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ABSTRACT

Virus capsids and crystalline surfactant vesicles are two examples of self-assembled shells in the nano- to micrometer size range. Virus capsids are particularly interesting since they have to sustain large internal pressures while encapsulating and protecting the viral DNA. We therefore study the mechanical properties of crystalline shells of icosahedral symmetry on a substrate under a uniaxial applied force by computer simulations. We predict the elastic response for small deformations, and the buckling transitions at large deformations. Both are found to depend strongly on the number of elementary building blocks N (the capsomers in the case of viral shells), the Föppl-von Kármán number γ (which characterizes the relative importance of shear and bending elasticity), and the confining geometry. In particular, we show that whereas large shells are well described by continuum elasticity-theory, small shells of the size of typical viral capsids behave differently already for small deformations. Our results are essential to extract quantitative information about the elastic properties of viruses and vesicles from deformation experiments.

INTRODUCTION

The formation of regular polyhedra is a frequently encountered strategy of nature to optimize self-assembled structures. Microscopic boron clusters (1), mesoscopic surfactant vesicles (2), vesicles formed from wheel-shaped molybdenum clusters (3), as well as about half of all known spherical virus particles (4) are all small self-assembled structures that have an underlying icosahedral symmetry. It is interesting that the overall three-dimensional structure of many viruses is so similar whereas they are built from different protein subunits. Two illustrative examples of well-known viruses are the tomato bushy stunt virus and the bacteriophage φ29. Tomato bushy stunt virus was the first virus for which the icosahedral structure was predicted (5) and later confirmed by virus crystallography (6). This virus consists of 180 protein subunits that aggregate into a virus shell of ~34-nm diameter that encapsulates the viral genome. The typical contact energy between the different subunits is in the range of 100-400 kJ/mol, which corresponds to several tens of k^sub B^T per bond. The bacteriophage φ29 is a small bacteria-infecting virus consisting of a head of 235 gp8 protein subunits-forming two icosahedral end caps and a cylindrical equatorial region-and a long flexible tail (7). It has been demonstrated by DNA packaging experiments using optical tweezers (8) that the genome of this virus is very tightly packed into the capsid by a molecular motor, leading to an internal pressure of ~50 atm (9). This high pressure results in a formidable injection force when the virus infects a host cell. Therefore, virus capsids must be mechanically very strong.

Different types of protein subunits aggregate under the appropriate conditions (ionic strength, pH, temperature) into stable virus capsids. However, the mechanically coherent shell often consists of only a single kind of protein subunit. This process of spontaneous aggregation is very similar to micellization in surfactant solutions and can be largely understood using self-assembly theory (10-12). Recently, several groups studied the formation of these icosahedral particles by computer simulation (13-15). More complex shapes can be obtained by introducing a spontaneous curvature that competes with the ratio of bending and stretching energy (13,16). The origin of the stability of the icosahedral shape as an approximation to a sphere lies in the fact that any regular triangulation of a smooth sphere requires an excess of at least 12 fivefold disclinations (17). Caspar and Klug (18) first showed that the organization of proteins in the viral shell is such that a few proteins each form hexavalent and pentavalent morphological units, the capsomers. Furthermore, the icosadeltahedral structure of a virus shell can then be characterized by two integers p and q such that the number of vertices (i.e., of the morphological units) is N = 10T + 2, the number of triangles is N^sub T^ = 20T, and the number of subunits is N^sub S^ = 3N^sub T^, where T = p^sup 2^ + pq + q^sup 2^, the so-called T-number of the virus. For many virus particles, relatively few subunits are involved so that T and N^sub T^ are small.

Given the intrinsic strength of the virus capsids that follows from experiments (8) as well as from calculations of protein-protein interactions (19), it is important to probe the mechanical properties of viruses directly by means of singleparticle experiments. The main question is then how to relate the experimentally accessible observable to the elastic constants of the virus capsid. Recently, controlled experiments using scanning-force microscopy have been used to measure the mechanical properties of (empty) bacteriophage φ29 capsids (20). Similar measurements on hollow spherical polyelectrolyte capsules (21) were reported in Lulevich et al. (22) and Fery et al. (23).

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