromolecular motions in a database framework: Developing mode concentration as a useful classifying statistic. Proteins: Structure, Function and Genetics 48: 682–695.

Levitt M, Sander C and Stern PS (1985) Protein normal‐mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. Journal of Molecular Biology 181: 423–447.

López‐Blanco JR, Aliaga JI, Quintana‐Ortí ES and Chacón P (2014) iMODS: internal coordinates normal mode analysis server. Nucleic Acids Research 42: W271–W276.

Lopéz‐Blanco JR and Chacón P (2013) iMODFIT: Efficient and robust flexible fitting based on vibrational analysis in internal coordinates. Journal of Structural Biology 184: 261–270.

Lopéz‐Blanco JR, Garzón JI and Chacón P (2011) iMod: Multipurpose normal mode analysis in internal coordinates. Bioinformatics 27: 2843–2850.

López‐Blanco JR, Reyes R, Aliaga JI et al. (2013) Exploring large macromolecular functional motions on clusters of multicore processors. Journal of Computational Physics 246: 275–288.

Lu M and Ma J (2005) The role of shape in determining molecular motions. Biophysical Journal 89: 2395–2401.

Ma J (2005) Usefulness and limitations of normal mode analysis in modeling dynamics of biomolecular complexes. Structure 13: 373–380.

Maguid S, Fernández‐Alberti S, Parisi G and Echave J (2006) Evolutionary conservation of protein backbone flexibility. Journal of Molecular Evolution 63: 448–457.

Meireles L, Gur M, Bakan A and Bahar I (2011) Pre‐existing soft modes of motion uniquely defined by native contact topology facilitate ligand binding to proteins. Protein Science 20: 1645–1658.

Miyashita O, Gorba C and Tama F (2011) Structure modeling from small angle X‐ray scattering data with elastic network normal mode analysis. Journal of Structural Biology 173: 451–460.

Noguti T and Go N (1983) Dynamics of native globular proteins in terms of dihedral angles. Journal of the Physical Society of Japan 52: 3283–3288.

Orellana L, Rueda M, Ferrer‐Costa C et al. (2010) Approaching elastic network models to molecular dynamics flexibility. Journal of Chemical Theory and Computation 6: 2910–2923.

Rueda M, Chacón P and Orozco M (2007) Thorough validation of protein normal mode analysis: a comparative study with essential dynamics. Structure 15: 565–575.

Setny P and Zacharias M (2013) Elastic network models of nucleic acids flexibility. Journal of Chemical Theory and Computation 9: 5460–5470.

Stein A, Rueda M, Panjkovich A, Orozco M and Aloy P (2011) A systematic study of the energetics involved in structural changes upon association and connectivity in protein interaction networks. Structure 19: 881–889.

Tama F and Brooks CL III (2006) Symmetry, form, and shape: Guiding principles for robustness in macromolecular machines. Annual Review of Biophysics and Biomolecular Structure 35: 115–133.

Tama F, Gadea FX, Marques O and Sanejouand YH (2000) Building‐block approach for determining low‐frequency normal modes of macromolecules. Proteins: Structure, Function and Genetics 41: 1–7.

Tama F, Miyashita O and BrooksCL III (2004) Normal mode based flexible fitting of high‐resolution structure into low‐resolution experimental data from cryo‐EM. Journal of Structural Biology 147: 315–326.

Tama F and Sanejouand YH (2001) Conformational change of proteins arising from normal mode calculations. Protein Engineering 14: 1–6.

Tirion MM (1996) Large amplitude elastic motions in proteins from a single‐parameter, atomic analysis. Physical Review Letters 77: 1905–1908.

Van Vlijmen HWT and Karplus M (2005) Normal mode calculations of icosahedral viruses with full dihedral flexibility by use of molecular symmetry. Journal of Molecular Biology 350: 528–542.

Yang L, Song G and Jernigan RL (2009a) Comparisons of experimental and computed protein anisotropic temperature factors. Proteins: Structure, Function and Bioinformatics 76: 164–175.

Yang L, Song G and Jernigan RL (2009b) Protein elastic network models and the ranges of cooperativity. Proceedings of the National Academy of Sciences of the USA 106: 12347–12352.

Zacharias M (2010) Accounting for conformational changes during protein‐protein docking. Current Opinion in Structural Biology 20: 180–186.

Zheng W and Brooks BR (2006) Modeling protein conformational changes by iterative fitting of distance constraints using reoriented normal modes. Biophysical Journal 90: 4327–4336.

Zheng W, Brooks BR and Thirumalai D (2006) Low‐frequency normal modes that describe allosteric transitions in biological nanomachines are robust to sequence variations. Proceedings of the National Academy of Sciences of the USA 103: 7664–7669.

Further Reading

Dill KA and Bromberg S (2003) Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology. New York and London: Garland Science.

Contact Editor close
Submit a note to the editor about this article by filling in the form below.

* Required Field

How to Cite close
López‐Blanco, José Ramón, Miyashita, Osamu, Tama, Florence, and Chacón, Pablo(Oct 2014) Normal Mode Analysis Techniques in Structural Biology. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0020204.pub2]