Recent advances in non-equilibrium statistical mechanics and single-molecule measurements have enabled the determination of equilibrium free energies from non-equilibrium work measurements for fluctuating systems ranging from biological molecules to quantum oscillators. However, for many important non-equilibrium processes, it is difficult or impossible to apply and measure the work required to drive the system through the relevant conformational changes. Here, we show that it is possible, with an appropriate extrapolation to infinite temporal scale and zero spatial scale, to determine equilibrium free energies, without work measurement, by analysing the stochastic trajectories of single biomolecules or other nanoscale, fluctuating systems as they spontaneously relax from a non-equilibrium initial state. We validate the method with simulations and demonstrate its application by determining the free-energy profile for DNA molecules in a structured nanofluidic environment with an experimental protocol that mimics many natural processes with energy injection followed by thermal relaxation.
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We thank M. Zwolak and J. Hubbard for careful reading of the manuscript and helpful comments.
The authors declare no competing interests.
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Supplementary Information, Supplementary Figures 1–16, Supplementary Table 1, Supplementary References 1–32
Supplementary Movie 1 shows a longer sequence of the motion of the same two molecules that are shown in Fig. 4b of the main text. The molecule on the top right moves primarily forward (left to right) into regions of lower free energy. The molecule on the bottom left moves backward two steps before moving forward. The thin vertical lines show the position of step edges. The lateral distance between step edges is 4 µm. The time step between frames of the movie is 5 s.
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Ross, D., Strychalski, E.A., Jarzynski, C. et al. Equilibrium free energies from non-equilibrium trajectories with relaxation fluctuation spectroscopy. Nature Phys 14, 842–847 (2018). https://doi.org/10.1038/s41567-018-0153-5