International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 22.1, pp. 537-538
|
Volume calculations are principally applied in measuring packing. This is because the packing efficiency of a given atom is simply the ratio of the space it could minimally occupy to the space that it actually does occupy. As shown in Fig. 22.1.1.8, this ratio can be expressed as the VDW volume of an atom divided by its Voronoi volume (Richards, 1974, 1985; Richards & Lim, 1994). (Packing efficiency also sometimes goes by the equivalent terms `packing density' or `packing coefficient'.) This simple definition masks considerable complexities – in particular, how does one determine the volume of the VDW envelope (Petitjean, 1994)? This requires knowledge of what the VDW radii of atoms are, a subject on which there is not universal agreement (see above), especially for water molecules and polar atoms (Gerstein et al., 1995; Madan & Lee, 1994).
Knowing that the absolute packing efficiency of an atom is a certain value is most useful in a comparative sense, i.e. when comparing equivalent atoms in different parts of a protein structure. In taking a ratio of two packing efficiencies, the VDW envelope volume remains the same and cancels. One is left with just the ratio of space that an atom occupies in one environment to what it occupies in another. Thus, for the measurement of packing, standard reference volumes are particularly useful. Recently calculated values of these standard volumes are shown in Tables 22.1.1.3 and 22.1.1.4 for atoms and residues (Tsai et al., 1999).
|
|
In analysing molecular systems, one usually finds that close packing is the default (Chandler et al., 1983), i.e. atoms pack like billiard balls. Unless there are highly directional interactions (such as hydrogen bonds) that have to be satisfied, one usually achieves close packing to optimize the attractive tail of the VDW interaction. Close-packed spheres of the same size have a packing efficiency of ∼0.74. Close-packed spheres of different size are expected to have a somewhat higher packing efficiency. In contrast, water is not close-packed because it has to satisfy the additional constraints of hydrogen bonding. It has an open, tetrahedral structure with a packing efficiency of ∼0.35. (This difference in packing efficiency is illustrated in Fig. 22.1.1.8b)
References
Chandler, D., Weeks, J. D. & Andersen, H. C. (1983). van der Waals picture of liquids, solids, and phase transformations. Science, 220, 787–794.Google ScholarGerstein, M., Tsai, J. & Levitt, M. (1995). The volume of atoms on the protein surface: calculated from simulation, using Voronoi polyhedra. J. Mol. Biol. 249, 955–966.Google Scholar
Madan, B. & Lee, B. (1994). Role of hydrogen bonds in hydrophobicity: the free energy of cavity formation in water models with and without the hydrogen bonds. Biophys. Chem. 51, 279–289.Google Scholar
Petitjean, M. (1994). On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects. J. Comput. Chem. 15, 1–10.Google Scholar
Richards, F. M. (1974). The interpretation of protein structures: total volume, group volume distributions and packing density. J. Mol. Biol. 82, 1–14.Google Scholar
Richards, F. M. (1985). Calculation of molecular volumes and areas for structures of known geometry. Methods Enzymol. 115, 440–464.Google Scholar
Richards, F. M. & Lim, W. A. (1994). An analysis of packing in the protein folding problem. Q. Rev. Biophys. 26, 423–498.Google Scholar
Tsai, J., Taylor, R., Chothia, C. & Gerstein, M. (1999). The packing density in proteins: standard radii and volumes. J. Mol. Biol. 290, 253–266.Google Scholar