research papers
Representation of viruses in the remediated PDB archive
^{a}RCSB Protein Data Bank, Department of Chemistry and Chemical Biology, Rutgers, The State University of New Jersey, 610 Taylor Road, Piscataway, NJ 088548087, USA, and ^{b}Macromolecular Structure Database–European Bioninformatics Institute, EMBL Outstation–Hinxton, Cambridge CB10 1SD, England
^{*}Correspondence email: cathy.lawson@rutgers.edu
A new scheme has been devised to represent viruses and other biological assemblies with regular
in the Protein Data Bank (PDB). The scheme describes existing and anticipated PDB entries of this type using generalized descriptions of deposited and experimental coordinate frames, symmetry and frame transformations. A simplified notation has been adopted to express the symmetry generation of assemblies from deposited coordinates and matrix operations describing the required point, helical or Complete correct information for building full assemblies, subassemblies and crystal asymmetric units of all virus entries is now available in the remediated PDB archive.Keywords: virus structures; Protein Data Bank; database integration; uniform curation; point symmetry; helical symmetry; biological assemblies.
1. Introduction
Recent improvements in structural biology methods have given rise to an increasing body of structural data for biological assemblies composed of tens to thousands of individual protein and/or nucleic acid polymer chains. Structures of such quaternary complexes or assemblies present many challenges for archival representation and validation, graphical display and analysis (Dutta & Berman, 2005).
Large biological assemblies are often composed of multiple copies of one or more polymer entities, with the arrangement of repeating units following a regular point or helical symmetry (Goodsell & Olson, 2000). The largest class of biological assemblies with regular symmetry currently represented in the Protein Data Bank (PDB) archive (Berman et al., 2000) are the icosahedral viruses, with approximately 250 structures determined either by Xray crystallography or cryoelectron microscopy (CryoEM; reviewed by Harrison, 2001; Chiu & Rixon, 2002; Lee & Johnson, 2003). A smaller group of virus entries have helical symmetry: approximately 30 structures determined mainly by fiber Xray diffraction methods (Marvin, 1998; Stubbs, 1999).
Other assemblies with regular et al., 2005), clathrin cages and chaperonins with dihedral symmetry (Fotin et al., 2006; Braig et al., 1994), ferritins with tetrahedral or octahedral symmetry (Johnson et al., 2005; Hamburger et al., 2005) and DNAprocessing enzymes with helical symmetry (Van Loock et al., 2003; Conway et al., 2004).
are also represented in the PDB. These include viral toxins with circular symmetry (TilleyAssemblies may have multiple embedded symmetries or adjacent symmetries. For instance, the icosahedral Paramecium bursaria chorella virus type 1 (PBCV1) algal virus shell has thousands of copies of a membraneembedded coat protein arranged with pseudocrystalline symmetry (Nandhagopal et al., 2002). The T4 tailed bacteriophage has fivefold, sixfold and helical symmetries aligned along a common axis (Leiman et al., 2003).
The PDB entries of icosahedral and helical viruses and a handful of other large biological assemblies with regular
were previously archived in an inconsistent manner and were prone to errors. To address these problems, we have developed a flexible scheme to represent assemblies with regular symmetry. The scheme involves four key elements: (i) a set of atomic coordinates representing the repeating unit, (ii) parameters defining the regular symmetry, (iii) an operations list containing regular symmetry operations plus any frame transformations (transformations between different coordinate frames) and (iv) a compact set of assemblygeneration instructions, with the possibility of defining multiple assemblies. Using this scheme, instructions may be given to build a full icosahedral virus in the deposited frame, a pentamer subassembly of the virus in the standard icosahedral point frame and the of the virus crystal in the standard spacegroup frame.This representation was developed to provide uniformity among virus structures within the PDB as part of a larger remediation project to remove legacy errors and improve the uniformity of the entire archive (Henrick et al., 2008). The representation has been fully implemented in the PDB exchange dictionary and has been incorporated in the remediated entries of over 280 structures, mainly viruses but also several nonvirus assemblies (Table 1). The new scheme will permit routine annotation of future entries with regular and complex symmetries and will also make it possible to more easily build and view such assemblies within graphical display programs.

2. Background: remediation of virus entries
A review of 250 icosahedral virus structure entries and 30 helical virus entries deposited into the PDB between 1984 and 2006 revealed three major issues to be addressed in remediation: missing or erroneous sets of transformation operations, inconsistency in coordinateframe representations and overly complex building instructions. For each issue, corrected information was gathered and validated in a systematic way.
For approximately 40% of virus entries, the set of matrix transformations needed to build up the full biological assembly either was absent or contained errors. Problem entries were identified by inspection of images generated via an automated script using the Multiscale Model module of Chimera (https://www.cgl.ucsf.edu/chimera/ ; Goddard et al., 2005; Pettersen et al., 2004). Corrected transformations were obtained from the Virus Particle Explorer database (VIPERdb; https://viperdb.scripps.edu ; Reddy et al., 2001; Natarajan et al., 2005; Shepherd et al., 2006) or the Protein server (PQS; https://pqs.ebi.ac.uk ; Henrick & Thornton, 1998). For helical viruses, parameters to construct representative matrix transformations were collected from PQS.
The atomic coordinates of virus entries have been archived in a variety of different coordinate reference frames. CryoEM structures and early crystal structures of icosahedral viruses are typically presented in one of two standard icosahedral reference frames. However, the recent trend for crystal structures is to deposit in the frame of the ). For each icosahedral virus, the transformation [P] that moves the deposited coordinates into the VIPER standard icosahedral frame was determined using the PDB2VIPER program (Shepherd et al., 2006) with minor modifications. 60 transformations [T_{m}], m = 1–60, were calculated for each assembly from a standard ordered set of icosahedral operations [I_{m}] (see §3.1.1 for definition),
(Fig. 1For 210 icosahedral virus crystal structures, transformations to the Crystal Contacts module of Chimera. Of 88 entries with deposited structure factors, 70 yielded R factors below 0.40 (56 below 0.30) using SFCHECK (Vaguine et al., 1999). Before remediation, only a handful of these entries yielded reasonable validation statistics.
frame were collected from author text remarks or primary citations, extracted from SCALE records, or set to identity, as appropriate. One transformation was defined for each independent particle in a crystal (NCS) operations defining crystal asymmetric units were determined automatically using software developed inhouse. Crystal packing was inspected using theFor the majority of virusstructure entries with atomic coordinates representing one regular symmetry (point or helical) e.g. an icosahedral pentamer, or a full crystal with one quarter or one half of a full virus capsid. In some of these cases coordinates were presumably duplicated for convenient viewing of a particular interface, but in others regular symmetry is only approximate and explicit coordinates are required to represent the unique part of a lower symmetry structure. For the PBCV1 virus (PDB code 1m4x ; Nandhagopal et al., 2002), atomic coordinates are only provided for a small fraction (1/28th) of one icosahedral containing three chains: a total of 3 × 28 × 60 chains and 16 284 240 atoms are required to build the complete capsid. In all of these special situations, symmetryparameter representation and instructions for building complete assemblies from selections of matrix operations, selections of coordinates and/or hierarchical application of transformation operations were defined on a casebycase basis.
application of regular symmetry operations is all that is required to build a full or representative assembly. However, several entries contain explicit atom coordinates for larger assemblies,3. Representation of complexes with regular symmetry
In order to archive the corrected information gathered in the virus remediation process, the PDB exchange dictionary was extended (https://mmcif.pdb.org ). New terms enable explicit definition of regular noncrystallographic point and helical symmetries and provide for definition of transformation operations and implementation of a compact notation for assembly generation. The new dictionary categories are used in conjunction with existing data items for crystal symmetry and logical groups of atomic coordinates. The resulting representation permits the description of biological assemblies with any regular symmetry and determined by any experimental method. An example of the representation in mmCIF format is provided as supplementary material.^{1}
3.1. Regular symmetry definitions
Regular symmetries include point, helical and crystal symmetries. Given parameters appropriate to the symmetry type and a standard reference frame with a defined relationship between symmetry axes and Cartesian coordinate axes, a complete set of symmetry operations can be defined for any ). Parameter and standard frame definitions used for point and helical symmetries are described below and follow the conventions for cryoEM structural studies proposed by Heymann et al. (2005).
and representative symmetry operations can be defined for any helical or crystal symmetry. The PDB follows standard definitions for crystal symmetry (Hahn, 20023.1.1. Point symmetry
The five point symmetries that can be adopted by biological assemblies are circular, dihedral, tetrahedral, octahedral and icosahedral, corresponding to Schöenflies symbols C, D, T, O and I, respectively. For structures with circular or dihedral symmetry, a circular symmetry parameter is required to define the number of repeats around the major symmetry axis. Examples include a viral toxin with C38 symmetry (Fig. 2c), a clathrin cage with D6 symmetry (Fig. 2d) and a fourlayer ring with D17 symmetry (Fig. 2e).
Standard frames and hierarchical order of symmetry operations for the point symmetries are defined in Table 2. In every case the symmetry center is at the origin and symmetry elements are aligned to major orthogonal coordinate axes. The icosahedral standard frame is identical to the VIPERdb frame, with twofolds aligned to the x, y, z axes and fivefolds closest to the z axis lying in the yz plane (Fig. 3). Icosahedral pointsymmetry operations are initiated by the application of fivefold symmetry around the vector (0, 1, φ), followed by application of tetrahedral symmetry operations. Where possible, the hierarchical order of symmetry operations follows the related P23 for tetrahedral symmetry, P432 for octagonal symmetry.
‡The icosahedral standard frame is identical to that utilized by VIPERdb (Reddy et al., 2001), but the hierarchy of symmetry operations follows tetrahedral symmetry after the application of fivefold symmetry. φ = [(5)^{1/2} + 1]/2. 
The VIPER database restricts the position of the primary icosahedral et al., 2005; Shepherd et al., 2006). The advantage of restricted placement is that the transformation from an arbitrary deposited frame into the standard frame {[P] in (1)} has one unique solution. We utilize the same boundaries, as illustrated in Fig. 3: for triangleshaped icosahedral asymmetric units (e.g. Fig. 2a) the center of mass must fall within the yellow outline, or for rhomboidshaped icosahedral asymmetric units (e.g. Fig. 2b) within the green outline. Restricted placement conditions for the primary center of mass are also defined for the other point symmetries (last row in Table 2).
center of mass within the icosahedral standard frame (Natarajan3.1.2. Helical symmetry
Symmetry parameters, standard frames, hierarchy of symmetry operations and . Polar and nonpolar helical symmetries closely follow the definitions for related circular and dihedral point symmetries.
placement for polar and nonpolar helical symmetries are defined in Table 3

Helical screw symmetry is defined using three parameters in order to allow an exact repeat: rotation around the helical axis for n subunit repeats, translation along the helical axis for n subunit repeats and number of subunit repeats divisor (n). For example, the fiberdiffraction structure of cucumber green mottle mosaic virus (CGMMV; Fig. 2f) with 49 subunits in three turns has a rotation per subunit repeat of 1080/49 degrees and translation per subunit repeat of 70.8/49 Å. When there is no exact repeat, rotation and translation is defined for a single subunit repeat with the divisor set to unity.
Two additional parameters define rotational symmetries of a helical assembly. The presence or absence of dyad symmetry perpendicular to the helical axis distinguishes nonpolar helical structures (two ends equivalent) and polar helical structures (each end unique). Circular symmetry is a positive integer that defines the number of subunit strands twisting in parallel about the helical axis. Circular symmetry is onefold for CGMMV (Fig. 2f) and fivefold for the filamentous phage illustrated in Fig. 2(g). Both of these helical viruses are polar.
Although not an essential parameter, the number of symmetry operations needed to generate a representative helical assembly should be defined. The number is arbitrary but should be large enough to represent the overall symmetry and all unique intersubunit interactions. It should also ideally be a multiple of the circular symmetry parameter, a multiple of 2 if dyad symmetry is present and a multiple of an odd number so that generated operations may be centered about the identity operation.
3.2. Transformation operations list
All transformation operations that may be applied to the deposited orthogonal angstrom coordinate positions are gathered into a single unified list. The list can include transformations to other orthogonal coordinate frames, as well as regular point, helical and crystal symmetry operations in the deposited frame. Inverse transformations (i.e. transformations from other frames/positions into the deposited frame/position) are not included, since they do not meet the criteria of being applicable to the deposited coordinates.
Each operation is identified by a unique ID and is represented as nineelement rotation matrix plus a threeelement translation vector. To convert to the more convenient 16element 4 × 4 matrix form, the rotation matrix is placed in the first three rows and columns and the translation vector becomes the first three elements of the fourth column. The fourth row is set to 0, 0, 0, 1. The resulting 4 × 4 matrix that operates on fourelement vectors is
3.2.1. Frame transformations
Assemblies in experimental orthogonal coordinate frames other than the deposited frame may be defined. The deposited frame can be any arbitrary orthogonal coordinate frame favored by the deposition authors, although a standard frame is preferred. The relationship between the deposited frame and standard point, helical, crystal and/or other frames is then explicitly defined by including frame transformations in the operations list.
3.2.2. Regular symmetry operations
Point, helical or crystal symmetry operations in the deposited frame of the entry may be included in the transformation list. By convention, pointsymmetry operations begin with the identity operation and the order of subsequent operations follows the hierarchy for the defined symmetry in the standard frame (e.g. fivefold, twofold, twofold, threefold for icosahedral symmetry; see Table 2). For point symmetries deposited in nonstandard frames, symmetry operations are calculated using (1) after determination of the frame transformation matrix [P] (see §3.1.1). This method ensures that relative spatial relationships among symmetryrelated asymmetric units are consistent across the database. For example, the pentamer subassembly of every remediated icosahedral virus entry may be built by applying the first five pointsymmetry operations. Helical symmetry operations are defined in a continuous run centered about the identity operation.
3.3. Assembly generation
Here, we describe the logic for generating complete macromolecular assemblies for a PDB entry containing minimal coordinates plus a set of regular presents an overview of generation of assemblies in multiple coordinate frames using the example of the icosahedral φX174 procapsid (PDB entry 1al0 ; Dokland et al., 1997), a structure determined by Xray crystallography with two independent virusparticle positions in the crystal Atomic coordinates were deposited in an alternate icosahedral frame.
operations. Fig. 4The assembly path begins at the top center of Fig. 4 with the deposited chains represented as enveloped ribbons and proceeds counterclockwise. The coordinates are moved into the standard icosahedral frame (upper left) by application of the frametransformation matrix [P]. The complete biological assembly (lower left) is produced in the standard icosahedral frame by the application of 60 pointsymmetry operations and is moved back to the deposited frame (bottom center) by the application of [Pinv], calculated as the inverse of matrix [P]. [X0] and [X1] are authorprovided transformations that place two independent copies of the virus assembly onto the cubic (I2_{1}3) body diagonal (lower right). A subset of operations defines the crystal (upper right).
Assembly definitions corresponding to the path in Fig. 4 are summarized in Table 4. Each definition includes a text description and a list of one or more operation expressions with associated coordinate selections. Operation expressions are given in a compact notation and specify matrices from the operations list, which includes frame transformations [P], [X0], [X1] and 60 icosahedral symmetry operations, labelled 1–60, calculated in the deposited frame, [P^{−1}][I_{m}][P]. An operation expression can be a commaseparated list (`1, 5, 9'), a dashdelimited range (`160') or a matrix multiplication involving two or more lists or ranges. For instance, `(X0)(120)' specifies the portion of the φX174 procapsid crystal belonging to the first independent virus particle and corresponds to the 20 transformations [X0][1], [X0][2], …, [X0][20]. Similarly, `(X1)(120)' specifies the portion of the crystal belonging to the second independent virus particle. The two specifications listed together define the full crystal (see bottom row of Table 4). Coordinate selections are given as lists of commaseparated coordinategroup identities (Bourne et al., 1997).

Complex cases such as the pseudocrystalline symmetry in icosahedral PBCV1 (PDB entry 1m4x ; Nandhagopal et al., 2002) can also be represented (Figs. 2h and 2i and Table 5). Three deposited chains represent 1/28th of the icosahedral point (yellow trimer in Fig. 2i). The operations list contains 60 pointsymmetry operations (`160') and 28 operations to build the icosahedral point (`6188'). The complete capsid (Fig. 2h) is built with 1680 operations specified by `(160)(6188)' applied to the three deposited chains. The pentasymmetron and trisymmetron subassemblies of PBCV1 described by Nandhagopal and coworkers are also readily defined via matrix selections (Fig. 2i, Table 5).

4. Discussion
Remediated entries for the viruses and other assemblies listed in Table 1 were released into the PDB archive on 31 July 2007 and are available by ftp or web interface from any of the wwPDB partners (RCSB PDB, EBI MSD, PDBj; see https://wwpdb.org and Berman et al., 2003). PDBformat files automatically generated from remediated mmCIFs hold much of the updated information, including corrected BIOMT matrices to build the full biological assembly and a text description of the regular symmetry. For crystal structures deposited in the crystal frame, operations to build the crystal are provided in MTRIX records. The mmCIF files or their PDBML translations should be consulted for the most complete machinereadable representations of these entries.
One immediate consequence of remediation is that routine visualization of complete biological assemblies of viruses is now possible. Biological unit files containing explicit coordinates for the full assembly are available in the PDB archive and can be viewed with a number of different software programs. However, the downloading, storage and manipulation of a biological unit file is inefficient compared with handling the equivalent representation in matrices and coordinates. PBCV1 virus (PDB entry 1m4x ) is the most extreme case: the compressed storage size for the biounit file with 5040 chains is 1000 times bigger than the mmCIF or PDB file with three chains and matrices (0.3 Gb versus 0.3 Mb). The Chimera Multiscale Module was designed specifically for displaying large assemblies and can calculate full assemblies on the fly from PDB BIOMT records (Goddard et al., 2005); examples of its use are shown in Figs. 2 and 4. Adoption of this mmCIF (or equivalently, PDBML) representation will further enhance the capabilities of visualization tools to display complex biological assemblies.
To optimally represent future entries of this type, we encourage the deposition of coordinates representing the minimal unique repeating unit along with a clear description of the symmetry, including all local, point, helical, twodimensional and/or threedimensional crystal parameters. A complete set of pointsymmetry operations or representative set of helical operations should be provided in the deposited frame, along with known transformations to other experimental frames. We anticipate that continued progress in development of Xray diffraction, cryoEM and other structural biology methods will result in many more examples of large biological assemblies with regular symmetry in years to come.
Acknowledgements
We are grateful to Vijay Reddy and Ian Borelli for providing the PDB2VIPER code, Tom Goddard for providing automated scripts to generate pictures of virus structures and for creating the crystalcontacts module in Chimera, Huanwang Yang for performing SFCHECK validations and Zukang Feng for updating entries with the remediated data. We also thank David Belnap, Bridget Carragher and Ron Milligan for helpful suggestions regarding symmetry definitions. The RCSB PDB is supported by funds from the National Science Foundation, the National Institute of General Medical Sciences, the Office of Science, Department of Energy, the National Library of Medicine, the National Cancer Institute, the National Center for Research Resources, the National Institute of Biomedical Imaging and Bioengineering and the National Institute of Neurological Disorders and Stroke. The EMBL–EBI MSD is supported by funds from the Wellcome Trust, the EU (FELICS, EXTENDNMR, EuroCarbDB and 3DEM), the BBSRC, the MRC and EMBL. CLL was supported in part by National Center for Research Resources grant award P20RR020647 to Wah Chiu.
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