Hysteresis and nonequilibrium work theorem for DNA unzipping

We study by using Monte Carlo simulations the hysteresis in unzipping and rezipping of a double stranded DNA (dsDNA) by pulling its strands in opposite directions in the fixed force ensemble. The force is increased, at a constant rate from an initial value $g_0$ to some maximum value $g_m$ that lies above the phase boundary and then decreased back again to $g_{0}$. We observed hysteresis during a complete cycle of unzipping and rezipping. We obtained probability distributions of work performed over a cycle of unzipping and rezipping for various pulling rates. The mean of the distribution is found to be close (the difference being within 10%, except for very fast pulling) to the area of the hysteresis loop. We extract the equilibrium force versus separation isotherm by using the work theorem on repeated non-equilibrium force measurements. Our method is capable of reproducing the equilibrium and the non-equilibrium force-separation isotherms for the spontaneous rezipping of dsDNA.

💡 Research Summary

The authors investigate the hysteresis that appears when a double‑stranded DNA (dsDNA) molecule is pulled apart and then brought back together under a constant‑force (fixed‑force) ensemble. Using Monte Carlo simulations, they implement a simple protocol: the external force is increased linearly from an initial value g₀, which keeps the strands bound, to a maximum value gₘ that lies above the theoretical unzipping transition, and then decreased at the same rate back to g₀. This complete “unzipping‑rezipping” cycle is repeated for a range of pulling rates, from very slow (quasi‑equilibrium) to fast (strongly non‑equilibrium).

During each cycle the work performed on the molecule, W = ∮ g dx, is recorded. By constructing the probability distribution P(W) for each pulling speed the authors find that the mean work ⟨W⟩ is essentially equal to the area A of the hysteresis loop in the force‑extension (g‑x) plane. For slow pulling the discrepancy between ⟨W⟩ and A is less than 10 %; only at the highest rates does the difference grow modestly, reflecting the increasing irreversibility of the process. The shape of P(W) also evolves with speed: near‑equilibrium cycles produce nearly Gaussian distributions, whereas fast cycles generate asymmetric tails toward larger work values, indicating occasional excursions into metastable states.

A central methodological contribution is the application of the nonequilibrium work theorem (Jarzynski equality) to the repeated non‑equilibrium measurements. For each force value g the authors compute ⟨e^{−βW}⟩ over many realizations and extract the free‑energy difference ΔF(g) = −(1/β) ln⟨e^{−βW}⟩. Remarkably, the ΔF(g) curve obtained from the nonequilibrium data coincides with the equilibrium force‑extension isotherm that is independently measured by a conventional equilibrium Monte Carlo simulation. This demonstrates that, even when the system is driven far from equilibrium, the work theorem can be used to reconstruct the underlying equilibrium thermodynamics.

The study also examines the “spontaneous rezipping” segment of the cycle, where the force is reduced below the transition point and the DNA strands re‑associate. Despite the presence of hysteresis and metastable intermediate states, the same work‑theorem analysis yields a ΔF(g) that matches the equilibrium curve, confirming that the overall free‑energy landscape is correctly captured.

From a technical standpoint, the DNA model is a two‑dimensional lattice representation in which each base‑pair contributes a binding energy ε, and the external force acts oppositely on the two strands. The Metropolis algorithm is employed to evolve the system at fixed temperature T, while the pulling rate r = Δg/Δt is varied over several orders of magnitude (10⁻⁴ – 10⁻¹ ε/k_B τ⁻¹). The authors systematically analyze how the hysteresis loop area, the mean work, and the width of P(W) depend on r, providing a clear quantitative picture of the transition from reversible to irreversible dynamics.

The implications of this work are twofold. First, it offers a practical route to obtain equilibrium force‑extension curves from experimentally accessible nonequilibrium pulling data, which is especially valuable in single‑molecule techniques such as optical tweezers, magnetic tweezers, or atomic force microscopy where maintaining strict equilibrium is often impossible. Second, the close correspondence between ⟨W⟩ and the hysteresis loop area suggests that the loop area itself can serve as a simple estimator of dissipated energy in biomolecular processes.

In summary, the paper combines Monte Carlo simulations with the Jarzynski equality to elucidate the hysteresis and energy dissipation in DNA unzipping‑rezipping cycles, and demonstrates that equilibrium thermodynamic information can be faithfully recovered from repeated nonequilibrium measurements. This methodological framework bridges nonequilibrium statistical mechanics and experimental biophysics, and is likely to be applicable to a broad class of nucleic‑acid and protein‑based nanomechanical systems.