# Nonlinear elasticity in biological gels

## Reference

Storm, C., Pastore, J.J., et al., Nature 435 (2005).

## Keywords

elasticity, polymer gel, polymer length scales

## Summary

This paper deals with elucidating the nonlinear elastic properties common to biological gels. As shown in Fig. 1, the shear moduli of various biological networks vary over orders of magnitude as a function of applied strain. The molecular structures responsible for the nonlinear elasticity are unknown, but the paper reports a molecular theory that accounts for the strain-stiffening in these biological networks.

The proposed model starts with the force of a single filament. To get from there to the bulk elastic properties of a network of filaments, several assumptions are made. First, the network is assumed to be isotropic, in which a pair of nodes are connected by independent semi-flexible filaments. It is also assumed that no torques are exerted at nodes, so that filaments can stretch or compress but cannot bend. In addition, the deformations are assumed to be affine. Lastly, all filament end-to-end lengths are assumed to be equal.

The experimental shear moduli along with the theoretical prediction is shown in Figure 2. Although the theoretical prediction agrees for low strains, it quickly becomes invalid at higher strains. To correct this, the authors eliminated the restriction that all filament end-to-end lengths are equal. Instead, they take a the distribution of end-to-end lengths to be the equilibrium distribution of end-to-end lengths of filaments of a given (constant) contour length. The results of this extended theory are shown in Figure 3, along with experimental results.

The extended theory shows better agreement with experiment, but there is still significant deviation, especially at higher strains. One could imagine that the reason for disagreement is the simplifying assumption that no torques are exerted at nodes. This assumption does not seem realistic, as TEM images have shown that filaments do indeed bend in response to external forces. Taking this into account would be difficult, but would probably give more accurate predictions of real biological systems.