Fluctuating Nonlinear Spring Model of Mechanical Deformation of Biological Particles

The mechanical properties of virus capsids correlate with local conformational dynamics in the capsid structure. They also reflect the required stability needed to withstand high internal pressures generated upon genome loading and contribute to the success of important events in viral infectivity, such as capsid maturation, genome uncoating and receptor binding. The mechanical properties of biological nanoparticles are often determined from monitoring their dynamic deformations in Atomic Force Microscopy nanoindentation experiments; but a comprehensive theory describing the full range of observed deformation behaviors has not previously been described. We present a new theory for modeling dynamic deformations of biological nanoparticles, which considers the non-linear Hertzian deformation, resulting from an indenter-particle physical contact, and the bending of curved elements (beams) modeling the particle structure. The beams’ deformation beyond the critical point triggers a dynamic transition of the particle to the collapsed state. This extreme event is accompanied by a catastrophic force drop as observed in the experimental or simulated force (F)-deformation (X) spectra. The theory interprets fine features of the spectra, including the nonlinear components of the FX-curves, in terms of the Young’s moduli for Hertzian and bending deformations, and the structural damage dependent beams’ survival probability, in terms of the maximum strength and the cooperativity parameter. The theory is exemplified by successfully describing the deformation dynamics of natural nanoparticles through comparing theoretical curves with experimental force-deformation spectra for several virus particles. This approach provides a comprehensive description of the dynamic structural transitions in biological and artificial nanoparticles, which is essential for their optimal use in nanotechnology and nanomedicine applications.Author Summary: Dynamic force experiments, which have become available to explore the physical properties of biological assemblies, oftentimes reveal results that are difficult to understand without theoretical framework. We employed a multiscale modeling approach—a combination of Molecular Dynamics simulations of atomic structures with Langevin simulations of coarse-grained models of virus shells—to characterize the degrees of freedom defining the deformation and structural collapse of biological particles tested mechanically. This enabled us to develop an analytical model that provides meaningful interpretation of force-deformation spectra available from single-particle nanoindentation experiments. The Fluctuating Nonlinear Spring (FNS) model of uniaxial particle’s deformation captures essential features of the force-deformation spectra as observed in nanomanipulations in vitro and in silico: initial non-linearity, then a subsequent force decrease transition due to structural collapse. Our theory uniquely combines the elements of continuum mechanics with the statistics of extremes, enabling one to gather mechanical and statistical characteristics of nanoparticles, which determine the Hertzian deformation of the particle’s protein layer, and bending deformation and structural damage to the particle structure. We have demonstrated how the FNS theory can accurately model the deformation of several viral shells, showing promising model applications for describing a variety of natural and synthetic nanoparticles.