Hierarchical Chain Model of Spider Capture Silk Elasticity

Spider capture silk is a biomaterial with both high strength and high elasticity, but the structural design principle underlying these remarkable properties is still unknown. It was revealed recently by atomic force microscopy that an exponential force-extension relationship holds both for capture silk mesostructures and for intact capture silk fibers [N. Becker et al., Nat. Mater. 2, 278 (2003)]. In this Letter a simple hierarchical chain model was proposed to understand and reproduce this striking observation. In the hierarchical chain model, a polymer is composed of many structural motifs which organize into structural modules and supramodules in a hierarchical manner. Each module in this hierarchy has its own characteristic force. The repetitive patterns in the amino-acid sequence of the major flagelliform protein of spider capture silk is in support of this model.

Figure 1

The hierarchical chain model. At each hierarchy level $h$ a structural module $M_h$ is composed of a tandem sequence of $m_h$ submodules $M_{h+1}$ of hierarchy level $h+1$. Under external stretching, $M_h$ responds by (i) adjusting the arrangements of its $m_h$ subunits and making them more aligned along the force direction, and (ii) extending these $m_h$ subunits. The thick broken lines between submodules of each hierarchy level indicate the existence of sacrificial bonds.

Figure 2

Exponential force-extension relationship for the hierarchical chain model. Equation (4) is used in the numerical calculation. The parameters are $f_0={10}^{-4}\mathrm{N}$, $\alpha =0.3$, and $\beta =2$ (the upper curve) or $\beta =1.75$ (the lower curve). Extension is in units of $L_0$. Symbols are experimental data from Fig. 4 of Ref. 3.

Figure 3

The force-extension relationship of a hierarchical chain is insensitive to the assumption made to the response $\Delta x_h(f)$ of Eq. (4), but is sensitive to the hierarchical scaling form of the characteristic force $f_h$. The solid lines are obtained by assuming $\Delta x_h(f)$ has the form of Eq. (8), while other parameters are the same as those in Fig. 2. The dotted line shows the change in the force-extension curve when additionally a power-law form of $f_h=f_0{h}^{\gamma }$ with $\gamma =3.0$ is assumed for $f_h$. The dashed lines are obtained by assuming Eq. (8) and Eq. (5), with $\beta$ fluctuating uniformly within $[1.25,2.25]$ (the lower curve) and within $[1.5,2.5]$ (the upper curve).