Minimal Bending Energies of Bilayer Polyhedra
aa r X i v : . [ c ond - m a t . s o f t ] J un Minimal Bending Energies of Bilayer Polyhedra
Christoph A. Haselwandter and Rob Phillips
Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA (Dated: June 16, 2018)Motivated by recent experiments on bilayer polyhedra composed of amphiphilic molecules, westudy the elastic bending energies of bilayer vesicles forming polyhedral shapes. Allowing for segre-gation of excess amphiphiles along the ridges of polyhedra, we find that bilayer polyhedra can indeedhave lower bending energies than spherical bilayer vesicles. However, our analysis also implies that,contrary to what has been suggested on the basis of experiments, the snub dodecahedron, ratherthan the icosahedron, generally represents the energetically favorable shape of bilayer polyhedra.
PACS numbers: 87.16.dm, 68.60.Bs
In an aqueous environment, amphiphilic moleculessuch as lipids are observed to self-organize into bilayervesicles [1, 2], thus forming the physical basis for cellmembranes. Bilayer vesicles generally exhibit shapeswith constant or smoothly varying curvature [3, 4]. Butin recent experiments [5, 6], bilayer vesicles with poly-hedral shape, consisting of flat faces connected by ridgesand vertices with high local curvature, have been ob-served. In these experiments, two types of oppositelycharged, single-tailed amphiphiles were used, with aslight excess of one amphiphile species over the other. Athigh temperatures, the amphiphiles were found to formspherical bilayer vesicles. However, provided that thenumber of excess, unpaired amphiphiles was tuned tosome optimal range, cooling the system below the chainmelting temperature yielded polyhedral bilayer vesicles.It was reported that the bilayer polyhedra were stableover weeks, and that their shape was consistently repro-duced upon thermal cycling. Furthermore, it was sug-gested [5, 6] that the observed polyhedra had icosahe-dral symmetry, although some uncertainty regarding thepolyhedral symmetry remained.What is the mechanism governing the formation andsymmetry of bilayer polyhedra? It was argued [5, 6]on the basis of the experimental phenomenology thatelastic contributions to the polyhedron free energy dom-inate over entropic or electrostatic contributions, andthat minimization of elastic bending energy alone de-termines the shape of bilayer polyhedra. In this Let-ter, we take these intriguing observations as our startingpoint and address the general problem of finding poly-hedral shapes with minimal bending energy. Questionsregarding the minimal energy shape of bilayer vesicles arecommonly answered using a variational approach [1–3].However, a given polyhedral shape is defined by the geo-metric parameters characterizing its vertices and ridges.Thus, polyhedra are inherently of singular nature, whichseverely restricts the applicability of variational calculus.We therefore employ a complementary method, in whichwe allow for ridges and vertices with arbitrary geometricproperties and, on this basis, calculate polyhedron bend-ing energies as a function of polyhedron symmetry. In the remainder of this Letter, we first consider the moststraightforward case of bilayers with uniform composi-tion, and then turn to the richer case in which there issegregation of excess amphiphiles.The solution of the two-dimensional equations of elas-ticity is a formidable challenge, and has only beenachieved for the vertices and ridges of polyhedra in cer-tain limiting cases [7–11]. Thus, in order to determinethe elastic energies of arbitrary polyhedral shapes, wemainly employ simple expressions based on the Helfrich-Canham-Evans free energy of bending [1–3], G = K b Z dS (cid:18) R + 1 R − H (cid:19) , (1)where K b is the bilayer bending rigidity, R and R arethe two principal radii of curvature, and H is the spon-taneous curvature. The resulting expressions for poly-hedron energies are intuitive and only involve a few pa-rameters but, ultimately, are purely phenomenological.We assess their validity by making comparisons to poly-hedron energies obtained for the aforementioned limit-ing cases of the equations of elasticity, which allow forstretching as well as bending deformations.Figure 1(a) shows schematic illustrations of a bilayerbending gradually (left panel) and sharply (right panel)along a ridge with dihedral angle α i . The first model isinspired by the electron micrographs of bilayer polyhedrain Refs. [5, 6, 12], while the second model provides a morefaithful representation of the polyhedral geometry. Basedon the picture presented in the left panel of Fig. 1(a), weapproximate ridges by a bilayer bending partially arounda cylinder of radius R = d/ ( π − α i ), where d is thearc length. Following the right panel, we discretize thebilayer with a lattice spacing b , and assume a harmonicpotential for the angle between adjacent bond vectors.Upon setting d = b one finds from either approach aridge energy similar to the expression used in Ref. [6], G r = ¯ K b π − α i ) l , (2)where ¯ K b = K b /b , l is the ridge length, and we haveassumed that H = 0. A simple expression for the bend- a b cd FIG. 1: (color online). Illustration of the contributions tothe elastic bending energies of polyhedra: (a) Side view of aridge with dihedral angle α i , (b) vertex with face angle β j ,(c) cross section of half of a pore around the tip of a cone (seeinset) with apex angle π − θ and radius r , (d) top-down view(left panel) and side view (right panel) of a pore composed ofstraight edges along each face. ing energy associated with closed bilayer vertices [seeFig. 1(b)] is obtained from Eq. (1) following analogoussteps, leading to the vertex energy G v = K b X j ( π − β j ) , (3)where β j denotes the face angle subtended by two ridgesmeeting at a given polyhedron vertex.Closed bilayer vertices may break up to form pores[5, 6, 12], which was suggested [5] as a mechanism foravoiding the curvature singularity associated with closedvertices. Figure 1(c,d) show two models for pores atthe vertices of polyhedra which, similarly as before, areinspired by the experimental images in Refs. [5, 6, 12][Fig. 1(c)] and a stricter interpretation of the polyhe-dral geometry of bilayer vesicles [Fig. 1(d)]. In our firstmodel [see Fig. 1(c)] we approximate the vertex of a givenpolyhedron by a cone with apex angle π − θ , where θ = π/ − arccos (1 − Ω / π ) for a solid angle Ω sub-tended by the polyhedron vertex. In our second modelwe assume that, along each face, the pore consists of astraight cylindrical edge [see Fig. 1(d), left panel], whichbends through an angle γ j across a ridge from one face to a neighboring face [see Fig. 1(d), right panel]. Inboth cases, the elastic pore energy G p can be evaluatedon the basis of Eq. (1) and is found [13] to depend onthe pore radius, the monolayer bending rigidity, K ⋆b , andthe monolayer spontaneous curvature, H ⋆ , as well as theface angle and solid angle characterizing the geometry ofa given polyhedron. For physically relevant parameterranges, G p increases with decreasing H ⋆ and Ω.We now turn to the presence of excess amphiphiles inbilayer polyhedra. The expulsion of excess amphiphilesfrom flat bilayers and the resulting molecular segregation,together with the high spontaneous curvature of single-tailed excess amphiphiles, are thought [5, 6] to have twoprincipal effects on the bending energies of polyhedra.On the one hand, excess amphiphiles can seed pores intobilayers [5, 6] and, thus, pores may have a role beyond re-ducing the elastic energy of polyhedron vertices. In par-ticular, it has been suggested [6] that excess amphiphilesproduce pores in the spherical bilayer vesicles from whichbilayer polyhedra originate upon cooling. On the otherhand, it has been found [6] that excess amphiphiles pref-erentially accumulate along the ridges of polyhedra. Asa result, molecular segregation can decrease the bendingenergy of the outer monolayer at ridges.The above observations suggest a simple description ofhow vertex and ridge energies are modified by the pres-ence of excess amphiphiles. Ideally, excess amphiphilesare arranged along ridges such that they induce ananisotropic spontaneous curvature commensurate withthe dihedral angle. Assuming such “perfect segregation”,only the bending energy of the inner layer must be con-sidered when computing ridge energies. We therefore ob-tain a lower bound on the modified ridge energy whichtakes a similar form as Eq. (2), but with the rescaled bilayer bending modulus ¯ K b replaced by the rescaled monolayer bending modulus ¯ K ⋆b = K ⋆b /b . Thus, pro-vided that the optimal amount of excess amphiphiles ispresent [5, 6], the ridge energy is lowered by a factor K ⋆b /K b , with the number of pores seeded into sphericalvesicles equal to or greater than the number of polyhe-dron vertices. Experiments [5, 6] and simulations [14]suggest that K ⋆b /K b / − .From the simple model for perfectly segregated bilayerpolyhedra described above one finds that the optimal ra-tio of the amphiphile species in excess to the total am-phiphile content is given by r I ≈ .
51 for the polyhedronsizes observed in experiments [5, 6]. This optimal valuefor r I is a direct result of amphiphile and polyhedrongeometry and, hence, does not depend on any elastic pa-rameters. The corresponding experimental estimate is r I ≈ .
57 [5, 6]. We expect that in experiments not allexcess amphiphiles are segregated along the ridges andvertices of polyhedra as a result of, for instance, entropicmixing within bilayer polyhedra or the formation of mi-celles [6]. Thus, our theoretical estimate for r I is in broadagreement with experimental observations.For certain limits of the equations of elasticity, approx-imate solutions corresponding to polyhedron vertex [7–9] and ridge [10, 11] energies have been obtained. Inparticular, it has been found [7] that 5-fold disclinationsin hexagonal lattices are accommodated for small lat-tice sizes through a stretching of lattice vectors. How-ever, for large enough lattice sizes it becomes energet-ically favorable to buckle out of the plane [7], in whichcase the energetics of the system are dominated by bend-ing. This behavior is characterized by a dimension-less quantity known as the F¨oppl-von K´arm´an numberΓ = Y R /K b , where Y is the two-dimensional Young’smodulus and R is the lattice size. As Γ → ∞ , pro-nounced ridges develop [8, 9] between the 5-fold disclina-tion sites of icosadeltahedral triangulations of the sphere.In this limit, the total elastic energy is dominated byridges and was determined in Refs. [10, 11] to be of theform K b ( π − α ) / l / f ( Y /K b ), with 0 . / f ( Y /K b ) / . − / for bilayer polyhedra [6, 13, 14]. Thus,the asymptotic expression for the ridge energy found inRefs. [10, 11] leads to a similar dependence on the dihe-dral angle and proportionality factor as in Eq. (2), whileincreasing sub-linearly with the ridge length l .The lowest energy states of icosadeltahedral triangu-lations of the sphere are found to resemble icosahedrafor Γ ' [8, 9], which corresponds to a vertex en-ergy greater than 8 K b with, for instance, a value 12 K b for Γ = 10 . This compares quite favorably with theestimate G v ≈ K b implied by Eq. (3) for the icosa-hedron. Moreover, we find [13] that our two models forpores predict similar ranges for the pore energy G p , withthe competition between pores and closed bilayer verticesgoverned by the ratio K ⋆b /K b . In particular, the afore-mentioned estimate K ⋆b /K b / − [5, 6, 14] implies thatclosed bilayer vertices will be unstable to the formation of(closed) pores, thus removing the singularity associatedwith polyhedron vertices. This is consistent with exper-imental observations [5, 6, 12] and allows adjustment ofthe volume of bilayer polyhedra for a fixed total area ornumber of amphiphiles.Ridges impose an energetic cost and, hence, one ex-pects that for a fixed area and dihedral angle the faces ofpolyhedra relax to form regular polygons. We thereforefocus here on the convex polyhedra with regular polygonsas faces, but we have also considered polyhedra with ir-regular faces [13]. The class of convex polyhedra withregular faces encompasses the five Platonic solids, thethirteen Archimedean solids, the two (infinitely large)families of prisms and antiprisms, and the 92 Johnsonsolids [15, 16]. It has been shown [17] that this list ex-hausts all convex polyhedra with regular faces. Thus,counting prisms and antiprisms as one solid each, thereare exactly 112 convex polyhedra with regular faces.With each of these polyhedra a specific set of parame-ters characterizing ridges, vertices, and faces is associ-ated, leading to distinct contributions to the total elastic a R p nm G G i Platonic solidsArchimedean solidsPrisms and antiprismsJohnson solidsSphere b R p nm G G i Platonic solidsArchimedean solidsPrisms and antiprismsJohnson solidsSphere
FIG. 2: (color online). Total elastic bending energies of theconvex polyhedra with regular faces, obtained for the case ofperfect segregation of excess amphiphiles, and bending energyof the sphere, normalized by the total bending energy of theicosahedron, G i , with (a) pores with r = 0 nm and (b) poreswith r = 20 nm at each polyhedron vertex. The snub dodec-ahedron corresponds to the bold curve minimizing bendingenergy in (a,b) for R p ≈
500 nm. bending energy.We have evaluated the total elastic bending energiesof all convex polyhedra with regular faces. As illus-trated in Fig. 2, polyhedron energies are compared byplotting elastic energy as a function of the polyhedronradius R p [8], which is related to the polyhedron area A via A = 4 πR p , such that, for each value of R p , allshapes have the same total area. While the quantita-tive details of the resulting energy curves depend on theparticular combination of the aforementioned expressionsfor ridge, vertex, and pore energies used, we find that allcurves share the same basic qualitative features. Con-sistent with a previous study [6], the icosahedron [seeFig. 3(a)] minimizes bending energy among the Platonicsolids. However, we also find that, in general, the icosahe-dron does not minimize bending energy among arbitrary a bc d FIG. 3: (color online). Image representations of (a) the icosa-hedron, (b) the snub dodecahedron, (c) the snub cube, and(d) the great rhombicosidodecahdron. The polyhedra in (b)and (c) are chiral. polyhedral shapes. In fact, for large enough polyhedronsizes, the snub dodecahedron [see Fig. 3(b)] is the poly-hedral shape minimizing bending energy among the con-vex polyhedra with regular faces, and the snub cube [seeFig. 3(c)] also has a lower energy than the icosahedronin this limit.Allowing for an optimal number of excess amphiphiles,we find (see Fig. 2) that polyhedra can have lower bend-ing energies than the sphere, but only if we permit molec-ular segregation along ridges as observed in Ref. [6].Segregation at pores, which was originally suggested inRef. [5] as a potential mechanism stabilizing polyhedralshapes, is not sufficient to produce polyhedra with bend-ing energies which are favorable compared to the sphere[13]. Indeed, if we assume that pores are closed, bi-layer polyhedra are energetically favorable for the exper-imentally observed polyhedron radius R p ≈
500 nm [seeFig. 2(a)], with the snub dodecahedron as the minimumenergy shape among the convex polyhedra with regularfaces. If we allow pores of a finite size, a sequence of poly-hedral shapes is obtained which minimize bending energyfor smaller polyhedron radii [see Fig. 2(b)]. The most no-table of these polyhedral shapes is the great rhombicosi-dodecahedron [see Fig. 3(d)], which surpasses the snubdodecahedron in bending energy at R p ≈
300 nm. How-ever, according to our analysis, the snub dodecahedronrepresents the minimum in bending energy among theconvex polyhedra with regular faces for the polyhedronsizes and pore sizes ( r ≈
20 nm) found experimentally[5, 6, 12].In summary, we have used Eq. (1) and a variety ofother expressions [7–11] to systematically evaluate thebending energies of bilayer polyhedra. We find that, con-trary to what has been suggested on the basis of exper-iments [5, 6], the snub dodecahedron and the snub cube generally have lower total elastic bending energies thanthe icosahedron. This result is consistent with severalcomplementary theoretical studies which suggest that theelastic energies of chiral shapes such as the snub dodec-ahedron and the snub cube can be favorable comparedto the icosahedron [18–20] and that, even if the icosa-hedral shape is imposed, the minimum energy structuremay still be chiral [21]. While we followed here the ex-perimental phenomenology [5, 6] and assumed that min-imization of bending energy governs the shape of bilayerpolyhedra, other contributions to the free energy, as wellas kinetic effects [14, 22], could, in principle, modify thepreferred polyhedral symmetry. In light of our results,we suggest revisiting the symmetry of bilayer polyhedra,and the thermodynamic or kinetic mechanisms poten-tially governing their formation and stability, in greaterexperimental detail.This work was supported by a Collaborative Inno-vation Award of the Howard Hughes Medical Insti-tute, and the National Institutes of Health through NIHAward number R01 GM084211 and the Director’s Pio-neer Award. We thank A. Agrawal, M. B. Jackson, W.S. Klug, R. W. Pastor, T. R. Powers, D. C. Rees, M. H.B. Stowell, D. P. Tieleman, T. S. Ursell, and H. Yin forhelpful comments. [1] D. Boal,
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