The Role of Symmetry in Conformational Changes of Viral Capsids: A Mathematical Approach

For many viruses, structural transitions of the viral protein containers, which encapsulate and hence provide protection for the viral genome, form an integral part of their life cycle. We review here two complementary mathematical models for the expansion of an icosahedral viral capsid. The first is based on a geometrical description of the capsid involving a library of point sets obtained by affine extensions of the icosahedral group, and allows us to characterize the space of the possible transition paths between the initial and the final state. In the second approach, the capsid is described as a union of rigid tiles that interact with each other and with the genomic material, placing emphasis on the energetic determinants of the transition event. Both models predict loss of icosahedral symmetry along the transition path, even though the final state is icosahedral.