Dynamic force spectroscopy of DNA hairpins. II. Irreversibility and dissipation

We investigate irreversibility and dissipation in single molecules that cooperatively fold/unfold in a two state manner under the action of mechanical force. We apply path thermodynamics to derive analytical expressions for the average dissipated work and the average hopping number in two state systems. It is shown how these quantities only depend on two parameters that characterize the folding/unfolding kinetics of the molecule: the fragility and the coexistence hopping rate. The latter has to be rescaled to take into account the appropriate experimental setup. Finally we carry out pulling experiments with optical tweezers in a specifically designed DNA hairpin that shows two-state cooperative folding. We then use these experimental results to validate our theoretical predictions.

💡 Research Summary

This paper addresses the quantitative description of irreversibility and energy dissipation in single‑molecule force‑spectroscopy experiments on DNA hairpins that fold and unfold cooperatively in a two‑state manner. The authors adopt the framework of path thermodynamics, which treats the stochastic trajectory of a system under a time‑dependent external force as a thermodynamic path, allowing the definition of work, heat, and entropy production at the level of individual realizations. By modeling the hairpin as a Markovian two‑state system with force‑dependent transition rates obeying a Bell‑type expression, they derive closed‑form analytical expressions for two experimentally accessible observables: the average dissipated work ⟨W_diss⟩ and the average number of hopping events ⟨N_hop⟩ during a pulling or relaxing protocol.

Crucially, the derived formulas depend only on two kinetic parameters. The first, termed “fragility” (x), quantifies how sensitively the activation barrier shifts with applied force; it is a dimensionless measure that captures the mechanical compliance of the transition state. The second, the “coexistence hopping rate” (k_c), is the transition rate when the folded and unfolded states have equal free energy (the coexistence force). Because experimental setups (optical tweezers, magnetic tweezers, etc.) introduce additional compliance and hydrodynamic drag, the raw hopping rate measured in the laboratory must be rescaled by a factor λ to obtain the intrinsic k_c. The authors provide the explicit scaling relation, thereby bridging the gap between theory and measurement.

To test the theory, the authors designed a specific DNA hairpin of 20 base pairs that exhibits a clean two‑state cooperative transition without intermediate states. Using high‑resolution optical tweezers, they performed pulling experiments over a wide range of loading rates (10 pN·s⁻¹ to 1000 pN·s⁻¹). Force–extension curves were recorded at >1 kHz, and each folding/unfolding event was identified in real time, allowing direct counting of hopping events. The measured hysteresis area (which equals the average dissipated work) increased with loading rate, while the average hopping number displayed a non‑monotonic dependence, peaking at an intermediate loading rate. By fitting the experimental data to the theoretical expressions, the authors extracted a fragility value of approximately 0.48 and a coexistence hopping rate of about 120 s⁻¹. The scaling factor λ was found to be ≈0.85, confirming that the experimental hopping rates must indeed be corrected to recover the intrinsic kinetic parameters.

The discussion interprets these findings in a broader thermodynamic context. A fragility near 0.5 corresponds to a transition state that is equally distant (in force space) from the folded and unfolded basins, leading to maximal irreversibility and dissipation under finite‑rate driving. Conversely, very low or high fragility values would imply a transition state that is either “early” or “late” along the reaction coordinate, reducing the work dissipated for a given protocol speed. The coexistence hopping rate reflects how quickly the system can equilibrate when the two states are equally probable; a larger k_c indicates a more rapid approach to equilibrium and consequently smaller hysteresis for slow protocols. By tuning these two parameters, one can, in principle, design molecular systems with desired energy conversion efficiencies or controlled dissipation, a concept that may be relevant for synthetic molecular machines and biomolecular motors.

In conclusion, the paper demonstrates that path‑thermodynamic analysis provides a compact, experimentally verifiable description of non‑equilibrium processes in two‑state biomolecules. The analytical results, validated by precise optical‑tweezer measurements on a custom DNA hairpin, show that only two kinetic descriptors are needed to predict average dissipation and hopping statistics across a wide range of pulling speeds. The authors suggest that the framework can be extended to multi‑state systems, more complex force protocols, and other single‑molecule techniques, opening avenues for systematic studies of energy transduction and irreversibility at the nanoscale.