# Why large icosahedral viruses need scaffolding proteins

^{a}Department of Physics and Astronomy, University of California, Riverside, CA 92521;^{b}Department of Pathogen Molecular Biology, Faculty of Infectious and Tropical Diseases, London School of Hygiene and Tropical Medicine, London WC1E 7HT, United Kingdom;^{c}Department of Physics and Astronomy, Iowa State University, Ames, IA 50011-3160;^{d}Ames Laboratory, Iowa State University, Ames, IA 50011

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Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved September 11, 2018 (received for review May 14, 2018)

## Significance

Despite a plethora of experimental data on the role of scaffolding proteins in the structure of viral shells, no theoretical/numerical explanations have been put forward to decipher the indisputable need of large shells for scaffolding proteins. This paper explains the underlying physical mechanisms for the formation of large viral shells and elucidates the “universal” role of scaffolding proteins in the formation of large spherical crystals.

## Abstract

While small single-stranded viral shells encapsidate their genome spontaneously, many large viruses, such as the herpes simplex virus or infectious bursal disease virus (IBDV), typically require a template, consisting of either scaffolding proteins or an inner core. Despite the proliferation of large viruses in nature, the mechanisms by which hundreds or thousands of proteins assemble to form structures with icosahedral order (IO) is completely unknown. Using continuum elasticity theory, we study the growth of large viral shells (capsids) and show that a nonspecific template not only selects the radius of the capsid, but also leads to the error-free assembly of protein subunits into capsids with universal IO. We prove that as a spherical cap grows, there is a deep potential well at the locations of disclinations that later in the assembly process will become the vertices of an icosahedron. Furthermore, we introduce a minimal model and simulate the assembly of a viral shell around a template under nonequilibrium conditions and find a perfect match between the results of continuum elasticity theory and the numerical simulations. Besides explaining available experimental results, we provide a number of predictions. Implications for other problems in spherical crystals are also discussed.

More than 50 y ago, Caspar and Klug (1) made the striking observation that the capsids of most spherical viruses display icosahedral order (IO), defined by 12 five-coordinated units (disclinations or pentamers) occupying the vertices of an icosahedron surrounded by hexameric units (Fig. 1). While many studies have shown that this universal IO is favored under mechanical equilibrium (2⇓–4), the mechanism by which these shells grow, circumventing many possible activation barriers and leading to the perfect IO, remains mainly unknown.

Under many circumstances, small icosahedral capsids assemble spontaneously around their genetic material, often a single-stranded viral RNA (5⇓⇓⇓–9). However, larger double-stranded (ds) RNA or DNA viruses require what we generically denote as the template: scaffolding proteins (SPs) or an inner core (10⇓⇓⇓⇓–15). The focus of this paper is on these large viruses that require a template for successful assembly.

The major difficulty in understanding the pathway toward IO is apparent from the results of the generalized Thomson problem, consisting of finding the minimum configuration for interacting M-point particles constrained to be on the surface of a sphere. Simulation studies show that the number of metastable states increases exponentially with M (16), and only with the help of sophisticated optimization algorithms at relatively small values of M (5, 17⇓–19) is it possible to obtain IO ground states. These situations, typical of spherical crystals, become even more difficult when considering the assembly of large capsids, in which once protein subunits are attached and a few bonds are made, it becomes energetically impossible for them to rearrange: Should a single pentamer appear in an incorrect location, IO assembly would fail.

The combined effect of irreversibility and the inherent exponentially large number of metastable states typical of curved crystals puts many drastic constraints on IO growth. The complexity of the problem may be visualized by the various viral shells illustrated in Fig. 1, characterized by a structural index, the *T* number (1, 20⇓–22)

A possible mechanism to successfully self-assemble a desirable structure might consist of protein subunits with chemical specificity, very much like in DNA origami (23) where structures with complex symmetries are routinely assembled. In viruses, however, capsids are built either from one or from a few different types of proteins, so specificity cannot be the driving mechanism leading to IO (9, 18, 21, 24, 25). In this paper, we show that a “generic” template provides a robust path to self-assembly of large shells with IO. This is consistent with many experimental data in that regardless of amino acid sequences and folding structures of virus coat and/or scaffolding proteins, due to the “universal” topological and geometrical constraints, large spherical viruses need scaffolding proteins to adopt IO (Fig. 1). Although the focus of this paper is on virus assembly, the implication of our study goes far beyond that and extends to many other problems where curved crystals are involved, a point that we we further elaborate in *Conclusion* (26, 27).

The distinct feature of spherical crystals is that their global structure is constrained by topology. More concretely, if **1** does not really restrict the number of disclinations during the growth process, as pentamers or other disclinations may be created or destroyed at the boundaries. For a complete shell, the easiest way to fulfill Eq. **1** is with 12

A minimal model for spherical crystals consists of a free energy**2** as a nonlocal theory of interacting disclinations, with free energy (34)**3** have been done for curved crystals without a boundary. In this paper, we provide the necessary formalism to include the presence of a boundary.

A discrete version of Eq. **2** is given by (26, 27, 34, 35)*Methods*, we associate dynamics to these models, which corresponds to following a local minimum energy pathway.

## Methods

### Discrete Model.

The growth of the shells is based on the following assumptions (6, 9, 19, 24): At each step of growth, a new trimer is added to the location in the boundary which makes the maximum number of bonds with the neighboring subunits. This is consistent with the fact that protein–protein attractive interaction is weak and a subunit can associate and dissociate until it sits in a position that forms a few bounds with neighboring proteins. These interactions eventually become strong for the subunits to dissociate and trimer attachment becomes irreversible (5). The attractive interactions between subunits, whose strength depends on electrostatic and hydrophobic forces, are implicit in the model. Note that pH and salt can modify the strength of protein–protein and protein–template interactions and thus the growth pathway.

A crucial step in the assembly process is the formation of pentamers, which occurs only if the local energy is lowered, as illustrated in Fig. 2. After the addition of each subunit or the formation of a pentamer, using the HOOMD package (36, 37), we allow the triangular lattice to relax and to find its minimum energy configuration (20).

The proposed mechanism follows a sequential pathway where trimers (T) attach to the growing capsid (**5** takes place, there is no possibility for correcting mistakes: If a pentamer forms in the incorrect location, IO is frustrated. With some additional assumptions about the dependence of

Two important parameters arise in discussing spherical crystals with the model in Eq. **4**. One is the Foppl von-Karman (FvK) number (26)

### Continuum Model.

We now consider the model given in Eq. **3** on a spherical cap with an aperture angle *B*). The Lame term (**3** can then be written as*SI Appendix*, section S1, we provide the detailed calculations. We note that approximate solutions of Eq. **9** are available under the assumption that the Laplacian is computed with a flat metric (38), which immediately leads to **1**. Therefore previous results (39) are limited to small curvatures or aperture angles (**3** to include boundaries proceeds by defining the stress tensor by the expression *B*), we use the metric

With the above definitions, the topological constraint in Eq. **1** is satisfied exactly for a sphere. The free energy in Eq. **3** then becomes

## Results

Consistent with the assumptions describing the dynamics of growth noted in the previous section, we consider the spherical cap (see Fig. 4*B*) with an aperture angle, which monotonically varies from **10** and compare it to the one with an additional new defect (local condition). Once the latter one is favorable, the new defect is added.

For small values of *SI Appendix*, Fig. S2, where the contour plots of the different elastic free energies as a function of the location of the first disclination, r are shown. As the shell grows, the appearance of a new disclination becomes energetically favorable; i.e., a new energy valley for the formation of a new disclination emerges, as illustrated in Fig. 4*B*, where we show the contour plots of total elastic energies for spherical caps with

Results from the discrete model in Eq. **4** are shown in Fig. 4*A*. Here again, the disclinations universally appear at the vertices of an icosahedron, in complete agreement with the analytical calculation. The simulations were performed for all values between *A* is commensurate with *SI Appendix*, Movie S1 illustrates the growth of a

Fig. 5 shows the stretching energy vs. N (number of subunits assembled) as a

For small values of *SI Appendix, section S3* for more details. At the beginning of the growth, the shells with different values of γ might follow different pathways and thus the number of hexamers might vary before the first few pentamers form. However, as the shell grows, the pentamers appear precisely at the same place, independently of γ. Note that the bending energy of the shells always grows linearly as a function of number of subunits for any γ (*SI Appendix*, Fig. S3).

## Discussion

Our results show that for large shells (

In the absence of the template, small spherical crystals (

A template can have a significant impact on the structure and symmetry of the shell. While a weak subunit–core attractive interaction has a minimal role in the shell shape, a very strong subunit–core interaction will override the mechanical properties of proteins. The subunits sit tightly on the template to form a sphere with no specific symmetry. We were able to observe large shells with IO only for *B*). Indeed, a strong bending energy is needed to overcome the shell adsorption. We find that without decreasing γ (increasing

The role of the inner core or the preformed scaffold layer presented above is very similar to the role of SPs, which assemble at the same time as the capsid proteins (CPs); i.e., the template grows simultaneously with the capsid (Fig. 6). In fact, one can think of the inner core as a permanent “inner scaffold” (15). For example, bacteriophage P22 has a *T* = 7 structure, but in the absence of scaffolding (Fig. 1, P22) often a smaller

## Conclusions

Our model establishes that successful self-assembly of components into a spherical capsid with IO requires a template that determines the radius of the final structure. This template is very nonspecific, and in its absence, protein subunits assemble into either smaller capsids or structures without IO.

Even though the focus of this study was on the impact of the preformed scaffolding layer, based on the experimental observations we conclude that the SPs, which assemble simultaneously with CPs (Fig. 6), play basically the same role as the inner core in the assembly of large icosahedral shells. Fig. 6 shows that in the absence of SPs, CPs of IBDV form a

The contribution of the SPs is twofold. The CPs of many viruses including bluetongue virus noted above do not assemble in the absence of SPs. On the one hand, it appears that SPs lower the energy barrier and help capsid subunits to aggregate. On the other hand, by forcing the CPs to assemble into a structure larger than their spontaneous radius of curvature, they contribute to preserving IO.

Examples of the role of templates in the formation of spherical crystals are not limited to viruses, but include crystallization of metals on nanoparticles (40), solid domains on vesicles (41, 42), filament bundles (38), and colloidal assemblies at water–oil interfaces (43). Nevertheless, it has been shown (44) that sufficiently rigid crystals grow as almost flat sheets free of defects, unable to assemble with IO. This regime, however, seems not to be accessible to viral capsids, as the hydrophobic interaction between monomers forces close-packing structures that are incompatible with grain boundaries.

This study sheds light at a fundamental scale on the role of the mechanical properties of building blocks and scaffolding proteins. The proposed mechanism is consistent with available experiments on viruses involving either scaffolding proteins or inner capsids. Further experiments will be necessary to validate many predictions of our described mechanism.

## Acknowledgments

The authors thank Greg Grason for many helpful discussions. S.L. and R.Z. were supported by NSF Grant DMR-1719550 and A.T. by NSF Grant DMR-1606336. P.R. is funded through a Senior Investigator Award from Wellcome Trust under Grant 100218. A.T. and R.Z. thank the Aspen Center for Physics where part of this work was done with the support of NSF Grant PHY-1607611.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: sli032{at}ucr.edu.

Author contributions: S.L., P.R., A.T., and R.Z. designed research; S.L., A.T., and R.Z. performed research; S.L., A.T., and R.Z. analyzed data; and S.L., P.R., A.T., and R.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1807706115/-/DCSupplemental.

Published under the PNAS license.

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