# Nonlinear elasticity in biological gels

## Reference

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

## Keywords

elasticity, polymer gel

## 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.

In polymer theory, there are three types of filaments, characterized by the persistence length and the contour length. The persistence length is a measure of stiffness: for pieces of polymer shorter than the persistence length, the molecules can be approximated as a flexible rod; for pieces longer than the persistence length, a random walk is more appropriate to describe the mechanics of the polymer. The contour length is simply the length of the polymer if stretched to its full length. The first two types of filaments are filaments for which the persistence length dominates the contour length, and vice verse. The third is one for which the two length scales are comparable. These filaments do not form loops and knows, but they are flexible enough to have thermal bending fluctuations.

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.