arXiv:0711.3527v1 [physics.pop-ph] 22 Nov 2007 hs tutrsta a nioaethda symmetry. icosadeltahedral of subset an large has explain a that to form of structures is to symmetries these paper and this i.e. principles of interior design aim the object The the ar- structure. nearly enclose to cage-like have a fully of that to domes) up so geodesic built proteins in range are struts fullerenes, and in objects viruses, atoms in these (carbon all domes elements reason that geodesic The identical and is symmetry. nm, and this 100 design for same the share viruses m) nm, 100 1 (fullerenes o ato t ihteioaerl”akoe.A Fig. ”backbone”. in shown icosahedral is the statement with this it) of representation of surface visual spherical part the a of (or triangulations as described cally ue fms ftes-ald”peia”(icosahedral) ”spherical” time so-called the the at known fea- of viruses the most explained of that theory tures first the constructed Klug a tutrscle csdlaer.EtnigFuller’s ideas Extending mathemati- design icosadeltahedra. of that called class domes structures larger achiral a cal that to clear belong discov- constructed became Fuller the it before fullerenes Even of ery symmetry. their retained image xiiini oteli er16.Goei domes Geodesic chiral 1967. not year were constructed them in Fuller Expo that of World Montreal the famous in for Most pavilion exhibition American USA. house to in used 1960’s was that in architecture to dominant Fuller alternative was an grid. as (hemi)spherical domes a constructed inter- forming of networks struts - connected domes geodesic popularizing an architect for was nized and who inventor Fuller designer, Buckminster American Richard Buck- after - molecule molecules fullerene to C the attributed or all minsterfullerene in of be abundant atoms surely most carbon can the of arrangement surprise symmetrical this amazingly of discovery part experimental Some their announcing paper a lished ato h cetfi omnt hnKroto when community scientific the of part itdaddiscussed and dicted I CSDLAERLGOEI DOMES GEODESIC ICOSADELTAHEDRAL II. ti nrgigta bet odffrn nsize in different so objects that intriguing is It csdlaerlgoei oe a emathemati- be can domes geodesic Icosadeltahedral lhuhtefleeemlclswr rvosypre- previously were molecules fullerene the Although 4 csdlaerlgoer ffleee,vrssadgeod and viruses fullerenes, of geometry Icosadeltahedral pcfi aeo csdlaerlgeometry. icosadeltahedral Caspar-Klug of d case fullerenes. in specific in faces Eul a rings and Using edges, hexagonal vertices, and structures. of pentagonal number these ico the of calculate The all to how of examination objects. by these details of representation icosadeltahedral nya 92Dnl apradAaron and Caspar Donald 1962 year in , ics h ymtyo ulrns iue n edscd geodesic and viruses fullerenes, of symmetry the discuss I .INTRODUCTION I. 1 eateto hoeia hsc,Jˇe tfnInstitut Joˇzef Stefan Physics, Theoretical of Department 1 60 twsqieasrrs o largest for surprise a quite was it , rt n olausnmdthis named colleagues and Kroto . 5 . 2 nttt fPyis ..Bx34 00 arb Croatia Zagreb, 10001 304, Box P.O. Physics, of Institute 3 ..termirror their i.e. , 2 otrecog- most tal et Antonio pub- 2 . Siber ˇ onshvn i ers egbrn ons l fthese called of of All are vertices points. other triangulations neighboring all the nearest neighbors, six make so nearest having five points sphere that illus- have a 1 points and icosahedron twelve triangulate an Fig. to are As ways there many ver- that pentagons. icosahedral are thick the there of as trates, points outlined neighboring are the havetices 1 points other Fig. the In all - points oeteol nsta have that be- ones points twelve only These the icosahedral points. twelve come nature the special icosahedral preserved, become the is vertices that surface triangulation so the spherical triangles of the with When Fig. tri- covered equilateral of twenty vertices. is corner of twelve left consisting and solid upper angles platonic the a in is 1) (shown Icosahedron 1. lentv ovnin h oe with domes ( The to ( convention. alternative denote correspond the would and chosen, convention ( convention turn our by other left structures the the icosadeltahedral Were of assume symmetry shall the I right follows, or to left need what turn we to definite, the needs be after ”jumper” To the one. whether first specify the with degrees 60 n fte,and them, of one olwn h oddm sue nyfricosadeltahedral for only dome. used geodesic the is dome In word projections spheres. the as following circumscribed considered in their be shown can on domes shape icosadeltahedra the of spiky fact In a 1. is Fig. which icosadeltahedron an symmetry as same the have the domes Nevertheless, geodesic icosadeltahedral all polyhedra. that necessarily (spiky) triangles aspherical requirement equilateral produces The be faces polyhedral equilateral. the not of are of consist triangles. are equilateral domes not geodesic all icosadeltahedral are the faces speaking Strictly whose polyhedron a is eaieitgr,dntdby denoted integers, negative hretlnsbtentopit nasphere), a (the on points geodesics two spherical between different lines two shortest neighboring along its or- ( directed from in for be pentagon performed a Except be of to center pentagon. need a reach that through to dome ”jumps” der a of numbers of as vertices of the thought be can These are aho h oe a ecaatrzdb w non- two by characterized be can domes the of Each ,2, 1, chiral ioadlaer ic l ftetinlsta they that triangles the of all since (icosa)deltahedra aethda ymtyi xlie in explained is symmetry sadeltahedral ∗ lsicto fvrssi lbrtdas elaborated is viruses of classification rstermo oyer,i sshown is it polyhedra, on theorem er’s ms n ubro tm,bnsand bonds atoms, of number and omes, hsmasta hi irriaehsdiffer- has image mirror their that means This . m mswti nfidfaeokof framework unified a within omes ,S-00Lulaa Slovenia Ljubljana, SI-1000 e, up ln h rtshrclgoei.In geodesic. spherical first the along jumps n ln nte n,mkn nageof angle an making one, another along m, icosadeltahedral )dm,tejmsne to need jumps the dome, 0) scdomes esic m five and six ers neighboring nearest n ,m n, ers neighbors. nearest m ntefollowing. the in Deltahedron . ,n m, oei the in dome ) 6= n oein dome ) ; ,n> n m, m ,n m, along ). 0 2 ent symmetry. A mirror image of (m,n) dome is (n,m) neighbors. This type of bonding is also present in the dome. This is illustrated in the upper-right corner of Fig. planes of carbon atoms in graphite (graphene planes). It 1 for (3, 2) dome. is called sp2 bonding, in contrast to sp3 bonding that From m and n one can calculate the number of trian- is characteristic of diamond (in diamond each of the gles in a dome. The number of triangles per one spheri- carbon atoms is bonded with three nearest neighboring cal segment bounded by three spherical geodesic passing atoms). There are many different structures that can through neighboring fivefold coordinated vertices (out- be made of carbon atoms connected with sp2 bonds, at lined on a (4, 4) dome in Fig. 1) is least conceptually (see e.g. Refs. 8,9). Quite a differ- 2 2 ent question is whether such structures can be exper- T = m + n + mn, (1) imentally obtained. The icosadeltahedral symmetry of so that the total number of triangles (or faces) in an geodesic domes is characteristic of a class of especially icosadeltahedron is symmetric fullerene molecules, sometimes called icosahe- dral fullerenes, or giant icosahedral fullerenes in case that f = 20T. (2) molecules contain more than about 100 carbon atoms. Buckminsterfullerene belongs to this class (its ”compan- T is called the triangulation number or simply the ion” molecule C that was discovered simultaneously2 T 70 T-number. It adopts special integer values, = does not, however). The symmetry of these carbon , , , , , , , ... m n 1 3 4 7 9 12 13 . Instead of and integers, the T- molecules can be obtained from icosadeltahedral domes number can be used to classify the icosadeltahedral sym- by placing carbon atoms in (bary)centers of every tri- metry. The problem with this choice is that it doesn’t angle in the dome. The newly obtained set of points m,n n,m discriminate between ( ) and ( ) domes. That is (carbon atoms) is then ordered so that each point is con- why the T-number is sometimes used in combination with nected with its three nearest neighbors, i.e. the carbon- laevo dextro words (left) and (right) to resolve this am- carbon bonds are established (this procedure is illus- , biguity. For example, (2 1) structure in our convention trated in the upper-right corner of Fig. 2). This ful- T laevo T l would be in this case denoted as = 7 or = 7 fills the basic chemical requirement for carbon atoms or simply T = 7, while (1, 2) structure would be denoted 2 7 in sp bonding electronic configuration. The thus ob- T dextro T d as = 7 or = 7 . From the known number of tained structure contains now twelve pentagonal carbon f polyhedron faces ( ) one can proceed to find the number rings (pentagons) and certain number of hexagonal car- of its vertices (v) and edges (e) by using Euler’s theo- 6 bon rings (hexagons), depending on the T-number of the rem on polyhedra which relates these nonzero integer dome. The starting dome and the fullerene-like poly- quantities as hedron that was obtained from it are called dual poly- v − e + f =2. (3) hedra. The (m,n) symmetry that was characteristic of the dome will also be characteristic of its dual fullerene- This equation is valid for polyhedra that are homeomor- like polyhedron, but now the ”jumping” that character- phic to the sphere, i.e. their topology is the same as izes icosadeltahedral symmetry is allowed only through that of a sphere, which is the case of interest to us. the centers of pentagons and hexagons (not along the In icosadeltahedral domes twelve vertices belong to five carbon-carbon bonds). I have used the term ”fullerene- edges - these are located at the vertices of an icosahe- like polyhedron” since the true fullerene molecules will in dron. All the other vertices belong to six edges, i.e. six general be different from the polyhedron obtained by a edges meet at those vertices. Each edge is bounded by simple mathematical dualization of the dome. The domes two vertices and all these fact together can be used to are icosadeltahedral triangulations of the sphere and this relate e and v as necessarily means that the lengths of triangular edges are not all the same. In particular, they are considerably 2e =5 · 12+6 · (v − 12) = 60+6(v − 12). (4) shorter in the neighborhood of icosahedral vertices, and In combination with Euler’s theorem, one obtains that this will pertain also in the dual polyhedra. This feature is also characteristic of Fuller’s geodesic constructions2. f v = +2=10T +2, (5) However, the fullerene molecules are more than mathe- 2 matical entities and their exact shape is determined by and energetics of carbon-carbon interactions. Carbon-carbon bonds are much easier to bend than to stretch8, so the 3f e = = 30T. (6) shape of the fullerene molecule will be such to keep the 2 nearest-neighbor carbon-carbon distances as uniform as possible and as close to their equilibrium value as possi- III. ICOSAHEDRAL FULLERENES ble (the equilibrium length of carbon-carbon bonds in in- finitely large graphene plane is about 1.42 A).˚ This means that large enough fullerenes will necessarily be aspherical, Fullerene molecules are carbon cages in which all car- looking more like an icosahedron with vertices slightly bon rings are either pentagonal or hexagonal and all car- above the centers of carbon pentagons as the molecules bon atoms make three covalent bonds with their nearest 3

FIG. 1: Gallery of icosadeltahedral geodesic domes for m > n and m < 5. For the sake of clarity, the back sides of geodesic domes are not shown, i.e. only half of the dome and four of twelve icosahedral vertices can bee seen. The upper-right corner of the figure contains comparison between chiral (3, 2) and (2, 3) domes. Note that they are mirror images. The spiky shape is a (3, 2) icosadeltahedron (all of its faces are equilateral triangles). get larger10. Figure 2 displays a gallery of icosahedral fullerenes is to ”unfold” them so that they become polyg- fullerenes. Their shape is not merely a mathematical onal pieces of graphene. Alternatively, one can also think construction obtained by dualization of a dome, but a about this procedure, illustrated in Fig. 3 as a way to true minimum of energy, calculated by using the realistic construct these molecules. The concave polygonal shape model of energetics of carbon sp2 bonding11 as described consisting of 20 equilateral triangles outlined by thick in Ref. 8. Note that the Buckminsterfullerene (1, 1) is lines is cut out from the graphene plane. The polygon perfectly spherical, i.e. all of its carbon atoms are equally is then creased along the edges shared by the triangles distanced from the geometrical center of the molecule. A and folded into a perfect icosahedron. The thus obtained high degree of sphericity is also present in (2, 0) fullerene, shape is still not a fullerene since the details of its shape but already in (2, 1) fullerene a clear icosahedral shape are wrong, but it has the same icosadeltahedral sym- of the molecule develops and this becomes more promi- metry and the same connectivity and number of carbon nent in larger molecules. There is a long standing debate atoms as the icosahedral fullerene does. The integers m (perhaps of academic value only) concerning the shape of and n that characterize the shape can now be interpreted asymptotic icosahedral fullerenes, i.e. those that contain as components of a two-dimensional vector A in a basis extremely large (infinite) number of carbon atoms. Stud- of graphene unit cell vectors a1 and a2 denoted in Fig. ies based on continuum elasticity of the icosadeltahedral 3, shells12,13 and on microsopic models of carbon-carbon A ma na , m,n> . bonding14 predict that the asymptotic shape is a perfect = 1 + 2 0 (7) icosahedron. The A vector is directed along the side of one of the A way to better comprehend the symmetry of twenty triangles making the icosahedron as illustrated in 4

FIG. 2: Gallery of icosahedral fullerenes for m > n and m < 5. For the sake of clearer representation the back sides of fullerenes are not shown. The carbon-carbon bond in all fullerenes is practically everywhere equal to 1.42 A,˚ and this can be used to estimate their size. The upper-right corner of the figure contains comparison between (1, 1) dome and (1, 1) icosahedral fullerene (Buckminsterfullerene, not to scale with other depicted fullerenes). Note that these polyhedra are dual to each other.

Fig. 3. In this convention, the unit cell vectors a1 and by a need to be chosen so that their vector product points 2 f f f . from the paper towards the reader. This reproduces the = 5 + 6 (9) jumping-to-the-left convention discussed in the previous Pentagonal and hexagonal faces are bounded by five and section. The unfolding described here can also be applied six vertices (atoms), respectively and each vertex (atom) to icosadeltahedra. The underlying lattice is triangular in belongs to exactly three faces. This means that that case. This procedure is very convenient for counting faces, edges and vertices and can be used to derive Eq. 5f5 +6f6 =3v. (10) (1). Combining these equations with Euler’s theorem in An important piece of information on fullerene Eq.(3) one obtains that molecules can be obtained from the Euler’s theorem on polyhedra. Since exactly three bonds (or polyhedron f5 = 12. (11) edges) finish at each of the carbon atoms (polyhedron This is obviously true for the icosahedral fullerenes dis- vertices) and the bond (edge) is shared by two atoms cussed so far, but the equation holds for general fullerenes (vertices), it follows that as long as they are topologically equivalent to a sphere (including e.g. C70). In other words, every network 2e =3v. (8) of pentagonal and hexagonal ring of carbon atoms with spherical topology necessarily has five pentagonal rings. A relation between the number of carbon atoms in By definition, the fullerenes contain only pentagonal and fullerenes and the number of hexagonal faces can also hexagonal faces (carbon rings). Let us denote the num- be obtained from the above consideration. It states that ber of pentagonal and hexagonal faces by f5 and f6, re- spectively. The total number of faces is obviously given v = 2(f6 + 10). (12) 5

the C80 signature may in fact correspond to (m = 2, n = 0) icosahedral fullerene. Larger (giant) fullerenes can be observed in experiments but it is difficult to pre- cisely determine their geometry and the spatial distri- bution of pentagonal carbon rings16. Carbon pentagons in graphitic samples have been observed using scanning A tunneling microscopy17.

IV. CASPAR-KLUG CLASIFICATION OF VIRUSES: THE T-NUMBER

Viruses are particles made of DNA or RNA molecule a2 (genome) protected by a coating made of proteins. Their size depends on the type of a virus7. The typical diameter A of a virus is about 50 nm. For example the diameter of a a1 herpes simplex virus is 125 nm, while polio virus is only 32 nmm in diameter7. The information that is required to produce the proteins of the coating is contained in the viral RNA or DNA molecule. Once the virus penetrates the cell wall, the viral genome is delivered to the cell and the production of viral proteins starts. The proteins are produced by a cellular molecular machinery called ribosomes that can assemble proteins from amino acids by ”reading” the information on the required sequence of amino acids that is coded in the viral RNA molecule. FIG. 3: Cut-and-fold construction of (2, 1) icosahedral This process is called translation by biologists. A discus- a a A fullerene. The vectors 1, 2 and discussed in the text sion of these marvelously complicated mechanisms would are denoted. The triangular faces of (1, 1), (2, 0), (3, 1), and lead us far astray from the subject of interest. More in- (4, 2) fullerene-like icosahedra are shown in the bottom of the figure. formation on virus ”life” cycle can be found in Ref. 7 and on translation and transcription (a process for obtaining the information-carrying RNA from the DNA molecule) This means that the number of carbon atoms in the in textbooks on cell biology and chemistry (see e.g. Ref. fullerene molecules is necessarily even. This, at first puz- 18). 19 zling, piece of information was observed already in the In 1956 Crick and Watson proposed that the spher- mass spectra of carbon clusters obtained by laser vapor- ical protein coating (or capsid) probably has a platonic ization from the graphitic sample15. Only signatures of polyhedral symmetry i.e. that it is built of identical pro- 5 clusters containing even number of carbon atoms were teins assembled in a polyhedral shell. Caspar and Klug detected which can be nicely explained by assuming that developed this notion further by noting that the quan- the clusters detected were in fact fullerenes. Let us now tity of information contained in the viral genome is quite specify the discussion of general fullerenes to the case of small, so that only one or perhaps two to three different icosahedral fullerenes. Total number of carbon atoms in proteins can be produced from it. They considered differ- these molecules is ent polyhedral shells made of identical proteins and de- duced that icosadeltahedral ordering provides a structure v = 20(m2 + mn + n2)=20T, (13) in which all of the proteins are in surroundings that are to a best approximation equal of all the choices consid- and the number of carbon-carbon bonds is ered. They called this the principle of quasi-equivalence. The geometry behind the principle of quasi-equivalence is e = 30T. (14) the one already discussed in the cases of icosadeltahderal As shown earlier, there are exactly twelve pentagonal car- geodesic domes and fullerenes. bon rings and In most of the ”spherical” viruses, the proteins are grouped in clusters (capsomers) of five (pentamers) and f6 = 10(T − 1) (15) six (hexamers). This is very often the case even if not all proteins are equal, i.e when capsid consists of several hexagonal carbon rings. types of proteins. In the assembled capsids, the twelve Fullerenes with number of carbon atoms larger than pentamers occupy the same spatial positions as carbon about 120 were not clearly observed in the experiments pentagons in fullerenes. The hexamers are equivalent to described in Ref. 2. Thus, in addition to C60, only hexagonal carbon rings in fullerenes. Thus, viruses can 6 be constructed by connecting each of the vertices in pen- triangular protein (denoted by 1) and two brighter kite- tagonal and hexagonal rings of the fullerenes with the shaped proteins (denoted by 2 and 3). This structure ring centers and interpreting the thus obtained divisions could be identified as belonging to T = 1 class, with only of the pentagons and hexagons as the dividing lines be- twelve ”pentons” [outlined by thick full lines in Fig. 4c)] tween the viral proteins. In fact, the fullerene vertices composed of five protein trimers (180 proteins in total). need not be connected to the centers of the rings but On the other hand, we could at least conceptually arrange to points lying on approximate normals to the pentago- the proteins in pentamers and hexamers as indicated by nal and hexagonal faces and passing through centers of thick dash-dotted lines in Fig. 4c). In this case, hexamers the faces. This corresponds to capping the pentagons would contain three pairs of 2- and 3-proteins from three and hexagons with pentagonal and hexagonal pyramids, trimers, while pentamers would consist of five 1-proteins respectively, but the capping can also be performed in from five different trimers. This would implicate that many other ways. The thus obtained polyhedron may be the shape belongs to T = 3 class of symmetry. Such called omnicapped fullerene [all (omni) of the fullerene viruses are called pseudo-T3 (or pT3) viruses. The prob- faces capped by pyramids or some other polyhedra]. This lem with the identification could be resolved on physical procedure is illustrated panels a) and b) of Fig. 4. The grounds - if the binding energy between the proteins of proteins are pieces of matter that have a certain equilib- the trimer is larger than between the proteins from dif- rium shape which is of course three-dimensional. Repre- ferent trimers, it makes sense to speak about the trimer sentation of protein capsomers by pyramids or any other as the basic building block and to identify the structure polyhedron is thus approximate. Any three dimensional as belonging to T = 1 class. However, the problem in the shape erected above the hexagon (pentagon) and having mathematical sense occurs when there are two T num- a six-fold (five-fold) symmetry with respect to rotations bers that can be divided without remainder (e.g. T =9 around the hexagon (pentagon) normal will serve as a and T = 3). In the most trivial case, every capsid with representation of a viral hexamer (pentamer). T-number T1 could be interpreted as a T = 1 capsid The classification of the symmetry of the capsid, how- consisting of T1-mers. In addition, every (m,m) capsid ever, does not depend on the shape of individual protein could in principle be thought of as (m, 0) capsid made but only on the characteristics of the arrangement of all of trimers. The important question is again whether the the proteins in the capsid (at least when all proteins are conceptually obtained protein multimers make any sense equal, see below). The symmetry of viruses is character- as the strongly bonded elementary units. ized in the same way as in the case of fullerenes: m and Caspar and Klug quasi-equivalence principle predicts n integers are counted by ”jumping” through the centers that there are only twelve pentameric capsomers, all of the capsomers and using the convention of turning left other being hexameric. In year 1982 it became clear that after the first m jumps. If m ≥ n, the virus is classified there are viruses (e.g. SV-40 virus from polyomavirus 2 2 as a member of Tlaevo = m + mn + n class (or simply genus) composed only of pentamers, but still retaining T ), and if otherwise, the virus is classified as a member of the icosadeltahedral symmetry of their arrangement20. Tdextro (or Td) class. The virus-like polyhedra depicted The concept of T-number is still valid in that case and in panels a) and b) of Fig. 3 both have T = 3 symmetry, the number of capsomers is still given by Eq. (16), but although the details of their shapes are quite different. the total number of proteins is no longer given by Eq. Total number of capsomers (c) in a T-class virus is ob- (17). For such viruses, the total number of proteins is viously the same as the number points in the icosadelta- hedral dome of T-symmetry, p =5c = 5(10T + 2). (18) c = 10T +2. (16) Viruses, as fullerenes, should be considered as phys- Total number of proteins in a virus (p) is a sum of 60 ical objects, i.e. their precise shape should be a re- proteins in 12 pentamers and 60(T −1) proteins in 10(T − sult of interactions acting between proteins themselves 1) hexamers. Alternatively (see Fig. 3) one can deduce and possibly between the proteins and viral DNA or that p for a virus of T-class should be the same as the RNA molecule. Shapes of viruses have recently been ex- number of faces in a dome of 3T-symmetry, i.e. plored using a generic model of shells with icosadeltahe- dral symmetry21,22,23. Application of physical methods p = 60T, (17) to understand the energetics and assembly of viruses has been an area of very lively research in recent years (for so that both approaches give the same answer. a partial but quite readable glimpse of the field see Ref. There may occur problems in identifying the symme- 24). try of the capsid when several proteins form a capsid, or This work has been supported by the Ministry of when building blocks of the capsid are not pentamers and Science, Education, and Sports of Republic of Croatia hexamers but trimers (clusters of three proteins). The through Project No. 035-0352828-2837, and by the Na- problem is illustrated by the shape in panel c) of Fig. tional Foundation for Science, Higher Education, and 4. The building block of this shape is a protein trimer Technological Development of the Republic of Croatia outlined by thick dashed lines. It consists of a darker through Project No. 02.03./25. 7

FIG. 4: Panels a) and b) represent polyhedral models of T = 3 viruses. These polyhedra can be termed as omnicapped truncated icosahedra or omnicapped Buckminsterfullerenes. The ”pentamers” are colored in a darker tone and borders between ”capsomers” are represented by thicker lines. The polyhedron in panel b) is quite similar to turnip yellow mosaic virus. Panel c) represents a model T = 1 (pT3) virus whose building block is a ”protein trimer” outlined by dashed lines.

∗ Electronic address: [email protected] 12 T.A. Witten and H. Li, Asymptotic shape of a fullerene 1 H. Kroto, Symmetry, space, stars and C60, Rev. Mod. ball, Europhys. Lett. 23, 51-55 (1993). Phys. 69, 703-722 (1997). 13 Z. Zhang, H.T. Davis, R.S. Maier, and D.M. Kroll, Asymp- 2 H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, and totic shape of elastic networks, Phys. Rev. B 52, 5404-5413 R.E. Smalley, C60 - Buckminsterfullerene, Nature (Lon- (1995). don) 318, 162-163 (1985). 14 A. Siber,ˇ Shapes and energies of giant icosahedral 3 D.L.D. Caspar, Deltahedral views of fullerene polymor- fullerenes: onset of ridge sharpening transition, Eur. Phys. phism, Phil. Trans. R. Soc. Lond. A 343, 133-144 (1993). J. B 53, 395-400 (2006). 4 G.J. Morgan, Hystorical review: Viruses, crystals and 15 R.F. Curl Jr., Dawn of the fullerenes: experiment and con- geodesic domes, Trends Biochem. Sci. 28, 86-90 (2003). jecture, Rev. Mod. Phys. 69, 691-702 (1997). 5 D.L.D. Caspar and A. Klug, Physical principles in the con- 16 J.Y. Huang, F. Ding, J. Jiao, and B.I. Yakobson, Real time struction of regular viruses, Cold Spring Harbor Symp. microscopy, kinetics, and mechanism of giant fullerene Quant. Biol. 27, 1-24 (1962). evaporation, Phys. Rev. Lett. 99, 175503-1 - 175503-4 6 A.S. Posamentier, Math wonders to inspire teachers and (2007). students (Association for supervision and curriculum de- 17 B. An, S. Fukuyama, K. Yokogawa, M. Yoshimura, M. velopment, 2003), pp. 195-197. Egashira, Y. Korai, and I. Mochida, Single pentagon in 7 T.S. Baker, N.H. Olson, and S.D. Fuller, Adding the third a hexagonal carbon lattice revealed by scanning tunneling dimension to virus life cycles: three-dimensional recon- microscopy, Appl. Phys. Lett. 78, 3696-3698 (2001). struction of icosahedral viruses from cryo-electron micro- 18 S.R. Bolsover, J.S. Hyams, E.A. Shephard, H.A. White, graphs, Microbiol. Mol. Biol. Rev. 63, 862-922 (1999). and C.A. Wiedemann, Cell biology: a short course, 2nd 8 A. Siber,ˇ Energies of sp2 carbon shapes with pentago- edition, Wiley (2004). nal disclinations and elasticity theory, Nanotechnology 17, 19 F.H.C. Crick and J.D. Watson, Structure of small viruses, 3598-3606 (2006). Nature 177, 473-475 (1956). 9 A. Siber,ˇ Continuum and all-atom description of the ener- 20 I. Rayment, T.S. Baker, D.L.D. Caspar, and W.T. Mu- getics of graphene nanocones, Nanotechnology 18, 375705- rakami, Polyoma virus capsid structure at 22.5 A˚ resolu- 1 - 375705-6 (2007). tion, Nature 295, 110-115 (1982). 10 One should be somewhat careful when using the terms pen- 21 J. Lidmar, L. Mirny L, and D.R. Nelson, Virus shapes and tagon (hexagon) and pentagonal (hexagonal) carbon rings buckling transitions in spherical shells, Phys. Rev. E 68, as synonyms. All of the carbon atoms in a pentagonal or 051910-1 - 051910-10 (2003). hexagonal ring do not necessarily lie in a plane, so that 22 T.T. Nguyen, R.F. Bruinsma, and W.M. Gelbart, Elastic- a pentagonal (hexagonal) carbon ring could in principle ity theory and shape transitions of viral shells, Phys. Rev. be built up of three (four) triangular pieces not lying in a E 72, 051923-1 - 051923-19 (2005). plane. The angles between these pieces are in general very 23 A. Siber,ˇ Buckling transition in icosahedral shells subjected small, so that for all practical purposes, the carbon atoms to volume conservation constraint and pressure: Relations in rings may be considered as belonging to the same plane. to virus maturation, Phys. Rev. E 73, 061915-1 - 061915-10 11 D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stu- (2006). art, B. Ni, S.B. Sinnott, A second-generation reactive em- 24 A. Zlotnick, Viruses and the physics of soft condensed mat- pirical bond order (REBO) potential energy expression for ter, Proc. Natl. Acad. Sci. 101, 15549-15550 (2004). hydrocarbons, J. Phys. Cond. Matt. 14, 783-802 (2002).