Beyond icosahedral symmetry in packings of proteinsin spherical shellsMajid Mosayebia,b,1, Deborah K. Shoemarkb,c, Jordan M. Fletcherd, Richard B. Sessionsb,c, Noah Lindena,1,Derek N. Woolfsonb,c,d,1, and Tanniemola B. Liverpoola,b,1

aSchool of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom; bBrisSynBio, Life Sciences Building, Bristol BS8 1TQ, United Kingdom;cSchool of Biochemistry, University of Bristol, Bristol BS8 1TD, United Kingdom; and dSchool of Chemistry, University of Bristol, Bristol BS8 1TS, UnitedKingdom

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved July 10, 2017 (received for review May 5, 2017)

The formation of quasi-spherical cages from protein buildingblocks is a remarkable self-assembly process in many natural sys-tems, where a small number of elementary building blocks areassembled to build a highly symmetric icosahedral cage. In turn,this has inspired synthetic biologists to design de novo proteincages. We use simple models, on multiple scales, to investigatethe self-assembly of a spherical cage, focusing on the regularityof the packing of protein-like objects on the surface. Using build-ing blocks, which are able to pack with icosahedral symmetry, weexamine how stable these highly symmetric structures are to per-turbations that may arise from the interplay between flexibilityof the interacting blocks and entropic effects. We find that, in thepresence of those perturbations, icosahedral packing is not themost stable arrangement for a wide range of parameters; ratherdisordered structures are found to be the most stable. Our resultssuggest that (i) many designed, or even natural, protein cagesmay not be regular in the presence of those perturbations and(ii) optimizing those flexibilities can be a possible design strategyto obtain regular synthetic cages with full control over their sur-face properties.

coarse-grained modeling | icosahedral symmetry | protein cage |self-assembly | protein design

Many examples of self-assembled quasi-spherical shells orcages are found in biology. Small ferritin cages, numerous

viral capsids, clathrin, and large carboxysomes in bacteria are allcages composed of protein subunits, and the resulting structuresare used for packaging and transport (1–4). With a few excep-tions, these spherical cages typically have highly ordered struc-tures with icosahedrally symmetric shells (2, 4–9), characterizedby 6 fivefold, 10 threefold, and 15 twofold symmetry axes. Biol-ogy, through evolution, has developed very efficient routes tomake icosahedral cages using a small number of elementary pro-tein building-block types. These elementary building blocks seemto assemble in a hierarchical manner (10–13). They first assembleto make larger assemblies (hexagons and pentagons) and subse-quently a spherical cage is formed from these oligomers.

Inspired by biology, there have been several attempts to designsynthetic protein cages either by taking protein engineering orde novo design approaches (14–17) or by reengineering naturalprotein cages (18). In some synthetic-cage assemblies, the aimis to form a monodispersed, highly ordered cage (17), whereasin others, the goal is to use simpler and more chemically acces-sible design rules to form spherical cages that are not necessar-ily symmetric (14). Synthetic cages could potentially be used formany applications such as targeted drug delivery, vaccine design,nanoreactors, and synthetic biology (19–24).

A better understanding of the assembly mechanisms in proteincages would make it possible efficiently to control the structuralproperties of the protein cage to best suit the particular appli-cation required. In that direction, a number of questions arise:What are the key design rules for synthetic self-assembly aim-ing for regular cages? How robust is the icosahedral symmetry of

the protein cage? Can stable cages be constructed without thissymmetry?

By combining several coarse-grained models on multiplescales, here we investigate the robustness of the icosahedral cageagainst structural imperfections, arising from the flexibility of theprotein building blocks, which could possibly occur in syntheticself-assembly pathways. To fix our ideas with a concrete exam-ple of a synthetic cage, we seek inspiration from the recentlydesigned self-assembled cages (SAGEs) (14), which use de novodesigned coiled-coil (CC) peptides as building blocks.

Self-Assembled Cage from CC PeptidesThe SAGE design comprises two, noncovalent, heterodimericand homotrimeric CC bundles. These are joined back to backusing disulfide bonds to make two complementary hubs, whichwhen mixed form honeycomb networks (Fig. 1A). The honey-comb network folds due to the intrinsic splay inherent withinthe designed hubs and leads to cages with a typical diameterof 100 nm. Thus, SAGE assembly is reminiscent of the forma-tion of clathrin, where hexagonal and pentagonal subunits areallowed to form due to the finite flexibility of the triskelia (8, 25,26). The melting temperature of the trimeric CC bundle (greenCCs in Fig. 1A) is higher than the melting temperature of theheterodimeric bundles, resulting effectively in a one-structural-unit assembly with a hub, which comes in two types dependingon the type of heterodimeric CCs linked to the trimer. Hexago-nal subunits formed from six flat hubs would tile a flat surface.One way of covering a spherical surface in a regular manner,without any holes, is to include 12 pentagonal subunits, hence

Significance

The design and construction of man-made structures at micro-scopic scales are one of the key goals of modern nanotech-nology. With nature as inspiration, synthetic biological build-ing blocks have recently been designed that self-assemble intoquasi-spherical shells or cages. Whereas many natural proteinbuilding blocks self-assemble into highly symmetric orderedshells (e.g., viruses), our study shows that surprisingly evena small amount of (unavoidable) flexibility in the syntheticbuilding blocks leads to stable disordered configurations. Ourwork provides a new design paradigm: Modulating the flex-ibilities of the components, one can control the regularity ofthe packing and, consequently, the surface properties of a syn-thetic cage.

Author contributions: M.M., N.L., D.N.W., and T.B.L. designed research; M.M. performedresearch; M.M., D.K.S., J.M.F., R.B.S., N.L., D.N.W., and T.B.L. analyzed data; and M.M.,D.K.S., J.M.F., R.B.S., N.L., D.N.W., and T.B.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. Email: t.liverpool@bristol.ac.uk, majid.mosayebi@bristol.ac.uk, n.linden@bristol.ac.uk, or d.n.woolfson@bristol.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706825114/-/DCSupplemental.

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giving the required Euler characteristic of 2. This strategy couldbe achieved by making use of the flexibility of the angle andbond potentials between hub pairs to allow the formation of therequired number of pentagonal units within the honeycomb net-work. However, allowing more flexible or less angular specificinteractions would enhance formation of other polygonal unitsin addition to the required number of pentagons and hexagons.Such polygons act as additional structural defects and can poten-tially remove the icosahedral symmetry of the cage. This compe-tition is a natural place for one to look for ways to optimize thecage structure.

Studying the self-assembly process of a system as large as aSAGE over the timescale that it forms in the test tube (up to min-utes) is not feasible with atomistic detail. Thus, coarse-grained(CG) modeling (5, 11, 27–36) is required to answer relevantquestions about the dynamics of the self-assembly as well as thestability and uniqueness of the final structure. We used a CC-level CG model of SAGEs to simulate directly the self-assemblyof hubs. The overall angular specificity of the hub pairs in the CC-level CG model arises from the finite range of attractive interac-tions between the patches and also from the stiffness of the per-manent bond. To facilitate formation of the honeycomb latticein numerical simulations, the angular specificity of the hub pairswas tuned to be more than the observed angular specificity in theatomistic simulations (SI Appendix).

Assembly of the Peptide Network Incorporates DefectsOur simulation results indicate that, even with our more angularspecific model, the occurrence of nonhexagonal defects on theself-assembly pathway is likely, as it can be observed from the

Fig. 1. SAGEs from de novo CC peptides. (A) Homotrimeric CC (green) andheterodimeric CC (acidic CC in red and basic CC in blue) bundles. Each CC ismade from 21 aa with a height of 3 nm. Two hub types are made by link-ing three homotrimeric acidic or basic CCs to a trimer bundle, using threedisulfide bonds. These hubs, when mixed, form a honeycomb lattice thatcloses due to the intrinsic splay between hub pairs (14). (B, Top) Two typesof molecules in the CC-level CG model of SAGEs. (B, Bottom) Attractive LJpatches are illustrated as small spheres that drive formation of trimers anddimers. (C) Snapshots of a partially assembled SAGE obtained after anneal-ing a mixture of preformed hubs. (D) The probability distribution of thepolygon angles ψ averaged over the simulation trajectory.

Fig. 2. Polygon-level CG model energy, E (Left) and free-energy, F (Right)landscapes at kT = 0.1ε (Top) and at kT = 0.3ε (Bottom) for an ideal spheri-cal packing (radius R = 1.603σ corresponding to an icosahedral packing withN = 32) composed of only hexagonal and pentagonal particles. The ratio ofactivities is zP/zH = exp(∆µ/kT) = 12/20. The colors vary from purple (smallvalues) to red (large values).

abundance of nonhexagonal subunits in Fig. 1C (for instance,see the pronounced peak for the squares in Fig. 1D). In addi-tion, the lifetime of those defect-rich structures is longer thanthe accessible timescale of our simulations (≈100 µs), suggestingthat their contribution might not be negligible and they might sig-nificantly alter the free-energy landscape of the system. However,even at this level of coarse graining, calculating the free energy ofa complete SAGE was not feasible due to computational limita-tions. We note that, consistent with our CG simulation results,the existence of nonhexagonal subunits on the SAGE surfacehas recently been confirmed experimentally in the atomic forcemicroscopy measurements of the silica-coated SAGEs.∗

Packing of Proteins on Spherical Shells: A Mesoscale ModelTo gain further insight into whether the system is able to escapefrom such long-lived metastable defected states and find its mostsymmetrical configuration, and also to measure the thermody-namic stability of the icosahedral cage, we used a mesoscalemodel. This was on the level of polygonal units on the SAGEsurface. Following the work of Zandi et al. (5), who introducea minimal model for virus capsids, we extended the model byconsidering different type polygons as Lennard–Jones (LJ) par-ticles with different diameters {σi}, i ∈{H ,P ,S , ...} (H , P , andS stand for hexagon, pentagon and square, respectively) that areallowed to move on the surface of the sphere or change type,while interacting via a truncated LJ interaction of strength ε.The packing of particles on spherical surfaces has been used alsoas a model for colloidosomes (37–42). For reference, we takeour ideal system as composed only of hexagonal (σH =σ) andpentagonal (σP =σ/[2 sin(π/5)]) particles. (We use the wordideal to refer to a system that is composed only of hexagonsand pentagons.) The value of σP was chosen such that pen-tagons have the same length per edge as hexagons. Therefore,a polygonal particle with n edges would prefer (energetically),on average, to have n neighbors. However, at the finite tem-peratures of our simulations, the system is allowed (by paying

*Galloway J, et al. (2017) Silicification of self-assembled peptide cages (SAGEs) revealsthe morphology of the peptide network nanostructure, Fifty-first ESBOC Symposiumon Chemical Synthetic Biology: Self-assembly, Encapsulation, and Delivery, May 19-21,2017, Newtown, Powys, UK.

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Fig. 3. (Top and Middle) Energy E, free-energy F, and the number of nonhexagonal particles as a function of normalized BOOs W6 and Q6 at kT = 0.15ε andR = 1.603σ for an ideal system composed of hexagons and pentagons (solid lines) and perturbed systems composed of hexagons, pentagons, and squareswith varying strength for the perturbation controlled by the relative activity of the squares: weak (zS/zP = 0.01, dashed lines), intermediate (zS/zP = 0.1,dotted lines), and strong (zS/zP = 1, dashed-dotted lines). (Bottom) Five different configurations are also shown: i–iv are typical ideal packings at kT = 0.15ε(i–iii) and kT = 0.3ε (iv), and v is a typical perturbed packing at kT = 0.15ε and zS/zP = 0.01. Copper, blue, and cyan particles represent hexagons, pentagons,and squares, respectively. See SI Appendix, Fig. S5 for a 3D interactive version of these snapshots.

an energy penalty) to adopt a configuration in which the num-ber of neighbors deviates from n . It has been shown that thisminimal model has the lowest free energies for packings withspecial numbers of particles, e.g., N =12, 32, 42, ... (or equiva-lently special radii) corresponding to icosahedral arrangementswith N − 12 hexagons and 12 pentagons (5, 43), and is consistentwith the Casper–Klug quasi-equivalence principle for sphericalvirus capsids (2). We extended this model to include structuraldefects that may exist in the form of other polygonal units onthe cage surface. In particular, we considered a perturbed sys-tem that also had squares (σS =σ/(2 sin(π/4))). We carried outMonte Carlo (MC) simulations in the grand-canonical ensem-ble at a fixed sphere radius R, temperature T , and chemicalpotentials {µi}.

We characterized the icosahedral symmetry of the particlepackings using Steinhardt’s bond orientational order (BOO)parameters Q6 and W6, originally defined to distinguish the localcrystallinity classes in liquids and glasses (44, 45) (Materials andMethods). BOOs are rotational invariants based on sphericalharmonics and are calculated from orientation of nonhexago-nal particles with respect to the center of the spherical packingwith values of {Q icos

6 ,W icos6 }= {0.6633,−0.16975} for an icosa-

hedral packing (packing i in Fig. 3) and {Q rand6 ,W rand

6 }≈{0, 0}for a totally random packing. We measured the free-energy pro-files as a function of the normalized BOOs Q6 =Q6/Q

icos6 and

W6 =W6/Wicos6 , using umbrella sampling (46) to facilitate the

measurement of less favorable states. The umbrella weights wereadjusted iteratively to have a uniform sampling as a function of

BOOs. The starting configuration at each stage was chosen to bethe final configuration from the previous iteration. At each tem-perature, we ran from 20 to 40 independent simulations, eachwith ∼3× 109 MC steps per particle, and obtained the resultingprofiles using the weighted histogram analysis method (47). Thisprocedure is repeated until the obtained free-energy profiles areconverged. We note that, at the lowest temperatures studiedhere, the identity-exchange MC moves were mostly rejected andthe dynamics became very slow.

We observed that the lowest-energy configurations lay atregions where the normalized BOOs were close to 1 (Fig. 2,Left). These are icosahedrally ordered arrangements that the sys-tem typically samples at low temperatures where entropic effectsare negligible. Nonicosahedral arrangements with even lowerenergies than icosahedral arrangement have been described inref. 43 for spherical packings composed of same-sized LJ parti-cles. In such low-energy arrangements, particles that do not have6 neighbors tend to cluster together. For the two system sizesthat we studied here (R=1.603σ and R=2.469σ), the optimumpackings with clustering, which are typically invariant againsta smaller number of symmetry operations compared with theicosahedral packing (SI Appendix, Fig. S6C), were local energyminima (with E >Eicos). This indicates that icosahedral symme-try is a more robust feature of our ideal binary-mixture packings.However, for much larger packings with 312 particles, we sawarrangements with E <Eicos in which pentagons cluster togetherin 12× 6 groups (SI Appendix, Fig. S6). For the ideal system,we found the icosahedral packing to be the most probablearrangement at low temperatures (kT . 0.2ε), in agreement with

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previous studies (5, 43) (Fig. 2, Top Right). At sufficiently largetemperatures (kT & 0.2ε), the entropy term becomes the domi-nant term in the free energy and thus, at the expense of energy,the system explores conformations with a larger number of pen-tagons (defects). Therefore, in the high-temperature regime, thefree-energy minimum of the system is in the disordered regionwith small Q6 and |W6| (Fig. 2, Bottom Right).

For the ideal system, the projected landscapes and the aver-age number of pentagons as a function of W6 and Q6 are illus-trated in Fig. 3 with solid lines. At kT =0.15ε, the lowest-energypackings are icosahedrally ordered with exactly 12 pentagons.Packings i, ii, and iii are the three most probable packings (i.e.,the lowest minima in the free-energy landscape) that the sys-tem samples at this temperature, with a high level of icosahe-dral order reflected by W6≈ 1. Whereas i represents a perfecticosahedral arrangement, ii and iii are packings with 13 or 14pentagons, respectively. These are packings in which one or twohexagons are replaced with pentagons, resulting in an increasein the average energy per particle and also smaller normal-ized BOOs. At even lower temperatures, the icosahedral basinbecomes the dominant global free-energy minimum, in agree-ment with previous studies (5, 43). Similarly, the peaks in thefree-energy landscape as a function of Q6 include conformationsin which a pentagon is replaced with a bigger hexagon; this sub-stitution is very unfavorable energetically. In larger packings, theoscillations in the icosahedrally ordered region of the landscapeshift to the right and become smaller in amplitude (SI Appendix).This is because, in large packings, small movements of a largernumber of particles can help to accommodate those substitutionsmore easily. It has been shown in many viral-capsid assemblystudies (11, 48, 49) that the subunits’ binding energy is on theorder of ≈6 kT . Therefore, kT =0.15ε is expected to be roughlyroom temperature when we interpret the results of our minimalmodel for virus capsids.

To investigate the stability of icosahedral packing in a per-turbed system, we also considered a packing composed of hexa-gons, pentagons, and squares. Our umbrella-sampling resultsfor the perturbed systems are shown with dashed, dotted, anddashed-dotted lines, in the ascending order of the strength ofthe perturbation, respectively, in Fig. 3. For all cases (i.e., forzS/zP ≥ 10−2), we observed that the existence of squares signifi-cantly changes the free-energy landscape of the system by shiftingthe most stable packing to the disordered region with small Q6

and |W6| (packing v in Fig. 3). It is interesting to note that even atzS/zP =10−3, the icosahedral order is effectively removed whileonly less than 5% of particles are squares (Fig. 4). This showsthat in the presence of structural defects (squares), the icosa-hedral arrangement becomes unstable. Moreover, other ener-

norm

alized

BOO

Fig. 4. Averaged normalized BOOs (Left) and the fraction of squares(Right) are plotted as a function of zS/zP for a perturbed system at kT =

0.15ε and R = 1.603σ. Error bars represent two standard deviations awayfrom the mean value obtained from 40 independent simulations.

Fig. 5. Order–disorder transition in the polygon-level CG model. Top plotsdisplay average normalized BOOs and the energy as a function of tempera-ture for the ideal packing (solid lines) and for the perturbed packing (dashedlines) with zS/zP = 1 and R = 1.603σ. Bottom plots show average number ofspecies as a function of temperature. Error bars represent two standard devi-ations away from the mean value obtained from 40 (20 for the ideal system)independent simulations.

getically favorable symmetric configurations with a smaller sym-metry group (e.g., the packing in SI Appendix, Fig. S6D with aD5h symmetry that is also responsible for the drop of E whenQ6→ 0) could be adopted more easily in the perturbed system(43). Comparison between the relative population of squares inthe CC-level CG simulations of the SAGEs and the number ofsquares in our perturbed packings suggests that for the SAGEszS/zP ≈ 10−1. We note that the flexibility of the protein buildingblock in the SAGE determines the relative activity of squaresand we expect it to vary for different protein cages. We alsoobserved, in the disordered region of the landscape with increas-ing temperature, an increase in the number of nonhexagonal par-ticles, which is also accompanied by a slower increase in the totalnumber of particles. A qualitatively similar behavior was alsoobserved for larger packings (SI Appendix).

To quantify the order–disorder transition in our packings wemeasured the average BOO parameters and the number ofspecies as a function of temperature. The results are illustratedin Fig. 5 for the ideal and perturbed packings. We observeda rapid increase in the total number of defects (extra pen-tagons and squares) at high temperatures. Upon introducingdefects (squares) into the packing, the midpoint of the transi-tion (i.e., the melting point) shifted to lower temperatures forboth Q6 and W6 transitions, indicating that the icosahedrallyordered packings became less stable for those perturbed sys-tems. Moreover, the melting curves were characterized by asharper transition in those perturbed packings. This is prob-ably due to the fact that, starting from a disordered configu-ration in the perturbed packings, a significantly larger numberof particles are required to rearrange or change type coopera-tively to reach the icosahedral arrangement. Therefore, for theperturbed packings, the order–disorder transition is character-ized by a larger level of cooperativity compared with the idealpacking.

DiscussionMany natural systems including viruses have evolved over mil-lions of years to form icosahedral shells from protein buildingblocks (2, 5). These provide the motivation for the design of syn-thetic self-assembling cages (14–16). Natural icosahedral proteincages are formed usually via a hierarchical self-assembly processof highly specific building blocks with at least two stages: First,two classes of mesoscale units combine, which then self-assemble

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further to highly monodispersed regular shells. Using modelson multiple scales, we have investigated the self-assembly ofquasi-spherical shells from simple elementary building blocks,focusing on the regularity of their packing on the surface ofthe spherical cage. Using building blocks that are able to packwith icosahedral symmetry, we examine how stable these highlysymmetric structures are to perturbations that may arise fromthe flexibility of the interacting blocks. This allows us to explorethe rather simpler self-assembly processes involving more flex-ible units that allow the formation of a variety of mesoscaleobjects and permitting structural defects. We find that they canalso form shell-like structures with few or no holes, howeverwith more variability in both the local structure on the shell andthe overall symmetry and size of the shells formed. In partic-ular, we find that by introducing a small number of structuraldefects, icosahedral packings are not the most stable structuresfor a wide range of parameters and that rather a disorderedstructure is found on the shell’s surface. For many applicationsin protein design, however, such a variability is not a handicapand this suggests that icosahedral packings need not necessar-ily be the aim for synthetic design of microscale containers forpackaging and transport. Indeed there are a number of proteinshells [e.g., the HIV virus capsid (50, 51)] that show disorderednonicosahedral surfaces, which seem also to be correlated withtheir greater variability. Our results are also relevant to colloido-somes that are fabricated by the self-assembly of colloidal parti-cles onto the interface of emulsion droplets (37, 38). Our workprovides a new design paradigm: We propose that by modulatingthe flexibilities of the components, one can control the regular-ity of the packing and, consequently, the surface properties of asynthetic cage.

Materials and MethodsCC-Level CG Model of SAGEs. We designed a CG model on the level of CCsto directly simulate the assembly of SAGEs. Each CC in this model is a rigidbody with LJ attractive patches on its surface that drive formation of CCdimers and trimers (52). A permanent bond also connects a trimeric CC to adimeric CC. Details of the model and its parameterization are explained inSI Appendix.

Polygon-Level CG Model. In this model, a polygon with n edges is describedas an LJ particle of diameter σα =σ/(2 sin(π/n)). The pairwise LJ interactionbetween polygons α and β, separated by a distance r, is described by

VLJ(r) = 4ε[(σαβ/r)12 − (σαβ/r)6

],

where σαβ = (σα +σβ )/2. The interaction is truncated at a cutoff distance1.5σαβ . We performed grand-canonical MC simulations (53) at the tempera-ture T and activities {zα}= {exp(µα/kT)}, where µα is the chemical poten-tial of species α, and k is the Boltzmann constant. All particles are restrictedto move on the surface of a sphere of radius R and are allowed to changetype. MC trial moves include (i) single-particle moves, (ii) deletion moves,(iii) addition moves, (iv) type-change moves, and (v) position swap of twodissimilar particles. Care has been exercised to ensure that MC moves satisfythe detailed balance condition. We performed extensive MC simulations fortwo system sizes R = 1.603σ and R = 2.469σ, corresponding to icosahedralpackings with N = 32 and N = 72 particles, respectively. We confirmed thatthese arrangements are energetically optimized configurations for the idealsystem as T→ 0.

BOO Parameters. We adopt the BOO of Steinhardt et al. (44) of l-fold sym-metry to characterize our packing by considering the complex vector

Qlm =1

N − NH

N−NH∑i=1

Ylm(ri),

where the Ylm are spherical harmonics and ri is the position of the non-hexagonal particle i relative to the center of the sphere. One can constructthe following rotational invariants

Ql =

4π

2l + 1

l∑m=−l

|Qlm|2

1/2

, [1]

Wl =

∑m1,m2,m3

m1+m2+m3=0

[l l l

m1 m2 m3

]Qlm1

Qlm2Qlm3

(∑lm=−l |Qlm|2

)3/2, [2]

where the bracket in the third-order invariant in Eq. 2 represents theWigner 3-j symbol. It has been shown that these invariants can effectivelybe used to distinguish various types of local order within glasses and liquids(44, 45). For our purpose, which is to distinguish only the icosahedralorder, it is sufficient to restrict our consideration to l = 6. In particular,W6 is very sensitive to the level of icosahedral order within the pack-ing (44); the more negative the W6 is, the higher the level of icosahe-dral order.

ACKNOWLEDGMENTS. The computational resources of the University ofBristol Advanced Computing Research Center and the BrisSynBio high-performance computing facility are gratefully acknowledged. M.M., D.K.S.,R.B.S., D.N.W., and T.B.L. are supported by BrisSynBio [a UK Biotechnol-ogy and Biological Sciences Research Council (BBSRC)/UK Engineering andPhysical Sciences Research Council (EPSRC) Synthetic Biology Research Cen-ter (BB/L01386X/1)]. R.B.S. and D.N.W. are funded by BBSRC LoLa GrantBB/M002969/1. D.N.W. is a Royal Society Wolfson Research Merit Awardholder (WM140008).

1. Theil EC (1987) Ferritin: Structure, gene regulation, and cellular function in animals,plants, and microorganisms. Annu Rev Biochem 56:289–315.

2. Caspar DL, Klug A (1962) Physical principles in the construction of regular viruses.Cold Spring Harbor Symp Quant Biol 27:1–24.

3. Edeling MA, Smith C, Owen D (2006) Life of a clathrin coat: Insights from clathrin andAP structures. Nat Rev Mol Cell Biol 7:32–44.

4. Tanaka S, et al. (2008) Atomic-level models of the bacterial carboxysome shell. Science319:1083–1086.

5. Zandi R, Reguera D, Bruinsma RF, Gelbart WM, Rudnick J (2004) Origin of icosahedralsymmetry in viruses. Proc Natl Acad Sci USA 101:15556–15560.

6. Bruinsma RF, Gelbart WM, Reguera D, Rudnick J, Zandi R (2003) Viral self-assembly asa thermodynamic process. Phys Rev Lett 90:248101.

7. Ivanovska IL, et al. (2004) Bacteriophage capsids: Tough nanoshells with complex elas-tic properties. Proc Natl Acad Sci USA 101:7600–7605.

8. Kirchhausen T, Owen D, Harrison SC (2014) Molecular structure, function, anddynamics of clathrin-mediated membrane traffic. Cold Spring Harb Perspect Biol 6:a016725.

9. Wagner J, Zandi R (2015) The robust assembly of small symmetric nanoshells. BiophysJ 109:956–965.

10. Reguera J, Carreira A, Riolobos L, Almendral JM, Mateu MG (2004) Role of interfa-cial amino acid residues in assembly, stability, and conformation of a spherical viruscapsid. Proc Natl Acad Sci USA 101:2724–2729.

11. Perlmutter JD, Hagan MF (2015) Mechanisms of virus assembly. Annu Rev Phys Chem66:217–239.

12. Baschek JE, Klein HC, Schwarz US (2012) Stochastic dynamics of virus capsid formation:Direct versus hierarchical self-assembly. BMC Biophys 5:22.

13. Corchero JL, Cedano J (2011) Self-assembling, protein-based intracellular bacterialorganelles: Emerging vehicles for encapsulating, targeting and delivering therapeu-tical cargoes. Microb Cell Fact 10:92.

14. Fletcher JM, et al. (2013) Self-assembling cages from coiled-coil peptide modules. Sci-ence 340:595–599.

15. King NP, et al. (2014) Accurate design of co-assembling multi-component proteinnanomaterials. Nature 510:103–108.

16. Indelicato G, et al. (2016) Principles governing the self-assembly of coiled-coil proteinnanoparticles. Biophys J 110:646–660.

17. Hsia Y, et al. (2016) Design of a hyperstable 60-subunit protein icosahedron. Nature535:136–139.

18. Sasaki E, Hilvert D (2016) Self-assembly of proteinaceous multishell structures medi-ated by a supercharged protein. J Phys Chem B 120:6089–6095.

19. Kostiainen MA, et al. (2013) Electrostatic assembly of binary nanoparticle superlat-tices using protein cages. Nat Nanotechnol 8:52–56.

20. Huang X, et al. (2007) Self-assembled virus-like particles with magnetic cores. NanoLett 7:2407–2416.

21. Kushnir N, Streatfield SJ, Yusibov V (2012) Virus-like particles as a highly efficientvaccine platform: Diversity of targets and production systems and advances in clinicaldevelopment. Vaccine 31:58–83.

22. Bode SA, Minten IJ, Nolte RJ, Cornelissen JJ (2011) Reactions inside nanoscale proteincages. Nanoscale 3:2376–2389.

23. Uchida M, et al. (2007) Biological containers: Protein cages as multifunctionalnanoplatforms. Adv Mater 19:1025–1042.

24. Putri RM, Cornelissen JJ, Koay MS (2015) Self-assembled cage-like protein structures.ChemPhysChem 16:911–918.

9018 | www.pnas.org/cgi/doi/10.1073/pnas.1706825114 Mosayebi et al.

Dow

nloa

ded

by g

uest

on

May

16,

202

0

PHYS

ICS

BIO

PHYS

ICS

AN

DCO

MPU

TATI

ON

AL

BIO

LOG

Y

25. Kirchhausen T (1993) Coated pits and coated vesicles – sorting it all out. Curr OpinStruct Biol 3:182–188.

26. Avinoam O, Schorb M, Beese CJ, Briggs JAG, Kaksonen M (2015) Endocytic sitesmature by continuous bending and remodeling of the clathrin coat. Science348:1369–1372.

27. Perotti LE, et al. (2016) Useful scars: Physics of the capsids of archaeal viruses. PhysRev E 94:012404.

28. Vacha R, Frenkel D (2011) Relation between molecular shape and the morphology ofself-assembling aggregates: A simulation study. Biophys J 101:1432–1439.

29. den Otter WK, Renes MR, Briels WJ (2010) Self-assembly of three-legged patchy par-ticles into polyhedral cages. J Phys Condens Matter 22:104103.

30. Villar G, et al. (2009) Self-assembly and evolution of homomeric protein complexes.Phys Rev Lett 102:118106.

31. Mannige RV, et al. (2015) Peptoid nanosheets exhibit a new secondary-structuremotif. Nature 526:415–420.

32. Glotzer SC, Solomon MJ (2007) Anisotropy of building blocks and their assembly intocomplex structures. Nat Mater 6:557–562.

33. Chen T, Zhang Z, Glotzer SC (2007) Simulation studies of the self-assembly of cone-shaped particles. Langmuir 23:6598–6605.

34. Morphew D, Chakrabarti D (2015) Hierarchical self-assembly of colloidal magneticparticles into reconfigurable spherical structures. Nanoscale 7:8343–8350.

35. Vernizzi G, Sknepnek R, Olvera de la Cruz M (2011) Platonic and Archimedean geome-tries in multicomponent elastic membranes. Proc Natl Acad Sci USA 108:4292–4296.

36. Vernizzi G, Olvera de la Cruz M (2007) Faceting ionic shells into icosahedra via elec-trostatics. Proc Natl Acad Sci USA 104:18382–18386.

37. Dinsmore AD, et al. (2002) Colloidosomes: Selectively permeable capsules composedof colloidal particles. Science 298:1006–1009.

38. Meng G, Paulose J, Nelson DR, Manoharan VN (2014) Elastic instability of a crystalgrowing on a curved surface. Science 343:634–637.

39. Fantoni R, Salari JWO, Klumperman B (2012) Structure of colloidosomes with tunableparticle density: Simulation versus experiment. Phys Rev E 85:061404.

40. Burke CJ, Mbanga BL, Wei Z, Spicer PT, Atherton TJ (2015) The role of curvatureanisotropy in the ordering of spheres on an ellipsoid. Soft Matter 11:5872–5882.

41. Bowick MJ, Cacciuto A, Nelson DR, Travesset A (2006) Crystalline particle packings ona sphere with long-range power-law potentials. Phys Rev B 73:024115.

42. Paquay S, Both GJ, van der Schoot P (2017) Impact of interaction range and curvatureon crystal growth of particles confined to spherical surfaces. arXiv:1703.00576.

43. Paquay S, Kusumaatmaja H, Wales DJ, Zandi R, van der Schoot P (2016) Energeticallyfavoured defects in dense packings of particles on spherical surfaces. Soft Matter12:5708–5717.

44. Steinhardt PJ, Nelson DR, Ronchetti M (1983) Bond-orientational order in liquids andglasses. Phys Rev B 28:784–805.

45. Lechner W, Dellago C (2008) Accurate determination of crystal structures based onaveraged local bond order parameters. J Chem Phys 129:114707.

46. Torrie G, Valleau J (1977) Nonphysical sampling distributions in monte carlo free-energy estimation: Umbrella sampling. J Comp Phys 23:187–199.

47. Kumar S, Rosenberg JM, Bouzida D, Swendsen RH, Kollman PA (1992) The weightedhistogram analysis method for free-energy calculations on biomolecules. I. Themethod. J Comput Chem 13:1011–1021.

48. Ceres P, Zlotnick A (2002) Weak protein-protein interactions are sufficient to driveassembly of hepatitis B virus capsids. Biochemistry 41:11525–11531.

49. Zlotnick A (2003) Are weak protein–protein interactions the general rule in capsidassembly? Virology 315:269–274.

50. Wilk T, et al. (2001) Organization of immature human immunodeficiency virus type1. J Virol 75:759–771.

51. Saxena P, et al. (2016) Virus matryoshka: A bacteriophage particle—guided molecularassembly approach to a monodisperse model of the immature human immunodefi-ciency virus. Small 12:5862–5872.

52. Zhang, Glotzer SC (2004) Self-assembly of patchy particles. Nano Lett 4:1407–1413.

53. Cracknell RF, Nicholson D, Quirke N (1993) A grand canonical monte carlo study ofLennard-Jones mixtures in slit shaped pores. Mol Phys 80:885–897.

Mosayebi et al. PNAS | August 22, 2017 | vol. 114 | no. 34 | 9019

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