Nonlinear and heterogeneous elasticity of multiply-crosslinked biopolymer networks

We simulate randomly crosslinked networks of biopolymers, characterizing linear and nonlinear elasticity under different loading conditions (uniaxial extension, simple shear, and pure shear). Under uniaxial extension, and upon entering the nonlinear regime, the network switches from a dilatant to contractile response. Analogously, under isochoric conditions (pure shear), the normal stresses change their sign. Both effects are readily explained with a generic weakly nonlinear elasticity theory. The elastic moduli display an intermediate super-stiffening regime, where moduli increase much stronger with applied stress sigma than predicted by the force-extension relation of a single wormlike-chain (G(wlc) similar to sigma(3/2)). We interpret this super-stiffening regime in terms of the reorientation of filaments with the maximum tensile direction of the deformation field. A simple model for the reorientation response gives an exponential stiffening, G similar to e(sigma), in qualitative agreement with our data. The heterogeneous, anisotropic structure of the network is reflected in correspondingly heterogeneous and anisotropic elastic properties. We provide a coarse-graining scheme to quantify the local anisotropy, the fluctuations of the elastic moduli, and the local stresses as a function of coarse-graining length. Heterogeneities of the elastic moduli are strongly correlated with the local density and increase with applied strain.