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Equilibrium free energies from non-equilibrium trajectories with relaxation fluctuation spectroscopy


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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Fig. 1: Schematic free-energy landscape illustrating ReFlucS.
Fig. 2: ReFlucS data analysis.
Fig. 3: Simulation results.
Fig. 4: Experimental system and results.
Fig. 5: Experimental free-energy profiles.


  1. 1.

    Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997).

    ADS  Article  Google Scholar 

  2. 2.

    Crooks, G. E. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721–2726 (1999).

    ADS  Article  Google Scholar 

  3. 3.

    Hummer, G. & Szabo, A. Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl Acad. Sci. USA 98, 3658–3661 (2001).

    ADS  Article  Google Scholar 

  4. 4.

    Liphardt, J., Dumont, S., Smith, S. B., Tinoco, I. & Bustamante, C. Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science 296, 1832–1835 (2002).

    ADS  Article  Google Scholar 

  5. 5.

    Collin, D. et al. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437, 231–234 (2005).

    ADS  Article  Google Scholar 

  6. 6.

    Harris, N. C., Song, Y. & Kiang, C. H. Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski’s equality. Phys. Rev. Lett. 99, 068101 (2007).

    ADS  Article  Google Scholar 

  7. 7.

    Shank, E. A., Cecconi, C., Dill, J. W., Marqusee, S. & Bustamante, C. The folding cooperativity of a protein is controlled by its chain topology. Nature 465, 637–640 (2010).

    ADS  Article  Google Scholar 

  8. 8.

    Gupta, A. N. et al. Experimental validation of free-energy-landscape reconstruction from non-equilibrium single-molecule force spectroscopy measurements. Nat. Phys. 7, 631–634 (2011).

    Article  Google Scholar 

  9. 9.

    Alemany, A., Mossa, A., Junier, I. & Ritort, F. Experimental free-energy measurements of kinetic molecular states using fluctuation theorems. Nat. Phys. 8, 688–694 (2012).

    Article  Google Scholar 

  10. 10.

    Saira, O. P. et al. Test of the Jarzynski and Crooks fluctuation relations in an electronic system. Phys. Rev. Lett. 109, 180601 (2012).

    ADS  Article  Google Scholar 

  11. 11.

    An, S. M. et al. Experimental test of the quantum Jarzynski equality with a trapped-ion system. Nat. Phys. 11, 193–199 (2015).

    Article  Google Scholar 

  12. 12.

    Jarzynski, C. Equalities and Inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Annu. Rev. Condens. Matter Phys. 2, 329–351 (2011).

    ADS  Article  Google Scholar 

  13. 13.

    Dufrene, Y. F. et al. Five challenges to bringing single-molecule force spectroscopy into living cells. Nat. Methods 8, 123–127 (2011).

    Article  Google Scholar 

  14. 14.

    Kopelevich, D. I., Panagiotopoulos, A. Z. & Kevrekidis, I. G. Coarse-grained kinetic computations for rare events: Application to micelle formation. J. Chem. Phys. 122, 044908 (2005).

    ADS  Article  Google Scholar 

  15. 15.

    Hayashi, K., Ueno, H., Iino, R. & Noji, H. Fluctuation theorem applied to F-1-ATPase. Phys. Rev. Lett. 104, 218103 (2010).

    ADS  Article  Google Scholar 

  16. 16.

    Beltran-Villegas, D. J., Sehgal, R. M., Maroudas, D., Ford, D. M. & Bevan, M. A. A Smoluchowski model of crystallization dynamics of small colloidal clusters. J. Chem. Phys. 135, 154506 (2011).

    ADS  Article  Google Scholar 

  17. 17.

    Edwards, T. D., Yang, Y. G., Beltran-Villegas, D. J. & Bevan, M. A. Colloidal crystal grain boundary formation and motion. Sci. Rep. 4, 6132 (2014).

    ADS  Article  Google Scholar 

  18. 18.

    Prinz, J. H. et al. Markov models of molecular kinetics: Generation and validation. J. Chem. Phys. 134, 174105 (2011).

    ADS  Article  Google Scholar 

  19. 19.

    Adib, A. B. Free energy surfaces from nonequilibrium processes without work measurement. J. Chem. Phys. 124, 144111 (2006).

    ADS  Article  Google Scholar 

  20. 20.

    Adib, A. B. Symmetry relations in chemical kinetics arising from microscopic reversibility. Phys. Rev. Lett. 96, 028307 (2006).

    ADS  Article  Google Scholar 

  21. 21.

    Hummer, G. & Kevrekidis, I. G. Coarse molecular dynamics of a peptide fragment: Free energy, kinetics, and long-time dynamics computations. J. Chem. Phys. 118, 10762–10773 (2003).

    ADS  Article  Google Scholar 

  22. 22.

    Sriraman, S., Kevrekidis, L. G. & Hummer, G. Coarse master equation from Bayesian analysis of replica molecular dynamics simulations. J. Phys. Chem. B 109, 6479–6484 (2005).

    Article  Google Scholar 

  23. 23.

    Hummer, G. Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica molecular dynamics simulations. New J. Phys. 7, 34 (2005).

    Article  Google Scholar 

  24. 24.

    Gear, C. W., Kaper, T. J., Kevrekidis, I. G. & Zagaris, A. Projecting to a slow manifold: Singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst. 4, 711–732 (2005).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Constable, G. W. A., McKane, A. J. & Rogers, T. Stochastic dynamics on slow manifolds. J. Phys. A 46, 295002 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Neupane, K., Manuel, A. P. & Woodside, M. T. Protein folding trajectories can be described quantitatively by one-dimensional diffusion over measured energy landscapes. Nat. Phys. 12, 700–703 (2016).

    Article  Google Scholar 

  27. 27.

    Zheng, W. W. & Best, R. B. Reduction of all-atom protein folding dynamics to one-dimensional diffusion. J. Phys. Chem. B 119, 15247–15255 (2015).

    Article  Google Scholar 

  28. 28.

    Best, R. B. & Hummer, G. Coordinate-dependent diffusion in protein folding. Proc. Natl Acad. Sci. USA 107, 1088–1093 (2010).

    ADS  Article  Google Scholar 

  29. 29.

    Socci, N. D., Onuchic, J. N. & Wolynes, P. G. Diffusive dynamics of the reaction coordinate for protein folding funnels. J. Chem. Phys. 104, 5860–5868 (1996).

    ADS  Article  Google Scholar 

  30. 30.

    Best, R. B. & Hummer, G. Diffusive model of protein folding dynamics with Kramers turnover in rate. Phys. Rev. Lett. 96, 228104 (2006).

    ADS  Article  Google Scholar 

  31. 31.

    Beltran-Villegas, D. J., Sehgal, R. M., Maroudas, D., Ford, D. M. & Bevan, M. A. Colloidal cluster crystallization dynamics. J. Chem. Phys. 137, 134901 (2012).

    ADS  Article  Google Scholar 

  32. 32.

    Sisan, D. R., Halter, M., Hubbard, J. B. & Plant, A. L. Predicting rates of cell state change caused by stochastic fluctuations using a data-driven landscape model. Proc. Natl Acad. Sci. USA 109, 19262–19267 (2012).

    ADS  Article  Google Scholar 

  33. 33.

    Stephens, G. J., de Mesquita, M. B., Ryu, W. S. & Bialek, W. Emergence of long timescales and stereotyped behaviors in Caenorhabditis elegans. Proc. Natl Acad. Sci. USA 108, 7286–7289 (2011).

    ADS  Article  Google Scholar 

  34. 34.

    Stephens, G. J., Johnson-Kerner, B., Bialek, W. & Ryu, W. S. Dimensionality and dynamics in the behavior of C. elegans. PLoS Comput. Biol. 4, e1000028 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Diaconis, P. & Rolles, S. W. W. Bayesian analysis for reversible Markov chains. Ann. Stat. 34, 1270–1292 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Bacallado, S., Chodera, J. D. & Pande, V. Bayesian comparison of Markov models of molecular dynamics with detailed balance constraint. J. Chem. Phys. 131, 045106 (2009).

    ADS  Article  Google Scholar 

  37. 37.

    Siettos, C. I., Graham, M. D. & Kevrekidis, I. G. Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation. J. Chem. Phys. 118, 10149–10156 (2003).

    ADS  Article  Google Scholar 

  38. 38.

    Chodera, J. D. & Pande, V. S. Splitting probabilities as a test of reaction coordinate choice in single-molecule experiments. Phys. Rev. Lett. 107, 098102 (2011).

    ADS  Article  Google Scholar 

  39. 39.

    Stavis, S. M., Geist, J., Gaitan, M., Locascio, L. E. & Strychalski, E. A. DNA molecules descending a nanofluidic staircase by entropophoresis. Lab Chip 12, 1174–1182 (2012).

    Article  Google Scholar 

  40. 40.

    Strychalski, E. A., Geist, J., Gaitan, M., Locascio, L. E. & Stavis, S. M. Quantitative measurements of the size scaling of linear and circular DNA in nanofluidic slitlike confinement. Macromolecules 45, 1602–1611 (2012).

    ADS  Article  Google Scholar 

  41. 41.

    Leith, J. S. et al. Free energy of a polymer in slit-like confinement from the Odijk regime to the bulk. Macromolecules 49, 9266–9271 (2016).

    ADS  Article  Google Scholar 

  42. 42.

    Chen, J. Z. Y. & Sullivan, D. E. Free energy of a wormlike polymer chain confined in a slit: Crossover between two scaling regimes. Macromolecules 39, 7769–7773 (2006).

    ADS  Article  Google Scholar 

  43. 43.

    Gore, J., Ritort, F. & Bustamante, C. Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements. Proc. Natl Acad. Sci. USA 100, 12564–12569 (2003).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Palassini, M. & Ritort, F. Improving free-energy estimates from unidirectional work measurements: Theory and experiment. Phys. Rev. Lett. 107, 060601 (2011).

    ADS  Article  Google Scholar 

  45. 45.

    Doob, J. L. The Brownian movement and stochastic equations. Ann. Math. 43, 351–369 (1942).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Carpenter, B. et al. Stan: A probabilistic programming language. J. Stat. Softw. 76, 1–29 (2017).

    Article  Google Scholar 

  47. 47.

    Stan Development Team Stan Modeling Language User’s Guide and Reference Manual, Version 2.11.0; (2016).

  48. 48.

    Stan Development Team RStan: the R interface to Stan, Version 2.10.1; (2016).

  49. 49.

    Grassia, P. S., Hinch, E. J. & Nitsche, L. C. Computer simulations of Brownian motion of complex systems. J. Fluid Mech. 282, 373–403 (1995).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Tang, J. et al. Revisiting the conformation and dynamics of DNA in slitlike confinement. Macromolecules 43, 7368–7377 (2010).

    ADS  Article  Google Scholar 

  51. 51.

    Stavis, S. M., Strychalski, E. A. & Gaitan, M. Nanofluidic structures with complex three-dimensional surfaces. Nanotechnology 20, 165302 (2009).

    ADS  Article  Google Scholar 

  52. 52.

    Strychalski, E. A., Levy, S. L. & Craighead, H. G. Diffusion of DNA in nanoslits. Macromolecules 41, 7716–7721 (2008).

    ADS  Article  Google Scholar 

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We thank M. Zwolak and J. Hubbard for careful reading of the manuscript and helpful comments.

Author information




D.R. performed the simulations. D.R., E.A.S. and S.M.S. wrote the manuscript. D.R., E.A.S. and S.M.S. analysed the data. D.R. and C.J. developed the theoretical and statistical approaches. All authors discussed the results and commented on the manuscript at all stages.

Corresponding authors

Correspondence to David Ross or Samuel M. Stavis.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary Information, Supplementary Figures 1–16, Supplementary Table 1, Supplementary References 1–32

Supplementary Movie 1

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).

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