Nonlinear finite-element analysis of nanoindentation of viral capsids

Recent atomic force microscope (AFM) nanoindentation experiments measuring mechanical response of the protein shells of viruses have provided a quantitative description of their strength and elasticity. To better understand and interpret these measurements, and to elucidate the underlying mechanisms, this paper adopts a course-grained modeling approach within the framework of three-dimensional nonlinear continuum elasticity. Homogeneous, isotropic, elastic, thick-shell models are proposed for two capsids: the spherical cowpea chlorotic mottle virus (CCMV), and the ellipsocylindrical bacteriophage $\varphi 29$. As analyzed by the finite-element method, these models enable parametric characterization of the effects of AFM tip geometry, capsid dimensions, and capsid constitutive descriptions. The generally nonlinear force response of capsids to indentation is shown to be insensitive to constitutive particulars, and greatly influenced by geometric and kinematic details. Nonlinear stiffening and softening of the force response is dependent on the AFM tip dimensions and shell thickness. Fits of the models capture the roughly linear behavior observed in experimental measurements and result in estimates of Young’s moduli of $\approx 280–360\mathrm{MPa}$ for CCMV and $\approx 4.5\mathrm{GPa}$ for $\varphi 29$.

Figure 1

Schematic of idealized capsid models. Dimensions are averaged from the results of structural studies of CCMV 4 and $\varphi 29$ 15.

Figure 2

(Color online) Finite-element meshes of CCMV (left) and $\varphi 29$ (right) capsids. Dimensions of the capsids are shown in Fig. 1. The AFM tip is modeled as a rigid hemispherical shell with $R=14\mathrm{nm}$, and the substrate as a rigid flat plate. (Capsids are shown not on the same scale.)

Figure 3

(Color online) Indentation of spherical model of CCMV with dimensions from Fig. 1, and a nonlinear Hookean constitutive response described by Eq. (2) with $E=250\mathrm{MPa}$ and $\nu =0.4$. Deformed capsid shapes are shown for several values of substrate displacement. Color contours indicate the von Mises stress.

Figure 4

Force response for indentation of spherical CCMV model from Fig. 3.

Figure 5

(Color online) At higher deformations, the apex of the CCMV capsid is observed to separate from the indenting AFM tip. Separation is preceded by a drop in contact pressure near the apex.

Figure 6

(Color online) Sensitivity to constitutive law. Contact force curves for linear and nonlinear Hookean, compressible Neo-Hookean, and compressible Mooney-Rivlin models. Models incorporating nonlinear strains are nearly coincident and differ significantly from the fully linear Hookean model. Representative capsid dimensions from Fig. 1 are used.

Figure 7

(Color online) Contact-force comparison for several AFM tip sizes. Representative capsid dimensions from Fig. 1 are used in combination with the same nonlinear Hookean constitutive description as in Fig. 3.

Figure 8

(Color online) Contact-force response of CCMV over a range of capsid thicknesses. (Top) Force vs AFM plate displacement for $E=250\mathrm{MPa}$ and $R_{\mathrm{av}}=11.8\mathrm{nm}$. (Bottom) Dimensionless renormalization of force response according to expected scaling from linearized thin-shell elasticity.

Figure 9

(Color online) Contact-force response of $\varphi 29$ over a range of thicknesses, showing the dimensionless renormalization of the force according to expected scaling from linear elasticity. The average of the two principal radii $(R=23.2\mathrm{nm})$ is used to scale the plate displacement.

Figure 10

(Color online) Indentation of ellipsocylindrical model of $\varphi 29$ with dimensions from Fig. 1, $E=1.5\mathrm{GPa}$, and $\nu =0.4$. Deformed capsid shapes are shown for several values of substrate displacement. Color contours indicate the von Mises stress.