Nonlinear Elasticity in the Interaction of Living Cells with their Mechanical Environment

Abstract

The elastic cytoskeleton contains molecular motors that produce mechanical forces by which cells attach to and pull on their surroundings. This mechanical interaction is responsible for many aspects of cellular function, from cell spreading and proliferation to stem-cell differentiation and tissue development. Both the cytoskeleton and the extracellular matrix comprise cross-linked, semi-flexible polymeric filaments, and as such they exhibit very nonlinear viscoelastic behavior that includes a power-law stiffening of the elastic moduli with increasing stress [1].Our theoretical work [2] is motivated by traction-force-microscopy experiments of cells that adhere to soft gels. These have shown that non-motile cells are dominated by force dipoles, comprised of equal and opposite contractile forces. However, the dependence of strain energy on the total dipole moment exhibits peculiar scaling laws which have not yet been explained [3].We consider active force-dipoles embedded in a nonlinear elastic medium, with constitutive relations inspired by fracture mechanics [4], which obey the strain-stiffening scaling laws of biopolymers. For strong nonlinearity, the differential shear modulus diverges at finite strain, and we may employ a small strain (but strongly nonlinear) expansion. We find that for a spherical force-dipole, strains change sign with distance, indicating that even around a contractile inclusion or molecular motor there is radial compression; it is only at long distance that one recovers the linear response in which the medium is radially stretched. The renormalization of the far-field strain field implies that the material's nonlinearity causes the active force dipole to be equivalent to one which is dramatically larger and stronger.[1] Gardel et al., Science 304, 1301 (2004).[2] Shokef and Safran, Phys. Rev. Lett. 108, 178103 (2012).[3] Pompe et al., Biophys. J. 97, 2154 (2009).[4] Knowles, Int. J. Fracture 13, 611 (1977).