Section 22.1.1.5. Application of geometry calculations: the measurement of packing

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).

Figure 22.1.1.8 | top | pdf | Packing efficiency. (a) The relationship between Voronoi polyhedra and packing efficiency. Packing efficiency is defined as the volume of an object as a fraction of the space that it occupies. (It is also known as the `packing coefficient' or `packing density'.) In the context of molecular structure, it is measured by the ratio of the VDW volume (, shown by a light grey line) and Voronoi volume (, shown by a dotted line). This calculation gives absolute packing efficiencies. In practice, one usually measures a relative efficiency, relative to the atom in a reference state: . Note that in this ratio the unchanging VDW volume of an atom cancels out, leaving one with just a ratio of two Voronoi volumes. Perhaps more usefully, when one is trying to evaluate the packing efficiency P at an interface, one computes , where p is packing efficiency of the reference data set (usually 0.74), is the actual measured volume of each atom i at the interface and is the reference volume corresponding to the type of atom i. (b) A graphical illustration of the difference between tight packing and loose packing. Frames from a simulation are shown for liquid water (left) and for liquid argon, a simple liquid (right). Owing to its hydrogen bonds, water is much less tightly packed than argon (packing efficiency of 0.35 versus ∼0.7). Each water molecule has only four to five nearest neighbours while each argon atom has about ten.

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).

Table 22.1.1.4 | top | pdf | Standard atomic volumes Tsai et al. (1999) and Tsai et al. (2001) clustered all the atoms in proteins into the 18 basic types shown below. Most of these have a simple chemical definition, e.g. `=O' are carbonyl carbons. However, some of the basic chemical types, such as the aromatic CH group (`'), need to be split into two subclusters (bigger and smaller), as is indicated by the column labelled `Cluster'. Volume statistics were accumulated for each of the 18 types based on averaging over 87 high-resolution crystal structures (in the same fashion as described for the residue volumes in Table 22.1.1.3). No. is the number of atoms averaged over. The final column (`Symbol') gives the standardized symbol used to describe the atom in Tsai et al. (1999). The atom volumes shown here are part of the ProtOr parameter set (also known as the BL+ set) in Tsai et al. (1999). Atom type Cluster Description Average volume (Å3) Standard deviation (Å3) No. Symbol >C= Bigger Trigonal (unbranched), aromatics 9.7 0.7 4184 C3H0b >C= Smaller Trigonal (branched) 8.7 0.6 11876 C3H0s Bigger Aromatic, CH (facing away from main chain) 21.3 1.9 2063 C3H1b Smaller Aromatic, CH (facing towards main chain) 20.4 1.7 1742 C3H1s >CH— Bigger Aliphatic, CH (unbranched) 14.4 1.3 3642 C4H1b >CH— Smaller Aliphatic, CH (branched) 13.2 1.0 7028 C4H1s —CH2— Bigger Aliphatic, methyl 24.3 2.1 1065 C4H2b —CH2— Smaller Aliphatic, methyl 23.2 2.3 4228 C4H2s —CH3   Aliphatic, methyl 36.7 3.2 3497 C4H3u >N—   Pro N 8.7 0.6 581 N3H0u >NH Bigger Side chain NH 15.7 1.5 446 N3H1b >NH Smaller Peptide 13.6 1.0 10016 N3H1s —NH2   Amino or amide 22.7 2.1 250 N3H2u   Amino, protonated 21.4 1.2 8 N4H3u =O   Carbonyl oxygen 15.9 1.3 7872 O1H0u —OH   Alcoholic hydroxyl 18.0 1.7 559 O2H1u —S—   Thioether or –S–S– 29.2 2.6 263 S2H0u —SH   Sulfhydryl 36.7 4.2 48 S2H1u

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)

The protein core is usually considered to be the atoms inaccessible to solvent i.e. with an accessible surface area of zero or a very small number, such as 0.1 Ų. Packing calculations on the protein core are usually done by calculating the average volumes of the buried atoms and residues in a database of crystal structures. These calculations were first done more than two decades ago (Chothia & Janin, 1975; Finney, 1975; Richards, 1974). The initial calculations revealed some important facts about protein structure. Atoms and residues of a given type inside proteins have a roughly constant (or invariant) volume. This is because the atoms inside proteins are packed together fairly tightly, with the protein interior better resembling a close-packed solid than a liquid or gas. In fact, the packing efficiency of atoms inside proteins is roughly as expected for the close packing of hard spheres (0.74).

More recent calculations measuring the packing in proteins (Harpaz et al., 1994; Tsai et al., 1999) have shown that the packing inside of proteins is somewhat tighter (by ∼4%) than that observed initially and that the overall packing efficiency of atoms in the protein core is greater than that in crystals of organic molecules. When molecules are packed this tightly, small changes in packing efficiency are quite significant. In this regime, the limitation on close packing is hard-core repulsion, which is expected to have a twelfth power or exponential dependence, so even a small change is energetically quite substantial. Furthermore, the number of allowable configurations that a collection of atoms can assume without core overlap drops off very quickly as these atoms approach the close-packed limit (Richards & Lim, 1994).

The exceptionally tight packing in the protein core seems to require a precise jigsaw puzzle-like fit of the residues. This appears to be the case for the majority of atoms inside of proteins (Connolly, 1986). The tight packing in proteins has, in fact, been proposed as a quality measure in protein crystal structures (Pontius et al., 1996). It is also believed to be a strong constraint on protein flexibility and motions (Gerstein et al., 1993; Gerstein, Lesk & Chothia, 1994). However, there are exceptions, and some studies have focused on these, showing how the packing inside proteins is punctuated by defects, or cavities (Hubbard & Argos, 1994, 1995; Kleywegt & Jones, 1994; Kocher et al., 1996; Rashin et al., 1986; Richards, 1979; Williams et al., 1994). If these defects are large enough, they can contain buried water molecules (Baker & Hubbard, 1984; Matthews et al., 1995; Sreenivasan & Axelsen, 1992).

Surprisingly, despite the intricacies of the observed jigsaw puzzle-like packing in the protein core, it has been shown that one can simply achieve the `first-order' aspect of this, getting the overall volume of the core right rather easily (Gerstein, Sonnhammer & Chothia, 1994; Kapp et al., 1995; Lim & Ptitsyn, 1970). This has to do with simple statistics for summing random numbers and the fact that the distribution of sizes for amino acids usually found inside proteins is rather narrow (Table 22.1.1.3). In fact, the similarly sized residues Val, Ile, Leu and Ala (with volumes 138, 163, 163 and 89 ų ) make up about half of the residues buried in the protein core. Furthermore, aliphatic residues, in particular, have a relatively large number of adjustable degrees of freedom per  ų, allowing them to accommodate a wide range of packing geometries. All of this suggests that many of the features of protein sequences may only require random-like qualities for them to fold (Finkelstein, 1994).

Measuring the packing efficiency inside the protein core provides a good reference point for comparison, and a number of other studies have looked at this in comparison with other parts of the protein. The most obvious thing to compare with the protein inside is the protein outside, or surface. This is particularly interesting from a packing perspective, since the protein surface is covered by water, and water is packed much less tightly than protein and in a distinctly different fashion. (The tetrahedral packing geometry of water molecules gives a packing efficiency of less than half that of hexagonal close-packed solids.)

Calculations based on crystal structures and simulations have shown that the protein surface has intermediate packing, being packed less tightly than the core but not as loosely as liquid water (Gerstein & Chothia, 1996; Gerstein et al., 1995). One can understand the looser packing at the surface than in the core in terms of a simple trade-off between hydrogen bonding and close packing, and this can be explicitly visualized in simulations of the packing in simple toy systems (Gerstein & Lynden-Bell, 1993a,b).