Structural studies of Acidianus tailed spindle virus reveal a structural paradigm used in the assembly of spindle-shaped viruses

Significance

Lemon- or spindle-shaped viruses are common in the archaeal domain of life, but structural studies have been limited by the intrinsic heterogeneity of these uniquely shaped virions. Using a combination of cryo-electron microscopy and X-ray crystallography to study Acidianus tailed spindle virus, a large tailed spindle virus, we have shed light on the architectural principles that underlie assembly of a spindle-shaped virus. The architecture suggests a metastable multistart helical assembly of variable radius that, through a remarkable transition to a more stable cylindrical assembly, could be used to drive genome ejection. The structural architecture is clearly relevant to spindle-shaped viruses in general and other members of the large tailed spindle virus superfamily in particular.

Abstract

The spindle-shaped virion morphology is common among archaeal viruses, where it is a defining characteristic of many viral families. However, structural heterogeneity intrinsic to spindle-shaped viruses has seriously hindered efforts to elucidate the molecular architecture of these lemon-shaped capsids. We have utilized a combination of cryo-electron microscopy and X-ray crystallography to study Acidianus tailed spindle virus (ATSV). These studies reveal the architectural principles that underlie assembly of a spindle-shaped virus. Cryo-electron tomography shows a smooth transition from the spindle-shaped capsid into the tubular-shaped tail and allows low-resolution structural modeling of individual virions. Remarkably, higher-dose 2D micrographs reveal a helical surface lattice in the spindle-shaped capsid. Consistent with this, crystallographic studies of the major capsid protein reveal a decorated four-helix bundle that packs within the crystal to form a four-start helical assembly with structural similarity to the tube-shaped tail structure of ATSV and other tailed, spindle-shaped viruses. Combined, this suggests that the spindle-shaped morphology of the ATSV capsid is formed by a multistart helical assembly with a smoothly varying radius and allows construction of a pseudoatomic model for the lemon-shaped capsid that extends into a tubular tail. The potential advantages that this novel architecture conveys to the life cycle of spindle-shaped viruses, including a role in DNA ejection, are discussed.

Viruses infecting the Archaea show unique genetic content and viral morphologies that have not been observed in Eukarya or Bacteria (1). Perhaps the most prominent of these unique morphologies are the spindle- or lemon-shaped capsids. Lemon-shaped capsids are found in the Fuselloviridae family that infect Crenarchaea and in the structurally related Salteroprovirus genus that infect Haloarchaea, where members of each of these viral groups typically package a 14- to 20-kb dsDNA genome (1–5).

Lemon-shaped capsids are also a defining morphology for the Bicaudaviridae family and other members of the large tailed spindle virus superfamily that typically package 50- to 75-kb dsDNA genomes (1, 5, 6). Unlike the smaller fuselloviruses and salteroproviruses, members of the large tailed spindle virus superfamily commonly show extended tubular tails emanating from one or both ends of the large lemon-shaped particle (7–11). Remarkably, in the case of Acidianus two-tailed virus (ATV), and more recently for Sulfolobus monocaudavirus 1, these tails have been shown to develop extracellularly (10–12). In the case of ATV, tail growth was accompanied by a concomitant decrease in the size of the spindle (10, 11). While a crystal structure of the monomeric form of the ATV major capsid protein (MCP) has been determined (13), spindle-shaped virions themselves have eluded structural characterization at resolutions sufficient to elucidate the molecular architecture of this unique virus morphology. This is in part due to the pleomorphic nature of these viruses, which makes them difficult to study by single-particle reconstruction (14, 15).

Using culture-independent methods, we recently identified Acidianus tailed spindle virus (ATSV) as a member of the large tailed spindle virus superfamily (6, 16). ATSV contains a 70,812-bp circular dsDNA genome harboring 96 ORFs packaged in a spindle-shaped capsid with a single tail extending from one end. The dimensions of both the spindle head (170 ± 50 nm long by 100 ± 30 nm wide) and the tail (20 ± 9 nm wide) are variable in size, but unlike ATV, in which the spindle heads contract as the tails extend (10, 11), a correlation between spindle volume and the tail length has not been seen. In this paper we report structural characterization of purified ATSV by cryo-electron microscopy (cryo-EM) as well as complementary studies of the ATSV MCP by X-ray crystallography. Combined, this work illuminates a striking paradigm for viral assembly relevant to multiple viral families in the archaeal domain of life.

Results

Initial Structural Studies of ATSV by Cryo-Electron Tomography.

Following upon our initial characterization of ATSV by negative-stain transmission EM (16), we turned to cryo-electron tomography (CET) to further characterize ATSV purified directly from environmental samples. Sections from a representative tomogram are shown in Fig. 1. As expected, segmentation of the tomogram results in a model with a lemon-shaped capsid and an extended tubular tail. In this specific example the lemon-shaped capsid is ∼100 nm wide and 200 nm in length, and the tail is ∼22 nm wide by 200 nm in length. Importantly, the tomographic studies show a smooth transition from the lemon-shaped head of the virus into the extended tail, with no discernible boundary between the head and the tail, suggesting a continuous structure that transitions from the head into the tail. This observation is consistent with the extracellular maturation of ATV, in which tail growth correlates with diminished head size, suggesting movement of the capsid proteins from the lemon-shaped head into the growing tail.

Fig. 1.

Low-resolution structure of the ATSV virion by CET. A shows a 0.9-nm-thick section through the center of the virion with continuous density extending from the lemon-shaped capsid into the cylindrically shaped tail. The map was skewed to place the long axis of the virus within the central plane of the map, contributing to the apparent missing wedge artifacts along the upper left of the viral surface. The circular structure (lower right) is due to an unrelated copurifying membranous structure. B shows the outer viral surface segmented by TomoSegMemTV (36). The missing upper and lower surfaces are again due to the tomographic missing wedge effect.

Remarkably, as opposed to individual short-exposure images in the 61-image tilt series acquired for CET, the higher contrast and resolution of longer-exposure, subframe-aligned micrographs of ATSV revealed regularly spaced serrations along the outer edge of the capsid (Fig. 2A). Because the serrations along the edge are clearly on the outside surface of the virus, this indicates a set of regularly spaced (∼4 nm), parallel striations running across the outer surface of the virus, roughly perpendicular to the spindle axis. Indeed, in several cases in which clearly damaged, partially empty capsids are visualized, these surface features are clearly seen extending across the virus (Fig. 2 B and C). This suggests a capsid composed of either a set of concentric rings or, more likely, a helical construction in which the helical strands show an apparent pitch of ∼4 nm and a variable radius that diminishes toward each end of the lemon-shaped capsid.

Fig. 2.

Cryo-electron micrographs of ATSV particles. (A) An intact ATSV particle showing a serrated surface on the outside edge of the spindle-shaped head. (B) A disrupted, partially empty virus particle. (C) Enlarged view of the boxed region in B in which the serrated edge is seen to extend into striations running across the surface of the virus particle. Ten-nanometer gold particles included as fiducials for tomographic reconstruction are visible as black circles. (Scale bars, 50 nm.)

Structure of the ATSV MCP.

In conjunction with the tomographic study we also pursued crystallographic studies of the previously identified ATSV MCP (ORF D135) (16). The 15-kDa protein, which runs as a monomer on a sizing column, was crystallized (Table S1) in space group P2 with five copies in the asymmetric unit. The structure was solved by multiwavelength anomalous diffraction at the selenium edge [Protein Data Bank (PDB) ID code 5EQW, Table S1]. The structure revealed a decorated four-helix bundle with a right-handed, antiparallel topology lacking overhand connections (see ref. 17 for topological definitions of four-helix bundles), with helix lengths of 13, 17, 21, and 21 residues (Fig. 3). A fifth helix of 13 residues is present at the C terminus and extends away from the helical bundle.

Fig. 3.

Structure of the MCP of ATSV. The MCP adopts a right-handed antiparallel four-helix bundle (α1– α4) with a fifth helix extending from the C terminus. The helical bundle is capped by a small three-stranded, mixed β-sheet and two extended loops (α3/β2 and β2/β3).

Visual inspection and DSSP (18) both indicate the presence of three β-type secondary structural elements, specifically a β-bridge (β1) and two short β-strands (β2 and β3). The β-strands are inserted between helices α3 and α4 and run parallel to each other with typical H-bond patterns between Val⁸⁰–Tyr⁸¹ of β2 and Ile⁹⁴–Ala⁹⁵ of β3. The β-bridge connects α1 and α2 and is formed by a conserved Gly²⁸–Tyr²⁹–cisPro³⁰ structure, with Tyr²⁹ forming the antiparallel β-bridge to Val⁸⁰ in β2. This results in a small three-stranded β-sheet that stacks end-on to α1 and α2 and laterally abuts α3 and α4. This capping structure also harbors two extended loops (α3/β2 and β2/β3) and a number of residues conserved among the MCPs of other large tailed spindle viruses (Fig. S1), including a GXGF motif in the α3/β2 loop where the conserved Phe⁷⁵ side chain is found in a surface-exposed conformation. Overall, this cap structure is stabilized by 14 intrasubunit H-bonds.

Structural Homology to the MCP of ATV.

A structural homology search (19) confirmed the MCP of ATV (13) as the closest structural homolog to the MCP of ATSV (Fig. S2). Like ATSV, the ATV MCP is a right-handed, up-and-down, four-helix bundle with a fifth helix at the C terminus. In addition, although not previously noted, the helical bundle in ATV is capped by a two-stranded antiparallel β-sheet. The first β-strand contains the conserved Gly–Tyr–cisPro structural motif we identified in the β-bridge (β1) of the ATSV structure, with the second strand analogous to β2 in ATSV. ATV also shares the extensive α3/β2 loop; however, in the ATV structure this loop is disordered. Differences include a proline-induced kink in α4, the angle of the fifth helix relative to the four-helix bundle, and the loss of the third strand in the antiparallel β-sheet of the cap structure (Fig. S1). However, the overall structure of the cap is largely conserved among the MCPs of ATV and ATSV, strongly suggesting the cap plays a critical structural and functional role in the large tailed spindle virus superfamily.

A Four-Start Helical Assembly.

The ATSV MCP forms a unique higher-order structure within the crystal. The five copies of the ATSV MCP in the asymmetric unit contribute to a half-turn of a larger helical assembly (Fig. 4A). In conjunction with this first asymmetric unit a neighboring asymmetric unit, related by twofold crystallographic symmetry and unit-cell translation, completes a full helical turn of 10 subunits with a pitch of 16.67 nm (Fig. 4B). Importantly, three successive unit-cell translations add three additional helical strands that pack within the extended pitch of the first helix, resulting in an enclosed, cylinder-shaped, four-start helical assembly that runs the length of the crystal (Fig. 4D). This cylinder-shaped structure has an outer diameter of ∼15 nm and an inner diameter of ∼6 nm. While a single helical strand shows a 16.67-nm pitch, the spacing between adjacent strands is one-fourth that, or 4.17 nm.

Fig. 4.

ATSV MCP forms an enclosed, cylinder-shaped, four-start helical assembly within the crystal. (A) The asymmetric unit containing five subunits forming a half-turn of a single helix. (B) Continuation into a neighboring asymmetric (twofold rotation plus four unit cell translations) gives a full helical turn. (C) Two full helical turns of single helix (red) that runs the length of the crystal. (D) Three successive unit cell translations give three additional helical strands (yellow, green, and blue) that pack within the extended pitch of the first helix to make a four-start helical assembly. (E) Surface rendering of the four-start helix. (F) View in E looking down the long axis, relative dimensions are indicated.

Subunit Interactions in the Multistart Helix.

Relative to the four-start helical assembly, within a single subunit α-helices α1 and α2 lie predominantly on the inner surface of the cylinder, while α3 and α4 form much of the exterior surface. Each subunit interacts with six adjacent protomers in which three different subunit interface types are utilized: “side by side,” “top to bottom,” and “diagonal” (Fig. 5). Utilizing pairs of reciprocal interactions, each subunit engages two subunits through each of these three interface classes. (Fig. 5A).

Fig. 5.

Subunit interactions. (A) Surface of the four-start helix showing a central subunit (yellow) interacting with six neighboring protomers with three types of interactions: side by side (orange/light yellow), top to bottom (green/salmon), and diagonal (light green/red). (B) The side-by-side interactions include significant van der Waals contacts between helices α2 and α5. In addition, the extended α3β2 loop inserts into a hydrophobic pocket between helices α4 and α5 at the subunit interface. The interdigitated nature of this interaction is repeated by a reciprocal set of interactions in the preceding set of side-by-side interactions, effectively forming a set of interlocking interactions that anchor successive subunits within a single helical strand. (C) The top-to-bottom interactions are less extensive and are mediated by the β2/β3 loop with the N and C termini of α3 and α4, respectively. H-bond interactions are indicated with dashed lines.

Intrastrand Interactions in the Multistart Helix.

The most substantial are the side by side, or intrastrand, interactions which mediate formation of an individual helical strand (Figs. 4C and 5B). These interactions are mediated by significant interactions between α1 and α2 of the first subunit and α1, α4, and α5 of the second. In addition, the α3–β2 loop extends into a crevice between helices α4 and α5 of the adjacent subunit, inserting the conserved Phe⁷⁵ side chain into a hydrophobic pocket (Leu¹⁶, Leu³⁴, Met¹¹², Ala¹¹⁶, Val¹²³, and Ala¹²⁴) (Fig. 5B). The interaction between the loop and the adjacent subunit is further strengthened by a strong H-bond between Arg⁷³ and Gln¹⁰⁵. When the reciprocal interactions on the other side of the subunit are also considered these side-by-side interactions bury a total surface area of 1,377 Ų in each subunit.

Interstrand Interactions in the Multistart Helix.

As expected for end-on interactions between four-helix bundles, the top-to-bottom interactions are less extensive, being mediated by residues in α3 and α4 of one subunit with the β2/β3 loop of the neighboring subunit (Fig. 5C). The six (>50%) buried residues at this interface form a total of three interstrand H-bonds, one interstrand salt bridge, and an interstrand hydrophobic interaction between Pro⁹¹ and Ile⁴⁹. The diagonal interactions are the least extensive, with five interfacial residues involved in one weak hydrogen bond and minor hydrophobic interactions (Fig. S3). In total, however, formation of the four-start helical assembly buries 1,980 Ų, or 30% of the subunit surface area, suggesting these interactions could be biologically relevant.

Modeling the ATSV Tail Structure.

The four-start helical assembly in the crystal shows obvious similarity to the morphology of the ATSV tail structure, suggesting the assembly in the crystal may be related to the overall structure of the tail. As described above, the diameter of the ATSV tail appears variable. However, considering the tomographic structure depicted in Fig. 1 with a tail diameter of ∼22 nm, the diameter of the four-start helix in the crystal (15 nm) would need to be enlarged 1.5-fold.

Multistart helical tail-sheath assemblies are well known in bacteriophages. Structurally, the best-studied of these is the phage T-4 contractile sheath, in which gp18 forms a six-start helical assembly (20, 21). Upon contraction, the T-4 sheath changes both diameter and pitch and thus provides a precedent for dynamic, multistart helical assemblies in viruses. Thus, one potential explanation for the differing diameters of the ATSV tail in Fig. 1 and the assembly in the crystal are conformational changes leading to changes in pitch and diameter. However, it seems most likely that the ATSV tail (as opposed to the lemon-shaped capsid) and the ATSV four-start helix in the crystal are each likely to represent low energy conformations. Thus, an alternative explanation might be considered, specifically that the viral tail is constructed of an n-start helix with n greater than four. As the diameter of a cylinder is linearly proportional to its circumference (C = πD), a six-start helix with geometric properties similar to those in the crystal could also account for the 22-nm-diameter tail seen in the virus. Further, a six-start helix for the ATSV subunit in which the number of subunits per turn, the radius, and the pitch are each increased by a factor of 1.5 (6/4) is easily modeled and largely preserves the subunit–subunit contacts, with the most substantial difference being a 12° change in the angle between adjacent subunits in the side-by-side interaction (Fig. 6).

Fig. 6.

A pseudoatomic model for ATSV. (A) A single helical strand of the major structural protein covering the lemon-shaped capsid and cylindrical tail. (B) A complete six-start helical model. (C) An enlarged view of the section indicated in B. (D) An enlarged view of the space filling pseudoatomic model indicated in D, showing three adjacent helical strands with an apparent pitch of 4 nm. (E) A ribbon representation of the model indicated in D, with a 4-nm pitch between adjacent strands.

Modeling the Lemon-Shaped Capsid.

As discussed above, the serrated appearance along the outside surface edge of ATSV and the arcs extending across the capsid surface are consistent with a helical construction for the lemon-shaped capsid. This suggests that the multistart assembly in the tail extends into the lemon-shaped head of ATSV (or vice versa), but with a smoothly varying radius. Preserving the relative local orientation of each subunit when fitting the helix on the lemon-shaped capsid is more involved, as it is not a simple rotation about the long axis of the virus and translation parallel to this axis. Instead, the rotation angle is dependent on the changing radius, and the relative orientation of the subunit is dependent upon the tangent or normal to the capsid surface, parameters that are unique to each point on the capsid surface. For this reason, we employed a parametric equation that describes the path of each helix on the radially symmetric viral surface. Then, at regular arc-length intervals along each helix a Frenet basis rotation/translation matrix $\left(F{F}_0^T\right)$ was applied to the coordinates of a protein subunit in standard orientation, moving its center of mass onto the capsid surface and rotating it such that the orientation of the subunit relative to the surface is maintained. Approximately 10,000 subunits are required to model the lemon-shaped capsid and cylindrical tail (Fig. 6), giving an approximate mass for the capsid of 160 MDa. With regard to the side-by-side interactions, the local environment throughout the model is similar to that seen in the crystal. Within the tail the top to bottom and diagonal interfaces are also maintained; however, the interstrand interactions do not stay in register as the strands move into the lemon-shaped capsid.

Discussion

Combined, the cryo-EM and crystallographic studies reported here clearly elucidate the major architectural principles employed in the construction of a large tailed spindle virus. The striations across the surface of ATSV revealed in the cryo-electron micrographs are consistent with a helical assembly for the lemon-shaped capsid of ATSV. A helical assembly is also suggested by the crystallographic structure of the ATSV MCP, in which a tail-like structure is built with a four-start helix. Combined, these observations imply a multistart helical construction of variable radius that extends through the lemon-shaped capsid and into the tail.

While this work has illuminated the key architectural principles used in constructing ATSV and presumably other large tailed spindle viruses, several critical details remain to be explored. These include the number of helical strands in the multistart helix, the polarity of these strands with regard to the head and tail of the virus, whether minor structural proteins are incorporated into the lemon-shaped capsid to regulate the register of neighboring helical strands or otherwise stabilize the capsid, the mechanism of assembly, and a detailed picture of conformational capsid dynamics during intra- or extracellular capsid maturation and tail growth. Similar to single-particle analysis, we are currently witnessing rapid growth in the achievable resolution limits of tomography (22–24). In conjunction with optimal specimen preparation and grid optimization, successful subtomographic averaging over a single viral capsid has the potential to provide greatly enhanced resolution, potentially allowing density-based placement of the crystallographic structure into the viral capsid, addressing many of the unresolved details.

Regarding assembly, when the His-tagged MCP of ATSV is expressed in Escherichia coli it purifies as a monomer, and preliminary experiments at both neutral and acidic pH to induce assembly have been unsuccessful. Notably, the ATV structural protein also purifies as a monomer and does not assemble in vitro or as a miultistart helix in the crystal (13). Heterologous expression of the major structural proteins from SIRV (25) and STIV (26) and the STIV turret proteins A223 and C381 (27) also result in monomers, and reports of in vitro assembly are absent from the literature in these cases as well, although STIV C381 does form pentamers within the crystal that nicely fit the subnanometer resolution cryo-EM density of the intact the virion. Thus, it is probable that minor structural proteins at either the head or tail of the virus will serve to nucleate and/or terminate capsid assembly and are also excellent candidates to facilitate viral attachment.

Critically, we believe the model presented here is consistent with the apparent movement of capsid material from the lemon-shaped head of the related virus, ATV, into the growing tails. The observed extracellular maturation of ATV suggests that the lemon-shaped, tailless particle is in a metastable state and that movement of capsid proteins into the growing cylindrical tail represents a transition to a lower energy state. In this regard, the interlocking side-by-side interactions seen within a single helical strand suggest the side-by-side interaction will be largely maintained during this transition, while the weaker interstrand interactions may be more fluid, potentially allowing individual rope-like strands to slide against or reorient with regard to each other. The net result is a viral morphology that shows spectacular capsid dynamics that include major lattice rearrangements.

Further, as capsid volume decreases it is likely that the DNA is compressed, leading to elevated internal pressure on the DNA, and that the increased pressure eventually halts tail growth. Within the context of the viral life cycle, subsequent attachment to the viral host through either the nose or tail fibers might potentially uncork the vessel, allowing the internal pressure to then drive DNA ejection into the host. Moreover, this should further the transition toward a cylindrically shaped capsid. Interestingly, transition from a lemon-shaped capsid to a cylindrical structure has been observed for the smaller haloarchaeal virus His1 (2, 15) in response to biochemical perturbations that release DNA. In addition, DNA release in His1 is osmotically suppressed at 10 atm of pressure, indicating that the His1 capsid does indeed place the His1 DNA under substantial pressure, and that the pressure contributes to DNA release in vitro (28, 29). Thus, His1, ATV, and Sulfolobus monocaudavirus 1 (SMV1) are each capable of undergoing substantial quaternary rearrangement during which lemon-shaped capsids make a transition toward a tubular structure (2, 10–12, 15), and it is likely this mechanism is common to spindle-shaped viruses in general. Further, it provides a strong rationale for nature’s repeated and common use of this structural motif in the archaeal viruses, specifically to package DNA in a metastable container set to drive ejection of viral DNA into the recipient host, and these systems could potentially be harnessed as delivery vehicles for a broad array of applications in biotechnology.

Methods

CET.

ATSV was purified directly from CH041 hot springs in the Crater Hills area of Yellowstone National Park, as previously described (16). Ten-nanometer-sized BSA-coated colloidal gold was added to purified ATSV [3.9 × 10⁷ genomes per µL by quantitative PCR (16)], and 4-µL volumes were applied to glow-discharged Quantifoil Mo 200 R 2/1 (Quantifoil Micro Tools GmbH) grids at room temperature and 70% humidity for 20 s. The grid was then blotted from both sides (3 s, blotting offset of −5 mm) and subsequently plunged into a cryogenic coolant of 1:1 liquid ethane/propane. EM was performed using a Titan Krios transmission electron microscope equipped with a Quantum postcolumn energy filter operated in zero-loss mode, a Volta phase plate (30, 31), and a K2 Summit direct electron detector operating in dose fractionation mode. For each tomogram a tilt series from −60° to +60° was recorded using SerialEM (32) with 2° tilt increments at a magnification factor of 64,000× (2.26 Å/pixel). The total electron dose per tomogram was 60 e/Ų with an applied defocus of 0.1 µm, which showed the first zero of each micrograph’s power spectrum approaching Nyquist frequency. Subframe alignment was performed with Digital Micrograph (Gatan Inc.) to compensate for beam-induced shifts during exposure. Each subframe-aligned tilt series was processed with IMOD (33). The tomographic volumes were reconstructed using the simultaneous iterative reconstruction techniques (34, 35) with the low-pass-filtered (high-frequency filter cutoff, 0.35, 0.05; sigma) tilt series. A representative tomogram was segmented using TomoSegMemTV (36). The tomogram and segmented volume were subsequently visualized with Chimera (37).

Crystallographic Structure Determination for the ATSV MCP.

Selenomethionine-incorporated ATSV MCP was purified as described (16) and crystallized by hanging-drop vapor diffusion using 2 µL of the MCP at 10 mg/mL in 10 mM Tris (pH 8.0) and 2 µL well solution (3.4 M NaNO₃, 0.1 M Na-acetate, pH 2.6, 10% glycerol, and 10% acetonitrile). Crystals were flash-frozen in liquid nitrogen before data collection. A three-wavelength dataset was collected on beamline 11.1 at the Stanford Synchrotron Radiation Lightsource (SSRL), and data were integrated, scaled, and reduced in space group P2 (Table S1) using HKL-2000 (38). The structure was determined by multiwavelength anomalous diffraction using Solve (39), and RESOLVE (40) was used to average over the fivefold noncrystallographic symmetry in the asymmetric unit. Subsequently, a higher-resolution native dataset was collected on SSRL beamline 9.1 and used for refinement with REFMAC5 (41) and Phenix (42). Manual model inspection and building were performed with Coot (43). Statistics on model quality (Table S1) were calculated using MolProbity (44). Structural comparisons were performed using the Dali server (45). Structural figures were generated with Chimera (37) and Pymol (46). The structure has been deposited in the PDB (ID code 5EQW).

Modeling the Tailed, Spindle-Shaped Viral Capsid.

A detailed description of the modeling procedure is provided in SI Methods. In brief, we first determined the radius of the virus along its entire length and generated a surface of revolution. We then constructed six equally spaced helices as follows. First, the azimuthal angle of the helices (θ), which varies with x, was solved by numerical quadrature for θ(x):

$$\frac{d\theta }{dx}=\frac{\sqrt{(y{\left(x\right)}^2-r_2^2\left(1+y^{\prime }{\left(x\right)}^2\right)}}{r_{2}y\left(x\right)}.$$

Each helix (i = 1, 2, …, 6) was then, in turn, described by the parametric equation

$${\mathit{r}}_{\mathit{i}}\left(x\right)=x\mathit{i}+y\left(x\right)\mathit{cos}\left({\theta }_i\left(x\right)\right)\mathit{j}+y\left(x\right)\mathit{sin}\left({\theta }_{\mathit{i}}\left(x\right)\right)\mathit{k}\mathrm{.}$$

Along each spiral, the position of the center of mass of the protein subunit was then calculated such that the arc-length spacing (∆s) between adjacent proteins (28.7 Å) relative to that in the crystal structure was preserved. For adjacent spirals, an offset of ∆s/2 was used. Finally, the Frenet basis rotation/translation matrix $\left(F{F}_0^T\right)$ was applied to the coordinates of a protein subunit moving its center of mass onto the capsid surface and rotating it such that the principal axes of the subunit relative to the local surface are maintained, thus replicating a local environment similar to that seen in the crystal structure.