Theory of morphological transformation of viral capsid shell during the maturation process in the HK97 bacteriophage and similar viruses

We consider the symmetry and physical origin of collective displacement modes playing a crucial role in the morphological transformation during the maturation of the HK97 bacteriophage and similar viruses. It is shown that the experimentally observed hexamer deformation and pentamer twist in the HK97 procapsid correspond to the simplest irreducible shear strain mode of a spherical shell. We also show that the icosahedral faceting of the bacteriophage capsid shell is driven by the simplest irreducible radial displacement field. The shear field has the rotational icosahedral symmetry group $I$ while the radial field has the full icosahedral symmetry $I_h$. This difference makes their actions independent. The radial field sign discriminates between the icosahedral and the dodecahedral shapes of the faceted capsid shell, thus making the approach relevant not only for the HK97-like viruses but also for the parvovirus family. In the frame of the Landau-Ginzburg formalism we propose a simple phenomenological model valid for the first reversible step of the HK97 maturation process. The calculated phase diagram illustrates the discontinuous character of the virus shape transformation. The characteristics of the virus shell faceting and expansion obtained in the in vitro and in vivo experiments are related to the decrease in the capsid shell thickness and to the increase of the internal capsid pressure.

Figure 1

(a, b) Spherical surface deformation by the radial displacement fields proportional to the icosahedral function $f_6^{}\left(\theta ,\phi \right)$. The amplitudes $D_{6,1}$ of the displacement fields in panels (a, b) have opposite signs. (c, d) Experimental realization of the capsid shapes induced by displacement fields (a, b). Viral capsid of the bacteriophage HK97 [12] with the icosahedral faceted shape (c), and viral capsid of the pathogenic human parvovirus B19 [26] (d) with the dodecahedral faceted shape.

Figure 2

Simplest irreducible shear field $(l=6)$ with the icosahedral symmetry and hexamer deformation in the HK97 procapsid. (a) Shear field on a spherical surface. Vectors of the field are given by segments, segment color intensity gradient indicates vector direction, and segment length is proportional to the field amplitude. For the sake of clarity the amplitude is exaggerated with respect to the sphere size. The field is antisymmetric with respect to spatial inversion. It is invariant with respect to all icosahedron rotations and vanishes at the global icosahedral symmetry axes. With good accuracy, in the vicinity of the fivefold axes spherical membrane regions rotate as a whole under the shear field action, amplitude of the field being proportional to the distance from the axis. (b) Global deformation of the sphere tessellation (Caspar-Klug “spherical lattice” triangulation) with $T=7$ by the icosahedral shear field shown in panel (a). Structure of the HK97 bacteriophage is commonly interpreted in terms of this tessellation containing 12 pentamers and 60 hexamers. (с) Distorted hexamer shape in the HK97 procapsid (structural data taken from [7]).

Figure 3

Phase diagram of the free energy functional with density (12). Full line 1 and dash-dotted line 2 divide the phase diagram in qualitatively different regions: the region of stability of the minimum with the spherical symmetry, the region of stability of the minimum with the icosahedral symmetry, and the region where two minima with different symmetries coexist. The minimum with $\eta =0$ (spherical shape state without faceting) exists only in the region above line 2. The state with the icosahedral shape $(\eta <0)$ appears below line 1. At dashed line 3 free energies of the spherical and the icosahedral faceted states are equalized. Below line 2 the minimum which corresponds to the spherical state disappears. It is replaced by the weak metastable minimum with $\eta <0$. Insets (a–c) show free energy density (13) as a function of the order parameter $\eta$ in several typical points. Corresponding points are given by the same letters in the phase diagram. Two qualitatively different thermodynamic paths of the morphological transformation are shown by arrowed lines. Line 4 (downward) corresponds to the shell thinning induced by the $p\mathrm{H}$ level decrease. Line 5 (from left to right) illustrates the result of the pressure difference increase. During the in vivo genome packaging into the capsid the thermodynamic path of the system passes between lines 4 and 5 and corresponds to the simultaneous capsid shell thinning and internal capsid pressure increase.