Simulation studies of a phenomenological model for elongated virus capsid formation

We study a phenomenological model in which the simulated packing of hard, attractive spheres on a prolate spheroid surface with convexity constraints produces structures identical to those of prolate virus capsid structures. Our simulation approach combines the traditional Monte Carlo method with a modified method of random sampling on an ellipsoidal surface and a convex hull searching algorithm. Using this approach we identify the minimum physical requirements for nonicosahedral, elongated virus capsids, such as two aberrant flock house virus particles and the prolate prohead of bacteriophage $\varphi 29$, and discuss the implication of our simulation results in the context of recent experimental findings. Our predicted structures may also be experimentally realized by the evaporation-driven assembly of colloidal spheres under appropriate conditions.

Figure 1

(Color online) Number density of points obtained by the method of generating random points on a prolate spheroid surface with (solid green circles) and without (solid red diamonds) the correction factor $g∕g_{\max }$, and the calculated average number density (blue line).

Figure 2

(Color online) $N=15$, the smallest prolate structure. (a) Hypothetical structure for aberrant FHV particle 1 with an aspect ratio of 1.35. Reprinted from Ref. 19. (b) Simulated prolate structure with scaffold prolate spheroid of aspect ratio 1.35. In virus capsids, a hexamer is a capsomer surrounded in its nearest-neighbor shell by six other capsomers and consists of six protein subunits. A pentamer is a capsomer surrounded by five other capsomers and consists of five protein subunits. The assembly contains one extra ring of three spheres representing hexamers (in gray) and 12 spheres representing pentamers (in gold).

Figure 3

(Color online) $N=17$, $T=1$, $Q=2$ prolate structure. (a) Physical model for the principles of elongated capsid construction for $T=1$, $Q=2$. Reprinted from Ref. 22. (b) Simulated prolate structure with scaffold prolate spheroid of an aspect ratio of 1.30. The assembly contains one linear ring of five spheres representing hexamers (in gray) and 12 spheres representing pentamers (in gold).

Figure 4

(Color online) $N=18$ prolate structure. (a) Hypothetical structure for aberrant FHV particle 2 with aspect ratio 1.50. Reprinted from Ref. 19. (b) Simulated prolate structure with scaffold prolate spheroid of an aspect ratio of 1.60. Two halves of $T=1$ icosahedral cap with two rings of spheres representing hexamers (each ring has 3 hexamers), or one belt of six spheres representing hexamers in a zigzag pattern (in gray). Twelve spheres representing pentamers are shown in gold.

Figure 5

(Color online) $N=42$, $T=3$, $Q=5$ prolate structure, (a) (left) Triangulation net of $T=3$, $Q=5$ for the (right) prolate prohead of bacteriophage $\varphi 29$. Reprinted from Ref. 24. (b) Physical models for (left) an icosahedral capsid with $T=3$ quasisymmetry; (middle) a prolate $T=3$, $Q=5$ capsid; (right) visualization of the elongated capsid with two icosahedral caps and the insertion of a row of ten hexamers in a zigzag pattern. (Reprinted from Ref. 22). (c) Simulated $T=3$, $Q=5$ prolate structure with scaffold prolate spheroid of aspect ratio 1.34. Two halves of $T=3$ icosahedral cap with a row of ten spheres representing hexamers in a zigzag pattern (in gray). Twelve spheres representing pentamers are shown in gold.