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. 531539

For geometric analysis, a protein consists of a set of points in three dimensions. This information corresponds to the actual data provided by the experiment, which are fundamentally of a geometric rather than chemical nature. That is, crystallography primarily tells one about the positions of atoms and perhaps an approximate atomic number, but not their charge or number of hydrogen bonds.
For the purposes of geometric calculation, each point has an assigned identification number and a position defined by three coordinates in a righthanded Cartesian system. (These coordinates will be based on the electron density for Xray derived structures and on nuclear positions for those derived from neutron scattering. Each coordinate is usually assumed to have an accuracy between 0.5 and 1.0 Å.) Normally, only one additional characteristic is associated with each point: its size, usually measured by a van der Waals (VDW) radius. Furthermore, characteristics such as chemical nature and covalent connectivity, if needed, can be obtained from lookup tables keyed on the ID number.
Our model of a protein, thus, is the van der Waals envelope – the set of interlocking spheres drawn around each atomic centre. In brief, the geometric quantities of the model of particular concern in this section are its total surface area, total volume, the division of these totals among the aminoacid residues and individual atoms, and the description of the empty space (cavities) outside the van der Waals envelope. These values are then used in the analysis of protein structure and properties.
All the geometric properties of a protein (e.g. surfaces, volumes, distances etc.) are obviously interrelated. So the definition of one quantity, e.g. area, obviously impacts on how another, e.g. volume, can be consistently defined. Here, we will endeavour to present definitions for measuring protein volume, showing how they are related to various definitions of linear distance (VDW parameters) and surface. Further information related to macromolecular geometry, focusing on volumes, is available from http://www.molmovdb.org/geometry/ .
Protein volume can be defined in a straightforward sense through a particular geometric construction called the Voronoi polyhedron. In essence, this construction provides a useful way of partitioning space amongst a collection of atoms. Each atom is surrounded by a single convex polyhedron and allocated the space within it (Fig. 22.1.1.1). The faces of Voronoi polyhedra are formed by constructing dividing planes perpendicular to vectors connecting atoms, and the edges of the polyhedra result from the intersection of these planes.
Voronoi polyhedra were originally developed by Voronoi (1908) nearly a century ago. Bernal & Finney (1967) used them to study the structure of liquids in the 1960s. However, despite the general utility of these polyhedra, their application to proteins was limited by a serious methodological difficulty. While the Voronoi construction is based on partitioning space amongst a collection of `equal' points, all protein atoms are not equal. Some are clearly larger than others. In 1974, a solution was found to this problem (Richards, 1974), and since then Voronoi polyhedra have been applied to proteins.
The simplest method for calculating volumes with Voronoi polyhedra is to put all atoms in the system on a fine grid. Then go to each grid point (i.e. voxel) and add its infinitesimal volume to the atom centre closest to it. This is prohibitively slow for a real protein structure, but it can be made somewhat faster by randomly sampling grid points. It is, furthermore, a useful approach for highdimensional integration (Sibbald & Argos, 1990).
More realistic approaches to calculating Voronoi volumes have two parts: (1) for each atom find the vertices of the polyhedron around it and (2) systematically collect these vertices to draw the polyhedron and calculate its volume.
In the basic Voronoi construction (Fig. 22.1.1.1), each atom is surrounded by a unique limiting polyhedron such that all points within an atom's polyhedron are closer to this atom than all other atoms. Consequently, points equidistant from two atoms lie on a dividing plane; those equidistant from three atoms are on a line, and those equidistant from four atoms form a vertex. One can use this last fact to find all the vertices associated with an atom easily. With the coordinates of four atoms, it is straightforward to solve for possible vertex coordinates using the equation of a sphere. [That is, one uses four sets of coordinates (x, y, z) and the equation to solve for the centre (a, b, c) and radius (r) of the sphere.] One then checks whether this putative vertex is closer to these four atoms than any other atom; if so, it is a real vertex.
Note that this procedure can fail for certain pathological arrangements of atoms that would not normally be encountered in a real protein structure. These occur if there is a centre of symmetry, as in a regular cubic lattice or in a perfect hexagonal ring in a protein (see Procacci & Scateni, 1992). Centres of symmetry can be handled (in a limited way) by randomly perturbing the atoms a small amount and breaking the symmetry. Alternatively, the `choppingdown' method described below is not affected by symmetry centres – an important advantage to this method of calculation.
To collect the vertices associated with an atom systematically, label each one by the indices of the four atoms with which it is associated (Fig. 22.1.1.2). To traverse the vertices on one face of a polyhedron, find all vertices that share two indices and thus have two atoms in common, e.g. a central atom (atom 0) and another atom (atom 1). Arbitrarily pick a vertex to start at and walk around the perimeter of the face. One can tell which vertices are connected by edges because they will have a third atom in common (in addition to atom 0 and atom 1). This sequential walking procedure also provides a way of drawing polyhedra on a graphics device. More importantly, with reference to the starting vertex, the face can be divided into triangles, for which it is trivial to calculate areas and volumes (see Fig. 22.1.1.2 for specifics).
In the procedure outlined above, all atoms are considered equal, and the dividing planes are positioned midway between atoms (Fig. 22.1.1.3). This method of partition, called bisection, is not physically reasonable for proteins, which have atoms of obviously different size (such as oxygen and sulfur). It chemically misallocates volume, giving excess to the smaller atom.
Two principal methods of repositioning the dividing plane have been proposed to make the partition more physically reasonable: method B (Richards, 1974) and the radicalplane method (Gellatly & Finney, 1982). Both methods depend on the radii of the atoms in contact (R for the larger atom and r for the smaller one) and the distance between the atoms (D). As shown in Fig. 22.1.1.3, they position the plane at a distance d from the larger atom. This distance is always set such that the plane is closer to the smaller atom.
Method B is the more chemically reasonable of the two and will be emphasized here. For atoms that are covalently bonded, it divides the distance between the atoms proportionaly according to their covalentbond radii: For atoms that are not covalently bonded, method B splits the remaining distance between them after subtracting their VDW radii:
For separations that are not very different to the sum of the radii, the two formulae for method B give essentially the same result. Consequently, it is worthwhile to try a slight simplification of method B, which we call the `ratio method'. Instead of using equation (22.1.1.1) for bonded atoms and equation (22.1.1.2) for nonbonded ones, one can just use equation (22.1.1.2) in both cases with either VDW or covalent radii (Tsai et al., 2001). Doing this gives more consistent reference volumes (manifest in terms of smaller standard deviations about the mean).
If bisection is not used to position the dividing plane, it is much more complicated to find the vertices of the polyhedron, since a vertex is no longer equidistant from four atoms. Moreover, it is also necessary to have a reasonable scheme for `typing' atoms and assigning them radii.
More subtly, when using the plane positioning determined by method B, the allocation of space is no longer mathematically perfect, since the volume in a tiny tetrahedron near each polyhedron vertex is not allocated to any atom (Fig. 22.1.1.3). This is called vertex error. However, calculations on periodic systems have shown that, in practice, vertex error does not amount to more than 1 part in 500 (Gerstein et al., 1995).
Because of vertex error and the complexities in locating vertices, a different algorithm has to be used for volume calculation with method B. (It can also be used with bisection.) First, surround the central atom (for which a volume is being calculated) by a very large, arbitrarily positioned tetrahedron. This is initially the `current polyhedron'. Next, sort all neighbouring atoms by distance from the central atom and go through them from nearest to farthest. For each neighbour, position a plane perpendicular to the vector connecting it to the central atom according to the predefined proportion (i.e. from the method B formulae or bisection). Since a Voronoi polyhedron is always convex, if any vertices of the current polyhedron are on the other side of this plane to the central atom, they cannot be part of the final polyhedron and should be discarded. After this has been done, the current polyhedron is recomputed using the plane to `chop it down'. This process is shown schematically in Fig. 22.1.1.4. When it is finished, one has a list of vertices that can be traversed to calculate volumes, as in the basic Voronoi procedure.
Voronoi polyhedra are closely related (i.e. dual) to another useful geometric construction called the Delaunay triangulation. This consists of lines, perpendicular to Voronoi faces, connecting each pair of atoms that share a face (Fig. 22.1.1.5).
Delaunay triangulation is described here as a derivative of the Voronoi construction. However, it can be constructed directly from the atom coordinates. In two dimensions, one connects with a triangle any triplet of atoms if a circle through them does not enclose any additional atoms. Likewise, in three dimensions one connects four atoms with a tetrahedron if the sphere through them does not contain any further atoms. Notice how this construction is equivalent to the specification for Voronoi polyhedra and, in a sense, is simpler. One can immediately see the relationship between the triangulation and the Voronoi volume by noting that the volume is the distance between neighbours (as determined by the triangulation) weighted by the area of each polyhedral face. In practice, it is often easier in drawing to construct the triangles first and then build the Voronoi polyhedra from them.
Delaunay triangulation is useful in many `nearestneighbour' problems in computational geometry, e.g. trying to find the neighbour of a query point or finding the largest empty circle in a collection of points (O'Rourke, 1994). Since this triangulation has the `fattest' possible triangles, it is the choice for procedures such as finiteelement analysis.
In terms of protein structure, Delaunay triangulation is the natural way to determine packing neighbours, either in protein structure or molecular simulation (Singh et al., 1996; Tsai et al., 1996, 1997). Its advantage is that the definition of a neighbour does not depend on distance. The alpha shape is a further generalization of Delaunay triangulation that has proven useful in identifying ligandbinding sites (Edelsbrunner et al., 1996, 1995; Edelsbrunner & Mucke, 1994; Peters et al., 1996).
When one is carrying out the Voronoi procedure, if a particular atom does not have enough neighbours the `polyhedron' formed around it will not be closed, but rather will have an open, concave shape. As it is not often possible to place enough water molecules in an Xray crystal structure to cover all the surface atoms, these `open polyhedra' occur frequently on the protein surface (Fig. 22.1.1.6). Furthermore, even when it is possible to define a closed polyhedron on the surface, it will often be distended and too large. This is the problem of the protein surface in relation to the Voronoi construction.
There are a number of practical techniques for dealing with this problem. First, one can use very high resolution protein crystal structures, which have many solvent atoms positioned (Gerstein & Chothia, 1996). Alternatively, one can make up the positions of missing solvent molecules. These can be placed either according to a regular gridlike arrangement or, more realistically, according to the results of molecular simulation (Finney et al., 1980; Gerstein et al., 1995; Richards, 1974).
More fundamentally, however, the `problem of the protein surface' indicates how closely linked the definitions of surface and volume are and how the definition of one, in a sense, defines the other. That is, the twodimensional (2D) surface of an object can be defined as the boundary between two 3D volumes. More specifically, the polyhedral faces defining the Voronoi volume of a collection of atoms also define their surface. The surface of a protein consists of the union of (connected) polyhedra faces. Each face in this surface is shared by one solvent atom and one protein atom (Fig. 22.1.1.7).
Another somewhat related definition is the convex hull, the smallest convex polyhedron that encloses all the atom centres (Fig. 22.1.1.7). This is important in computergraphics applications and as an intermediary in many geometric constructions related to proteins (Connolly, 1991; O'Rourke, 1994). The convex hull is a subset of the Delaunay triangulation of the surface atoms. It is quickly located by the following procedure (Connolly, 1991): Find the atom farthest from the molecular centre. Then choose two of its neighbours (as determined by the Delaunay triangulation) such that a plane through these three atoms has all the remaining atoms of the molecule on one side of it (the `plane test'). This is the first triangle in the convex hull. Then one can choose a fourth atom connected to at least two of the three in the triangle and repeat the plane test, and by iteratively repeating this procedure, one can `sweep' across the surface of the molecule and define the whole convex hull.
Other parts of the Delaunay triangulation can define additional surfaces. The part of the triangulation connecting the first layer of water molecules defines a surface, as does the part joining the second layer. The second layer of water molecules, in fact, has been suggested on physical grounds to be the natural boundary for a protein in solution (Gerstein & LyndenBell, 1993c). Protein surfaces defined in terms of the convex hull or water layers tend to be `smoother' than those based on Voronoi faces, omitting deep grooves and clefts (see Fig. 22.1.1.7).
In the absence of solvent molecules to define Voronoi polyhedra, one can define the protein surface in terms of the position of a hypothetical solvent, often called the probe sphere, that `rolls' around the surface (Richards, 1977) (Fig. 22.1.1.7). The surface of the probe is imagined to be maintained at a tangent to the van der Waals surface of the model.
Various algorithms are used to cause the probe to visit all possible points of contact with the model. The locus of either the centre of the probe or the tangent point to the model is recorded. Either through exact analytical functions or numerical approximations of adjustable accuracy, the algorithms provide an estimate of the area of the resulting surface. (See Section 22.1.2 for a more extensive discussion of the definition, calculation and use of areas.)
Depending on the probe size and whether its centre or point of tangency is used to define the surface, one arrives at a number of commonly used definitions, summarized in Table 22.1.1.2 and Fig. 22.1.1.7.
The area of the van der Waals surface will be calculated by the various area algorithms (see Section 22.1.2.2) when the probe radius is set to zero. This is a mathematical calculation only. There is no physical procedure that will measure van der Waals surface area directly. From a mathematical point of view, it is just the first of a set of solventaccessible surfaces calculated with differing probe radii.
The solventaccessible surface is convex and closed, with defined areas assignable to each individual atom (Lee & Richards, 1971). However, the individual calculated values vary in a complex fashion with variations in the radii of the probe and protein atoms. This radius is frequently, but not always, set at a value considered to represent a water molecule (1.4 Å). The total SAS area increases without bound as the size of the probe increases.
Like the solventaccessible surface, the molecular surface is also closed, but it contains a mixture of convex and concave patches, the sum of the contact and reentrant surfaces. The ratio of these two surfaces varies with probe radius. In the limit of infinite probe radius, the molecular surface becomes convex and attains a limiting minimum value (i.e. it becomes a convex hull, similar to the one described above). The molecular surface cannot be divided up and assigned unambiguously to individual atoms.
The contact surface is not closed. Instead, it is a series of convex patches on individual atoms, simply related to the solventaccessible surface of the same atoms. In complementary fashion, the reentrant surface is also not closed but is a series of concave patches that is part of the probe surface where it contacts two or three atoms simultaneously. At infinite probe radius, the reentrant areas are plane surfaces, at which point the molecular surface becomes a convex surface. The reentrant surface cannot be divided up and assigned unambiguously to individual atoms. Note that the molecular surface is simply the union of the contact and reentrant surfaces, so in terms of area MS = CS + RS.
The detail provided by these surfaces will depend on the radius of the probe used for their construction.
One may argue that the behaviour of the rolling probe sphere does not accurately model real hydrogenbonded water. Instead, its `rolling' more closely mimics the behaviour of a nonpolar solvent. An attempt has been made to incorporate more realistic hydrogenbonding behavior into the probe sphere, allowing for the definition of a hydration surface more closely linked to the behaviour of real water (Gerstein & LyndenBell, 1993c).
The definitions of accessible surface and molecular surface can be related back to the Voronoi construction. The molecular surface is similar to `timeaveraging' the surface formed from the faces of Voronoi polyhedra (the Voronoi surface) over many water configurations, and the accessible surface is similar to averaging the Delaunay triangulation of the first layer of water molecules over many configurations.
There are a number of other definitions of protein surfaces that are unrelated to either the probesphere method or Voronoi polyhedra and provide complementary information (Kuhn et al., 1992; Leicester et al., 1988; Pattabiraman et al., 1995).
The definition of protein surfaces and volumes depends greatly on the values chosen for various parameters of linear dimension – in particular, van der Waals and probesphere radii.
For all the calculations outlined above, the hardsphere approximation is used for the atoms. (One must remember that in reality atoms are neither hard nor spherical, but this approximation has a long history of demonstrated utility.) There are many lists of the radii of such spheres prepared by different laboratories, both for single atoms and for unified atoms, where the radii are adjusted to approximate the joint size of the heavy atom and its bonded hydrogen atoms (clearly not an actual spherical unit).
Some of these lists are reproduced in Table 22.1.1.1. They are derived from a variety of approaches, e.g. looking for the distances of closest approach between atoms (the Bondi set) and energy calculations (the CHARMM set). The differences between the sets often come down to how one decides to truncate the Lennard–Jones potential function. Further differences arise from the parameterization of water and other hydrogenbonding molecules, as these substances really should be represented with two radii, one for their hydrogenbonding interactions and one for their VDW interactions.

Perhaps because of the complexities in defining VDW parameters, there are some great differences in Table 22.1.1.1. For instance, the radius for an aliphatic CH (>CH=) ranges from 1.7 to 2.38 Å, and the radius for carboxyl oxygen ranges from 1.34 to 1.89 Å. Both of these represent at least a 40% variation. Moreover, such differences are practically quite significant, since many geometrical and energetic calculations are very sensitive to the choice of VDW parameters, particularly the relative values within a single list. (Repulsive core interactions, in fact, vary almost exponentially.) Consequently, proper volume and surface comparisons can only be based on numbers derived through use of the same list of radii.
In the last column of the table we give a recent set of VDW radii that has been carefully optimized for use in volume and packing calculations. It is derived from analysis of the most common distances between atoms in smallmolecule crystal structures in the Cambridge Structural Database (Rowland & Taylor, 1996; Tsai et al., 1999).
A series of surfaces can be described by using a probe sphere with a specified radius. Since this is to be a convenient mathematical construct in calculation, any numerical value may be chosen with no necessary relation to physical reality. Some commonly used examples are listed in Table 22.1.1.2.

The solventaccessible surface is intended to be a close approximation to what a water molecule as a probe might `see' (Lee & Richards, 1971). However, there is no uniform agreement on what the proper water radius should be. Usually it is chosen to be about 1.4 Å.
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. Closepacked spheres of the same size have a packing efficiency of ∼0.74. Closepacked spheres of different size are expected to have a somewhat higher packing efficiency. In contrast, water is not closepacked 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 Å^{2}. 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 closepacked 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 hardcore 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 closepacked limit (Richards & Lim, 1994).
The exceptionally tight packing in the protein core seems to require a precise jigsaw puzzlelike 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 puzzlelike packing in the protein core, it has been shown that one can simply achieve the `firstorder' 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 Å^{3} ) 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 Å^{3}, 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 randomlike 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 closepacked 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 tradeoff between hydrogen bonding and close packing, and this can be explicitly visualized in simulations of the packing in simple toy systems (Gerstein & LyndenBell, 1993a,b).
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