Section 22.1.2.3. Estimation of binding energies

As previously introduced, hydrophobic energies result primarily from the increased entropy of water molecules at the macromolecule–solvent interface and can be estimated from the accessible surface area. A number of different constants relating area to free energy of transfer from a hydrophobic to aqueous environment have been proposed in the range of 67 to 130 J mol⁻¹ Å⁻² (Reynolds et al., 1974; Chothia, 1976; Hermann, 1977; Eisenberg & McLachlan, 1986), but if a single value is to be used for all of the protein surface, the consensus among crystallographers has been about 80 J mol⁻¹ Å⁻² (Richards, 1985).

There are two widely used enhancements of the basic method. Atomic solvation parameters (ASPs, Δσ) remove the assumption that all protein atoms have equal potential influence on the hydrophobic free energy. Eisenberg & McLachlan (1986) determined separate ASPs for atom types C, N/O, O^.., N⁺ and S (treating hydrogen atoms implicitly) by fitting these constants to the experimentally determined octanol/water relative transfer free energies of the 20 amino-acid side chains of Fauchere & Pliska (1983), assuming standard conformations of the side chains. A much improved free energy change of solvation can then be estimated from , where the summation is over all atoms with accessible area A and is specific for the atom type. Their estimates of ASPs are given in Table 22.1.2.1. Use of ASPs rather than a single value for all atoms makes substantial differences to the estimated free energies of association of macromolecular assemblies (Xie & Chapman, 1996). Through calculation of the overall energy of solvation, calculations with ASPs also allow discrimination between proposed structures that are correctly folded (with hydrophobic side chains that are predominantly internal) and those that are not (Eisenberg & McLachlan, 1986).

The work of Sharp et al. (1991) indicates that hydrophobicity depends not only on surface area, but curvature. Sharp et al. were trying to reconcile long-apparent differences between microscopic and macroscopic measurements of hydrophobicity (Tanford, 1979). Microscopic measurements, the basis of all of our preceding discussions, are derived from the partitioning of dilute solutes between solvents. Macroscopic values can come from the measurements of the surface tension between a liquid bulk of the molecule of interest and water. Macroscopic values for aliphatic carbons are much higher, ~302 J mol⁻¹ Å⁻². Postulating that the entropic effects at the heart of hydrophobicity depended on the number of water molecules in contact with each other at the molecular surface (Nicholls et al., 1991), Sharp et al. pointed out that not all surfaces were equivalent. Relative to a plane, concave solute surfaces would accommodate fewer solvent molecules neighbouring the molecular surface, whereas convex surfaces would accommodate more. Their treatment could be considered to be a second-order approximation to the number of interfacial solvent molecules, compared to the prior first-order consideration of only area.

To calculate the curvature of point a on the accessible surface (relative to that of a plane), a sphere of twice the solvent radius is drawn (Nicholls et al., 1991). This represents the locus of the centres of solvent molecules that could be in contact with a solvent at a. A curvature correction, c, is the proportion of points on the spherical surface that are inside the inaccessible volume, relative to that for a planar accessible surface (). In calculating the free energy of transfer, each element of the accessible area is multiplied by its curvature correction. When this is done, the increasingly convex surfaces of small aliphatic molecules account for most of the discrepancy between microscopic and macroscopic hydrophobicities (Nicholls et al., 1991). Furthermore, it emphasizes that, just by their shape, concave surfaces can become relatively hydrophobic. This has been clearly illustrated with GRASP surface representations (see below) in which the accessible surface is coloured according to the local curvature (Nicholls et al., 1991). Consideration of curvature also indicates that the energy of macromolecular association is slightly less than it would otherwise be due to the generation of a concave collar at the interface between two binding macromolecules (Nicholls et al., 1991).

In a molecular association in which (as is often the case) hydrophobic interactions dominate, the binding energy can be estimated from the surfaces of the individual molecules that become buried upon association (Richards, 1985). The buried area is simply the sum of the surfaces of the two molecules (calculated independently) minus the surface of the complex, calculated as if one molecule. Usually, all heteroatoms are regarded as equivalent, and the buried area is multiplied by a uniform constant, say 80 J mol⁻¹ Å⁻² (Richards, 1985). It is only slightly more complicated to use the different ASPs (Eisenberg & McLachlan, 1986) for different atom types and/or to account for curvature (Nicholls et al., 1991). It should be noted that in many crystal structures, the distinction between atom types in some side chains remains indeterminate, e.g. N and C in histidines, O and O^.. in carboxylates, and N and N⁺ in arginines. In such cases, average values of the two ASPs can be used (Xie & Chapman, 1996). Such energy calculations have been put to several uses, including attempts to predict assembly and disassembly pathways for viral capsid assemblies (Arnold & Rossmann, 1990; Xie & Chapman, 1996, and citations therein).

Which are the amino acids most likely to interact with other molecules? It is reasonable to expect them to be surface-accessible. In determining which residues are most surface-exposed, it is necessary to partition molecular or accessible surfaces between atoms. Contact surfaces (Fig. 22.1.2.1) are atom specific. Re-entrant or accessible surfaces can be divided among surface atoms by proximity. Surface areas can then be summed over the atoms in a residue. Accessible surface areas are sometimes reported as accessibilities (Lee & Richards, 1971) – fractions of a maximum where the standard is evaluated from a tripeptide in which the residue of interest is surrounded by glycines. A different approach to assessing surface exposure is to ask what is the largest molecular fragment that could contact a given atom. This is commonly assayed by determining the largest sphere that can be placed tangentially to the van der Waals surface without intersecting any other atom. An alternative approach to locating functionally important surface regions was proposed in the mid-1980s, but is currently not used very often. The local irregularity of surface texture was characterized through measurement of the fractal dimension (Lewis & Rees, 1985).

Substrates, drugs and ligands often bind in clefts or pockets that are concave in shape. Conversely, it is the most exposed convex regions that are likely to be antigenic. The surface shape can be determined by placing a large (say 6 Å radius) sphere at each vertex of the polyhedral molecular surface. If more than half of the sphere's volume overlaps the molecular volume, then the surface is concave, while if less than half, the surface is convex.

Are there similarities in the shapes of surfaces at the interfaces of macromolecular complexes? For example, are there similarities between the shapes of evolutionary-related antigens or the hypervariable regions of antibodies that bind to them? Quantitative comparison of surface topologies is far from trivial, with questions of 3D alignment, the metrics to be used in quantifying topology etc. In addition to real differences between molecules, their surfaces may appear to differ due to the resolutions at which their structures were determined. Gerstein (1992) has proposed that comparisons be made in reciprocal space so that correlations can be judged as a function of resolution. Coordinates are aligned. Spherical Gaussian functions are placed at each atom, and an envelope is calculated at some threshold value and modified to remove cavities. Gerstein found that comparison of the envelope structure-factor vectors, obtained by Fourier transformation, led to a plausible classification of the hypervariable regions of known antibody structures.