Investigation of an Idealized Virus Capsid Model by the Dynamic Elasticity Apparatus

Abstract

The three-dimensional dynamic theory of elasticity is applied to investigate the mechanical properties of virus capsid. The idealized model of the virus is based on the 3D boundary-value problem of mathematical physics formulated in spherical coordinate system for the steady-state oscillation process. The virus is modeled as a hollow elastic sphere filled by acoustic medium and is located in different acoustic medium. The stated boundary-value problem is solved with the help of the integral transform method and method of the discontinuous solutions. As a result, the exact solution of the problem is obtained. The numerical calculations of the virus elastic characteristics are carried out.The development of virus mathematical model is necessary to representing the effect of parameters variation on the behavior of a virus as a dynamic system. Such mathematical models based on the reasonable biological assumptions were obtained earlier within three main interdisciplinary approaches: 1) the hydrodynamic approach [23,21,30]; 2) the approach with the use of the numerical methods for solving the nonlinear problems of hydrodynamics and elasticity [16,18,28,29,36]; 3) the approach based on the linear elasticity models [15,34,36]. The mentioned models allowed to obtain many important characteristics of the virus, but they could not fully solve the problem of investigation of the virus as a 3D elastic object. In this paper the authors first propose to use the complete system of motion equations of linear elasticity for representation of virus wave field. It allows to take into consideration the virus 3D structure and to obtain its new qualitative characteristics.The morphology of icosahedral viruses ranges from highly spherical to highly faceted, and for some viruses a shape transition occurs during the viral life cycle. This phenomena is predicted from continuum elasticity, via the buckling transition theory by Nelson [22], in which the shape is dependent on the Foppl -von Karman number g , which is a ratio of the two-dimensional Young's modulus, Y , and the bending modulus ae : g= 2 / YR ae (R is the virus radius). However, until now, no direct calculations have been performed on atomic-level capsid structures to test the predictions of the theory.The elasticity and mechanical stability of empty and filled viral capsids under external force loading are studied in a combined analytical and numerical approach. Quantitative measurements of the mechanical response of nanosized protein shells (viral capsids) to large-scale physical deformations were reported in [29]. These measurements were compared with theoretical descriptions from continuum modeling and molecular dynamics. In [22] it was shown that the icosahedral packings of protein capsomeres proposed by Caspar and Klug for spherical viruses become unstable to faceting for sufficiently large virus size. A model, based on the nonlinear physics of thin elastic shells, produced excellent one-parameter fits in real space to the full three-dimensional shape of large spherical v...