Viral Capsids and Self Assembly Viral Capsids

MAE 545: Lecture 22 (12/10) Viral capsids and self assembly Viral capsids

helical symmetry icosahedral symmetry complex symmetry

Hepatitis B virus conical shaped capsid proteins are wrapped (spherical caps usually around the viral RNA still icosahedral) (e.g. Tobacco Mosaic Virus) Poxvirus

Note: many viruses also HIV have envelope (lipid bilayer) around the capsid

2 Icosahedral capsids

Brome mosaic Bacteriophage Cowpea Chlorotic L-A virus Dengue virus virus X 174 Mottle virus

Tobacco Human Hepatitis Sindbis virus Bacteriophage Necrosis virus B Virus Simian virus HK97

Colors correspond to different proteins.

3 Generating Vectors for the Lattice Structures of Tubular and Conical Viral Capsids 129 number, to form with icosahedral symmetry. The quasi-equivalence is demonstrated in the experiment results reported in [30]. There are also many viruses that have tubular or lozenge-like capsids. Mature HIV-1 cores have cone- shaped capsids [1, 11]. Unlike the icosahedral viral capsids, the structures of tubular and conical capsids are not yet fully understood. There have been models for the lattice structures of tubular and conical capsids, see [17, 21] and references therein. However, there is a aw in the Nguyen model [21], which missed the requirement on the cone height and could result in incomplete cones if the model is inappropriately applied. In this paper, we propose new models for tubular and conical capsids in a unied fashion based on an extension of the Capser-Klug quasi-equivalence theory. The new models are easier than the existing models. When applied to the HIV-1 (5,7)-cone (5 pentagons in the narrow end and 7 pentagons in the broad end), the capsid properties derived from our models show good agreement with published experimental data. This demonstrates the correctness and usefulness of the new models. The rest of this paper is organized as follows. Section 2 briey reviews the concepts of the T-number and generating vector for the icosahedral viral capsid. Section 3 presents a mathematical model for tubular viral capsids using two generating vectors and three parameters. Section 4 presents a model for conical viral capsids that uses three generating vectors and four parameters. Further details on the (5,7)- and (4,8)-cones are examined. According to the Euler theorem, there are 3 other possible cone angles for a hexagonal lattice, but models for the (1,11)-,(2,10)-, and (3,9)-cones are not investigated, since they are rarely seen in nature without overlapping in a spiral fashion [13]. Section 5 compares modeling results to published experimental data on the HIV-1 conical capsid. Section 6 concludes the paper with some remarks.

Generating Vector and T-number for IcosahedralCaspar-Klug Viral Capsids classiﬁcation

This section briey reviews the concepts of the T-number and generating vectorof for icosahedral an icosahedral viral cap- capsids sid. This will be helpful for understanding the models for tubular and conical viral capsids to be discussed in the following sections. About half of the virus species are found to have an icosahedral capsid [17]. The geometric structure (sym- Assembled Icosahedral frame on a metry and periodicity) of icosahedralGenerating capsids can be well characterizedvector by the Caspar-Klug quasi-equivalence icosahedron theory [7]. hexagonal lattice (h=1,k=2) By the Euler theorem, for a convex polyhedron made of hexagons and pentagons, there are exactly 12 pen- (h=1,k=2), T=7 tagons. When these 12 pentagons are evenlyA~ = distributed,h~a1 + an icosahedronk~a2 forms,130 which canFarrah be circumscribed Sadre-Marandi et al. into a sphere.

Figure 2: Left: A lattice with (h, k)=(, ) and T = h + hk + k = . The dotted lines indicate where to cut the lattice to ~ ~ ~ Figure 1: Two basis vectors a, a and~a one=(1 generating, 0) vector A~afor a=(1 hexagonal/2, lattice.p3/2) In thisfold illustration, it. Right: Theh = icosahedral, k = and capsid obtained from folding the lattice shown in the left panel. PentagonsSoccer are shown ball in red. An hence T = h + hk+ k = . Note: the two1 hexagons where the starting2 and ending points ofexample the generating for T = vectoris the~A widelyreside studied will bacteriophage HK97 [18]. be replaced by two pentagons when the lattice is folded into an icosahedron. (h=1,k=1), T=3 T number: T = A~ 2 = h2 + k2 + hk

Unauthenticated Download12 Date pentamers | 12/13/14 3:07 PM (vertices of icosahedron) 10 (T-1) hexamers 4

Figure 3: A commonly adopted approach for illustrating icosahedral viral capsids, see also [21] Figure 1. Top: The centers of the pentagons form a coarse triangular mesh; Bottom Left: Two basis vectors and one generating vector with (h, k)=(, ) and T = h + hk + k = ; Bottom Right: The lattice folds into an icosahedron with the pentagon centers being the triangle vertices. Authors’ note: However, the folded pentagons and the flat triangles shown in (Bottom Right) need careful reading. Actually all pentagons are planar objects. The coarse triangles are used to locate the pentagons. The hexagons do not lie on the virtual triangles.

To understand the concepts of the generating vector and T-number, we start with a at hexagonal lattice consisting entirely of identical hexagons. As shown in Figures 1 and 2, we choose the center of one hexagon as the origin and set the lengths of the basis vectors ~a, ~a as . It is obvious that the angle between ~a and ~a is and hence their inner product (dot product) is ~a , ~a = . The generating vector, as shown in Figures h i

Unauthenticated Download Date | 12/13/14 3:07 PM However, biological solutions of the Thomson problem were invented by viruses ~2-3 billion years ago …..

Caspar-Klug classiﬁcation of icosahedral capsids Ordering on a sphere: ‘geometric frustration’ forces at least twelve However, biological solutions of the Thomson problem 5-fold coordinatedwere sites invented (= “disclinations”) by viruses ~2-3 billioninto the years ground ago ….. state…

Ordering on a sphere: ‘geometric frustration’ forces at least twelve 5-fold coordinated-Icosadeltahedral sites (= “disclinations”) solutions ofinto the the Thomson ground state… problem for intermediate particle numbers are exhibited by the capsid shells of virus-Icosadeltahedral structures solutions for of‘magic the Thomson numbers’ problem forof protein subunits indexedintermediate byparticle pairs numbers of integers are exhibited (P,Q) by the capsid shells of virus structures for ‘magic numbers’ of protein Duality betweensubunits hexagonal indexed by pairs of integers (P,Q) and triangular lattice Pentamers D. Caspar & A.D. CasparKlug, & A. Klug, correspond to Simian virus SV40 Cold Spring Harbor Symp. Simian virus SV40 Cold Spring Harboron Quant. Symp. Biology 27, 1 (1962) vertices with 5 (P, Q) = (1,2)on Quant. Biology 27, 1 (1962) neighbors: (P, Q) = (1,2) (,PQ )= (1,0) NparticlenumberPQ= =+++10(22 PQ ) 2 T = 1 (,PQ13 )= (3,1) 5-fold disclinations 22 (,PQ )= (1,0) NparticlenumberPQ= =+++10( PQ ) 2 (h, k)= (1, 0) (3, (,1)PQ )= (3,1) 5 Caspar-Klug classiﬁcation of icosahedral capsids

Bacteriophage Cowpea chlorotic Simian virus SV40 X 174 mottle virus

T =1 T =3 T =7 (h =1,k = 0) (h =1,k = 1) (h =1,k = 2) Capsids with Caspar-Klug numbers of type (h,0) and (h,h) have mirror symmetry, while other (h,k) structures are chiral! Why are viral capsids very often icosahedral?

6 Euler’s polyhedron formula V = number of vertices V E + F =2 E = number of edges F = number of faces tetrahedron octahedron

V =4 V =6 E =6 E = 12 F =4 F =8

cube icosahedron

V =8 V = 12 E = 12 E = 30 F =6 F = 20

7 Euler’s polyhedron formula V = number of vertices V E + F =2 E = number of edges F = number of faces For arbitrary polyhedron constructed with triangles the numbers of vertices, edges and faces are related: 3F each face has 3 edges E = 2 each edge is shared between 2 faces

Nz number of vertices with z neighbors

V = Nz total number of vertices z 1X E = zNz total number of edges 2 z X

V E + F =2 12 = Nz(6 z) z X 8 However, biological solutions of the Thomson problem were invented by viruses ~2-3 billion years ago …..

Ordering onEuler’s a sphere: ‘geometricpolyhedron frustration’ formula forces at least twelve 5-fold coordinated sites (= “disclinations”) into the ground state… 12 = N (6 z) z -Icosadeltahedralz solutions of the Thomson problem for intermediateX particle numbers are exhibited by the capsid shells of virus structures for ‘magic numbers’Caspar-Klug of protein subunits indexed by pairs of integers (P,Q) tetrahedron octahedron icosahedron (h=3, k=1), T=13

D. Caspar & A. Klug, Simian virus SV40 Cold Spring Harbor Symp. on Quant. Biology 27, 1 (1962) (P, Q) = (1,2)

22 (,PQ )= (1,0) NparticlenumberPQ= =+++10( PQ ) 2 (,PQ )= (3,1)

N3 =4 N4 =6 N5 = 12 N5 = 12 N = 10(T 1) 6 N6 = 120 3-fold 4-fold 5-fold 5-fold disclinations disclinations disclinations disclinations

9 RESEARCH LETTER

charge of a sphere, whereas a cylinder can be defect-free with a require- a contribution of 6(2p/6) regardless of the size of the loop. If sufficient ment of 0 total charge. curvature is enclosed, the angular stress generated by the disclination is Topology constrains the total charge, but it is energetics that deter- screened on the outside. Within the patch the screening is incomplete 1 mines the number and the arrangement of each charge. In elasticity and leads to an energetic cost. For later use we define V~ p=3 G dA, theory15, disclinations appear as discrete interacting charges and the the integrated curvature in units of disclinations. Ð Gaussian curvature as a charge density, both acting as sources of stress: The crystals we create consist of poly(methyl methacrylate) (PMMA) particles (,2 mm diameter) that are bound to an oil–glycerol 1 4 + x~Gx{ qad x{xa 1 interface and repel each other. By index-matching the oil to the Y ðÞ a ðÞ ðÞ glycerol, we can image the full surfaces (Fig. 1, Methods). The first X 1 1 surfaces we investigate are domes (truncated spheres), created by where G~ is the Gaussian curvature (with R1 and R2 the principal R1 R2 droplets sitting on a coverslip, with a circular contact line. The contact radii of curvature), q is the charge of the disclination at x , d represents a a angle, controlled by treating the glass surface, determines the solid a Dirac delta function, Y is Young’s modulus and x is the Airy stress angle of the spherical droplet. function—and we will refer to the partial or complete cancellation of The domes (Fig. 3) exhibit both disclinations and scars as previously the effects of curvature and stress by topological charges (r.h.s. of seen on spherical surfaces. However, for domes, the net disclination equation (1)) as screening. For curvature concentrated at a point, as charge on the surface can vary as the dome inflates from a disk to a full in our model paper disclinations of Fig. 2c, the screening can be per- sphere. In Fig. 3b we show the topological charge that is on the surface of 1 2 2 fect, the lattice is stress free and the energy E~ 2Y + x dA~0 the dome and detached from the boundary as a function of V.Wefind (where we have assumed no bending energy). For smooth surfaces, the the intuitive result that the detached charge varies approximately line- Ð ÀÁ screening is more subtle. Geometry provides some insight: consider a arly with V. That is, for a full sphere there are 12 pentagons (1p/3)s, for geodesic triangle drawn on a curved surface, for example, a sphere. a hemisphere there are 6 pentagons (1p/3)s and on smaller fractions Curvature causes lines to diverge or converge, affecting the angles at the two remain approximately proportional. Note that the topological the vertices: the sum of the external angles will differ from 2p by requirement of a total of 6 (1p/3)s is satisfied at all times by com- Dh 5 #GdA. The same applies to any closed loop formed by connecting pensating charges on the boundary. lattice sites and serves as a measure of angular strain. If instead, a Negative curvature surfaces lack the simplicity and familiarity that SUPPLEMENTARY INFORMATION RESEARCH disclination is encircled by the loop, by definition this adds/removes we associate with positive curvature surfaces such as the sphere. For Topological charge of disclinations a 5-fold disclination ef remove a wedge of angle ⇡ / 3 ,

z-fold disclination is associated total topological charge for with a topological charge closedb convex polyhedra ⇡ qz = (6 z) 4⇡ = Nzqz 3 z X Note: dipoles of opposite charges (e.g. 5-fold and 7- fold disclinations)WWW.NATURE.COM /NAproduceTURE | 3 dislocations defects. 10

dA 4⇡ = = N q R R z z 1 2 z I X

Excess angle

Triangles on Excess angle over the a sphere edge of a curved surface

↵ dA ↵ + + > ⇡ ✓ = = N q R R z z 1 2 z Z X

11 RESEARCH LETTER

charge of a sphere, whereas a cylinder can be defect-free with a require- a contribution of 6(2p/6) regardless of the size of the loop. If sufficient mentRESEARCH of 0 total charge.LETTER curvature is enclosed, the angular stress generated by the disclination is Topology constrains the total charge, but it is energetics that deter- screened on the outside. Within the patch the screening is incomplete 1 minescharge the of number a sphere, and whereas the arrangementa cylinder can be of defect-free each charge. with a In require- elasticitya contributionand leads of to6 an(2p energetic/6) regardless cost. of theFor size later of use the loop. we define If sufficientV~ p=3 G dA, theoryment15, disclinationsof 0 total charge. appear as discrete interacting charges and thecurvaturethe integrated is enclosed, the curvature angular stress in units generated of disclinations. by the disclination is GaussianTopology curvature constrains as a charge the total density, charge, both but it acting is energetics as sources that deter- of stress:screened on the outside. Within the patch the screening is incomplete Ð The crystals we create consist of poly(methyl1 methacrylate) mines the number and the arrangement of each charge. In elasticity and leads to an energetic cost. For later use we define V~ p=3 G dA, 15 (PMMA) particles (,2 mm diameter) that are bound to an oil–glycerol theory , disclinations1 4 appear as discrete interacting charges and the the integrated curvature in units of disclinations. + x~Gx{ qad x{xa 1 interface and repel each other. By index-matchingÐ the oil to the Gaussian curvatureY as a charge density, both acting as sources of stress: The crystals we create consist of poly(methyl methacrylate) ðÞ a ðÞ ðÞ glycerol, we can image the full surfaces (Fig. 1, Methods). The first X (PMMA) particles (,2 mm diameter) that are bound to an oil–glycerol 1 1 1 4 surfaces we investigate are domes (truncated spheres), created by where G~ is the Gaussian+ x~Gx curvature{ qad (withx{xRa 1 and R2 the principal1 interface and repel each other. By index-matching the oil to the R1 R2 Y ðÞ a ðÞ ðÞ glycerol,droplets we can sitting image on the a coverslip, full surfaces with (Fig. a 1, circular Methods). contact The line. first The contact radii of curvature), qa is the charge ofX the disclination at xa, d represents 1 1 surfacesangle, we controlled investigate are by domes treating (truncated the glass spheres), surface, created determines by the solid a Diracwhere deltaG~ function,is the Gaussian Y is Young’s curvature modulus (with R and1 andxRis2 the the principal Airy stress R1 R2 droplets sitting on a coverslip, with a circular contact line. The contact radii of curvature), q is the charge of the disclination at x , d represents angle of the spherical droplet. function—and we willa refer to the partial or completea cancellation ofangle, controlled by treating the glass surface, determines the solid a Dirac delta function, Y is Young’s modulus and x is the Airy stress The domes (Fig. 3) exhibit both disclinations and scars as previously the effects of curvature and stress by topological charges (r.h.s. ofangle of the spherical droplet. function—and we will refer to the partial or complete cancellation of seen on spherical surfaces. However, for domes, the net disclination equation (1)) as screening. For curvature concentrated at a point, as The domes (Fig. 3) exhibit both disclinations and scars as previously the effects of curvature and stress by topological charges (r.h.s. of charge on the surface can vary as the dome inflates from a disk to a full in our model paper disclinations of Fig. 2c, the screening can be per-seen on spherical surfaces. However, for domes, the net disclination equation (1)) as screening. For curvature concentrated at a point, as sphere. In Fig. 3b we show the topological charge that is on the surface of 1 2 2 charge on the surface can vary as the dome inflates from a disk to a full fect,in the our lattice model is paper stress disclinations free and the of Fig. energy 2c, theE~ screening2Y can+ x be per-dA~0 the dome and detached from the boundary as a function of V.Wefind 2 sphere. In Fig. 3b we show the topological charge that is on the surface of (wherefect, we the have lattice assumed is stress no free bending and the energy). energy E~ For1 smooth+2x surfaces,dA~0 the the intuitive result that the detached charge varies approximately line- 2Y Ð ÀÁ the dome and detached from the boundary as a function of V.Wefind screening(where is we more have assumed subtle. Geometry no bending provides energy). For some smooth insight: surfaces, consider the the a intuitive resultV that the detached charge varies approximately line- 1p Ð ÀÁ arly with . That is, for a full sphere there are 12 pentagons ( /3)s, for geodesicscreening triangle is more drawn subtle. on Geometry a curved provides surface, some for insight: example, consider a sphere. a arly witha hemisphereV. That is, for there a full are sphere 6 pentagons there are 12 ( pentagons1p/3)s and (1p on/3)s, smaller for fractions Curvaturegeodesic causes triangle lines drawn to diverge on a curved or converge, surface, for affecting example, the a sphere. angles ata hemispherethe two there remain are approximately 6 pentagons (1p proportional./3)s and on smaller Note fractions that the topological the vertices:Curvature the causes sum lines of to the diverge external or converge, angles affecting will differ the from angles 2 atp bythe tworequirement remain approximately of a total proportional. of 6 (1p/3)s Note is that satisfied the topological at all times by com- the vertices: the sum of the external angles will differ from 2p by requirement of a total of 6 (1p/3)s is satisfied at all times by com- Dh 5 GdA. The same applies to any closed loop formed by connecting Dh# 5 #GdA. The same applies to any closed loop formed by connecting pensatingpensating charges charges on the boundary. on the boundary. latticelattice sites sites and and serves serves as as a a measure measure of of angular angular strain. strain. If instead, If instead, a a NegativeNegative curvature curvature surfaces surfaces lack the simplicity lack the andsimplicity familiarity and that familiarity that disclination is encircled by the loop, by definition this adds/removes we associate with positive curvature surfaces such as the sphere. For Bucklingdisclination is encircledinstability by the loop,for bydisclinations definition this adds/removes we associate with positive curvature surfaces such as the sphere. For SUPPLEMENTARY INFORMATION RESEARCH a ef remove a wedge a strain ✏ qz ef ⇠ of angle ⇡ / 3 , stretching energy E A Y ✏2 s ⇠ ⇥ ⇥ E YR2q2 r s ⇠ z

0 Y = 2D Young’s modulus b b bending energy (for the corresponding cone)

R  E dA b ⇠ (r/q )2 z-fold disclination is Z z  associated with a Eb rdr 2 ⇠ (r/qz) topological charge c Z 2 WWW.NATURE.COM/NATURE | 3 Eb qz ln(R/a) ⇡ ⇠ d qz = (6 z) c 3 buckling favorable for R & Rb /Y ⇠ d 12 p

Figure 2 | Disclinations and pleats in a hexagonal lattice. a, Disclinations in a uncharged pairs of seven- and five-fold disclinations, can also be made by hexagonal lattice are topological defects that result from an extra (right panels) folding and gluing hexagonal paper. A set of three closely spaced, aligned or missing (left panels) 60u crystalline wedge that matches their topological dislocations (7-5,7-5,7-5) are shown on an approximately relaxed sheet. This is a charge of 1/2(2p/6) marked here by cream/brick circles, respectively. At the grain boundary which vanishes at the centre of the sheet—a ‘pleat’. Note that Figure‘core’ 2 | ofDisclinations the disclination and is a corresponding pleats in a hexagonal five-fold (or lattice. seven-fold) a, Disclinations coordinated innegative a uncharged curvature pairs emanates of seven- from the and vanishing five-fold point disclinations, of the pleat, as can evidenced also be made by hexagonalparticle. lattice In flat are space, topological disclinations defects produce that a large result amount from an of stress extra in (right a crystal, panels)both byfolding the buckling and gluing of the hexagonal sheet and by paper. the 30u Adivergence set of three of parallel closely lines spaced, aligned or missingdisrupting (left orientational panels) 60u order.crystallineb, This wedge stress can that be matches relieved by their buckling topological the impingingdislocations from the (7-5,7-5,7-5) top. e, A stress-free are shown pleat can on be an achieved approximately by allowing relaxed steps sheet. This is a chargecrystal of 1 to/2 create(2p/6) a curved marked surface. here Five-fold by cream/brick disclinations circles, are sources respectively. of positive At theout ofgrain the surface. boundary The pleat which retains vanishes the property at the that centre width of theis added sheet—a along ‘pleat’. the Note that ‘core’Gaussian of the disclination curvature, seven-fold is a corresponding disclinations five-fold of negative (or curvature. seven-fold)c,This coordinatedpleat lengthnegative in proportion curvature to emanates the linear from density the of vanishing dislocations. pointf, The of top the of pleat, the as evidenced coupling can be intuitively understood by making disclinations out of paper (see Chrysler building in New York consists of four vertical pleats on a cylinder. Here particle.Supplementary In flat space, Information disclinations for instructions produce a and large cut-outs). amount Paper of stress can be in bent a crystal,the pleatsboth are by formed the buckling from dislocations of the sheet in a square and by field. the Counting 30u divergence from the of top, parallel lines disruptingmuch more orientational easily than order. it can beb, stretched/compressed, This stress can be relieved so it bends, by resulting buckling in thea steps 3,impinging 4, 5, 6 are approximately from the top. equallye, A stress-free spaced at 8.4 pleat m and can form be a achieved cone with by no allowing steps crystalsurface to create free of a stress, curved with surface. all the Five-fold Gaussian curvature disclinations concentrated are sources at the of positiveGaussianout curvature. of the surface. A gradient The in pleat linear retains dislocation the property density is that achieved width by is added along the Gaussianlocations curvature, of the disclinations. seven-fold Neighbouring disclinations crystal of negative planes divergecurvature. on thesec,This spacingpleat the second length and in proportion first step at 9.4 to m the and linear 9.8 m density spacing, of resulting dislocations. in a spikef, The top of the couplingsurfaces, can matching be intuitively the geodesics understood of the by curved making surface. disclinationsd, Dislocations, out of paper (seewith negativeChrysler Gaussian building curvature in New crowning York consists the cone. of four vertical pleats on a cylinder. Here Supplementary Information for instructions and cut-outs). Paper can be bent the pleats are formed from dislocations in a square field. Counting from the top, much948 more | NATURE easily than | it VOL can be 468 stretched/compressed, | 16 DECEMBER 2010 so it bends, resulting in a steps 3, 4, 5, 6 are approximately equally spaced at 8.4 m and form a cone with no surface free of stress, with all the Gaussian curvature©2010 concentratedMacmillan Publishers at the Limited.Gaussian All rights curvature.reserved A gradient in linear dislocation density is achieved by locations of the disclinations. Neighbouring crystal planes diverge on these spacing the second and first step at 9.4 m and 9.8 m spacing, resulting in a spike surfaces, matching the geodesics of the curved surface. d, Dislocations, with negative Gaussian curvature crowning the cone.

III. NUMERICAL SIMULATION OF FLAT MEMBRANES v'3 k=p = (3.8) We consider a 2D triangular lattice of atoms with unit ' mean spacing. Any deformation maps the lattice points and calculate the 2D Young's Inodulus and Poisson ratio r, into new locations r,'. We deSne a stretching energy using (2.12), ' " ""='e r—'b — 2 l 1)', (3.1) F, , g-(lr. Ko — —e, o. =—. (3.9) &a, b& v'3 1016 H. S. SEUNG AND DAVID R. NELSON 38 where the sum is over distinct nearest-neighbor pairs of To construct membranes with defects, we start out atoms a and b. This can be reexpressed in terms of a with a hexagonal region of the lattice. The hexagon is all R ~ Rb,E we would expect hnear as expected from (5.7a). To corn stretching energy density U„ composed of six equilateral triangular wedges. If we ex- Mit h 11-H ad resu ltts we extract a slo Fdiscrete ] cise one wedge and attach the exposed we obtain a & edges, 1.159 2 ~ U (3.2) ' ' 1 R, wh'hic are most likel to positive disclination. The negativeF(R,disclinationa, KO)=P is con- 1 +F;„(~,KO), (6.11) a 2 276 R proximation to thee continuum limit, structed by cutting a slit and inserting a seventh wedge. where U, is the local sum over nearest neighbors b of The dislocation is a hexagon with a line of atoms running ' Kpb where the numerical constants foror positiveosi and negative — atom a, from a vertex to the origin removed, or equivalently, a Rb 127+10, (6.13) disclinations haveave been taken from (6.8). A positive and negative disclination ' ' separated by a single ' = —, ( — — proximation for F;„cancan be obtainedb from th e energy of a l 1) U, g r, rb l (3.3) lattice spacing. Sample configurations are depicted in where the errorr is estimated from the b Fig. 1. Hat disclination of size Rbb~ Th e agreement with Eq. (5.10c) is rath- To derive the continuum limit, imagine that some con- In Sec. II we minimized the stretching energy function-Es tinuous deformation map r~r' is given which matches al F,[u;~] to obtain the equations of plane elasticity b (6.12) Fromrom the assertion that the dislocatio ' our discrete map r, ~r, =r, +u, (r, ) at the lattice theory. We have shown that the continuum limit of our fi't td 1 of'in formation can be tracted by seal- '""" ' points. Let g; be the metric tensor induced by the defor- function F, of 2N coordinates SoF(where;„'isNjustis the~ timesnum- a constant of order uni s. sing (5.3) with A=r 1, e a ant mation ber of atoms) is precisely this functional.F/a hasHencebeenminimiz-plottedo e versus the rescaled varia e energy in the region R (r(R 2~ '""" should give the same results for large N. r' ing FS si ive isclinations, showin g; =8;r' B. . (3.4) ' F(R i,R2, &,,KO)p—0 =F(AR , AR 2, A,a.,KO). (6.14) There are two algorithms for minimizationmd d off a functiontio n onyo 1 offRR/R as gg R, —, which utilize information: gradient Then we can make the approximation gradient conjugate We apply this relation to inextension and variable metric. ' The gradient method is conjugate Two views of a bucklede diislocation are shown in Fi . H ' — 'b]'" superior for our purposes because it10.requiresWhenenlesst~iestorage.energies are plotted on I r. rb l =[g;,(r. )r.'br. g (4.16) th t tll ootaI energy must be given b Like most methods of multidimensional minimization, it ' ig. a), the slope of each cucurve is monotoni- = +2u, 'br, is performed as a series of one-dimensional minimiza-' I [5; j(r, )]r, b ] cally decreasin g.. Thiis means that the enernergy of a buckled F(a, ~,ir, Ko ) =a@oo (6.15) tions. Its special feature is that it dislocationconstructs adivergesseriesrgesofslows ower thanh any lo arith =[1+2u;j(r, )r,'b,'b] " where C „=limR /~, R) is a "noninterfering, or "conjugate" directions.th to Minimiza-of 'ol td „e(~, en o and other elastic =1+u;j(r, )r.br,'b, (3.5) rithmically, this result suffices to @GAL was defined in Eq. (4.16). The n at any finite tern eratur a, ' in the ouout side some r since the strain tensor is defined by the equation c ure tth at the energies as m pot gy d' region and the undeformed distance is uni- fo in o the total ener gy of g;J =5;j+2u;J, l r,b l s an s as R ~ Do, which i i a = a /R )~ by choosing A, =a /R in ty. As the index b ranges over the nearest neighbors of 1 ibl . Tll e buc kl' radii versus ir/K for y — mg atom a in Eq. (3.3), the unit ~ector r, b =r, rb steps over Bucklingtion are graphed instabilityin Fig.ig. 122. ThThe plot isforapproximately disclinations the vertices dp of a regular hexagon. This fact enables us F(R ~ a , Ko)=F(AR, ao, Air K )=F= Q, ao, KK to evaluate the stretching energy density p 5 l l I I I I I I I I I I I I II 6 ' 1014 H. S. SEUNG AND DAVID R. NELSON 38 U, = —, (u, dpjp) a g (6.16) P=1 R + Kp/K =2 Fdiscrete 2 ) 8 ~ij ~kl(5ij 5kl+5ik5jl+5il5jk b x (a,Ko/2=P) 1 4 —', = iT Kp/K=2( 1 — . = (2u,j+ukk ), (3.6) g n nii) . (6.1) &a,P) ' Ko/k= & where the strain tensor is understood to be evaluated at The sum is over nearest-neighbor pairs of normals n and r, . Thus the continuum limit of (3.2) becomes E n&, and the normals have been assigned with a consistent 1 '"""= U = — U( —) choice of orientation. To derive the continuum limit, we FS 2'g 0 Jd  a suppose that the shape of the membrane is parametrized (b) by the map x(tr;) (i =1,2). The coordinate frame for the E d r(2u, . ) . (3.7) (c) 8 j+ukk 2 2 Es YR qz g;, =e;..e,e , andan the second fundamental form by 2 (b) ' Eb q ln(R/a) Comparing with (2.6), we can read off the 2D Lame FIG. l.⇠(a) Flat dislocation. (b) Flat positiven disclinationIn the continuum limit the difference z 0, =, 8 FIG. 5. (a) Buckled positive disclination (Ko/K=2000). (b) coefficients (s =2~/6). (c) Flat negative disclination (s = —2m/6). ⇠ n~n —npn& sshouldOU become the gradient of the normal vector Buckled negative disclination (Ko /K =2000). field, Q Fb 2x——dS g'jt};nQt} n. . (6 2) p f I I I I I I I I i I I I I II io-' ioo ioI for negative disclinations. The positive disclination result Substituting t);n=Q,"ek allows us to express the bending agrees with (4.24} very well, and the negative disclination energy in terms of R/Rb 0;, result is only about 10%%uo less than the upper bound (4.26). = 'ir "g In Fig. 7 are graphed the energies of buckled disclina- Fb , f dS—g Q;kQ, i . (6.3) FIG. 9.. PPositive disclinationn enerenerg&esies vs the rescaled variable FIG.tions 10.versusTwo Rviewsfor variousof a buckucvaluesed dislocationof Ko/K. withEach curve Using Rthe/Rb.identity e'e&kR—b5j5'k —5'k5j,/Yor, equivalently, Eobranches/Fc = 100. off from the flat disclination curve at the critical ⇠ buckling radius Rb. The smooth way these curves join (6.4) p H.S.onto Seungthe andflat disclinationD.R. Nelson,curve suggests a weakly first- we rewrite the integrand of (6.3) as order or possibly continuous buckling transition. Figure 13 PRA8 shows 38,that 1005the (1988)radii of buckling are linear in Qir/Kos g QikQji (g Qik } +e ernng Qjig Qik [as predicted in (5.7b)], with slopes =(Q') +e"E „Qi Q;"=H 2K, — (6.5} where H =trtQkI is the mean curvature and

I I I I I I I K =det t Qk ) is the Gaussian curvature. Our final expres- sion for the continuum limit is + positive disclination — Fb=2K dS H 2E (6.6) ~ negative disclination OO This is nothing but the Helfrich energy (4.2) with bending O —— O rigidity K and Gaussian rigidity KG — K. The relation- a ship between K and the microscopic parameter K is most O O easily derived by bending a triangular lattice into the O shape of a cylinder. The sum in (6.1) can then be calculated by hand easily, and compared with the integral in F (6.6). This procedure yields

v'3 ++ K= K (6.7) ++ 2 ++ + + Buckled positive and negative disclinations are depict- + ed in Fig. 5. A check with the inextensional calculations of Sec. IV results from simulating a positive disclination with Ko/ir so large that Rb is much less than a lattice spacing. Figure 6 is a semilogarithmic plot of F/K versus R for buckled positive and negative disclinations with = 2000 EojK=2000. The energies behave like R o I I I I I I I F =1.159Kln (6.8a) 1 5 10 a

for positive disclinations and r R FIG. 6. Semilogarithmic plot of energy (in units of K) vs R Fb =2.276K 1n (6.8b) for buckled disclinations. Here Ko/k=2000, which places these a systems in the inextensional limit. Buckling instability for spherical VIRUS SHAPES ANDshells BUCKLING with TRANSITIONS 12 5-fold IN . . . disclinations PHYSICAL REVIEW E 68, 051910 ͑2003͒ h = k =2 h = k =4 2 VIRUS SHAPES AND BUCKLING TRANSITIONS IN . . . 45 PHYSICAL REVIEW E 68176, 051910 ͑2003͒

/R ⇡ ⇡ ↵

2 VIRUS SHAPES AND BUCKLING TRANSITIONS IN . . . PHYSICAL REVIEW E 68, 051910 ͑2003͒ R ⌦

2 Basic structure of viral capsids h = k =6 h = k =8 393 694 ⇡ ⇡

asphericity FIG. 4. Mean-square asphericity as a function of YR2/␬ for many different icosahedral shells. The inset shows a ͑6,6͒ structure with Foppl-von Ka´rma´n number ␥ϭYR2/␬Ϸ400. The arrow marks the location of the buckling transition in ﬂat space.

2 theFIG. resistance 4. Mean-square of the2 capsid asphericity to mechanical as a function deformation of YR /␬ canfor FIG. 5. Numerically calculated shapes with (h,k) indices ͑2,2͒, degrade.many different= AsYR we icosahedral shall/ see, shells. a theory The inset of shows buckled a ͑6,6 crystalline͒ structure ͑4,4͒, ͑6,6͒, and ͑8,8͒ for fixed ˜␬ϭ0.25 and fixed spring constant with Foppl-von Ka´rmań number ␥ϭYR2/␬Ϸ400. The arrow marks order on spheres also allows estimates of important macro- ⑀ϭ1. The Foppl-von Ka´rmań numbers for these shapes are ␥Ϸ45, the location of the buckling transition in flat space. 2 FIG. 4. Mean-squarescopic asphericity elastic parameters as a of function the capsid shell of fromYR structural/␬ for 176, 393, and 694. data on the shape anisotropy. Estimates of quantities such as FIG. 5. Numerically calculated shapes with (h,k) indices ͑2,2͒, Simianmany different virus icosahedral the shells. resistance The of the inset capsidshows to mechanical a ͑6,6 deformation͒ structure can with ᐉϭ0,6,10,12,16,18,... ͓23͔. Although any parameter the bending rigidityParamecium and Young’s modulus of a empty viral Note: viral˜ ϭcapsids may degrade. As we shall2 see, a theory of buckled crystalline ͑4,4͒, ͑6,6͒, and ͑8,8ˆ ͒ forˆ fixedˆ ␬ 0.25 and fixed spring constant ´ ´ shell might allow an understanding of deformations due to set of the form ͕Q6 ,Q10 ,Q12 ,...͖ could be consistent with with Foppl-von Karman numberorder on spheres␥ϭYR also allows/␬Ϸ estimates400. The of important arrow macro- marks ⑀ϭ1. The Foppl-von Ka´rmań numbers for these shapes are ␥Ϸ45, SV40 loading with DNABursana or RNA ͓4͔. Although some aspects of an icosahedral symmetry, all buckled objects describable by scopic elastic parameters of the capsid shell from structural 176, 393, and 694. have non-zero the location of the bucklingvirus transitionstructure may be in accounted flat space. for by the physics of shell the theory of elastic shells in fact lie on a universal curve data on the shape anisotropy. Estimates of quantities such as theory, weChlorella should emphasize Virus that other features could be parametrizedspontaneous by the value of YR2 /␬curvature!. Deviations from this R 25nm the bending rigidity and Young’s modulus of a empty viral with ᐉϭ0,6,10,12,16,18,... ͓23͔. Although any parameter driven by the need for cell recognition, avoidance of immune curve would presumably describe biological features such as shell might allow an understanding of deformations due to set of the form ͕Qˆ ,Qˆ ,Qˆ ,...͖ could be consistent with ⇡ response, etc. the protrusionsFIG. of 5.6 the10 Numerically adenovirus12 in Fig. 2.calculated shapes with (h,k) indices ͑2,2͒, the resistance of the capsidloading with to DNA mechanical or RNA ͓4͔. Although deformation some aspects can of an icosahedral symmetry, all buckled objects describable by A summary ofR our investigations93nm of buckling14 transitions in In Sec. II, we describe our theoretical results for disclina- virus structure may be accounted for by the physics of shellJ. Lidmarthe theory͑4,4 et͒ of ,al., elastic͑6,6 PRE͒ shells, and 68, in͑ fact8,8 051910 lie͒ onfor a universal fixed (2003)˜␬ curveϭ0.25 and fixed spring constant degrade. As we shalla see, spherical a geometry theory͑discussed of buckled in detail in Sec. crystalline II͒ is shown tion buckling in the icosadeltahedral2 spherical shells pro- theory, we shouldFigure 1: emphasizeExamples⇡ of capsids that with icosahedral other symmetry features [1]. The could images were be created withparametrized the by the value of YR /␬. Deviations from this in Fig. 4. As illustratedmolecular ingraphics Fig. 4 program͑see UCSF also Chimera Fig. using 5͒ dataicosahedral from cryo-electron microscopyposedϭ by Caspar and Klug as models´ of viruses´ ͓7͔. The driven by the need forand cellX-ray di recognition,↵raction. avoidance of immune curve⑀ would1. presumably The Foppl-von describe biological Karma featuresn numbers such as for these shapes are ␥Ϸ45, order on spheres also allows estimates of important macro- energy, mean-square aspherity, and spherical harmonic con- shellsresponse, do etc. indeed become aspherical as the ‘‘Foppl-von the protrusions of the adenovirus in Fig. 2. ´ ´ 2 tent of176, these 393,shells are and determined 694. as a function of the Foppl- scopic elastic parametersKarmaA summary ofn number’’ the of capsid our␥ϭ investigationsYR /␬ shellincreases of from buckling from values structuraltransitions of order in In Sec. II, we describe our theoretical results for disclina- 2 von Ka´rmań number, discussed above, unitya spherical to YR geometry/␬ӷ1. The͑discussed mean square in detail ‘‘asphericity’’ in Sec. II͒ is͑ showndevia- tion buckling in the icosadeltahedral spherical shells pro- data on the shape anisotropy. Estimates of quantities4 such as tionin Fig. from 4. As a illustrated perfect spherical in Fig. 4 shape͑see also͒ departs Fig. 5͒ significantlyicosahedral posed by Caspar and Klug as models of viruses ͓7͔. The 2 ϭ ␥ϭYR2/␬. ͑7͒ the bending rigidity andfromshells Young’s zero do whenindeedYR becomemodulus/␬ exceeds aspherical 154,of aas the empty the location ‘‘Foppl-von of viral the energy,with mean-squareᐉ 0,6,10,12,16,18 aspherity, and spherical,... harmonic͓ con-23͔. Although any parameter bucklingKa´rmań number’’ transition␥ inϭ flatYR2 space/␬ increases͓13͔. Fits from of buckled values of viruses order tent of these shells are determined as a function of the Foppl- ˆ ´ ˆ ´ ˆ shell might allow an understandingorunity crystalline to YR2/ vesicles␬ӷ1. The to of this mean deformationsuniversal square curve‘‘asphericity’’ would allowdue͑devia- an to vonMost Kaset viruses´rmań of number, have the either formdiscussed Foppl-von͕ above,Q6 Ka,Qrma10n number,Q12 ␥,...Շ150 ͖ could be consistent with experimentaltion from a determination perfect spherical of the shape ratio͒ departsY/␬. More significantly quantita- ͑implying a close to spherical shape͒ or 200Շ␥Շ1500 ͑no- loading with DNA or RNA ͓4͔. Although some aspects of an icosahedral symmetry,2 ´ ´ all buckled objects describable by tivefrom information zero when onYR the2/␬ buckledexceeds shape 154, canthe location be obtained of the by ticeably buckled͒. Higher␥ϭ vonYR Ka/␬.rman numbers describing͑7͒ virus structure may beexpandingbuckling accounted transition the radius for inR flat(␪ by, space␾) in the͓13 spherical͔. physics Fits of harmonics, buckled of viruses shell objectsthe with theory very sharp of corners elastic cannot shells be obtained in for fact vi- lie on a universal curve or crystalline vesicles to this universal curve would allow an Mostruses viruses with RՇ have0.2 either␮m composed Foppl-von of Ka´ finitermań size number proteins.␥Շ150 Of 2 theory, we should emphasizeexperimental determination that otherϱ ofᐉ the features ratio Y/␬. More could quantita- be ͑course,implyingparametrized very a close high to von spherical Ka´rma by´ shapen the numbers͒ or value 200areՇ␥Շpossible1500 of ͑YRno- for /␬. Deviations from this spherical vesicles with crystalline order composed of much driven by the need for celltive information recognition,R͑␪,␾ on͒ϭ the͚ buckled avoidance͚ Q shapeᐉmY ᐉm can͑␪,␾ beof͒, obtained immune͑ by5͒ ticeablycurve buckled would͒. Higher von presumably Ka´rmań numbers describe describing biological features such as expanding the radius Rᐉϭ(␪0 ,m␾ϭϪ) inᐉ spherical harmonics, objectssmaller with lipid very molecules sharp͓ corners19,20͔. cannot be obtained for vi- response, etc. rusesWethe with haveR protrusionsՇ studied0.2 ␮m the composed scaling of of of the finitethe curvature adenovirus size proteins.C at Of the in Fig. 2. and studying the rotationallyϱ ᐉ invariant quadratic invariants course,creases formedvery high after von the Ka shells´rmań buckle numbers for largeare possible␥. We even- for A summary of our investigationsallowed for viruses or of vesicles buckling with icosahedral transitions symmetry, in sphericaltually recover vesiclesInSec. the with scaling II, crystalline proposed we describe order and composed studied our in of other much theoretical ge- results for disclina- R͑␪,␾͒ϭ ͚ ͚ QᐉmY ᐉm͑␪,␾͒, ͑5͒ namely, ᐉϭ0 mϭϪᐉ smallerometries lipid by molecules Lobkovsky,͓19,20 Witten,͔. and collaborators ͓16,17͔, a spherical geometry ͑discussed in detail in Sec. II͒ is shown tion buckling in7 the icosadeltahedral spherical shells pro- butWe only have for studiedvery large the␥, scaling␥у10 of, appropriate the curvature forC theat buck- the ᐉ creasesled icosahedralposed formed after vesicles by the Caspar shellsdescribed buckle in and Ref. for large͓20 Klug͔.␥. We as even- models of viruses ͓7͔. The in Fig. 4. As illustratedand in studying Fig. 4 the͑see rotationally also1 invariant Fig. 5 quadratic͒ icosahedral invariants ˆ 2 tuallyIn Sec. recover III, wethe discuss scaling briefly proposed the and relevance studied of in our other work ge- to allowed for virusesQᐉϭͱ or vesicles͚ with͉Q icosahedralᐉm͉ /Q00 symmetry,͑6͒ energy, mean-square aspherity, and spherical harmonic con- shells do indeed becomenamely, aspherical2ᐉϩ1 m asϭϪᐉ the ‘‘Foppl-von ometriesicosahedral by viruses Lobkovsky, from the Witten, library and of collaborators those whose structures͓16,17͔, Ka´rmań number’’ ␥ϭYR2/␬ increases from values of order but onlytent for ofvery these large ␥, ␥ shellsу107, appropriate are determined for the buck- as a function of the Foppl- ᐉ 051910-3led icosahedral vesicles described in Ref. ͓20͔. 2 1 unity to YR /␬ӷ1. The meanˆ square ‘‘asphericity’’2 ͑devia- Invon Sec. III, Ka we´ discussrmań briefly number, the relevance discussed of our work to above, Qᐉϭͱ ͚ ͉Qᐉm͉ /Q00 ͑6͒ tion from a perfect spherical shape2ᐉϩ͒ 1departsmϭϪᐉ significantly icosahedral viruses from the library of those whose structures 2 2 from zero when YR /␬ exceeds 154, the location of the051910-3 ␥ϭYR /␬. ͑7͒ buckling transition in flat space ͓13͔. Fits of buckled viruses or crystalline vesicles to this universal curve would allow an Most viruses have either Foppl-von Ka´rmań number ␥Շ150 experimental determination of the ratio Y/␬. More quantita- ͑implying a close to spherical shape͒ or 200Շ␥Շ1500 ͑no- tive information on the buckled shape can be obtained by ticeably buckled͒. Higher von Ka´rmań numbers describing expanding the radius R(␪,␾) in spherical harmonics, objects with very sharp corners cannot be obtained for viruses with RՇ0.2 ␮m composed of finite size proteins. Of ϱ ᐉ course, very high von Ka´rmań numbers are possible for spherical vesicles with crystalline order composed of much R͑␪,␾͒ϭ ͚ ͚ QᐉmY ᐉm͑␪,␾͒, ͑5͒ ᐉϭ0 mϭϪᐉ smaller lipid molecules ͓19,20͔. We have studied the scaling of the curvature C at the and studying the rotationally invariant quadratic invariants creases formed after the shells buckle for large ␥. We even- allowed for viruses or vesicles with icosahedral symmetry, tually recover the scaling proposed and studied in other ge- namely, ometries by Lobkovsky, Witten, and collaborators ͓16,17͔, but only for very large ␥, ␥у107, appropriate for the buck- ᐉ led icosahedral vesicles described in Ref. ͓20͔. 1 ˆ 2 In Sec. III, we discuss briefly the relevance of our work to Qᐉϭͱ ͚ ͉Qᐉm͉ /Q00 ͑6͒ 2ᐉϩ1 mϭϪᐉ icosahedral viruses from the library of those whose structures

051910-3 Icosahedral viral capsids

Topology requires certain number of disclinations

12 = N (6 z) z z X 5-fold disclinations have lower energy then 4- fold and 3-fold disclinations. 7-fold and 8-fold disclinations would have to be compensated by additional 5-fold disclinations.

12 5-fold disclinations want to be as far away as possible, which produces structures with icosahedral symmetry.

15 Bacteriophage T4 infecting bacteria

After attachment to head Before attachment bacteria the virus tail to bacteria the contracts and the hard inner core tube pierces virus tail is in the tail extended state. through bacteria cell wall. Then viral DNA enters the core tube 50nm cell through the tube. 16 Contraction of Bacteriophage T4 tail

Contraction of the head virus tail is achieved by movement of dislocations (5-fold + tail 7-fold disclination) core 50nm through the tail. tube

movement of dislocations modiﬁes the crystal orientation on tail sheet