Principles of Protein Structure - G. Schultz 1982

Interactions Determining Protein Structure
Molecular Packing

The energy of Hydrogen Bonds is utilized efficiently. Following a General Overview of the non-covalent forces contributing to the Stability of the folded polypeptide chain, we will attempt to evaluate their actual efficacy. Studies of protein structures show that approximately 90% of all internal polar groups form hydrogen bonds [17]. This is consistent with the high content of Secondary Structure observed in Proteins (Chapter 5) and indicates that virtually all the potential hydrogen-bonding energy is indeed expended on its formation.

High packing density is consistent with the significant role of dispersion forces. The efficiency of dispersion forces can be judged by the interior packing density of a protein, as packing density reflects the number of existing contacts. Packing density is defined as The ratio of the intrinsic molecular volume (the volume enclosed within the Van der Waals envelope of the molecule, which in turn is determined by the van der Waals radii of the surface atoms) to the volume actually occupied by the molecule in space. In crystals of small molecules, the actually occupied volume is defined by the unit Cell, while the volume of the molecule within the van der Waals envelope can be determined from the atomic coordinates in the crystal and their van der Waals radii.

To determine the overall packing density of a protein molecule, the van der Waals volume of the extended chain should be divided by the volume of the folded chain, i.e., the volume of the native protein molecule. However, it is advisable first to determine the local packing density, which may vary within a protein depending on specific structural features. By integrating over all local packing densities, one can then obtain the average packing density of the protein.

The Voronoi polyhedron allows for the determination of local packing density. Local packing densities were first determined by Richards [63]. For this purpose, the polypeptide chain was divided into small atomic groups, such as methyl, methylene, amide, hydroxyl, and others containing one heavier atom and up to three hydrogen atoms. Aromatic rings and the guanidinium group of Arg were also treated as separate groups. The centers of these groups were determined from X-Ray Diffraction data. Space was then partitioned into Voronoi polyhedra, as shown in Fig. 3.4. The local packing density at a given center is the ratio of the atomic group volume to its corresponding occupied volume, i.e., the volume of the surrounding Voronoi polyhedron. To avoid errors associated with surface accounting, a monolayer of H₂O molecules located On the surface was included in the consideration. Thus, only protein atoms, but not the actual H₂O molecules, were included in the full Voronoi polyhedron. This is illustrated in Fig. 3.4, where A is the center of a protein atom, and B, C, and D are the centers of actual H₂O molecules (see also Fig. 1.9). This Procedure simultaneously allows for the Determination of the molecular surface area. It is best to use the monolayer of surface Water molecules that contains their maximum number. However, since determining such an optimal layer is difficult, the chosen layer typically contains fewer water molecules than the optimal number. The resulting error introduced by this is negligible.

Proteins are packed as densely as good molecular crystals. The observed local packing densities in proteins range from 0.68 to 0.82. Low density is found in active sites [63, 64], which Supports the hypothesis of Active Site mobility. Hydrophobic cores in the center of the protein exhibit high density*. The average packing density of a protein is about 0.75 (the packing density of regular hard spheres is 0.74). Crystals of small molecules held together by van der Waals forces typically feature values from 0.70 to 0.78, averaging 0.74. Glasses, oils, or exceptionally soft van der Waals crystals (or certain crystals built through directional bonds, such as the hydrogen bonds of ordinary ice, Fig. 3.2) have packing densities below 0.70 and even below 0.60. Thus, proteins are indeed packed as densely as small molecules in molecular crystals; naturally, dispersion energy must play a major role for them.

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Fig. 3.4. Partitioning of space into polyhedra around atoms.

For a given set of atoms, space can be partitioned into irregular polyhedra with atoms at their vertices; one such polyhedron is shown in this figure. As Voronoi (1908) showed, the four bisecting planes between each pair of atoms intersect at a single point—vertex V. These planes divide the tetrahedron into four unequal parts. The part adjacent to atom A is highlighted by the hatched planes. A given atom A can belong to an arbitrary number of tetrahedra. To determine the total polyhedron around A, the construction described above must be performed for all such tetrahedra, and all parts adjacent to A must be summed. After repeating this procedure for all atoms (surface atoms of the set are excluded), space is completely partitioned. The packing density of a given atom is defined as the ratio of the volume within the van der Waals envelope to the volume of the surrounding polyhedron. Richards [63] extended this procedure to protein molecules, which contain several types of atoms with different van der Waals radii. In this case, the Voronoi procedure, which yields exact results for identical atoms, is inapplicable in its original form. In its modified form, which yields satisfactory results, bisecting planes are drawn through points that divide the distance between atoms, for example between A and B, proportionally to the ratio of the van der Waals radii of A to B or, if A and B are covalently bonded, in accordance with the ratio of their covalent radii. As a result of such displacements, the four planes no longer intersect at point V, but form a small tetrahedron near V. The relative volume of this "error" tetrahedron is only about 1%, and it can be neglected unless the radius ratio exceeds 1.5. To keep these ratios at the level of 1.0, Richards incorporated all (small) H atoms into the atomic groups.

* Packing density must not be confused with electron density or mass density. Hydrophobic cores, like all Hydrocarbons, have a significantly lower mass density than polar regions.

The observed packing densities indicate that Kauzmann's oil-drop metaphor should not be taken literally. The interior of a protein is more akin to a crystal. This is also supported by the low compressibility of proteins compared to oil [65, 66]: k_oil/k_protein ≈ 20.

Packing is conserved during evolution. METABOLISM/2.html">THE CONCEPT OF tight protein packing and, consequently, the steric constraints associated with it must be reconciled with The formation of A large number of directional hydrogen bonds by virtually all polar groups within its molecule. In this regard, It is worth recalling that in ordinary ice, it is precisely the formation of linear hydrogen bonds that results in a packing density of only 0.58. The high structural optimization in proteins confirms the hypothesis that evolutionary changes have affected the protein interior far less frequently than its surface. Furthermore, There is a distinct tendency toward mutual compensation of internal changes [67], for example, an Ile → Val substitution may be accompanied by a neighboring Gly → Ala change that restores the lost methyl group.

Packing density can serve as a criterion in structure prediction. Packing density provides an opportunity to independently verify the results of computational modeling of chain folding (Section 8.6) and structure prediction from known Amino acid sequences by comparison with phylogenetically related proteins (Chapter 9). This criterion is not limited merely to comparisons with average packing densities. The volume occupied by side chains (the sum of the volumes of all atomic groups constituting a given chain) is specific to each side chain [63, 68] and varies by only approximately 5%. Therefore, the volume occupied by each side chain can be utilized as an additional criterion in Protein Structure Prediction.