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.

💡 Research Summary

The paper tackles the longstanding challenge of reconstructing the equilibrium free‑energy landscape (FEL) of a protein from non‑equilibrium mechanical unfolding data. Using an off‑lattice model consisting of 46 residues with Lennard‑Jones and bonded interactions, the authors simulate pulling experiments where the termini are separated at constant velocity, thereby generating force‑extension curves that serve as the raw non‑equilibrium work measurements.

Two conceptually distinct approaches are employed. The first is an extended Jarzynski equality (EJE). The classic Jarzynski relation states that the exponential average of the work performed along many non‑equilibrium trajectories equals the equilibrium free‑energy difference. The authors adapt this formula to the case of a time‑dependent external force that directly controls the reaction coordinate (the end‑to‑end distance). For each pulling trajectory they compute the work (W = \int F_{\text{ext}},dx) and then evaluate (\Delta F(x) = -k_{\mathrm B}T \ln \langle e^{-\beta W}\rangle) as a function of the instantaneous distance (x). By repeating the pulling at several temperatures (0.25, 0.30, 0.35 ε/k_B) and at different pulling speeds, they obtain a statistically converged estimate of the FEL.

The second approach is based on the inherent‑structure (IS) formalism. In this framework the potential‑energy surface is partitioned into basins surrounding each local minimum (the inherent structures). Each basin contributes a term (e^{-\beta E_i}) multiplied by a vibrational factor (\Omega_i(\beta)) that accounts for harmonic fluctuations around the minimum. The authors generate a large ensemble of ISs by quenching configurations sampled during equilibrium molecular‑dynamics runs, compute the Hessian matrix for each minimum, and thus evaluate the vibrational entropy. The free energy as a function of the reaction coordinate is then reconstructed by summing the contributions of all ISs whose end‑to‑end distance falls within a narrow bin.

The central result is a striking quantitative agreement between the FEL obtained from EJE and that derived from the IS analysis, especially in the temperature window surrounding the folding transition temperature (T_f \approx 0.30,\epsilon/k_B). Both methods reveal two prominent free‑energy barriers: the first at an end‑to‑end distance of roughly (5\sigma), corresponding to the disruption of an α‑helical segment; the second near (12\sigma), associated with the formation or breaking of β‑sheet contacts. The barrier heights decrease with increasing temperature, reflecting the expected softening of the landscape in the unfolded regime.

Beyond the methodological validation, the study provides physical insight into how mechanical pulling probes the intrinsic thermodynamics of folding. The structural transitions observed during pulling map directly onto thermodynamic signatures (e.g., peaks in the specific heat) that characterize the equilibrium folding–unfolding transition. Consequently, the work demonstrates that non‑equilibrium single‑molecule experiments, when combined with rigorous statistical‑mechanical analysis, can faithfully reconstruct equilibrium free‑energy profiles and elucidate the underlying folding mechanisms.

In conclusion, the paper establishes that (i) the extended Jarzynski equality is a reliable tool for extracting equilibrium FELs from a realistic number of pulling trajectories, (ii) the inherent‑structure formalism provides an independent, structure‑based route to the same information, and (iii) mechanical unfolding experiments can be interpreted in terms of the protein’s intrinsic thermodynamic landscape, thereby bridging the gap between non‑equilibrium single‑molecule measurements and equilibrium folding theory.