Buckling transitions and soft-phase invasion of two-component icosahedral shells

What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component “invades” the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.

Figure 1

Structure of a rigid crystalline icosahedral shell with the vertices “invaded” by the soft material (the image courtesy of Vliegenthart, unpublished). The triangulated lattice mesh of the rigid and soft materials is shown in magenta red and green colors, respectively.

Figure 2

Deformation map of a partial disk into a dome for the defect of charge $q=1$. This corresponds to a cut-out planar azimuthal angle of $2\pi /6$, as for icosahedral shells. Some model parameters are denoted in the plot.

Figure 3

Symmetric arrangements of topological defects of a given charge $q$ on sphere that results in the energy minimum.

Figure 4

Scaled in-plane strain energies of the dome given by Eq. (22) computed for $q=\{1,2,3,4,6\}$ (almost fully superimposing curves of different colors), collapse onto a single universal curve. The values of $A_q$ from Table 1 were used here. The quadratic law for the bottom curve is Eq. (25), illustrating a rather close agreement with the exact result (22).

Figure 5

Normalized total elastic energy of a closed crystalline surface as a function of the FvK number $\gamma$ before (a) and after (b) the finite-mesh adjustment of parameters, given by Eqs. (42) and (44). Both the main plots and insets are in log-linear scale. The dotted dark blue curves show the results of computer simulations performed in Ref. [18]. The bending and strain energy of the sphere is the dashed green curve (see the legend). The energy of the cylindrical segments at ${\beta }_{\text{cyls}}\to {\beta }_T,$ when the faces of the shell are flat, is the dashed red curve, calculated from Eq. (67) upon minimization with respect to $x_{\text{dome}}$ (it contains also the strain in the domes and all bending contributions). The results of the approach of circular cones and spherical domes are shown as the dashed light blue curves. The insets show the crossing of the energy curves occurring at moderate $\gamma$ values.

Figure 6

Variable cones of noncircular cross section: cylindrical segments with the cone surfaces visualized for $q=\{1,2,3,4\}$ defining the cylindrical segment angle (see Sec. 2c1 for details). The first row are the standard or pure cones, the structures in the middle row contain half-formed cylindrical segments , while the structures in the last row are partial cylindrical segments with flat faces between them. The length of the boundary remains the same in all panels. The empty spaces left on top of the cones are to be filled in the model with the corresponding domes.

Figure 7

Construction of the $n=0$ (blue) and $n=1$ (red) cylindrical segments for $q=1$ defect with the maximal tubular angle. The slope angle $\alpha (q=1)$ is given by $arccos\left[sin\left(\pi /6\right)/sin\left(\pi /5\right)\right]$.

Figure 8

Two cylindrical segments forming the edges of a variable cone. The blue inclined plane contains the bases of the arcs tilted by an angle ${\theta }_{\text{face}}$. The face angle ${\theta }_{\text{face}}$ is denoted as well [see Eq. (56) for details].

Figure 9

The square-root factor from the mean curvature for the variable cone, Eq. (58), with a linear fit computed as function of parameter ${\beta }_{\text{rel}}$ given in (64), shown for several values of the defect charge $q$. The inset shows the standard deviation denoted as $s_d$ from the linear fit.

Figure 10

Energy curves for variable cones computed for several ${\beta }_{\text{rel}}$ values using Eq. (66) shown with the simulation data of Lidmar et al. [18] (the blue dots).

Figure 11

Scaling of the parameters ${\beta }_{\text{rel}}$ (a) and $x_{\text{dome}}$ (b) with the FvK number. The red solid lines are the asymptotes (70) and (71), while the data points are the results of Ref. [18].

Figure 12

Energy variation of the shell plotted using the simplified variable-cone analytical expression (75) (the thin solid red curve). The simulations data of Lidmar et al. [18] are the blue points.

Figure 13

(a) Filling configurations by the soft material in the $\{\mathrm{\Gamma }-\mathrm{\Lambda }\}$ plane, with the different colors filling the regions of occupation in stages I, II, and III, as denoted in the plot (see also Table 4 for details). The two red dots in (a) designate the values of parameters used to compute the results presented in (b) and (c), as indicated in the legend. The boundary curves $\mathrm{\Lambda }\left(\mathrm{\Gamma }\right)$ between the stages I, II, and III of invasion illustrated in (a) are explained in Table 4. (b), (c) The minimal energy curves $\mathcal{E}\left(n\right)$ as given by Eqs. (86, 87, 88) plotted versus the number $n=1,2,\cdots ,12$ of defect-centered segments filled by the soft material, plotted for the model parameters $\mathrm{\Gamma }$ in Eq. (89), $\mathrm{\Delta }$ in Eq. (90), and $\mathrm{\Lambda }$ in Eq. (91) as indicated in the legend. The fraction $f$ in (b) and (c) increases as the soft material continuously fills the shell. In (c) this filling happens in a cone up to a point when the filling stalls, while $f$ increases up to a level when the next cone starts filling in the energy-minimum state, etc. Different colors of the curves in (b) and (c) denote additional different sectors being invaded or filled by the soft material, while the dotted lines in (c) denote the energy increase (not realizable in the ground state of the system). The dashed black horizontal lines in (c) extending into the vertical gray-bluish strips are the stalling intervals of invasion.

Figure 14

The same quantities as in Fig. 13, with the same notations for the curves in the respective panels, but for the case when the bending moduli of the two phases are close, at $\mathrm{\Delta }=0.9$. Here, both energy plots in (b) and (c) stall for only very small sectors (see the thin shaded regions): one of them is in stage II and the other one is in stage III. Note that when $\mathrm{\Lambda }<\mathrm{\Lambda }{m}_{23}$ the minimum is in stage III and the energy is always negative.

Figure 15

Buckling of a spherical shell into the domes connected by the cylindrical segments and flat triangular faces. This dome-face-ridge simplified model for FvK numbers above buckling is a “predecessor” of the variable-cone model (see Sec. 3c1 for details).

Figure 16

Viral capsid and asymmetric unit of Cowpea Mosaic Virus (CPMV), as reproduced from Ref. [32] (where the detailed explanation of the components is given). We acknowledge the permission from Elsevier to reuse this figure without charges.

Figure 17

Pentagonal “stargate” on one vertex of the mimivirus. The image is reproduced from Ref. [40].

Figure 18

Fullerenes of increasing generation-number demonstrate the polyhedral faceting transition, with $n_{\text{full}}=2$, 6, and 10 structures shown in the plot (from left to right). The color scheme quantifies the displacements in atomic positions with respect to an ideally flat graphene triangle (see Ref. [7] for more details). The image is reproduced from Ref. [7], with permission from the PCCP Owner Societies.