Theory of force-extension curves for modular proteins and DNA hairpins

We study a model describing the force-extension curves of modular proteins, nucleic acids, and other biomolecules made out of several single units or modules. At a mesoscopic level of description, the configuration of the system is given by the elongations of each of the units. The system free energy includes a double-well potential for each unit and an elastic nearest-neighbor interaction between them. Minimizing the free energy yields the system equilibrium properties whereas its dynamics is given by (overdamped) Langevin equations for the elongations, in which friction and noise amplitude are related by the fluctuation-dissipation theorem. Our results, both for the equilibrium and the dynamical situations, include analytical and numerical descriptions of the system force-extension curves under force or length control and agree very well with actual experiments in biomolecules. Our conclusions also apply to other physical systems comprising a number of metastable units, such as storage systems or semiconductor superlattices.

Figure 1

Normalized FECs for $N=8$ (solid black) and $N=20$ (dashed red). Zero length corresponds to having half of the units unfolded at $F=F_c$. There are $N+1$ branches in the metastability region $|\phi /{\phi }_0|<1$, with the number of unfolded units, $J$, increasing from left to right. The first $\left(J=0\right)$ and last $\left(J=N\right)$ branches are independent of $N$. Note that the branches become denser as $N$ increases, and also the up-down and left-right symmetry thereof. These symmetries stem from the simple form of the Landau-like free energy (18), and thus they are not present if a more realistic potential is considered; see Appendix pp1.

Figure 2

(top) First-order transition in the length for a quasistatic increase of the force. We use different colors for the stable parts of the branches (solid black) and the metastable parts (dashed red). The first branch $J=0$ is swept until the critical force $\phi =0$ is reached. Then all the modules unfold simultaneously and the system goes directly (arrow) to the completely unfolded branch $J=N=20$ (bottom). Hysteresis cycles under force-controlled conditions for a $N=20$ system. The lines correspond to simulations of the Langevin equations (6) for the temperature $T=0.02$ and different rates of variation of the force, namely $|dF/dt|=3\times {10}^{-k}$, with $k=2$ (dot-dashed red), $k=3$ (dotted green), $k=4$ (solid blue), and $k=5$ (dashed orange). The same rates of variation of the force are considered for the very low temperature $T=2\times {10}^{-5}$. All the curves are superimposed and thus they are plotted with the same symbols (black dots). For the higher temperature, the area of the hysteresis cycle decreases with the rate, approaching the behavior for a quasistatic process.

Figure 3

(top left) Equilibrium force rips in the $F-L$ curve for a system with $N=8$ domains. We use different colors for the stable parts of the branches (solid black), metastable parts (dotted red), and the force rips (black arrows). The system follows the solid black curve in a quasistatic pulling process, with a series of first-order transitions in the force (marked by the arrows). At the $J\mathrm{th}$ transition, the force changes from $f_J^{-}$ [over the $(J-1)\mathrm{st}$ branch] to $f_J^{+}$ (over the $J\mathrm{th}$ branch). These forces $f_J^{\pm }$ increase with the number of unfolded units, as observed in AFM experiments with modular proteins, even though all the units are perfectly identical in the model. (top right) Hysteresis cycle for a system composed of $N=8$ modules. The dimensionless temperature $T=0.02$, and the rate of variation of the length is $|dL/dt|=1.2\times {10}^{-3}$ ($\dot{L}>0$, solid blue; $\dot{L}<0$, dashed red). (bottom left) The same as in the top-right panel, but for a smaller rate $|dL/dt|=1.2\times {10}^{-6}$. Aside from thermal fluctuations, the system almost sweeps the equilibrium curve. (bottom right) The same plot as in the bottom-left panel, but for $T=2\times {10}^{-5}$. Thermal fluctuations are so small that the system approaches the $T=0$ behavior, in which the branches are swept up to the end of the metastability region.

Figure 4

(top panel) Zoom of the metastability region for two chains with $N=8$ (solid line) and $N=20$ (dashed line). The size of the rips decreases as the number of units $N$ increases, and it vanishes as $N\to \infty$. (bottom panel) Decrease of the force rips with the number of units $N$. We plot the size of the rips $\mathrm{\Delta }F=f_J^{-}-f_J^{+}$ for the “central” transition with $J=(M+1)/2$, scaled with the factor $N/F_c$ (circles). The limiting value $6\sqrt{3}$, which represents the power law behavior given Eq. (29), is shown with a solid line. It is observed that the system approaches rapidly this asymptotic behavior, being very close to it for $N\gtrsim 20$.

Figure 5

Stable stationary wavefront with increasing profile from $u^{\left(1\right)}$ to $u^{\left(3\right)}$ (corresponding to ${\eta }^{\left(1\right)}$ and ${\eta }^{\left(3\right)}$, respectively) pinned at a particular point $j=J$ of an infinitely long chain, for $k=1.615$. The specular reflection of this pinned wave with respect to the center of the chain $j=N/2$ gives a pinned wave with decreasing profile from ${\eta }^{\left(3\right)}$ to ${\eta }^{\left(1\right)}$.

Figure 6

FECs for a system with $N=8$ modules. (top) Stable stationary branches for $k=0.055$, quite similar to those for $k=0$ (see Fig. 1). (bottom) Stable stationary branches, each corresponding to a wavefront pinned at a different site $j=J,J=1,...,8$, for $k=0.55$. The completely folded and unfolded branches are basically unchanged, but the size of the intermediate branches is considerably reduced. Here $L$ refers to the physical length (24b) that vanishes at $F=0$.

Figure 7

Second derivative of the potential for the folded unit at the domain wall up to order $k$, Eq. (39), as a function of the normalized force $\phi /{\phi }_0$. It is clearly seen that the stable part of the branch, with $a^{\prime \prime }>0$, decreases with $k$. For $k=0.5$ the size of the branch is reduced by $20\%$, approximately, from its maximum size $\left(k=0\right)$, consistently with the behavior observed in Fig. 6.

Figure 8

(top) FEC obtained by adiabatically increasing the length of the system. The local maxima of the branches are close to the corresponding upper ends of the equilibrium branches in Fig. 6. (bottom) Hysteresis loop obtained by adiabatic force sweeping the force-extension diagram of a chain of identical units at zero temperature. In both cases, $k=0.55$. $L$ is the physical length that vanishes at $F=0$, defined in Eq. (24b).

Figure 9

FEC for a DNA hairpin as in Fig. 5 but with $N=40$ and disorder as in Eq. (40) (strength of the potential with $\beta =0.5$ and $k=1$). (top) Hysteresis under length-controlled conditions. The upper (lower) curve corresponds to adiabatically increasing (decreasing) length, with a rate $|dL/dt|=1.2\times {10}^{-3}$ or smaller. (bottom) Hysteresis under force-controlled conditions. Similarly, the upper (lower) curve corresponds to adiabatically increasing (decreasing) force, with a rate $|dF/dt|=3\times {10}^{-3}$ or smaller. In the plots, $L$ is the physical length introduced in Eq. (24b).

Figure 10

Same as in Fig. 9 but with additional white noise of temperature $T=0.02$. Thermal noise may suppress and blur solution branches. In both panels, the dashed curves correspond to the same rate as in Fig. 9, while in the solid ones the rate has been reduced by a factor ${10}^{-3}$. Again, $L$ is the physical length introduced in Eq. (24b).

Figure 11

FECs for the BGMKUF potential, with $N=8$ (solid red) and $N=15$ (dashed blue). There are $N+1$ branches in the metastability region $F_m<F<F_M$, with the number of unfolded units $J$ increasing from left to right. The first $\left(J=0\right)$ and last $\left(J=N\right)$ branches are independent of $N$, as in Fig. 1. Note the asymmetry of the branches with respect to the critical force $F_c$ (dot-dashed line).

Figure 12

Unfolding-refolding cycle for a modular protein with eight units with free energies given by the BGMKUF potential. The dot-dashed lines correspond to the equilibrium branches of the FEC; see Fig. 11. There are sharp force rips in the unfolding process, each corresponding to the unfolding of one of the units (jumps between neighboring branches). In the refolding process, there are no sharp peaks until the length has almost completely relaxed, $L\lesssim 0.2$.

Figure 13

Unfolding-refolding cycle for a modular protein with eight units with free energies given by the BGMKUF potential and a finite value of the cantilever stiffness. The force $F$ is plotted against the end-to-end distance of the molecule $L\left(\mathbf{\eta }\right)$. The equilibrium branches of the FEC (see Fig. 11) are the dot-dashed lines. Both the unfolding and refolding curves are very similar to those in Fig. 12, except for the force rips in the unfolding process not being perfectly vertical as a consequence of the imperfect length control.