Reconstructing the free energy landscape of a polyprotein by single-molecule experiments

The mechanical unfolding of an engineered protein composed of eight domains of Ig27 is investigated by using atomic force microscopy. Exploiting a fluctuation relation, the equilibrium free energy as a function of the molecule elongation is estimated from pulling experiments. Such a free energy exhibits a regular shape that sets a typical unfolding length at zero force of the order of 20 nm. This length scale turns out to be much larger than the kinetic unfolding length that is also estimated by analyzing the typical rupture force of the molecule under dynamic loading.

💡 Research Summary

In this work the authors combine atomic‑force‑microscopy (AFM) based single‑molecule force spectroscopy with an extended version of the Jarzynski equality (JE) to reconstruct the equilibrium free‑energy landscape (FEL) of a polyprotein composed of eight tandem repeats of the Ig27 domain from human titin. The experimental protocol consists of pulling the protein from a gold substrate with a constant velocity ranging from 200 to 2000 nm s⁻¹ while recording the force‑extension curves. The characteristic saw‑tooth pattern of the curves reflects the sequential unfolding of individual Ig27 modules; the distance between successive peaks is 22–26 nm, consistent with the known contour length increase of a single domain.

First, the authors analyse the rupture force (f*) as a function of loading rate using the Bell–Evans kinetic model (Eq. 2). By fitting the data they obtain a kinetic transition length xu = 0.30 ± 0.07 nm and a zero‑force unfolding time τ₀, values that agree with previous measurements (≈0.25 nm). This xu is interpreted as a purely kinetic parameter describing the distance over which the force must act to trigger the transition, not the physical distance to the free‑energy barrier.

Second, they apply the extended JE (Eq. 3) to the non‑equilibrium work performed on the molecule during pulling. Because the exponential average ⟨e⁻ᵝᴡ⟩ becomes numerically unstable when the work exceeds a few hundred kBT, they introduce a constant shift Δ (0, 1000, 2000, 3000, 4000 kBT) and evaluate ⟨e⁻ᵝ(W+Δ)⟩, thereby reconstructing the FEL piecewise. For the slowest pulling speeds (200 and 400 nm s⁻¹) the resulting FELs collapse onto a single curve, confirming that the method yields a pulling‑rate‑independent equilibrium quantity. At the highest speed (2000 nm s⁻¹) deviations appear at larger extensions, reflecting the increasing distance from equilibrium.

The reconstructed FEL, F₀(ℓ), displays a single minimum at ℓ = 0 (the folded state) and a series of equally spaced cusps separated by Δℓ ≈ 20 nm. Each cusp corresponds to the unfolding of one Ig27 domain. When a constant external force f is applied, the landscape tilts according to F(ℓ,f) = F₀(ℓ) − fℓ. At f ≈ 50 pN the second minimum becomes deeper than the first, indicating that under a steady force more than one domain can be unfolded simultaneously. The distance from the folded minimum to the first cusp at zero force (≈20 nm) is two orders of magnitude larger than the kinetic xu, highlighting that xu and the physical barrier distance are distinct quantities.

The authors compare the average work required to unfold a single domain (⟨W*⟩ ≈ 405 kBT) with the free‑energy value at the corresponding extension (F₀(ℓ*) ≈ 200 kBT). Since ⟨W*⟩ > F₀(ℓ*), the JE provides a more accurate estimate of the equilibrium free energy than a direct work measurement, as expected from its theoretical foundation.

In conclusion, the study demonstrates that AFM pulling combined with the extended Jarzynski equality can map the full equilibrium free‑energy landscape of a multi‑domain protein over the entire range of molecular extension. It also clarifies the conceptual difference between the kinetic transition length xu obtained from rupture‑force analysis and the physical distance to the free‑energy barrier inferred from the FEL. The authors suggest that the same methodology can be extended to heterogeneous proteins composed of different domains, offering a powerful tool for quantitative thermodynamic characterization of biomolecules at the single‑molecule level.