Statistical Physics of Viral Capsids with Broken Symmetry

We present a model to understand quantitatively the role of symmetry breaking in assembly of macromolecular aggregates in general, and the protein shells of viruses in particular. A simple dodecahedral lattice model with a quadrupolar order parameter allows us to demonstrate how symmetry breaking may reduce the probability of assembly errors and, consequently, enhance assembly efficiency. We show that the ground state is characterized by large-scale cooperative zero-energy modes. In analogy with other models, this suggests a general physical principle: the tendency of biological molecules to generate symmetric structures competes with the tendency to break symmetry in order to achieve specific functional goals.

Figure 1

Construction of a CK $T=9$ icosahedron from an ($h=3$, $k=0$) lattice vector creating an equilateral triangle. The perpendicular height of the triangle is taken to be 1. Blue: pentons. Gray: penton-bordering hexons. Red: face-centered hexons. Bottom left: lattice triangle constructed from an ($h=3$, $k=0$) lattice vector of a stretched hexagonal lattice where the perpendicular height of the triangle is increased by a factor $\lambda$. Second column: example of hexamer protein configurations in the unstretched-mature (top) and stretched-immature (bottom) capsid. These examples refer to the Head II and Prohead II $T=7$ HK97 virus (ViperDB [3]).

Figure 2

Symbolic representation of an icosahedral CK shell constructed from stretched hexons. (a) The solid lines show the stretch direction at the centers of the twenty faces of the icosahedron, which span the dual dodecahedral lattice. The solid lines project onto one of the three edges of the dodecahedron that connect a site to its three neighbors. (b) Depiction of the four distinct bond patterns for pairs of neighboring order parameters.

Figure 3

Configurations used to compute energy tables: the orientation of the prestretch on the face-centered hexamer in the middle is rotated over 120 deg to compute the different types of interaction energies. The nonzero bond energies follow from the relations $E_2-E_1={\epsilon }_1=0.043$ and $E_3-E_2={\epsilon }_4=0.030$.

Figure 4

Nonzero bond energies ${\epsilon }_1$ and ${\epsilon }_4$ in units of the bending energy $\kappa$ as a function of the stretch $\lambda$. The corresponding capsid morphology is shown below the graph. Notice that, since the order parameter distribution is not icosahedral symmetric, it follows that the capsid morphologies are not strictly icosahedral or dodecahedral symmetric.

Figure 5

A zero-energy mode: six zero energy states that can be transformed into each other by the rotation of the order parameter on one site. Type 2 bonds are shown thick and red, while type 3 are thin and black. The bond that is about to be moved is indicated by a dot.

Figure 6

Energy spectrum of the lattice model for ${\epsilon }_1=0.04$ and ${\epsilon }_4=0.03$ in units of the bending energy. Horizontal axis right: states ordered from low to high energy. Horizontal axis left: degeneracy level. Vertical axis: energy in units of the bending energy.