How does the size of a system affect its thermodynamic irreversibility? A deft experiment that observes the unfolding and refolding of a single molecule of RNA provides insights into the question at a small scale.
When vigorously stirred, a cup of coffee will heat up; a cup of hot coffee, however, will never stir a spoon (Fig. 1). This type of irreversibility is a cornerstone of macroscopic thermodynamics. But does it still hold when the spoon is extremely small? Newly discovered relationships between heat, work and energy have revealed unexpected features of small systems when they are driven far from equilibrium — relationships that are becoming increasingly relevant as experiments probe ever-tinier systems, including the nanoscale machinery of living cells. On page 231 of this issue, Collin et al.1 describe a single-molecule experiment that not only validates one prominent recent postulate, known as the Crooks fluctuation theorem, but also provides a new method for quantifying the difference in equilibrium free energy — the useful work expected to be extracted from a system — between two biomolecular states.
The Crooks fluctuation theorem2 (CFT) describes the exchange of energy between a system and its environment in forward and reverse processes. It generalizes the so-called Jarzynski equality3, which relates equilibrium free energy to the work done in multiple, non-equilibrium measurements. The application of the CFT to a single-molecule experiment can be understood by considering an idealized ‘dashpot’ — a plunger that can move through a viscous fluid held at a constant temperature. The force required to move a large, slow-moving plunger is proportional to its speed; all the mechanical work performed is lost, irretrievably, to the environment as heat. At a given speed, the work required to move the plunger between two well-defined states is always the same: viewed microscopically, the resistance to the motion is just the statistically averaged effect of many collisions between the plunger and the fast-moving molecules of the fluid.
If the plunger is very small, however, this average is performed over far fewer collisions, and the random, brownian movements of the plunger become significant. When an excess of molecular collisions occurs in the direction in which the plunger is moving, the work done by the operator is reduced. This work can on rare occasions be reduced to zero, or even become negative — meaning that the system performs work on the operator. The mechanical work needed to move a small plunger is no longer a precise, deterministic variable; rather, it is described by a statistical distribution.
The CFT stipulates that the probability of exerting a given amount of mechanical work on the plunger in the conventional, forward process, divided by the probability of getting the same amount of work back from the reverse process, depends exponentially on the amount of work lost to the environment. This implies that the likelihood of dissipating virtually no work is the same in both the forward and reverse directions. Because the amount of work dissipated increases as a system gets larger, in a sufficiently small system the likelihood that ‘dissipation-free’ events will occur becomes large enough for them to be observed directly.
Collin et al.1 studied the mechanical unfolding and refolding of individual RNA molecules, applying and measuring forces using optical tweezers — a focused laser beam that acts as an ultra-sensitive spring to trap and hold small particles. The two ends of an RNA molecule, folded into junctions and hairpin shapes, were each linked to a small bead, one held by a micropipette and the other by the optical tweezers. In each experimental cycle, the beads were moved apart to produce an unzipping or unfolding transition, and then moved back together to allow refolding.
By integrating the force over the change in length of the molecule through many cycles, Collin and colleagues1 obtained separate statistical distributions for the work done on the molecule during the forward and reverse transitions. They used these to verify that the CFT holds. By finding the energy at which the two distributions crossed for a range of speeds of bead movement, the authors were able to evaluate the free energy of transition between the folded and unfolded state of an individual RNA hairpin. They also demonstrated the robustness of the CFT by using their analysis to quantify the equilibrium free energy accurately even when the beads were moved apart too fast for the system to respond — in other words, when the system was taken far from equilibrium.
Collin and colleagues' work clearly verifies that the CFT can be applied to small biomolecular systems. But what are its wider implications for single-molecule research? The method provides a unique approach to quantifying the free energy in two-state systems — even in those systems far from equilibrium, where the transition between the two states requires very different amounts of work depending on its direction (a phenomenon known as hysteresis). This is important, as it is often the drifting baseline of a measuring instrument that limits how slowly, and thus how close to equilibrium, an experiment can be performed.
The CFT method is unlikely, however, to be suitable for all molecular systems and probe techniques. Its success in Collin and colleagues' experiments is due in part to certain features of the RNA hairpins and junctions used that made force a relatively smooth function of displacement between the well-defined initial (folded) and final (unfolded) states. Specifically, the free-energy landscape between these states resembles a ‘staircase’ of small energetic steps of short duration spread over an easily observable distance of many nanometres. The nonlinear, elastic behaviour of the polymer linkers and unfolded RNA chain also ensured that forward and reverse responses could be easily aligned in the presence of instrument drift. These attributes, together with the exquisite sensitivity of the optical tweezers — in regard to both the force applied and the displacement caused — allowed accurate integration for calculating the work during both unfolding and refolding. It also allowed for the subtraction of energetic contributions from the optical tweezers and chemical linkers, a necessary step in precisely determining the free-energy difference of the molecular transition. Extending Collin and colleagues' technique to other biomolecular transitions is thus sure to present fresh challenges.
Even so, this exciting experiment has provided insight into the way that recently developed fluctuation theories can be applied to transitions in biomolecular systems. Not only does physics teach us new ways to understand single-molecule measurements, but such measurements also feed back into theory, extending our knowledge of thermodynamics to systems that are small and out of equilibrium. Stir a cup of coffee with a small enough spoon, and the coffee might just stir you.
Collin, D. et al. Nature 437, 231–234 (2005).
Crooks, G. E. Phys. Rev. E. 60, 2721–2726 (1999).
Jarzynski, C. Phys. Rev. Lett. 78, 2690–2693 (1997).
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Physical Review E (2014)