Normal modes of symmetric protein assemblies. Application to the tobacco mosaic virus protein disk.

We use group theoretical methods to study the molecular dynamics of symmetric protein multimers in the harmonic or quasiharmonic approximation. The method explicitly includes the long-range correlations between protein subunits. It can thus address collective dynamic effects, such as cooperativity between subunits. The n lowest-frequency normal modes of each individual subunit are combined into symmetry coordinates for the entire multimer. The Hessian of the potential energy is thereby reduced to a series of blocks of order n or 2n. In the quasiharmonic approximation, the covariance matrix of the atomic oscillations is reduced to the same block structure by an analogous set of symmetry coordinates. The method is applied to one layer of the tobacco mosaic virus protein disk in vacuo, to gain insight into the role of conformational fluctuations and electrostatics in tobacco mosaic virus assembly. The system has 78,000 classical, positional, degrees of freedom, yet the calculation is reduced by symmetry to a problem of order 4,600. Normal modes in the 0-100 cm-1 range were calculated. The calculated correlations extend mainly from each subunit to its nearest neighbors. The network of core helices has weak correlations with the rest of the structure. Similarly, the inner loops 90-108 are uncorrelated with the rest of the structure. Thus, the model predicts that the dielectric response in the RNA-binding region is mainly due to the loops alone.