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 x. (b) Normal mode analysis: harmonic approximation of the potential energy surface. For any biological systems, 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 analysis consists of describing motions of a biological molecule in normal mode coordinates q, which requires transforming Cartesian coordinates into normal mode coordinates. In this example of two coordinate system, normal mode coordinates are independent, but not Cartesian coordinates. (d) Simple normal mode vector for the water molecule. Each arrow represents the direction of motion each atom will undergo as obtained from normal mode theory. Normal mode predicts three distinct motions for the water molecule: symmetric and asymmetric stretching mode and a bending mode, which is in agreement with experimental observation.

Figure 3. Normal mode analysis applied to a biological molecule: in such a case, a large number of modes (3n–6) with n being the number of atoms is obtained. As with the water molecule, stretching motions are observed and such motions have a very high-frequency (spring with a high force constant); those are fast motions with small amplitude of displacement, and they are localized. On the other hand, biological molecules also experience very-low frequency (spring with a small force constant) motions, which correspond to slow motions with large displacements and involve a large number of atoms. Representative atomic displacements for these motions are shown for two proteins. The arrows represent the direction and amplitude of motions of an atom. The lowest frequency normal mode of the adenylate kinase shows that most of the atoms are moving with a large amplitude, while a high-frequency normal mode of the cytochrome c reveals that only very few atoms are moving together, with a very small amplitude of displacement.

Figure 4. Conformational change pathway of the maltose binding protein from its open conformation to closed one. Starting with the open conformation, the protein was iteratively displaced using a combination of normal modes to generate the intermediate structures. This approach gives tentative models that could be tested experimentally.