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 forcedepth 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.

A theoretical model for F-actin remodeling in vascular smooth muscle cells subjected to cyclic stretch

“…Many families of CSK proteins contribute to overall cell stiffness, but F-actin tends to dominate in many cases (Pourati et al, 1998;Wu et al, 1998;Smith et al, 2003;Na et al, 2004;Huang et al, 2005) and thus was the focus herein. There is increasing information available on the elasticity of F-actin, including isolated filaments, networks having different degrees of crosslinks, and stress fibers (e.g., Liu and Pollack, 2002;Gardel et al, 2004;Deguchi et al, 2006).…”

“…Indeed, our previous modeling results showed that removal of intermediate filaments and microtubules reduced smooth muscle cell stiffness less than 10% (Na et al, 2004). Herein, therefore, we focus exclusively on F-actin for simplicity and illustrative purposes.…”

“…To compute Γ(W) and Σ(W) using the results of Humphrey et al (1991), we need the strain energy function W in terms of invariants I i of the right Cauchy-Green tensor C. Because the proposed strain energy function for CSK remodeling is written in terms of filament stretch α k (equation 6), which is related to C, we can easily find the requisite relations between derivatives with respect to the invariants I i and those in terms of the components of C (Na et al, 2004). These results are used without restatement here.…”

“…Detailed analysis of an AFM indentation superimposed on a finite equibiaxial stretch was described previously (Na et al, 2004). Briefly, the force versus indentation depth (P-θ) relationship is (Beatty and Usmani, 1975;Green et al, 1952):…”

Indentation Versus Tensile Measurements of Young's Modulus for Soft Biological Tissues

“…80 With very small indentations, the working end of the pyramid has been modeled as a sphere 81 especially when the sharp tip has been made blunt. [82][83][84] More typically, pyramidal indenters are assumed to be cones, especially in the case of cell indentation studies using AFMs. 67,85 We emphasize the pyramidal indenter here, due to a citation error, which has occurred when referencing the rigid cone solution.…”