Normal mode analysis of oligomeric proteins: Reduction of the memory requirement by consideration of rigid geometry and molecular symmetry

Static and dynamic hidden symmetries of icosahedral viral capsids

Viral shells self-assemble from identical proteins, which tend to form equivalent environments in the resulting assembly. However, in icosahedral capsids containing more than 60 proteins, they are enforced to occupy not only the symmetrically equivalent locations but also the quasi-equivalent ones. Due to this important fact, static and dynamic symmetries of viral shells can include additional hidden components. Here, developing the Caspar and Klug ideas concerning the quasi-equivalence of protein environments, we derive the simplest hexagonal tilings, that in principle could correspond to the local protein order in viral shells, and apply the resulting theory to nucleocytoplasmic large dsDNA viruses. In addition, analyzing the dynamic symmetry of the P22 viral shell, we demonstrate that the collective critical modes responsible for the protein reorganization during the procapsid maturation are approximately equivalent to the normal modes of the isotropic spherical membrane with O(3) symmetry. Furthermore, we establish the relationship between the dynamic symmetry of the P22 procapsid and the protein arrangement regularities that appear only in the mature capsid.

Diversity and Identity of Mechanical Properties of Icosahedral Viral Capsids Studied with Elastic Network Normal Mode Analysis

We analyze the mechanical properties and putative dynamical fluctuations of a variety of viral capsids comprising different sizes and quasi-equivalent symmetries by performing normal mode analysis using the elastic network model. The expansion of the capsid to a swollen state is studied using normal modes and is compared with the experimentally observed conformational change for three of the viruses for which experimental data exist. We show that a combination of one or two normal modes captures remarkably well the overall translation that dominates the motion between the two conformational states, and reproduces the overall conformational change. We observe for all of the viral capsids that the nature of the modes is different. In particular for the T=7 virus, HK97, for which the shape of the capsid changes from spherical to faceted polyhedra, two modes are necessary to accomplish the conformational transition. In addition, we extend our study to viral capsids with other T numbers, and discuss the similarities and differences in the features of virus capsid conformational dynamics. We note that the pentamers generally have higher flexibility and propensity to move freely from the other capsomers, which facilitates the shape adaptation that may be important in the viral life cycle.

The mechanism and pathway of pH induced swelling in cowpea chlorotic mottle virus

Normal mode analysis based on a simplified energy function was used to study the swelling process of the icosahedral virus, cowpea chlorotic mottle virus (CCMV). Native state virus particles (coat proteins) of this T=3 icosahedral virus have been shown to undergo a large conformational change to a swollen state when metal ions are removed or the pH is raised. A normal mode analysis based on the native state capsid showed one preferential direction, a breathing mode, that explains the majority of the structural rearrangement necessary to bring the native structure close to the swollen state. From the native form of CCMV, the structure can be displaced along the direction of a single breathing mode by different amounts to create several candidate swollen structures and a putative pathway for virus expansion. The R-factor between these predicted swollen capsid structures and experimental electron density from cryoelectron microscopy (cryo-EM) measurements is then calculated to indicate how well each structure satisfies the experimental measurements on the swollen capsid state. A decrease of the crystallographic R-factor value from approximately 72% to approximately 49% was observed for these simple incremental displacements along the breathing mode. The simultaneous displacement of the native structure along other relevant (symmetric, non-degenerate) modes produce a structure with an R-factor of 45%, which is further reduced to 43.9% after minimization: a value in good accord with models based on the EM data at 28 A resolution. Based on the incrementally expanded structures, a pathway for the swelling process has been proposed. Analysis of the intermediate structures along this pathway indicates a significant loss of interactions at the quasi-3-fold interfaces occurs in the initial stages of the swelling process and this serves as a trigger for the compact to swollen transition. Furthermore, the pH dependent swelling appears to be triggered by the titration of a single residue with an anomalous pK(a) value in the unswollen particle.