The buried surface model

Given the atomic structure of a macromolecular assembly, defining the interface between two components A and B may be viewed as a problem of geometry in space. The simplest definition is based on distance: the AB interface is the set of atoms or chemical groups i of A and j of B, which satisfy the condition d _ij<d ₀. In most implementations, d ₀ depends on the atomic or group radii r _i and r _j, and on a cutoff value r ₀ in the range 0·5–2 Å:

(1)

The interface can also be defined as the surface buried between the two components (Chothia & Janin, Reference Chothia and Janin1975). Given the solvent accessible surfaces of A, B and the AB pair, any point of the accessible surface of A or B that does not belong to the accessible surface of AB is part of the interface. The solvent accessible surface is at one probe radius (r _W=1·4–1·5 Å for a water probe) of the molecular surface, and its construction may implement the rolling sphere algorithm of Lee & Richards (Reference Lee and Richards1971), or an analytical algorithm. The buried surface area, or interface area, that represents the interface size, is computed as:

(2)

where ASA_A, ASA_B and ASA_AB are the accessible surface areas of A, B and the AB pair. In that model, the atoms and residues that lose ASA in AB are part of the interface. The BSA is essentially proportional to their number, and also to the number of atom pairs that satisfy Eq. (1) with r ₀=2r _W.

Several servers, including PP at University College London, and PISA at the European Bioinformatics Institute (see Section 2.3), return the areas of the solvent accessible (ASA) and of the buried surfaces (BSA) of individual polypeptide chains, residues and atoms, when atomic coordinates are entered. These servers, and many publications, quote a BSA ‘per subunit’, that is BSA/2 in Eq. (2). This assumes implicitly that A and B bury the same surface area, which is exact only for interfaces with C ₂ symmetry. The difference between the contributions of the two partners to the BSA depends on the shape of the interface. In most cases, it is only a few percents, but it may exceed 10% when a convex surface is in contact with a concave surface; examples are protease inhibitors binding to proteases (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999) and RNA binding to proteins (Bahadur et al. Reference Bahadur, Zacharias and Janin2008).

The Voronoi model

An alternative definition of interfaces is based on the Voronoi diagram and a related construction, the alpha-complex. The Voronoi diagram associates to each atom its Voronoi cell, the convex polyhedron that contains all points of space closer to that atom than to any other atom. Its first application to proteins was developed by Richards (Reference Richards1974) to evaluate the volume occupied by individual atoms or chemical groups. Because it does not account for the different sizes of the chemical groups, the Voronoi diagram is at present replaced in most applications by the closely related power diagram (Gellatly & Finney, Reference Gellatly and Finney1982; Aurenhammer, Reference Aurenhammer1987). Instead of the Euclidean distance |ax| between a point x and an atom centered in a, this diagram uses the power distance p(x) of x with respect to the ball of radius r that represents the atom or chemical group:

(3)

The Voronoi cell of an atom then comprises all points of space that have a power distance to that atom less than to any other atom. Its facets belong to the radical plane, which contains the intersection of the spheres if they do intersect.

The Voronoi (or power) diagram offers a natural definition of contacts: two atoms are in contact if and only if their Voronoi cells share a facet. Labeling molecules A and B with different colors, the set of ‘bicolor’ facets shared by atoms of A and B forms the AB Voronoi interface (Fig. 3). However, the atoms on the molecular surface all have unbounded or poorly defined Voronoi cells, a problem that may be circumvented by placing solvent molecules on the surface (Soyer et al. Reference Soyer, Chomilier, Mornon, Jullien and Sadoc2000; Dupuis et al. Reference Dupuis, Sadoc, Jullien, Angelov and Mornon2005), or solved in a mathematically more rigorous manner by using an extension of the power diagram. This extension, called the alpha complex (Edelsbrunner & Mucke, Reference Edelsbrunner and Mucke1994), makes use of a remarkable property of the power diagram: it is invariant when a constant α is added to all squared radii, r ², in Eq. (3). For a given value of α, the alpha complex is built like a power diagram, except that one restricts the Voronoi cell of each atom to its associated ball and seeks intersections between these restricted regions. Thus, a facet between two atoms is not part of the alpha complex if the associated balls do not intersect, or if the facet lies outside the intersection.

Fig. 3. The Voronoi model of a protein–protein interface. The interface between hen egg white lysozyme (gray ribbon) and the Fv fragment of antibody D1.3 (heavy chain in red, light chain in blue) is drawn as a set of Voronoi facets that belong to the alpha-complex for α=0. Unbounded facets have been removed by the procedure of Cazals et al. (Reference Cazals, Proust, Bahadur and Janin2006). The interface is heavily hydrated with the water molecules drawn as gray spheres. PDB entry 1vfb (Bhat et al. Reference Bhat, Bentley, Boulot, Greene, Tello, Dall'Acqua, Souchon, Schwarz, Mariuzza and Poljak1994). Courtesy of F. Cazals (Sophia-Antipolis, France).

Applications of the alpha complex to macromolecules have been reviewed by Poupon (Reference Poupon2004). The interfaces defined in this manner may still contain a few unbounded facets that Edelbrunner and collaborators (Ban et al. Reference Ban, Edelsbrunner and Rudolph2004; Headd et al. Reference Headd, Ban, Brown, Edelsbrunner, Vaidya and Rudolph2007) removed through an elaborate iterative retraction procedure. The algorithm of Cazals et al. (Reference Cazals, Proust, Bahadur and Janin2006), implemented on the Intervor server, does it on purely geometric criteria. Both procedures have been applied to a set of protein–protein interfaces that had been previously analyzed with the buried surface approach by Chakrabarti & Janin (Reference Chakrabarti and Janin2002). The Voronoi interface area, calculated as the sum of the areas of the bicolor facets, correlates linearly with the BSA in both cases, but the correlation is much better with the procedure of Cazals et al. (Reference Cazals, Proust, Bahadur and Janin2006) than that of Ban et al. (Reference Ban, Edelsbrunner and Rudolph2004) (R ²=0·98 vs. 0·85). A remarkable result is that about 13% of the atoms that share bicolor facets do not contribute to the BSA, mostly because they are not solvent accessible in the free molecules. Thus, the Voronoi interface comprises significantly more atoms, and especially more main chain atoms, than the buried surface.

Chemical and amino acid compositions

The chemical composition of an interface is commonly characterized by the relative abundance of the different types of atoms or amino acid residues. Atom types are often defined as non-polar/polar/charged, or main chain/side chains. The latter distinction is useful because the protein main chain contributes significantly to most macromolecular interfaces, and biochemical approaches based on site directed mutagenesis tend to ignore that contribution. In addition to protein atoms, the interface may comprise water molecules (see Section 3.4.3).

A composition may be evaluated by counting interface atoms or residues, or by measuring their contribution to the BSA. The two definitions are nearly equivalent for atoms, but with residues, the area-based composition reflects their size as well as their abundance. In either case, the fractions may be converted into propensities:

(6)

where f _i is the number or area fraction of type i at the interface, f _i°, the corresponding number in a reference set that can be the whole protein, its interior or its surface. If the reference is the protein surface, f _i° may be either a fraction of the number of solvent-accessible atoms or residues, or a fraction of their contribution to the ASA. Then, p _i>0 implies that interfaces are enriched in atoms or residues of type i relative to the protein surface in contact with the solvent; p _i<0 implies that the interfaces are depleted in type i. An interface commonly comprises only a few tens of residues, and propensities calculated on individual interfaces or on small sets with 20 residue types have poor statistics. Grouping types with similar properties may improve the statistics, but this will mask at least one interesting feature: Arg and Lys are usually counted as similar, yet protein–protein interfaces are depleted in Lys, but not Arg.

Hydrophobicity

Hydrophilicity/hydrophobicity is an important physical chemical property of the protein surface, and possibly the one most relevant to its propensity to interact with a ligand. It can be estimated at the atomic level: aliphatic and aromatic groups in proteins are non-polar, and therefore hydrophobic; O- and N-containing groups are polar and hydrophilic. When they interact with a ligand, all these groups become partly or fully dehydrated, and groups of the ligand also, allowing solvent molecules to gain degrees of freedom. Their entropy increases, causing the hydrophobic effect to favor association. The system enthalpy also changes, and it can do so in either direction, because van der Waals and polar interactions made by surface groups with the solvent are replaced by protein–ligand interactions. The whole process can be viewed as a transfer of the ligand from water to the protein environment. With hydrocarbons, which make similar van der Waals interactions with water and other solvents, and no polar interaction, the free energy of transfer scales linearly with the BSA. Calibrations based on their relative solubility in water and organic solvents yield a slope of 20–30 cal mol⁻¹ Å⁻² (Chothia, Reference Chothia1974; Vajda et al. Reference Vajda, Weng and DeLisi1995). Site-directed mutagenesis experiments discussed in Section 4.1.3 suggest that this correctly represents the contribution of the hydrophobic effect to protein–protein interaction, although the contribution of individual hydrophobic groups may depend on the structural context.

Polar interactions and hydration

Electrostatic interactions implicate polar groups containing oxygen or nitrogen atoms. These atoms bear a partial or a full charge, and they form H bonds, much stronger than van der Waals interactions. Electrostatics plays an essential role in all processes involving proteins, DNA or RNA, but its evaluation requires elaborate calculations that are highly sensitive to the solvent model (Sheinerman et al. Reference Sheinerman, Norel and Honig2000). The H bonds between protein groups, or between the protein and a ligand, have an energy similar to those made with water, and the balance depends on the details of their geometry and their environment. Program HBPLUS (McDonald & Thornton, Reference McDonald and Thornton1994) is commonly used to detect H bonds in protein structures; with a donor–acceptor distance cutoff of 3·9 Å, the default value in its implementation by the PP server of University College London, it retains H bonds that may be much weaker than that those between water molecules in the liquid, where the oxygen–oxygen distance is 2·8 Å. H bonding is mostly a dipole–dipole interaction, but electrostatics also involves long-range interactions between charged groups, and weaker interactions involving aromatic groups or water bridges. Although efficiently screened by salts under most conditions, long-range electrostatic interactions may play an active part in the kinetics of association (Schreiber et al. Reference Schreiber, Shaul and Gottschalk2006).

The representation of electrostatics in force fields is not accurate enough to give reliable values of the energy balance of these interactions in a water environment. In practice, all-atom force fields perform best if they are calibrated on actual protein structures. An example is the Rosetta force field, which has yielded excellent models of several proteins and protein–protein complexes (Bradley et al. Reference Bradley, Misura and Baker2005; Schueler-Furman et al. Reference Schueler-Furman, Wang, Bradley, Misura and Baker2005). Energy refinement with an all-atom force field is computationally expensive, albeit well suited for distributed computing (Das et al. Reference Das, Qian, Raman, Vernon, Thompson, Bradley, Khare, Tyka, Bhat, Chivian, Kim, Sheffler, Malmström, Wollacott, Wang, Andre and Baker2007). It is usually left for late steps of structure predictions, and earlier steps rely instead on knowledge-based statistical potentials, of which TASSER (Zhang et al. Reference Zhang, Arakaki and Skolnick2005) is an example. Knowledge-based potentials, originally developed to model protein folding, are now commonly used for protein design (Poole & Ranganathan, Reference Poole and Ranganathan2006). They can also be applied to protein interactions (Keskin et al. Reference Keskin, Bahar, Badretdinov, Ptitsyn and Jernigan1998), but the tendency in recent years has been to develop potentials that are specifically designed for that purpose (Mintseris et al. Reference Mintseris, Pierce, Wiehe, Anderson, Chen and Weng2007), and to complement the statistical approach with machine-learning procedures that can handle parameters not easily converted into potentials (Block et al. Reference Block, Paern, Hüllermeier, Sanschagrin, Sotriffer and Klebe2006; Zhu et al. Reference Zhu, Domingues, Sommer and Lengauer2006; Ofran & Rost, Reference Ofran and Rost2007; Bernauer et al. Reference Bernauer, Bahadur, Rodier, Janin and Poupon2008).

The interface between two proteins, or between a protein and a nucleic acid in a complex, is never completely dehydrated. This can be seen in high-resolution X-ray structures that report water molecules: many are located at molecular interfaces. Rodier et al. (Reference Rodier, Bahadur, Chakrabarti and Janin2005) consider as interface waters all crystallographic waters located within 4·5 Å of atoms of both subunits; nearly all are within H-bond distance of at least one protein polar atom. Their number increases with the interface size, but also with the resolution of the X-ray structure, which strongly suggests that solvent is under-reported in most studies.

Size and topology of the interfaces

Sets of non-obligate protein–protein complexes of known X-ray structure have been assembled by Janin & Chothia (Reference Janin and Chothia1990), Jones & Thornton (Reference Jones and Thornton1996), Lo Conte et al. (Reference Lo Conte, Chothia and Janin1999) and recently by Hwang et al. (Reference Hwang, Pierce, Mintseris, Janin and Weng2008). Table 2 reports average properties of the protein–protein interfaces in a set of 70 complexes adapted from Lo Conte et al. (Reference Lo Conte, Chothia and Janin1999) by Chakrabarti & Janin (Reference Chakrabarti and Janin2002). The table compares their interfaces to those of homodimers and to crystal contacts. The interface size measured by the BSA has a mean value of 1910 Å². An average of 204 atoms contribute to that BSA; they belong to 57 amino acid residues, that is about 28 residues per component protein. When site-directed mutagenesis experiments are performed on some of these complexes, many of the residues that lose ASA can be changed to Ala without much effect on the dissociation constant; only a fraction, termed ‘hotspots’, yield large changes in K _d. In the human growth hormone (HGH)/HGH receptor system for instance, 31 interface residues were mutated on the receptor, but only 11 mutants showed a significant loss of affinity for the hormone (Clackson & Wells, Reference Clackson and Wells1995). Similar results have been obtained in other systems (Bogan & Thorn, Reference Bogan and Thorn1998; DeLano, Reference DeLano2002). The discrepancy with the geometric definition of the interfaces is due in part to the limits of the alanine-scanning method: it does not test all the residues (Ala and Gly are almost always omitted), and ignores the interactions made by the main chain, which contributes 20% of the BSA and a majority of the interface H bonds (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999). Nevertheless, the discrepancy is real, and relevant to the topology of the interfaces and the core/rim model discussed below.

Table 2. Properties of protein–protein interfaces

^a Data from Chakrabarti & Janin (Reference Chakrabarti and Janin2002) on a subset of the complexes of Lo Conte et al. (Reference Lo Conte, Chothia and Janin1999).

^b Data from Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2003) ; the set of Dey et al. (unpublished data) was derived from the PiQSi database (Lévy, Reference Lévy2007) as described in the text.

^c Homodimers described in PiQSi (Lévy, Reference Lévy2007) as being in equilibrium with the monomer, based on the literature.

^d Pairwise interfaces in crystals of monomeric proteins. The first mean BSA value and the S.D. are for the 1320 interfaces in the 152 crystal forms analyzed by Janin & Rodier (Reference Janin and Rodier1995). All other numbers are for 188 interfaces with BSA >800 Å² that were selected among those 1320 interfaces by Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004).

^e Score obtained by summing over the whole interface the propensity of individual residues to occur at the interface of homodimers (Bahadur et al. Reference Bahadur, Chakrabarti, Rodier and Janin2004).

^f Number of polypeptide segments per 1000 Å² of BSA. A separate dataset of 204 structures has been used for protein–protein complexes (Pal et al. Reference Pal, Chakrabarti, Bahadur, Rodier and Janin2007).

^g Data from Rodier et al. (Reference Rodier, Bahadur, Chakrabarti and Janin2005).

^h Mean values in subsets that comprise 52 protein components of the complexes (excluding antigen–antibody complexes), 121 homodimers and 102 monomeric proteins in crystal contacts. s is the mean Shannon entropy in aligned sequences. Data from Guharoy & Chakrabarti (Reference Guharoy and Chakrabarti2005).

Lo Conte et al. (Reference Lo Conte, Chothia and Janin1999) noted that all the protein–protein complexes in their set bury more than 1100 Å², and that a majority has a BSA in the range of 1200–2000 Å². They retained that range as defining ‘standard-size’ protein–protein interfaces, as opposed to the ‘small’ interfaces with BSA <1200 Å², and the ‘large’ interfaces with BSA >2000 Å². An interface with a BSA of more than 1200 Å² implicating about 60 atoms and 16 residues per component, allows the stable, specific association of two proteins. Nearly all antibody–antigen and many enzyme–inhibitor complexes have standard-size interfaces. The barnase–barstar complex mentioned above does, and also several protease–inhibitor complexes with K _d well below 10⁻¹²m. Antigen–antibody complexes are not quite as stable: they typically have a nanomolar K _d and half-lives of a few hours (Braden & Poljak, Reference Braden, Poljak and Kleanthous2000; Sundberg & Mariuzza, Reference Sundberg and Mariuzza2002).

The recent set of complexes assembled by Hwang et al. (Reference Hwang, Pierce, Mintseris, Janin and Weng2008) confirms and extends these results. The interface size distribution in that set is shown in Fig. 5. The mean BSA and its standard deviation are essentially the same as in Table 2; 7% of the interfaces are small, 55% are standard size and 38% are large. Three complexes in the new set have a BSA near 1000 Å², and one buries only 810 Å². It occurs in a complex of the Cbl-b ubiquitin ligase with ubiquitin (PDB entry 2oob). Ubiquitin binding promotes the dimerization of the Cbl-b ligase (Peschard et al. Reference Peschard, Kozlov, Lin, Mirza, Berghuis, Lipkowitz, Park and Gehring2007), which buries additional surface, but several other complexes involving ubiquitin also have a small BSA. They are short-lived like the redox complexes involved in electron transfer, also listed in Fig. 5. Their interfaces are of standard size or smaller (Crowley & Carrondo, Reference Crowley and Carrondo2004); the smallest, with a BSA of 830 Å², occurs in a ternary assembly (PDB entry 2 mta). Taken together, the BSA data suggest that the minimum protein surface that must be buried to form a functional complex is of the order of 900 Å² and implicates no more than 12 residues on each partner. The ubiquitin ligase/ubiquitin and the redox complexes illustrate biological processes that depend on short-lived interactions associated with a minimal size interface. Such processes are still poorly represented in the PDB in spite of their importance (Noreen & Thornton, Reference Noreen and Thornton2003b).

Fig. 5. Interface size in transient protein–protein complexes. Histogram of the BSA in the set of 124 protein–protein complexes assembled by Hwang et al. (Reference Hwang, Pierce, Mintseris, Janin and Weng2008), and the set of 11 redox protein complexes of Crowley & Carrondo (Reference Crowley and Carrondo2004). The Hwang set comprises 25 antigen–antibody complexes, 35 enzyme/inhibitor or substrate complexes and 64 complexes of other types. The mean value of the BSA is 1910 Å² in that set, and 1290 Å² in the redox set.

With few exceptions, the small and standard-size interfaces have a simple topology: they comprise a single pair of recognition patches as defined by the clustering algorithm of Chakrabarti & Janin (Reference Chakrabarti and Janin2002), and a single connected component in the Voronoi model of Cazals et al. (Reference Cazals, Proust, Bahadur and Janin2006). The antibody–antigen interface shown in Fig. 3 above is an example of those. Standard-size interfaces are also mostly planar, except in protease–inhibitor complexes where the inhibitor offers a convex surface to the concave active site of the protease (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999). In contrast, the large interfaces usually consist of multiple pairs of recognition patches, at least one of them of standard size (Chakrabarti & Janin, Reference Chakrabarti and Janin2002). An example is the Gα/Gβγ interface of transducin (Fig. 4): the blue patch is standard size, and the red patch is small. Individual patches may be planar, but that is unlikely for the whole interface.

The proteins that carry more than one recognition patch on their surface often have the capacity to undergo conformational changes of large amplitude. In transducin, the red patch implicates the N-terminal segment of Gα. This segment is disordered in the free subunit, and forms an α helix in the complex (Lambright et al. Reference Lambright, Sondek, Bohm, Skiba, Hamm and Sigler1996). Thus, it undergoes a disorder-to-order transition during association. In other systems, the proteins may have domains, each bearing a recognition patch, that move one relative to the other when the complex forms. In the set assembled by Hwang et al. (Reference Hwang, Pierce, Mintseris, Janin and Weng2008), the structure of the free components is known along with that of the complex, and the amplitude of the conformation changes can be estimated by least-square superposition. Figure 6 indicates that the RMSD of the C-alpha atoms tends to increase with the BSA. It is smaller than 1·8 Å in all the complexes with small interfaces, and in 90% of those with standard-size interfaces. This confirms that rigid-body recognition is the preferred mechanism for assembling complexes with standard-size, single-patch interfaces. On the other hand, the RMSD is more than 1·8 Å for 40% of the complexes with large interfaces. Large multi-patch interfaces allow flexible recognition to occur, and we may assume that the additional buried surface and atomic interactions pay for the cost of the conformational change (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999; Janin et al. Reference Janin, Rodier, Chakrabarti and Bahadur2007).

Fig. 6. Interface size and conformation changes. In the set of Hwang et al. (Reference Hwang, Pierce, Mintseris, Janin and Weng2008), the free components of the complexes have a known structure that can be compared to their structure in the complex by rigid-body least-square superposition. The residual RMSD of the C-alpha atoms is plotted against the BSA of each complex. The dotted lines are drawn at RMSD=1·8 Å and BSA=2000 Å². The categories are the same as in Fig. 5.

Composition, packing and hydration

The interfaces of non-obligate complexes have been examined from the viewpoint of their chemical composition. Table 2 shows that the fractional contribution of non-polar (carbon-containing) groups to the BSA is f _np=58% on average. This value is almost the same as for the ASA of the average monomeric protein (f _np=57%; Miller et al. Reference Miller, Janin, Lesk and Chothia1987), which reflects the fact that the protein surface involved in an interface remains solvent accessible until the components of the complex meet. Thus, it cannot have very different physical–chemical properties from the remainder of the protein surface. The interfaces of antigen–antibody complexes (f _np=51%) tend to be more polar than the average accessible surface, and those of protease–inhibitor complexes (f _np=61%) are less polar (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999). The interfaces comprise fewer charged groups, and therefore, more neutral polar groups, than the protein surface; the charged groups contribute 14% of the BSA in Table 2 instead of 19% to the ASA (Miller et al. Reference Miller, Janin, Lesk and Chothia1987). The polar fraction also governs the capacity of interface atoms to form H bonds. On average, transient complex interfaces contain 10 H bonds, about 1 per 190 Å² of BSA (Table 2), defined with the geometric criteria of HBPLUS (McDonald & Thornton, Reference McDonald and Thornton1994). The number of H bonds, n _HB, in individual interfaces increases with the BSA, but the correlation is mediocre (R ²=0·84). Moreover, high-resolution X-ray structures tend to display more H bonds, which suggests that the surface density calculated on the whole sample is underestimated.

A large majority of the atoms at protein–protein interfaces is still accessible to the solvent in the complex. The fraction of fully buried interface atoms (zero ASA), f _bu, is only 34% on average (Table 2), and it never exceeds 50%. Measurements of the Voronoi volume indicate that the buried interface atoms pack as densely as the atoms of the protein core. This implies that the interfaces are close-packed, a statement that can be extended to 60% of the interface atoms by including the crystallographic water molecules in the construction of the Voronoi diagram. In most cases, the volumes are within 1% of those of the protein core (Lo Conte et al. Reference Lo Conte, Chothia and Janin1999).

Residual water plays an important role in the packing. It is abundant at the interfaces: they contain an average of 20 water molecules, that is 10 water molecules per 1000 Å² of BSA (Table 2). This amounts to about 10% of the hydration water present before association, a water molecule covering about 10 Å² of the protein surface on average. This estimate of the residual hydration of the interfaces is certainly below the reality, because X-ray structures do not report solvent in a consistent way, and the surface density of interface water increases with the resolution even faster than for H bonds (Rodier et al. Reference Rodier, Bahadur, Chakrabarti and Janin2005). Water molecules make many polar interactions and H bond bridges across the interface. The balls representing water in Figs 3 and 4 illustrate two different patterns of interface hydration that can be described as ‘wet’ and ‘dry’ (Janin, Reference Janin1999). The antigen–antibody interface (Fig. 3) is wet, that is it is hydrated all through its surface. The Gα/Gβγ transducin interface (Fig. 4) is split into two distinct patches, both dry: many water molecules line their edges, but few if any penetrate inside.

Although the interface residues of a non-obligate complex are surface residues in the free components, there are some noticeable features in their amino acid composition. Relative to the accessible protein surface, the interfaces are depleted in Glu, Asp and Lys, and enriched in Met, Tyr and Trp. The propensities, shown in Fig. 7 c together with those of protein–nucleic acid complexes discussed in Section 4.5, are generally weak, but they become more apparent once the interfaces are dissected into a core and a rim as in Fig. 7 a. The rim, made of residues in which none of the interface atoms are fully buried, has a composition close to the protein accessible surface. The core comprises all the buried interface atoms, and 55% of all interface residues on average. It is enriched in aromatic residues, and to a lesser extent, in aliphatic residues, but not in Val, Ala and Pro. Val is indifferent, whereas Ala and Pro have a negative propensity for the interface core. Albeit aliphatic, the side chains of these residues are more constrained than in Ile, Leu, Met or the aromatic residues, and it may be that the flexibility of the longer side chains makes them more amenable to the needs of the interface formation. Arginine, far from being disallowed like the other charged residues, contributes 10% of the BSA in both the core and the rim, and also 10% of the ASA (Chakrabarti & Janin, Reference Chakrabarti and Janin2002).

Fig. 7. Residue propensities for interfaces vs. the protein surface. The propensities [Eq. (6)] are derived from the relative contributions of the 20 amino acid types to the BSA of the interfaces and the ASA of the proteins in the same set. (a) Core and rim of the interfaces of protein–protein complexes. Data from Chakrabarti & Janin (Reference Chakrabarti and Janin2002). (b) The core of the homodimers interfaces is compared to crystal packing interfaces. Data from Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004). (c) The interfaces of protein–DNA (Nadassy et al. Reference Nadassy, Wodak and Janin1999) and protein–RNA complexes (Bahadur et al. Reference Bahadur, Zacharias and Janin2008) are compared to those of protein–protein complexes (Chakrabarti & Janin, Reference Chakrabarti and Janin2002).

The core/rim model of protein–protein recognition

Interface residues make inter-molecular interactions. They are structurally and possibly functionally important, and therefore subject to a selection pressure that should be detected in homologous sequences. Several methods for predicting protein–protein contacts rely on identifying patches of surface residues conserved in evolution (Lichtarge et al. Reference Lichtarge, Bourne and Cohen1996; Armon et al. Reference Armon, Graur and Ben-Tal2001; Valdar & Thornton, Reference Valdar and Thornton2001; Lichtarge & Sowa, Reference Lichtarge and Sowa2002; Ma et al. Reference Ma, Elkayam, Wolfson and Nussinov2003). The conservation signal is usually weak, but like the amino acid composition signal, it can be enhanced by splitting the interfaces into core and rim regions. Guharoy & Chakrabarti (Reference Guharoy and Chakrabarti2005) estimated the effect of evolutionary selection by measuring Shannon entropies in sets of aligned sequences. In protein–protein complexes, the mean value of the normalized entropy of the interface residues was s=0·65 in the core and s=0·80 in the rim (Table 2). Figure 8 shows how the core and the rim of an interface are distributed on the surface of two proteins, and compares that distribution with the sequence entropy. The proteins are the inhibitor component of an enzyme–inhibitor complex, and the subunit of a homodimer.

Fig. 8. The core/rim model of protein–protein interfaces. Top row: The protease component of the elastase–ovomucoid inhibitor complex (1ppf; Bode et al. Reference Bode, Wei, Huber, Meyer, Travis and Neumann1986). Bottom row: The subunit of the malate dehydrogenase homodimer (1bmd; Kelly et al. Reference Kelly, Nishiyama, Ohnishi, Beppu and Birktoft1993). (a, c) The core (red) comprises the residues that contain interface atoms with zero ASA, the rim (blue), the residues where all the interface atoms are accessible (Chakrabarti & Janin, Reference Chakrabarti and Janin2002). (b, d) Shannon entropies of interface residues in aligned sequences [Eq. (7)]; red stands for low entropies (maximum conservation); blue denotes high entropies (maximum divergence). Figure made with GRASP (Nicholls et al. Reference Nicholls, Sharp and Honig1991), courtesy of M. Guharoy (Calcutta, India).

The picture of the interface conservation derived from the Shannon entropy is coherent with the results of site-directed mutagenesis. Clackson & Wells (Reference Clackson and Wells1995) noted that, in the HGH/HGH receptor system, the residues of the receptor that show large changes of affinity upon mutation to Ala are grouped at the center of the interface. In general, hotspot residues tend to cluster and be sheltered from the solvent (Bogan & Thorn, Reference Bogan and Thorn1998; Halperin et al. Reference Halperin, Wolfson and Nussinov2004). Guharoy & Chakrabarti (Reference Guharoy and Chakrabarti2005) observed that in 14 complexes, there is a correlation between the contribution of the interface residues to the BSA and ΔΔG _d, the change in the free energy of dissociation of the complex upon their mutation to Ala. The correlation is significant for the interface core, but not the rim, and it yields a slope of 26–38 cal mol⁻¹ Å⁻², consistent with the estimates of Chothia (Reference Chothia1974) and Vajda et al. (Reference Vajda, Weng and DeLisi1995) based on the solubility of hydrocarbons. Sundberg et al. (Reference Sundberg, Urrutia, Braden, Isern, Tsuchiya, Fields, Malchiodi, Tormo, Schwarz and Mariuzza2000) and Li et al. (Reference Li, Huang, Swaminathan, Smith-Gill and Mariuzza2005) introduced a series of different size side chains at two sites of the interface of antibody–lysozyme complexes: a site at the center of the interface, the other at its periphery. In both series, ΔΔG _d is proportional to the change in the BSA, but the slope is twice as large at the central site than at the peripheral site (46 vs. 21 cal mol⁻¹ Å⁻²).

Other features of the interfaces support the core/rim distinction, for instance the ‘residue depth’ of Chakravarty & Varadarajan (Reference Chakravarty and Varadarajan1999), who find it to be correlated to ΔΔG _d values, or the distribution of the conserved ‘dry residues’ of Mihalek et al. (Reference Mihalek, Res and Lichtarge2007). Taken together, the site-directed mutagenesis and the sequence conservation data suggest that, in a complex, protein–protein recognition exerts most of its selection pressure on the interface core. The data fit an ‘O ring’ model in which the interface has a core of buried atoms and (partly) buried residues, surrounded by a rim of residues whose atoms remain solvent accessible (Bogan & Thorn, Reference Bogan and Thorn1998; Lo Conte et al. Reference Lo Conte, Chothia and Janin1999). The model applies as such to a single patch standard-size interface, for instance the antigen–antibody interface of Fig. 3, which is the one that was mutated by Sundberg et al. (Reference Sundberg, Urrutia, Braden, Isern, Tsuchiya, Fields, Malchiodi, Tormo, Schwarz and Mariuzza2000), or the enzyme–inhibitor interface of Fig. 8 a. In a multi-patch interface like that of transducin (Fig. 4), or the HGH/HGH receptor complex where the receptor is a homodimer and bears one recognition patch per subunit, each patch makes its own O ring.

To conclude, we wish to stress that the core/rim model, and the mutant data themselves, should not be interpreted to say that the rim plays no part in the interaction. The model and the data only state that the rim residues contribute less to the free energy of dissociation than the core residues, or that their contribution does not depend heavily on the nature of their side chain.

Interface size and stability in homodimers

Subunit interfaces have been analyzed in several sets of oligomeric proteins that comprise tetramers, hexamers and larger assemblies (Argos, Reference Argos1988; Janin et al. Reference Janin, Miller and Chothia1988; Jones & Thornton, Reference Jones and Thornton1995, Reference Jones, Thornton and Kleanthous2000; Ponstingl et al. Reference Ponstingl, Kabir, Gorse and Thornton2005), but most of the available data concern homodimers, and we will limit our analysis to that category. Table 2 lists the mean values of the interface parameters in three separate sets of homodimers. That of Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2003) was manually assembled and comprises 122 proteins. A second set more than twice that size, and a third set of 19 ‘weak’ dimers were recently compiled by our group (Dey et al. unpublished data) with the help of an early version of the PiQSi database (Lévy, Reference Lévy2007); they are non-redundant at the 35% sequence identity level, and the QS of the proteins in them was confirmed by checking the literature. The weak dimers are homodimers reported to be in equilibrium with the monomers in the publications cited in PiQSi. Some of the homodimers in the first two sets may also be weak in that sense, but the scanned publications did not say. Noreen & Thornton (Reference Noreen and Thornton2003a, Reference Noreen and Thorntonb) also assembled a set of weak dimers, but due to different criteria of choice, the present list shares only two entries with theirs.

The most obvious feature that distinguishes the homodimer interfaces in Table 2 is their large BSA, which on average is twice that of the non-obligate complexes. As the number of interface atoms and interface residues increases linearly with the BSA, the factor of 2 also applies to atoms and residues. However, the average BSA is of little significance given the very large standard deviation. Figure 9 shows that individual values spread from less than 1000 Å² to more than 6000 Å²; the largest in the set is 14 300 Å². The BSA distribution is essentially the same in the Bahadur set and the larger new set, but it is very different in the weak dimer set. The mean BSA of the weak dimers is 1620 Å², consistent with the data of Noreen & Thornton (Reference Noreen and Thornton2003b), who report an average of 740 Å² for the buried surface area per subunit (that is, BSA/2) in their set of homodimers in equilibrium with monomers. In Fig. 9, most of the interfaces with BSA <1000 Å² belong to the weak dimers. The smallest, with a BSA near 750 Å², occur in cytochrome c6 (1c6o) and ferredoxin (1sj1), two proteins that may not be actual homodimers in the cell. The functional state of cytochrome c6 is likely to be a heterodimer (Schnackenberg et al. Reference Schnackenberg, Than, Mann, Wiegand, Huber and Reuter1999); that of ferredoxin is unknown.

Fig. 9. Interface size in homodimers. Histogram of the BSA in the three sets of homodimers reported in Table 2; the weak dimers are in equilibrium with the monomers.

Chemical and amino acid composition

Non-polar (carbon-containing) groups contribute f _np=65% of the BSA of homodimer interfaces, significantly more than to the accessible protein surface (f _np=57%), or to the interfaces in complexes; the weak dimer interfaces (f _np=62%) are in between. The hydrophobic character of the homodimer interfaces is associated in Table 2 with a lesser contribution of the neutral polar groups, but not of the charged groups, to the BSA. Nevertheless, homodimer interfaces do not, in general, form large hydrophobic patches (Larsen et al. Reference Larsen, Olson and Goodsell1998), and they contain many H bonds and water molecules. Table 2 reports an average of 18–19 H bonds and 44 water molecules per interface, which makes the surface density of the H bonds and the water molecules in homodimer interfaces similar to that of the complexes. The data also suggest that the H bond density is less in the weak dimers, but the statistics are poor because of the small size of that set.

The amino acid composition of the homodimer interfaces has often been analyzed (Jones & Thornton, Reference Jones and Thornton1995, Reference Jones and Thornton1996; Tsai et al. Reference Tsai, Lin, Wolfson and Nussinov1997; Bahadur et al. Reference Bahadur, Chakrabarti, Rodier and Janin2003, Reference Bahadur, Chakrabarti, Rodier and Janin2004). It follows the same trends as the atomic composition: the interfaces are enriched in aliphatic and aromatic residues, on average by a factor of about 2, and depleted by the same factor in charged residues other than Arg. When the relative contributions of the residue types to the BSA and the ASA are expressed as propensities, the observation made above for the complexes remains valid: the propensities are much more marked for the residues of the interface core, which contain the buried interface atoms, than the interface rim. Figure 7 b reports the propensities to be at the interface core in the Bahadur set. A comparison with Fig. 7 a shows that they generally resemble those in complexes, but the aliphatic residues Ala, Val and Leu are more abundant. Leu and Arg are the largest contributors to the cores of homodimer interfaces, each with 10–11% of the BSA. In the complexes, Tyr and Arg both contribute 10% of the core BSA; Leu is third, but with only 6% (Chakrabarti & Janin, Reference Chakrabarti and Janin2002; Bahadur et al. Reference Bahadur, Chakrabarti, Rodier and Janin2003).

The propensity of an atom or a residue to be at an interface vs. the protein surface may be viewed as one-body statistical potential. The R _p score of Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004), derived simply by summing the propensities over all the residues of an interface, has a positive and large mean value in homodimers (Table 2). It is also positive, but much lower, in the complexes and the weak dimers, in line with their relative stabilities. Two-body potentials can be derived statistically by counting pairwise contacts across the interfaces (Moont et al. Reference Moont, Gabb and Sternberg1999; Glaser et al. Reference Glaser, Steinberg, Vakser and Ben-Tal2001; Ofran & Rost, Reference Ofran and Rost2003; Saha et al. Reference Saha, Bahadur and Chakrabarti2005). When residue–residue contacts are analyzed in different types of protein–protein interfaces, the homodimers show a peculiar feature: pairs containing identical residues are much more frequent than expected on a random basis. This results from contacts made by twofold-related residues along the symmetry axis, which make up 12% of the BSA. After correcting for the relative abundance of each residue type, Leu comes to be by far the largest contributor to the pairwise contacts (13%), followed by Phe, Val and Arg (Saha et al. Reference Saha, Bahadur and Chakrabarti2005). The leucine zippers are a well-known example of how Leu/Leu contacts may contribute to the stability of a homodimer.

Atomic packing and sequence conservation

The oligomeric proteins are generally known to assemble while the subunits fold, and their interfaces can be expected to be close-packed like the subunits' interior. Ponstingl et al. (Reference Ponstingl, Kabir, Gorse and Thornton2005) observe that the Voronoi volumes of the interface atoms of oligomeric proteins are within a few percent of the reference volumes of Tsai et al. (Reference Tsai, Taylor, Chothia and Gerstein1999), but this applies only to the buried atoms, which are a minority (f _bu=35–36% in Table 2). Buried atoms are even fewer at the interfaces of weak dimers (f _bu=28%). Table 2 lists two other parameters related to the atomic packing: the L _D packing index of Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004) and the S _c shape complementarity score of Lawrence & Colman (Reference Lawrence and Colman1993). Both take very similar mean values in homodimers and in complexes, which supports the idea that all these interfaces are close-packed. On the other hand, the weak dimer interfaces have low values of f _bu and of L _D, and they may be poorly packed.

Guharoy & Chakrabarti (Reference Guharoy and Chakrabarti2005) analyzed the conservation of the interface residues in homodimers, and obtained a result very similar to what they had seen in complexes (Table 2): the mean Shannon entropy of interface residues is significantly less than the average in the whole polypeptide chain, and it is lower for the residues of the interface core than the rim. In that study, the mean number of residues in the core was the same in both types of interfaces, and the core represented a majority of the residues. This suggests that the core/rim model of interfaces is also valid for homodimers, but the correlation with hotspots is difficult to establish in their cases. Most are permanent assemblies much less amenable to a quantitative study than a non-obligate complex. The monomers are in general not observed, and K _d values cannot be measured. The measurement can, in principle, be done on the weak dimers, but mutagenesis data are comparatively scarce on those.

Size and composition of crystal packing interfaces

Protein crystals are held by the same forces that stabilize other macromolecular assemblies, and one may compare the interfaces between pairs of molecules in crystals to those of the complexes and oligomeric proteins (Janin & Rodier, Reference Janin and Rodier1995; Carugo & Argos, Reference Carugo and Argos1997; Dasgupta et al. Reference Dasgupta, Iyer, Bryant, Lawrence and Bell1997). A feature then strikes the eye: their small size. In the set of 1320 pairwise interfaces assembled by Janin & Rodier (Reference Janin and Rodier1995), the mean BSA is 570 Å². Crystal packing nevertheless buries much protein surface, because each molecule has 8–10 neighbors, and it makes pairwise interfaces with all. The average crystal packing interface represents only one-third of that of a complex, and one-sixth of that of a homodimer. The great majority buries less than 800 Å² and implicates fewer than 13 residues per protein molecule. It is often assumed that such interfaces cannot be biologically meaningful, but there are some larger ones in crystals. The BSA values approximate an extreme value distribution, in which large values are much more frequent than in a Gaussian distribution. Moreover, the interfaces with twofold symmetry tend to be larger than those created by other types of crystal symmetry operations (Janin, Reference Janin1997). About 10% of the crystal packing interfaces are similar in size to those of the complexes, and a majority of those have twofold symmetry. They form ‘crystal dimers’, which are artifacts of crystallization and do not exist in solution, yet they may be difficult to distinguish from biological homodimers on the basis of the crystal structure alone.

Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004) selected 188 large crystal packing interfaces by keeping only those with a BSA >800 Å² in the set of Janin & Rodier (Reference Janin and Rodier1995). They have an average BSA of 1510 Å², comparable to standard-size interfaces in complexes, and they comprise about the same average number of atoms and of residues (Table 2). Their chemical composition is also similar to that of the biologically significant interfaces, except for a slight excess of charged groups. Their amino acid composition is reported in Fig. 7 b in the form of propensities that are all small (<0·5) in absolute value, but follow the same pattern as homodimers and complexes: there is a mild depletion in Lys, and a mild excess of aliphatic and aromatic residues. The R _p score, which is the sum of the propensities of individual interface residues to be at a homodimer interface and has a positive mean value in complexes, is negative on average at crystal contacts (Table 2).

Crystal packing interfaces have a comparatively low average density of H bonds: 1 per 280 Å² of BSA, instead of 1 per≈200 Å² in complexes and homodimers (Table 2). This may be related to their low fraction of buried atoms: f _bu=21% vs. 34–36%; and their high level of hydration: 15 waters per 1000 Å² of interface area vs. 10–11 in homodimers and complexes. Their topology is another cause for these differences. A standard-size interface in a complex is in general single-patch and compact, and a crystal packing interface of the same size is almost always fragmented. This can be seen in Fig. 10, which compares the interfaces of a biological homodimer (κ-bungarotoxin, PDB entry 1kba) and a crystal dimer (pokeweed antiviral protein, PDB entry 1qci). The two are of similar size, but the second is poorly packed; it buries very few atoms and its L _D index is low. In Table 2, the fragmentation of crystal packing interfaces shows in the low mean value of L _D and in the number of polypeptide segments, which is greater on average at a crystal packing interface than at an interface of the same size in a complex; homodimer interfaces implicate an even smaller number of segments in proportion of their size.

Fig. 10. The atomic packing of a specific and a non-specific interface. The κ-bungarotoxin homodimer (1kba) and the pokeweed antiviral protein crystal dimer (1qci) form interfaces of a similar size, with a BSA of almost 1000 Å², but only the first is biologically significant. Each subunit of either dimer contains about 50 interface atoms, drawn here as spheres in the plane of the twofold axis. Their packing is very different, as are the values of the fraction of buried atoms (f _bu=28% vs. 7%) and the L _D index (31 vs. 13).

Biological assemblies vs. crystal artifacts

These differences may be used to identify crystal and biological dimers in PDB entries. The interface size alone correctly distinguishes homodimers from monomers in 85% of the cases (Ponstingl et al. Reference Ponstingl, Henrick and Thornton2000). Additional information can be drawn from the literature, residue conservation or physical–chemical properties of the interfaces as discussed above (Section 2.3). In the sample of Bahadur et al. (Reference Bahadur, Chakrabarti, Rodier and Janin2004), where all the crystal packing interfaces have a BSA >800 Å², size is not a powerful discriminator, but it can be combined with three parameters that tend to take higher values in biological homodimers than crystal dimers: the fractions of non-polar groups (f _np) and buried atoms (f _bu), and the number of H bonds (n _HB). Thus, an interface has an 88% probability of belonging to a biological homodimer if it satisfies one of the two conditions:

and it has the same probability to be due to the crystal packing if neither condition is met (Janin et al. Reference Janin, Rodier, Chakrabarti and Bahadur2007).

The molecular contacts in crystals are unspecific and not biologically meaningful, with a few interesting exceptions (Janin, Reference Janin1997). Unlike specific interfaces, they should exert no evolutionary pressure on the regions of the protein surface implicated, an assumption supported by a comparison of aligned sequences. Table 2 reports the normalized Shannon entropy of the residues in the large crystal packing interfaces. Its mean value is s≈1, meaning that they evolve at the same rate as the average residue in the polypeptide chain to which they belong. Moreover, the core and the rim have nearly identical s, the core and rim residues being defined here by the buried atoms as in the other types of interfaces even though many crystal packing interfaces do not have a proper core, because they bury few atoms and are highly fragmented. In addition, proteins often come in several crystal forms that use different contacts. Pancreatic ribonuclease, for instance, crystallizes in at least six forms. Several contain the same large interface that forms a dimer in solution under the conditions that lead to crystallization, but the other crystal packing contacts essentially implicate the whole protein surface (Crosio et al. Reference Crosio, Janin and Jullien1992).

Symmetry

Icosahedral virus capsids are multi-subunit assemblies, a category that constitutes most of the cellular machines, but is poorly represented in the PDB at present (Dutta & Berman, Reference Dutta and Berman2005). The capsids encapsulate and protect the viral genome, and they are frequently implicated in the recognition and infection of target cells. They are very large objects with molecular weights of millions, and a symmetry that greatly helps in their study, so that atomic structures could be solved three decades ago for tomato bushy stunt virus (Harrison et al. Reference Harrison, Olson, Schutt, Winkler and Bricogne1978) and satellite tobacco necrosis virus (Liljas et al. Reference Liljas, Unge, Jones, Fridborg, Lövgren, Skoglund and Strandberg1982). Here, we can compare the subunit interfaces in capsids to those found in binary assemblies in terms of their structural and physical chemical properties, and discuss the process of capsid self-assembly.

The virus capsids were considered spherical until Crick & Watson (Reference Crick and Watson1956) proposed that they have the symmetry of the cubic I (icosahedral) point group. This was demonstrated by the electron microscopy studies of Caspar & Klug (Reference Caspar and Klug1962), who introduced the rule of quasi equivalence and the lattice triangulation number T. Quasi equivalence allows the icosahedral asymmetric unit (IAU) to contain T subunits, and the whole capsid, 60T subunits instead of the 60 implied by the point group. The subunits in the IAU are related by inexact or ‘quasi’ symmetries. They have identical sequences, but different conformations, and they make different contacts with their neighbors (Rossmann & Johnson, Reference Rossmann and Johnson1989). X-ray structures are available for a number of capsids with the T=1 lattice exemplified by satellite tobacco necrosis virus, the T=3 lattice exemplified by tomato bushy stunt virus, and also the pT3 (pseudo T=3) lattice exemplified by rhinovirus (Rossmann et al. Reference Rossmann, Arnold, Erickson, Frankenberger, Griffith, Hecht, Johnson, Kamer, Luo, Mosser, Rueckert, Sherry and Vriend1985). The pT3 capsids resemble the T=3 capsids, but the subunits in their IAU have different sequences. The PDB also reports the X-ray structures of a few capsids with larger T numbers, or with lattices that do not follow the rule of quasi equivalence, for instance the very large and complex capsids of Bluetongue virus (Grimes et al. Reference Grimes, Burroughs, Gouet, Diprose, Malby, Ziéntara, Mertens and Stuart1998) and phage PhiX174 (Dokland et al. Reference Dokland, Bernal, Burch, Pletnev, Fane and Rossmann1999). The VIPER database of the Scripps Research Institute (Shepherd et al. Reference Shepherd, Borelli, Lander, Natarajan, Siddavanahalli, Bajaj, Johnson, Brooks and Reddy2006) offers a particularly convenient access to some 250 virus structures obtained by X-ray and electron microscopy.

Subunit interfaces in capsids

Bahadur et al. (Reference Bahadur, Rodier and Janin2007) have assembled a non-redundant dataset of 49 viral capsids that includes 11 with lattice T=1, 17 with lattice T=3, 10 with lattice pT3 and 11 others. In each capsid, a unique set of interfaces between pairs of polypeptide chains can be identified. Icosahedral symmetries repeat these unique interfaces 60 times, except those with twofold (I ₂) symmetry, which are repeated 30 times. There is an average of 16 unique interfaces per capsid; one-third occurs within the IAU, another third has I ₂, I ₃ or I ₅ symmetry, the remainder, a quasi-symmetry.

The average pairwise interface in that set has a BSA of 1750 Å², similar to transient complexes, but the range of sizes is large. Bahadur et al. (Reference Bahadur, Rodier and Janin2007) split the pairwise interfaces into three size categories. The small interfaces (BSA <800 Å²), which make up 40% of the sample, contribute only 7% of the BSA. The remainder belongs to medium-size (800–2000 Å²) and large interfaces (more than 2000 Å²). The T=1 capsids contain in general three unique interfaces with a BSA >800 Å², one of each of the I ₂, I ₃ and I ₅ symmetry types; T=3 capsids contain 5–9, including 3 between the chains in the IAU; pT3 capsids contain 6–16.

In a virus capsid, each subunit has many neighbors and buries a large fraction of its surface in contacts with them. The average number of neighbors is 7 in T=1 and T=3 capsids. In these capsids, the subunit contacts implicate about 60% of all residues and bury about 45% of the subunit ASA. In pT3 capsids, the number of neighbors increases to 12; the fraction of the residues that are at interfaces, to 73%; and the fraction of the ASA that is lost, to 60%. The complexity of the capsid assembly and the multiplicity of interfaces of all sizes are illustrated in Fig. 11 for rhinovirus (PDB entry 1 aym; Hadfield et al. Reference Hadfield, Lee, Zhao, Oliveira, Minor, Rueckert and Rossmann1997). Its pT3 capsid contains four different polypeptide chains, one of which (chain A in gray) forms homopentamers about the icosahedral fivefold axes, whereas chains B (blue) and C (cyan) form heterohexamers about the threefold axes. Chain A covers much of the outer surface of the capsid, yet the bottom part of Fig. 11 shows that in two opposite orientations, most of its surface is involved in one or several of the 13 different interfaces it makes with its neighbors.

Fig. 11. Capsid assembly and subunit interfaces in rhinovirus. Rhinovirus (1aym; Hadfield et al. Reference Hadfield, Lee, Zhao, Oliveira, Minor, Rueckert and Rossmann1997) has a pT3 capsid with four polypeptide chains in the IAU. Top: The capsid and its icosahedral lattice; chain A in gray forms pentamers about the fivefold axes; chain B in blue and chain C in cyan form heterohexamers about the threefold axes. Bottom: The molecular surface of chain A is drawn in two orientations 180° apart in the plane of the I ₅ axis (dashed line), and colored according the subunit contacts. Chain B is drawn as a gray tube, chain C as a black tube. In the capsid, chain A has 13 neighbors with which it makes five large, four medium-size and four small interfaces. The large interfaces are with chain B (blue surface) and chain C (green), two A chains in the A pentamer (pink and brown) and chain C′, threefold related to C (cyan). The medium size interfaces involve the small chain D (yellow) and symmetry-related chains B′, B″ and C″. Chain A5″, related to A by a 144° rotation about I ₅, makes a small interface (red) that implicates only one residue of A. Figure adapted from Bahadur et al. (Reference Bahadur, Rodier and Janin2007).

An important feature of the capsid assembly cannot occur in binary complexes and homodimers: the pairwise interfaces between subunits overlap, and many atoms or residues are part of more than one. On average, 15% of the interface atoms and 24% of the interface residues are in contact with two neighboring subunits, 2% of the atoms and 5% of the residues with three, and there are extreme cases where a residue is in contact with seven subunits (Bahadur et al. Reference Bahadur, Rodier and Janin2007).

Composition and topology

Capsid interfaces resemble homodimer interfaces in their non-polar character (f _np=63%), irrespective of their size (Bahadur et al. Reference Bahadur, Rodier and Janin2007). The fraction of buried atoms (f _bu) is only 29% in the pairwise interfaces, but it increases to 36% in the whole assembly. The difference comes from atoms in contact with more than one neighboring subunit; they can have a non-zero ASA in each of the pairwise interfaces, and a zero ASA in the capsid. The large capsid interfaces have f _bu and L _D packing indexes similar to those of homodimers, which implies that they are well packed. In contrast, the interfaces with BSA <2000 Å² bury very few atoms (f _bu=11–23%), and their L _D index is low. Like crystal packing contacts, they may be loosely packed.

The interface core, which contains the buried atoms, represents half of the residues in pairwise interfaces, and two-thirds in the whole capsid. The high proportion of core residues affects the amino acid composition of the capsid interfaces. As in homodimers and complexes, the core is enriched in aliphatic residues and depleted in Asp, Glu and Lys, but not Arg. On the other hand, the capsid interfaces contain fewer aromatic and more neutral polar residues: Ser, Thr, Asn, Gln and Pro contribute 32% of the BSA in capsids vs. only 21% in homodimers.

Capsid interfaces contain an average of 7 H bonds. The surface density of 1 H bond per 250 Å² of BSA is less than for the homodimers in Table 2, but the difference is probably an artifact of the comparatively low resolution of many of the viral X-ray structures: the interface H bond density is 1 per 200 Å² in the 17 capsids structures with 2·8-Å or better resolution.

The topology of capsid interfaces was analyzed with the clustering method of Chakrabarti & Janin (Reference Chakrabarti and Janin2002). Most of the small interfaces are single-patch, medium size interfaces form one to three pairs of patches and large interfaces three or more. Interfaces of equivalent size in complexes have a similar number of patches. When the interface residues are distributed into segments of the polypeptide chain following Jones & Thornton (Reference Jones and Thornton1997) and Pal et al. (Reference Pal, Chakrabarti, Bahadur, Rodier and Janin2007), the mean number of segments is 3·9 per chain, and that of interface residues per segment is 6·4, again like in the complexes in Table 2. However, the averaging hides a great disparity between the small interfaces, often fragmented into very short segments, and larger ones that tend to contain a small number of long segments. In T=3 and pT3 capsids, some interfaces contain segments of up to 47 residues located at the N or C terminus of the polypeptide chain. These tail segments adopt extended conformations and are heavily involved in subunit contacts. In T=3 capsids, the tails are preferentially involved in the twofold and quasi-sixfold interfaces; in pT3 capsids, they contribute primarily to the interfaces in the IAU (Bahadur et al. Reference Bahadur, Rodier and Janin2007).

Residue conservation

The subunit contacts hold the capsid together and play a major role in the process by which it assembles itself. They should therefore impose stringent evolutionary constraints on the protein sequence. Bahadur & Janin (Reference Bahadur and Janin2008) used the Shannon entropy to estimate residue conservation in sets of aligned capsid protein sequences. The mean value of s normalized to the average value in each polypeptide chain, is close to 1 for the interface residues, which reflects the fact that they constitute two-thirds of the polypeptide chain. Nevertheless, there are very significant differences within a chain. As in other systems, the residues of the protein interior are much better conserved than surface residues (s=0·7 vs. 1·6), and the residues of the interface core, much better conserved than the rim (s=0·8 vs. 1·2). Moreover, the capsids contain residues involved in several interfaces, with no equivalent in homodimers or binary complexes. They are abundant (34% of all interface residues), and their mean entropy s=0·8 shows that they are better conserved than the residues involved in only one interface.

The sequence conservation needs not be homogeneous within a given interface. An example is the interface between chains A and C of the pT3 rhinovirus (the green surface in Fig. 11). It is well conserved as a whole (s=0·9), but chain A has a highly divergent C tail with s=2·0 and a highly conserved N tail with s=0·6, and both tails contribute to the interface. The C tails are more divergent than the N tails in most pT3 capsids, but in general, the tails have the same rate of evolution as the parts of the polypeptide chains that form the globular core of the subunits (Bahadur & Janin, Reference Bahadur and Janin2008).

A plausible mechanism for capsid assembly

The self-assembly of a virus capsid comprises the folding of its subunits, their association and maturation steps that often involve conformation changes and covalent modifications. This elaborate process may implicate nucleic acids, accessory viral proteins and host chaperones, in addition to the protein subunits (Steven et al. Reference Steven, Trus, Booy, Cheng, Zlotnick, Caston and Conway1997, Reference Steven, Heymann, Cheng, Trus and Conway2005; Liljas, Reference Liljas1999; Caspar, Reference Caspar1980). Nonetheless, it can be amazingly fast: bacteriophage T4 goes through a complete cycle of infection, replication and lysis in 15 min; the capsid assembles in minutes in spite of the large number of polypeptide chains that constitute it (Leiman et al. Reference Leiman, Kanamaru, Mesyanzhinov, Arisaka and Rossmann2003). This implies that capsid self-assembly must go through a series of low-order steps and intermediate species. These species, called capsomers, should resemble small oligomeric proteins.

Prevelige et al. (Reference Prevelige, Thomas and King1993) have analyzed the in vitro self-assembly of bacteriophage P22, and Xie & Hendrix (Reference Xie and Hendrix1995), that of bacteriophage HK97. The T=7 lattice of these two capsids can be viewed as comprising pentamers with I ₅ symmetry and hexamers with Q ₆ quasi-symmetry, in a 1:5 ratio. In solution, the HK97 subunits form pentamers and hexamers that interconvert slowly and reassemble most efficiently when they are in the same 1:5 ratio as in the capsid. All models of the capsid self-assembly assume that the capsomers keep their structure during the process, and thus, they can be identified in the capsid itself (Johnson & Speir, Reference Johnson and Speir1997; Reddy et al. Reference Reddy, Giesing, Morton, Kumar, Post, Brooks and Johnson1998; Dokland, Reference Dokland2000; Zlotnick, Reference Zlotnick2005). Bahadur et al. (Reference Bahadur, Rodier and Janin2007) postulate in addition that the capsomers contain the largest interfaces, and that capsomer–capsomer association makes use of the medium-size interfaces in priority. The many small interfaces that are seen in the capsids may contribute to the stability of the final product, but they are unlikely to play a role in its formation.

In the case of HK97, this postulate suffices to determine an assembly pathway compatible with the data of Xie & Hendrix (Reference Xie and Hendrix1995). The largest interfaces are the I ₅ and Q ₆ interfaces, and they build the hexamers and pentamers seen in solution. The next largest are of medium size; they occur between hexamers or between hexamers and pentamers in the capsid, and in solution, they should allow hexamers to associate in pairs or with a pentamer (Bahadur et al. Reference Bahadur, Rodier and Janin2007). Recently, Stockley et al. (Reference Stockley, Rolfsson, Thompson, Basnak, Francese, Stonehouse, Homans and Ashcroft2007) have analyzed the assembly of bacteriophage MS2 by electrospray ionization-mass spectrometry. They find that the T=3 capsid of MS2 dissociates into symmetric dimers, some of which become asymmetric upon addition of an RNA stem–loop fragment. Symmetric and asymmetric dimers then associate into dimer hexamers, whereas no pentamer is formed. These observations are compatible with the model of Bahadur et al. (Reference Bahadur, Rodier and Janin2007) : MS2 has large I ₂ and Q ₂ interfaces that build a symmetric and an asymmetric dimer, respectively. The next largest are the Q ₃ interfaces building the dimer hexamers; they are of medium size, and larger than the I ₅ interface needed to build pentamers.