On atomic force microscopy and the constitutive behavior of living cells

Abstract

Atomic force microscopy (AFM) is one of many new technologies available to study the mechanical properties and mechanobiological responses of living cells. Despite the widespread usage of this technology, there has been little attempt to develop new theoretical frameworks to interpret the associated data. Rather, most analyses rely on the classical Hertz solution for the indentation of an elastic half-space within the context of linearized elasticity. In contrast, we propose a fully nonlinear, constrained mixture model for adherent cells that allows one to account separately for the contributions of the three primary structural constituents of the cytoskeleton. Moreover, we extend a prior solution for a small indentation superimposed on a finite equibiaxial extension by incorporating in this mixture model for the special case of an initially random distribution of constituents (actin, intermediate filaments, and microtubules). We submit that this theoretical framework will allow an improved interpretation of indentation force-depth data from a sub-class of atomic force microscopy tests and will serve as an important analytical check for future finite element models. The latter will be necessary to exploit further the capabilities of both atomic force microscopy and nonlinear mixture theories for cell behavior.

Fig. 1

Deflection image of a vascular smooth muscle cell isolated from the rat skeletal muscle arterioles. The AFM probe was scanned across the cell surfaces at a speed of 40 µm/s, with a tracking force of approximately 400 pN. The image was collected using Nanoscope IIIa Software

Fig. 2

Local spherical coordinate system used for microstructurally motivated, phenomenological constitutive relations of a living cell. The arrow represents the orientation of an individual filament belonging to any of the three primary families of cytoskeletal filaments. The function R^k(ϕ, θ) quantifies the distribution of all such orientations

Fig. 3

Four configurations for an adherent cell, including an assumed materially isotropic cell in a non-adherent, traction-free reference configuration. Adherence and active spreading likely change the material symmetry from isotropic to anisotropic via an affine deformation-dependent mechanism and possibly active remodeling (not considered explicitly)

Fig. 4a, b

Illustrative (possible) distributions of cytoskeletal filaments given by a von Mises–Fisher distribution function (see Eq. 34). a κ=3, β= 0, γ=π/4. b κ=7, β=0, γ=π/4

Fig. 5a, b

Combined out-of-plane indentation and in-plane equibiaxial stretch for a exponential-type filaments and b linear-type filaments, each indented by a flat-ended circular cylinder with radius 30 nm. Each line corresponds to different in-plane stretches, from lower to upper, of μ=1.00 to 1.12 in steps of 0.02. All results are non-dimensional to emphasize the qualitative responses, not the specific values

Fig. 6a, b

Combined indentation and equibiaxial in-plane stretch, μ=1.2, for the exponential-type filament behavior. a Effects of four different indenter tips: flat-ended circular cylinder with radius a=30 nm (dash-dotted line, which is linear), sphere with radius a=30 nm (dashed line), cone with tip angle 2Φ=75° (dotted line), and blunted cone with tip angle 2Φ=75° and radius a=30 nm (solid line). The nonlinear responses (for all but the flat-ended indenter) are due, in part, to the nonlinearly increasing contact area between the indenter and cell. b Effects of cytoskeleton disrupting drugs with blunted cone, with 2Φ=75° and a=30 nm: control (solid line), actin filament disrupting drugs (dotted line for disruption of all actin filaments, and dash-dotted line disruption of half the actin filaments), and microtubule and intermediate filament disrupting drugs (dashed line). We assume that, if the filament is disrupted completely, the mass fraction φ of the corresponding filament is zero. Note that the predicted behavior is dominated by the actin. All results are non-dimensional to emphasize the qualitative responses, not the specific values