Model for Stretching and Unfolding the Giant Multidomain Muscle Protein Using Single-Molecule Force Spectroscopy

Single-molecule manipulation has allowed the forced unfolding of multidomain proteins. Here we outline a theory that not only explains these experiments but also points out a number of difficulties in their interpretation and makes suggestions for further experiments. For titin we reproduce force-extension curves, the dependence of break force on pulling speed, and break-force distributions and also validate two common experimental views: Unfolding titin Ig domains can be explained as stepwise increases in contour length, and increasing force peaks in native Ig sequences represent a hierarchy of bond strengths. Our theory is valid for essentially any molecule that can be unfolded in atomic force microscopy; as a further example, we present force-extension curves for the unfolding of RNA hairpins.

Figure 1

Schematic comparing three titin domains in nature and in our model. (a) A realistic backbone structure for two folded and one partially unfolded titin domain based on NMR data [8]. The dimensions of a domain are $\sim 3.4\times 1.2\times 0.8\mathrm{nm}$; a completely unfolded domain has a contour length of 28.4 nm. (b) A model representation of the same three domains. Here thick (straight) lines represent an intact effective bond, and thin (curved) lines represent extensible segments.

Figure 2

Nonequilibrium theory of identical linked titin domains. (a) Nonequilibrium force-extension curves for four domains connected to a cantilever with a 200 nm linking segment. Solid curves show the effect of increasing pulling speed: (top to bottom) $v={10}^4$, ${10}^3$, and ${10}^2\mathrm{nm}/\mathrm{s}$. Dashed curves are force-extension traces for wormlike chains of increasing contour length. Parameters: $L_p=0.4\mathrm{nm}$, $L_c=28.4\mathrm{nm}$, $k_c=46\mathrm{pN}/\mathrm{nm}$, $Q(0)=960\mathrm{meV}$, and $w_0=2\times {10}^9{\mathrm{s}}^{-1}$. (b) Break (unfolding) force versus pulling speed for the extension of a single titin domain [compare with the rightmost peaks in panel (a)]. The agreement between theory (crosses) and experiment [17] (squares) is excellent. In equilibrium, domains unfold freely; thus, the unfolding force is guaranteed to vanish in the limit of slow pulling speed. (c) Theoretical break-force distributions for a single domain [compare again with the rightmost peaks in panel (a)]. Because unfolding events are probabilistic, a distribution of unfolding forces is observed experimentally [panels (a) and (b) show the most likely behavior]; here we present the full distribution of unfolding forces for three pulling speeds ($v={10}^2$, ${10}^3$, and ${10}^4\mathrm{nm}/\mathrm{s}$ from left to right). (d) Break-force distribution for $v=6\times {10}^2\mathrm{nm}/\mathrm{s}$. Both experimental [17] (squares) and theoretical data (solid curve) have been normalized to unity.

Figure 3

Survival of the fittest: Nonequilibrium force peaks of a native titin segment showing a hierarchy of bond-breaking barriers [$Q(0)=800–1300\mathrm{meV}$, other parameters as in Fig. 2]. Nonequilibrium theory (solid curve) is compared with experiment [1] (dashed curve) showing excellent agreement.

Figure 4

Force-extension curves obtained via our equilibrium theory with parameters typical for RNA hairpins: $V_0=1.5\mathrm{eV}$, $L_p=1.0\mathrm{nm}$, and $L_c=18\mathrm{nm}$ [25]. In both panels, dashed (solid) curves are obtained for a stiff (soft) cantilever, i.e., $k_c\gg (\ll )\partial f/\partial L$, within the Helmholtz (Gibbs) ensemble for an isolated molecule. For RNA hairpins, the soft and stiff limits are attainable with $k_c=0.01\mathrm{pN}/\mathrm{nm}$ and $k_c=10\mathrm{pN}/\mathrm{nm}$, respectively. (a) A typical force-extension curve for a single RNA hairpin showing the effect of finite cantilever stiffness: Dotted-dashed and dotted curves represent $k_c=1$ and $0.1\mathrm{pN}/\mathrm{nm}$, respectively. (b) Predicted force-extension curves for $N=4$ hairpins connected in series, for different linker lengths: 100 (left side) and 500 nm (right side).