Alternative Explanation of Stiffening in Cross-Linked Semiflexible Networks

Strain stiffening of filamentous protein networks is explored by means of a finite strain analysis of a two-dimensional network model of cross-linked semiflexible filaments. The results show that stiffening is caused by nonaffine network rearrangements that govern a transition from a bending-dominated response at small strains to a stretching-dominated response at large strains. Filament undulations, which are key in the existing explanation of stiffening, merely postpone the transition.

Figure 1

Average stress ($\overline{\tau }$) versus strain ($\Gamma$) response for networks with ${\overline{l}}_{\mathrm{b}}=2.3\times {10}^{-4}$ and $\mu /L=1.6\mathrm{MPa}$ at three different densities $\overline{\rho }$ and for straight (${\overline{l}}_{\mathrm{p}}\gg 1$) and undulated (${\overline{l}}_{\mathrm{p}}=1$) filaments. The error bars have a length of 2 times the standard deviation in strain at a given $\Gamma$ for ten different realizations at each density. The squares correspond to three instances during deformation for which the network geometry is shown in Fig. 2.

Figure 2

(a) Initial, $\Gamma =0$; (b) intermediate, $\Gamma =0.08$, and (c) large strain, $\Gamma =0.24$, network configurations for a typical $\overline{\rho }=13$ realization close to the average response shown in Fig. 1 (squares). For an animation see Ref. 25.

Figure 3

Initial conformation of a $\overline{\rho }=13$ network with undulated filaments corresponding to ${\overline{l}}_{\mathrm{p}}=1$.

Figure 4

The deviation from affine behavior, $\Delta A$, as a function of strain, $\Gamma$, for ${\overline{l}}_{\mathrm{b}}=2.3\times {10}^{-4}$ and $\mu /L=1.6\mathrm{MPa}$, at three different densities and for straight (${\overline{l}}_{\mathrm{p}}\gg 1$) and undulated (${\overline{l}}_{\mathrm{p}}=1$) filaments.

Figure 5

Shear stiffness as a function of strain for networks with straight filaments having ${\overline{l}}_{\mathrm{b}}=2.3\times {10}^{-4}$ or $8.3\times {10}^{-4}$.