Strength limit of entropic elasticity in beta-sheet protein domains

Elasticity and strength of individual beta-sheet protein domains govern key biological functions and the mechanical properties of biopolymers including spider silk, amyloids, and muscle fibers. The worm-like-chain (WLC) model is commonly used to describe the entropic elasticity of polypeptides and other biomolecules. However, force spectroscopy experiments have shown pronounced deviations from the ideal WLC behavior, leading to controversial views about the appropriate elastic description of proteins at nanoscale. Here we report a simple model that explains the physical mechanism that leads to the breakdown of the WLC idealization in experiments by using only two generic parameters of the protein domain, the H-bond energy and the protein backbone’s persistence length. We show that a rupture initiation condition characterized by the free energy release rate of H-bonds characterizes the limit of WLC entropic elasticity of beta-sheet protein domains and the onset of rupture. Our findings reveal that strength and elasticity are coupled and cannot be treated separately. The predictions of the model are compared with atomic force microscopy experiments of protein rupture.

Figure 1

(Color online) Example force extension profile of a beta-sandwich protein structure, as obtained from experimental analyses (rupture marked with an “×”) [11]. Force-extension profiles of I27 domain shown in subplot (a) reveal a two-step process of unfolding, corresponding to rupture of two separate clusters of H-bonds (plot redrawn based on data from Ref. 11). The curves reveal that the WLC model alone is not capable of describing the entire deformation range, and show that there exist a transition from one WLC curve to a second one at a force level of approximately $110\mathrm{pN}$ (that is, two sets of WLC parameters are required to describe the observed behavior, where the point of transition from one to the other cannot be predicted).

Figure 2

(Color online) Schematic representation of model setup used in the beta-strand strength model (BSSM). Subplot (a) illustrates the characteristic structure of a beta-sheet protein domain. Subplot (b) shows a simple beta-sheet structure, the fundamental building block of larger beta-sheet rich protein structures. Subplot (c) illustrates the schematic of the BSSM consisting of a single polypeptide stabilized by a linear array of H-bonds with a free end at the left-hand side (corresponding to the unattached random coil protein part). The part of the polypeptide chain that is attached to the substrate (part drawn as straight line) is assumed to be relaxed before rupture occurs.

Figure 3

(Color online) BSSM model predictions and direct comparison with experimental results of rupture mechanics of the titin I27 domain. Subplot (a) shows the elastic curve and peak force for different sizes of H-bond (“HB”) clusters, with rupture marked by an “×.” Extension and relaxation processes are described by the WLC equation [Eq. (1)], and the fracture force is predicted by Bell theory [Eq. 2] or for larger clusters by the Griffith energy balance condition [Eq. (3)]. At rupture, the force remains constant and extension is due to the increased contour length released by the rupture of H-bonds. The three force regimes predicted by Eqs. (4, 5) are illustrated. Subplot (b) shows the rupture strength of a beta sheet (point of deviation from the WLC model), as a function of the number of interstrand H-bonds, $N$. While the Bell model (homogeneous shear assumption) predicts a continuous increase in force with increasing cluster size, BSSM predicts saturation after three hydrogen bonds in agreement with experimental data from I27 [geometry shown in Fig. 1b]. With a minimal number of constant parameters (persistence length of the polypeptide ${\xi }_P$ and H-bond dissociation energy $E_{\mathrm{HB}}$) and structural information (H-bond cluster size $N$), BSSM is able to predict the strength of different domains of a beta protein during an unfolding experiment. The dotted circle predicts the strength according to the Bell model; apparently it is much too large compared with the experimental observation. Subplot (c) illustrates the force-extension behavior of I27 under external loading, experimental data [11] (inset) and direct comparison with elastic behavior predicted by BSSM theory (blue and red curves). We note that the length scales in the experimental results [shown in the inset of (c)] have been normalized by the number of tandem repeats stretched experimentally, to obtain the extension length scales corresponding to rupture of a single domain. The shaded area corresponds to the energy dissipated by the rupture of two H-bonds. Note the indication of points $A$ and $B$ in this plot and in subplot (b), for the geometry shown in Fig. 1b.

Figure 4

(Color online) Comparison of the height of the overall energy barrier as a function of the number of H-bonds in a beta strand, $N$. Calculation of the height of the effective energy barrier for six H-bonds loaded in shear, based on experimental data, Bell theory, BSSM, and MD simulation results are shown in subplot (a). The results show close agreement between our model, experimental and computational predictions. As predicted by BSSM, the maximum height of the energy barrier that can be achieved by a cluster of H-bonds is limited, and therefore the Bell prediction fails for large clusters. Subplot (b) illustrates that Bell model overshoots experimentally observed energy barriers significantly (point shown as dotted square), for cases beyond the critical cluster size $N_{\mathrm{cr}}$. The experimental value observed suggests an asymptotical behavior with constant energy barrier for $N>N_{\mathrm{cr}}$. The BSSM prediction agrees well with this observation (points $A$ and $B$ are indicated to relate to the results shown in Fig. 3).