Reconstructing the Free-Energy Landscape of a Mechanically Unfolded Model Protein

The equilibrium free-energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the “folding transition” we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce equilibrium properties of the examined system. Moreover, for the studied model the structural transitions induced by pulling can be related to thermodynamical aspects of folding.

Figure 1

Free-energy profile $f$ as a function of the end-to-end distance $\zeta$, obtained by Eq. (1) for various pulling velocities: from top to bottom $v_p=5\times {10}^{-2}$, $1\times {10}^{-2}$, $5\times {10}^{-3}$, $5\times {10}^{-4}$, and $2\times {10}^{-4}$. In the inset, an enlargement of the curve for $v_p=5\times {10}^{-4}$ at low $\zeta$ is reported. Each curve has been obtained by averaging over 160–240 repetitions of the same pulling protocol at $T=0.3$. The letters refer to the $f(\zeta )$ values for the configurations shown in Fig. 2 and the (blue) dashed lines to the STs’ locations.

Figure 2

Pulled configurations at $T=0.3$: the NC (a) has ${\zeta }_0\sim 1.9$; the others are characterized by $\zeta =6.8$ (b), $\zeta =16.8$ (c), and $\zeta =27.1$ (d).

Figure 3

Free-energy profile $f(\zeta )$ as obtained by Eq. (1) for various temperatures: $T=0.2$ (magenta stars), 0.4 (blue plus signs), 0.5 (red squares), 0.6 (green triangles), and 0.7 (orange circles). In the inset an enlargement is reported at small $\zeta$. The data refer to $v_p=5\times {10}^{-4}$.

Figure 4

Free-energy profiles $f(\zeta )$ and $f_{\mathrm{IS}}(\zeta )$ as a function of the elongation $\zeta$ for $T=0.3$. The black solid line refers to the reconstruction in terms of the EJE, while the red dashed one corresponds to $f_{\mathrm{IS}}$ for a set of pulling experiments with $v_p=2\times {10}^{-4}$. The blue dot-dashed line is the $f_{\mathrm{IS}}(\zeta )$ obtained in terms of the ISs of the TDB. In the insets are reported the reconstructed $V_{\mathrm{IS}}(\zeta )$ (lower panel) and $R_{\mathrm{IS}}(\zeta )$ (upper panel) by employing ISs in the PDB.