Nonlinear Elasticity of Composite Networks of Stiff Biopolymers with Flexible Linkers

Motivated by recent experiments showing nonlinear elasticity of in vitro networks of the biopolymer actin cross-linked with filamin, we present an effective medium theory of flexibly cross-linked stiff polymer networks. We model such networks by randomly oriented elastic rods connected by flexible connectors to a surrounding elastic continuum, which self-consistently represents the behavior of the rest of the network. This model yields a crossover from a linear elastic regime to a highly nonlinear elastic regime that stiffens in a way quantitatively consistent with experiment.

Figure 1

Schematic figure of an isotropic stiff polymer network with highly compliant cross-linkers. The inset illustrates the proposed nonuniform deformation of the cross-linkers on a single filament in a sheared background medium.

Figure 2

The shear modulus $G$ normalized by the linear modulus $G_0$ as a function of strain $ϵ$ (1D) or $\gamma$ (3D) for simple cross-linkers and WLC cross-linkers in the 1D and 3D versions of the linear medium hairy rod model. In this plot we have chosen $K_{\mathrm{EM}}=100k_{\mathrm{cl}}$ as an example. The upper left inset shows the force-extension curve of a simple cross-linker (dashed blue curve) and of a WLC cross-linker (solid black curve). The lower right inset shows the normalized shear modulus as a function of strain $\gamma$ for various ratios of $L/{\ell }_0$ calculated with the self-consistent model.

Figure 3

Differential modulus $K=d\tau /d\gamma$ normalized by the linear modulus $G_0$ as a function of stress $\sigma$ normalized by ${\sigma }_c$ for the self-consistent (Self.-Con.) model. We also plot $K/G_0$ for the linear medium (Lin. Med.) model and a model for rigidly cross-linked semiflexible polymer networks (Rig. C.-L.). The black line indicates a slope of 1. The inset shows the reduced tension profile $\varphi$ along the rod, normalized by the midpoint tension $\tau$ in Eqs. (4, 5).