Landau Theory and the Emergence of Chirality in Viral Capsids
Sanjay Dharmavaram, Fangming Xie, William Klug, Joseph Rudnick, Robijn Bruinsma
LLandau Theory and the Emergence of Chirality in Viral Capsids
Sanjay Dharmavaram , Fangming Xie , William Klug , Joseph Rudnick , Robijn Bruinsma , ∗ Department of Physics and Astronomy, Department of Mechanical and Aerospace Engineering, Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095, USA and Department of Physics, University of Science and Technology of China, Hefei, Anhui, China
We present a generalized Landau-Brazovskii theory for the solidification of chiral molecules on aspherical surface. With increasing sphere radius one encounters first intervals where robust achiraldensity modulations appear with icosahedral symmetry via first-order transitions. Next, one en-counters intervals where fragile but stable icosahedral structures still can be constructed but only by superposition of multiple irreducible representations. Chiral icoshedral structures appear viacontinuous or very weakly first-order transitions. Outside these parameter intervals, icosahedralsymmetry is broken along a three-fold axis or a five-fold axis. The predictions of the theory arecompared with recent numerical simulations.
Landau theory is a central method for the analysis ofphase transitions and spontaneous symmetry breaking.When complex fluids solidify, the emergent density mod-ulations often adopt a variety of contending symmetries.Landau theory has been a powerful tool for the analysisof the phase diagrams [1]. In this letter, the Landau the-ory of the solidification of chiral molecules confined to aspherical surface is applied to the solidification of viralcapsids [2, 3]. Capsids are spherical protein shells thatsurround viral genome molecules. Spherical viral capsidsusually – though not always – have icosahedral symme-try [4]. On the other hand, numerical studies of systemsof interacting particles confined to spherical surfaces findthat icosahedral symmetry must contend with a varietyof other symmetries [5],[6–9].According to Landau theory, the density modulation ofthe ordered phase on a spherical surface should transformunder the symmetry group G of the isotropic fluid phaseaccording to a single irreducible representation (“irrep”)of G . The symmetry group G of the ordered phase mustbe an isotropy subgroup of G . Because the capsid pro-teins, or “subunits”, are chiral molecules (formed fromthe L-isomers of amino-acids) G should be the chiralgroup SO(3) of rotations without inversion. The densitymodulation of the ordered phase must then be expand-able in the basis set of the 2 l + 1 spherical harmonics Y ml for a particular l value, itself determined by minimizationof the Landau free energy. As shown in refs. [10, 11], ifone demands that the ordered phase has chiral icosahe-dral symmetry, so with G the chiral icosahedral group I , then the expansion coefficients can be obtained fromgroup theory. This leads to the icosahedral spherical har-monics Y h ( l )[12]. Irreps of SO(3) with icosahedral sym-metry can be constructed from even spherical harmonicsfor l = 6 j + 10 k with j, k ∈ { , , , · · · } and from oddspherical harmonics for l = 15 + 6 j + 10 k [13]. If I isthe isotropy subgroup then only odd values of l are al-lowed. This means that the smallest icosahedral capsids[4] should have surface density modulations proportionalto Y h ( l = 15) [10, 11] (maxima and minima of the density modulation correspond here to the locations of subunits,whose density is odd under inversion).The cubic non-linear term in the free energy densityintegrates to zero for odd l , which means that the solidifi-cation transition should be a continuous transition . Thisis surprising since solidification normally is a first-ordertransition [1]. This suppression of the first-order acti-vation barrier by chirality would have important conse-quences for viral assembly. Models for protein-by-proteincapsid assembly under equilibrium conditions encounteractivation energy barriers that can be very large com-pared to the thermal energy [14]. If the solidificationtransition for chiral molecules on a spherical surface in-deed has no activation barrier, then this could be an im-portant argument in favor of collective descriptions ofcapsid assembly [2, 3, 15] as opposed to the protein-by-protein assembly models.In this letter we re-examine the Landau theory for so-lidification on a spherical surface of molecules with chiralinteractions [10, 11] now without requiring the densitymodulation to have icosahedral symmetry and withoutrequiring it to be odd under inversion. Following otherLandau theories of chiral complex liquids [16, 17], chi-rality is introduced here by including the lowest-ordernon-zero pseudo-scalar invariant of the order parameterin the free energy. The Landau-Brazovskii (LB) free en-ergy density [1, 18] for solidification generalized to chiralsystems is then F = (cid:90) S (cid:20) (cid:18) (cid:16) (∆ + k o ) ρ (cid:17) + r ρ + u ρ + v ρ + .... (cid:19) + χ (cid:18) ∇ ρ · ( ∇∇ ρ ) · ( n × ∇ ρ ) (cid:19)(cid:21) dS. (1)The integral is over a spherical surface S . The first termis the standard LB free energy density. ∆ is the Laplace-Beltrami operator on a spherical surface and k the ratioof the radius of the sphere and the characteristic lengthscale of the density modulation. Lengths are measured a r X i v : . [ phy s i c s . b i o - ph ] J un in units of the sphere radius. The second part of the firstterm is an expansion of the free energy density in powersof the scalar invariants of the density modulation ρ with r , u , and v the lowest-order expansion coefficients ( u isnegative and v is positive). The second term of Eq.1 isthe lowest-order pseudo-scalar that can be constructedfrom ρ whose integral over a closed surface is non-zero[19]. Here, n is a unit vector normal to the sphere surfaceand χ is a parameter that measures the strength of chiralsymmetry breaking [20].Since the spherical harmonics Y ml are eigenfunctionsof the ∆ operator with eigenvalue − l ( l + 1), one can di-vide up the k axis into segments l < k < ( l + 1) so that in each segment the quadratic term in the freeenergy for that particular value of l first changes sign asone decreases the control parameter r [21]. The expan-sion coefficients of the density in the basis of Y ml are ob-tained by minimizing the full free energy, which results in2 l + 1 coupled non-linear equations (cid:104) Y ml | ∂F ( ρ ) ∂ρ (cid:105) = 0 [22].For a particular solution of these equations to be stable,the eigenvalues of the Hessian matrix of second deriva-tives for that solution must be positive, apart from threeeigenvalues that are zero associated with global rotation[23]. The first segment along the k axis that supportsa non-trivial icosahedral subgroup is the one for l = 6.The 13 coupled equations (cid:104) Y m | ∂F ( ρ ) ∂ρ (cid:105) = 0 are solved by ρ = ξ Y h (6). Here Y h (6) = Y + (cid:114)
711 ( Y − Y − ) . (2)is the first non-trivial icosahedral spherical harmonic.The order parameter amplitude ξ obeys the cubic equa-tion rξ + ˜ uξ + ˜ wξ = 0 , (3)with ˜ u = u (50 √ / √ π ) and ˜ w = 2145 w/ (1564 π ).Equation 3 has the standard form of a first-order phasetransition. The relevant eigenvalues of the 13 ×
13 Hessianmatrix of the ρ solution branch are positive when theuniform state is unstable [24]. The coefficient χ does notenter in Eq. 3 because the pseudoscalar invariant is zeroin any of the 2 l + 1 dimensional subspaces [25]. Similarfirst-order solidification transitions to a stable icosahe-dral state take place in the subsequent l = 10 and l = 12segments. If one identifies maxima of the density modula-tion with “capsomers” (groups of 5 or 6 subunits) ratherthen subunits themselves [7] then l = 6 corresponds to a T = 1 capsid and l = 10 and l = 12 to T = 3, respec-tively, T = 4 capsids [26].The next icosahedral intervals are for l =15, 16, 20 and22. The icosahedral harmonics remain solutions of theprojection equations (cid:104) Y ml | ∂F ( ρ ) ∂ρ (cid:105) = 0 but their Hessiannow has negative eigenvalues [27]. Stable icosahedralstates that involve these icosahedral harmonics still can be constructed but only by going beyond conventionalLandau theory and include multiple irreps of SO(3). Asan example, in the enlarged product space of the l = 15and l = 16 subspaces (denoted by “15 × ρ ( ζ, ξ ) = ζ Y h (15) + ξ Y h (16) , (4)The free energy has an extremum when the two-component order-parameter ( ζ, ξ ) obeys a pair of coupledcubic equations [28]:( c + r ) ξ + u ξ + u ζ + v ξ + v ξζ − χ (3 a ξ ζ + a ζ ) = 0 , (5a)( c + r ) ζ + u ξζ + v ζ + v ζξ − χ ( a ξ + 3 a ζ ξ ) = 0 , (5b)Numerical solution at the midpoint k = 16 betweenthe l = 15 and l = 16 stability intervals shows that inthat case the two components ( ζ, ξ ) of the resulting or-der parameter are nearly proportional to each other andcomparable in magnitude [29]. The chiral term in thefree energy does not vanish in the enlarged 15 ×
16 sub-space and now enters in these equations. For χ = 0,there are two degenerate solutions, which we denote byD and L, that transform into each other under inver-sion (with ζ → − ζ ). For χ (cid:54) = 0 but small, the freeenergy of the D and L branches shifts by an amount∆ F c = χ (cid:0) a ξ ζ + a ξ ζ (cid:1) where ( ζ , ξ ) is the χ = 0solution [30]. Since ∆ F c is odd in ζ the degeneracy be-tween the D and L branches under inversion is lifted byan amount proportional to χ . In the capsomer interpre-tation, the 15 ×
16 superposition state corresponds to the– intrinsically chiral – T=7 shell while it correspond inthe subunit interpretation to the T=1 shell. In the lat-ter case, the individual subunits are now neither odd noreven under inversion as is in fact the case for actual chiralmolecules [31].The stability of the superposition state is a sensitivefunction of the dimensionless radius k . Increasing the k parameter from the stability center of the l = 15 segmentat k (cid:39) . l = 16segment at k (cid:39) .
5, one finds that for decreasing r the uniform state initially does not transform directlyinto an icosahedral state. Instead, a stable superposi-tion state appear only for values of r that are well be-low the stability limit of the uniform phase, as shownin Fig.1 A. Starting in this stable icosahedral state, andnow increasing r , groups of three-fold degenerate eigen-values λ ∝ ( r − r ∗ ( k )) change sign at a first threshold r = r ∗ ( k ) (blue dot). Three corresponding eigenvectorsof the Hessian with 2-fold, 3-fold, and 5-fold axial sym-metry are shown in Fig.2. For r close to the stabilitylimit of the uniform phase, one encounters a second in-stability point along the now unstable 15 ×
16 branch (reddot), where groups of four degenerate eigenvalues change - - - - -
500 0 r51015 || ρ || - - - - - -
140 r2.53.03.5 || ρ || - - - - - -
200 r246810 || ρ ||- - - - -
256 r0.20.40.60.81.01.2 || ρ || - - - - - || ρ || A BC D l = FIG. 1. Stability diagrams of the 15 ×
16 chiral icosahedralstates for different values of the dimensionless radius k . Hor-izontal axis: control parameter r . Vertical axis: order param-eter amplitude || ρ || = ( ξ + ζ ) / . Solid blue: stable section.Dashed red: unstable section. (A): k (cid:39) . k (cid:39) .
9, (C): k = 16, (D): k (cid:39) .
5. The unstable section is now mostlya pure l = 16 state.FIG. 2. (top) Three unstable eigenvectors of the Hessian ofthe 15 ×
16 chiral icosahedral states with, respectively, C , C ,and C rotational symmetry (left to right) . (bottom) Com-peting groundstates of 72 one-state Lennard-Jones particleson a sphere with (left to right) D , D and D symmetry.Blue spheres: 5-fold coordinated particles. Red spheres: 6-fold coordinated particles. From ref.[9] with permission. sign. The corresponding eigenvectors have tetrahedralor D symmetry (see Fig.3). As k increases towardsthe boundary point k = 16, both instability points slidedown the stability curve and disappear around k = 15 . k values, between 15.90 to 15.96a stable 15 ×
16 branch appears directly from the uniformphase. Remarkably, within our numerical precision thetransition indeed is continuous as shown in Fig.1 (B).For k slightly larger than 15.96. There still is a di-rect transition into the 15 ×
16 branch but the transitionis now weakly first order (see Fig.1 C). As k increases FIG. 3. (top) Unstable eigenvectors with tetrahedral and D symmetry. (bottom) Tetrahedral and D groundstates of 72Lennard-Jones particles on a sphere. The two and three foldaxes are perpendicular for D . From ref.[9] with permission. towards the center of the l = 16 stability segment at k (cid:39) .
5, the icosahedral state again becomes progres-sively less stable as shown in Fig.1 D [32].The transition from the icosahedral state to one withlower symmetry can be analyzed by classical Landau the-ory. The symmetry group G is now the chiral icosahe-dral group I , which has one one-dimensional (“1D”) ir-rep, two 3D irreps, a 4D and a 5D irrep [33]. The firstinstability in Fig.1 (A) is associated with one of the two3D irreps, denoted by 3, while the second is associatedwith the 4D irrep. The isotropy subgroups of I associ-ated with the 3D irrep 3 are C , C , and C . A naturalorthogonal coordinate system in the 3D space of irrep 3is one with the (1 , ,
1) direction along a three-fold sym-metry direction, the (0 , ,
1) direction along a two-folddirection [34]. The order parameter is then a 3D vector (cid:126)η = ( η , η , η ) in this space. Using symmetry arguments[35] the Landau energy δF ( (cid:126)η ) can be shown to have theform: δF ( (cid:126)η ) ∝ λ | η | + V | η | + W | η | + ∆ √ η η η + ∆ √
560 ( η + η + η ) + ∆4 (cid:0) η ( η − η ) + cyclic perm. (cid:1) (6)to sixth order in η . Here, λ is the eigenvalue thatchanges sign at the transition while V and W are posi-tive constants [36]. A continuous transition takes placeat λ = 0. For ∆ = 0, the Landau energy is isotropicwith arbitrary rotations in (cid:126)η space connecting degener-ate states. For ∆ negative, the minimum of δF ( (cid:126)η ) liesalong the C direction and for ∆ positive along the C direction. For negative λ , the system thus has eithera three-fold or a five-fold axis. The second instabilityin Fig.1 (top left) is associated with the 4D irrep, whichhas T and D as its isotropy subgroups. The correspond-ing eigenvectors are shown in Fig.3. The δF ( η ) for the4D irrep contains cubic invariants in η . This means thaticosahedral symmetry breaking in this case is discontinu-ous. Finally, an instability driven by the 5D irrep wouldhave D , D and D as its isotropy subgroups.Recently, numerical simulations of the groundstate of72 “capsomer particles” on a spherical surface [7, 9] –corresponding to a T=7 shell in the capsomer inter-preation – that interact via Lennard-Jones (LJ) interac-tion. Depending on parameters, they found an icosahe-dral groundstate as well as competing groundstates with D , D , D , and T symmetry (see Figs.2 and 3). Theydid not encounter however the C and C states. Thissuggests that for a LJ particle system, icosahedral sym-metry is broken by the 4D and 5D irreps rather than the3D irrep of LB theory.What are the implications of these results for viral as-sembly? In the capsomer picture, a T = 1 capsid ap-pears via a robust first-order solidification transition andis quite stable. The capsid is chiral only in so far as cap-somer particles are chiral. In the subunit picture, the T = 1 capsid appears as a mixed 15 ×
16 state that ismuch less stable. It requires careful fine-tuning of the di-mensionless radius k . The ordering transition of the sub-unit fluid is continuous or very weakly first-order. Chi-rality is now directly expressed in the subunit densitymodulation, which is neither even nor odd under inver-sion. There is apparently no unique Landau descriptionand experiment may have to determine which one is ap-propriate. The T=7 bacteriophage virus HK97 [37] maybe an ideal “testbed” to test the theory. In the presenceof high salt, Prohead 1 capsids of HK97 dissociate intointerconverting pentamers and hexamers that are quitestable [38] so the capsomer picture is appropriate. Thekinetics of Prohead 1 reassembly indicates that it is ahighly cooperative process with some phase coexistencebetween capsids and capsomers [38], possibly consistentwith the weakly first-order transition picture of Fig.1 B.Inducing point mutations on the genes for the capsid pro-teins could destabilize the fragile icosahedral state andreveal the instabilities of Fig.2. Next, the theory predictsthat intrinsically chiral icosahedral capsids, such as T=7and T=13, should be very prone to undergo a transitioninto a lower symmetry state with C or C symmetry.The capsid should have an oval deformation for brokenicosahedral symmetry because the corresponding eigen-vectors are nearly odd under inversion. This has not yetbeen reported for HK 97 but capsids of the HIV virus dohave an oval shape.Finally, we note that the necessity of using of multi-ple irreps becomes less puzzling by noting that in thelimit that the sphere radius goes to infinity, the densitymodulation should reduce to the superposition of threeperiodic density waves whose wavevectors make an angleof 120 degrees with respect to each other (a hexagonalcrystal). The expansion of a plane wave into sphericalharmonics requires an infinite series of l terms so an in-finite series of irreps of SO(3) would be required to de-scribe solidification in the limit of large sphere radii. ACKNOWLEDGMENTS
We would like to thank Vladimir Lorman and Alexan-der Grosberg for helpful discussions, the NSF for supportunder DMR Grant No. 1006128 and the Aspen Centerfor Physics for hosting a workshop on the physics of viralassembly. ∗ [email protected][1] E. Kats, V. Lebedev, and A. Muratov, Physics Reports , 1 (1983).[2] R. Garmann, M. Comas-Garcia, A. Gopal, C. Knobler,and W. Gelbart, J. Mol. Biol. , 1050 (2013).[3] R. F. Bruinsma, M. Comas-Garcia, R. F. Garmann, andA. Y. Grosberg, Phys. Rev. E , 032405 (2016).[4] D. L. Caspar and A. Klug, in Cold Spring Harbor sym-posia on quantitative biology , Vol. 27 (Cold Spring HarborLaboratory Press, 1962) pp. 1–24.[5] J. M. Voogd, PhD thesis, Universiteit van Amsterdam,1994.[6] S. Fejer, D. Chakrabarti, and D. Wales, ACS Nano ,219 (1988).[7] R. Zandi, D. Reguera, R. Bruinsma, W. Gelbart, andJ. Rudnick, Proc Natl Acad Sci U S A. , 15556 (2004).[8] H. Kusumaatmaja and D.J Wales, Phys. Rev. Lett. ,165502 (2013).[9] S. Paquay, H. Kusumaatmaja, D. Wales, R. Zandi,and P. van der Schoot, http://arxiv.org/abs/1602.07945[cond-mat.soft] (2016).[10] V.L. Lorman and S.B Rochal, Physical Review B ,224109 (2008).[11] V.L. Lorman and S.B. Rochal, Physical review letters ,185502 (2007).[12] F. Klein, Lectures on the Icosahedron and the Solution ofthe Fifth Degree (Dover, 2003).[13] M. Golubitsky, I. Stewart, et al. , Singularities and groupsin bifurcation theory , Vol. 2 (Springer Science & BusinessMedia, 2012).[14] R. Zandi, P. van der Schoot, D. Reguera, W. Kegel, andH. Reiss, Biophys. J. , 1939 (2006).[15] O. Elrad and M. Hagan, Physical Biology , 045003(2010).[16] P. de Gennes and J. Prost, Physics of Liquid Crystals (Oxford, 1993).[17] W. Helfrich and J. Prost, Phys. Rev. A. , 3065 (1988).[18] S. Brazovskii, Zh. Eksp. Teor. Fiz , 175 (1975).[19] This term, which is similar to the chiral term of theHelfrich-Prost theory for curved chiral surfaces [17], isfurther discussed in section S1 of supplementary mate-rial.[20] To show that the uniform phase of this free energy hasSO(3) symmetry rather than O(3) it is necessary to com-pute higher order density correlation function in the pres-ence of thermal fluctuations.[21] The stability threshold is r l ( k ) = − (cid:0) k − l ( l + 1) (cid:1) .[22] The explicit form of the equations is provided in sectionS2 of supplementary material.[23] Hessian matrices were diagonalized numerically usingMathematica. [24] See last paragraph of section S2 of supplementary mate-rial.[25] See section S3 of supplementary material.[26] The corresponding density modulations are shown in sec-tion S4 of the supplementary material.[27] It is shown analytically in section S5 of supplementarymaterial that Y h (15) is unstable for any value of u and v .[28] c = ( k − × , c = ( k − × , a /
16 = − . a /
16 = − . u = 0 . u , u = 0 . u , v = 1 . w , v . w , u = 0 . u , v = 0 . w , and v = 0 . w .[29] The resulting density modulations are shown in sectionS6 of the supplementary material and compared with theresult of a two-state MC simulation of T=7 CK capsidscomposed of 72 capsomers [7].[30] There is also a shift of the two branches by a term thatis even in ζ .[31] This is illustrated in Fig. S4 of the supplementary mate-rial section S6. Similar results hold for larger values of l .For example, the 21 ×
22 branch produces a chiral T=3 shell in the subunt picture and a T=13 shell in the cap-somer picture. The 20 ×
21 branch produces a chiral T=2non-CK icosahedral shell.[32] The instability sequence is not symmetric around k =16: the unstable icosahdral branch now mostly is a pure l = 16 icosahedral harmonic with the 15 ×
16 state onlyappearing close to the stability limit of the icosahedralstate.[33] R. Hoyle, Physica D , 261 (2004).[34] The direction η = 0 and η = 1 / − / ) η is thenalong a 5-fold axis.[35] See section S7 of Supplementary Material supplementarymaterial.[36] The terms proportional to W and ∆ in Eq.6 are gener-ated by sixth and higher order terms in ρ in the LB freeenergy density Eq. 1.[37] J. Conway, W. Wikoff, N. Cheng, R. Duda, R. Hendrix,J. Johnson, and A. Steven, Science , 744 (2001).[38] Z. Xie and R. Hendrix, J. Mol. Biol.253