Key steps in a viral life-cycle, such as self-assembly of a protective protein container or in some cases also subsequent maturation events, are governed by the interplay of physico-chemical mechanisms involving various spatial and temporal scales. These salient aspects of a viral life cycle are hence well described and rationalised from a mesoscopic perspective. Accordingly, various experimental and computational efforts have been directed towards identifying the fundamental building blocks that are instrumental for the mechanical response, or constitute the assembly units, of a few specific viral shells. Motivated by these earlier studies we introduce and apply a general and efficient computational scheme for identifying the stable domains of a given viral capsid. The method is based on elastic network models and quasi-rigid domain decomposition. It is first applied to a heterogeneous set of well-characterized viruses (CCMV, MS2, STNV, STMV) for which the known mechanical or assembly domains are correctly identified. The validated method is next applied to other viral particles such as L-A, Pariacoto and polyoma viruses, whose fundamental functional domains are still unknown or debated and for which we formulate verifiable predictions. The numerical code implementing the domain decomposition strategy is made freely available.

Self-assembly refers to the spontaneous organization of individual building blocks into higher order structures. It occurs in biological systems such as spherical viruses, which utilize icosahedral symmetry as a guiding principle for the assembly of coat proteins into a capsid shell. In this study, we characterize the self-assembling protein nanoparticle (SAPN) system, which was inspired by such viruses. To facilitate self-assembly, monomeric building blocks have been designed to contain two oligomerization domains. An N-terminal pentameric coiled-coil domain is linked to a C-terminal coiled-coil trimer by two glycine residues. By combining monomers with inherent propensity to form five- and threefold symmetries in higher order agglomerates, the supposition is that nanoparticles will form that exhibit local and global symmetry axes of order 3 and 5. This article explores the principles that govern the assembly of such a system. Specifically, we show that the system predominantly forms according to a spherical core-shell morphology using a combination of scanning transmission electron microscopy and small angle neutron scattering. We introduce a mathematical toolkit to provide a specific description of the possible SAPN morphologies, and we apply it to characterize all particles with maximal symmetry. In particular, we present schematics that define the relative positions of all individual chains in the symmetric SAPN particles, and provide a guide of how this approach can be generalized to nonspherical morphologies, hence providing unprecedented insights into their geometries that can be exploited in future applications.

Viruses with icosahedral capsids, which form the largest class of all viruses and contain a number of important human pathogens, can be modelled via suitable icosahedrally invariant finite subsets of icosahedral 3D quasicrystals. We combine concepts from the theory of 3D quasicrystals, and from the theory of structural phase transformations in crystalline solids, to give a framework for the study of the structural transitions occurring in icosahedral viral capsids during maturation or infection. As 3D quasicrystals are in a one-to-one correspondence with suitable subsets of 6D icosahedral Bravais lattices, we study systematically the 6D-analogs of the classical Bain deformations in 3D, characterized by minimal symmetry loss at intermediate configurations, and use this information to infer putative viral-capsid transition paths in 3D via the cut-and-project method used for the construction of quasicrystals. We apply our approach to the Cowpea Chlorotic Mottle virus (CCMV) and show that the putative transition path between the experimentally observed initial and final CCMV structures is most likely to preserve one threefold axis. Our procedure suggests a general method for the investigation and prediction of symmetry constraints on the capsids of icosahedral viruses during structural transitions, and thus provides insights into the mechanisms underlying structural transitions of these pathogens.

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on threedimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.