Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies

Abstract

Atomic force microscopes and optical tweezers are widely used to probe the mechanical properties of individual molecules and molecular interactions, by exerting mechanical forces that induce transitions such as unfolding or dissociation. These transitions often occur under nonequilibrium conditions and are associated with hysteresis effects—features usually taken to preclude the extraction of equilibrium information from the experimental data. But fluctuation theorems1,2,3,4,5 allow us to relate the work along nonequilibrium trajectories to thermodynamic free-energy differences. They have been shown to be applicable to single-molecule force measurements6 and have already provided information on the folding free energy of a RNA hairpin7,8. Here we show that the Crooks fluctuation theorem9 can be used to determine folding free energies for folding and unfolding processes occurring in weak as well as strong nonequilibrium regimes, thereby providing a test of its validity under such conditions. We use optical tweezers10 to measure repeatedly the mechanical work associated with the unfolding and refolding of a small RNA hairpin11 and an RNA three-helix junction12. The resultant work distributions are then analysed according to the theorem and allow us to determine the difference in folding free energy between an RNA molecule and a mutant differing only by one base pair, and the thermodynamic stabilizing effect of magnesium ions on the RNA structure.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Force–extension curves.
Figure 2: Test of the CFT using an RNA hairpin.
Figure 3: Free-energy recovery and test of the CFT for non-gaussian work distributions.
Figure 4: Use of CFT to extract the stabilizing contribution of Mg 2+ to the free energy of the S15 three-helix junction (wild type).

References

  1. Evans, D. J., Cohen, E. G. & Morriss, G. P. Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404 (1993)

    Article  ADS  CAS  Google Scholar 

  2. Gallavotti, G. & Cohen, E. G. D. Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995)

    Article  ADS  CAS  Google Scholar 

  3. Ciliberto, S. & Laroche, C. An experimental test of the Gallavotti-Cohen fluctuation theorem. J. Phys. IV 8(Proc. 6), 215–220 (1998)

    Google Scholar 

  4. Evans, D. J. & Searles, D. J. The fluctuation theorem. Adv. Phys. 51, 1529–1585 (2002)

    Article  ADS  Google Scholar 

  5. Wang, G. M., Sevick, E. M., Mittag, E., Searles, D. J. & Evans, D. J. Experimental demonstration of violations of the second law of thermodynamics for small systems and short timescales. Phys. Rev. Lett. 89, 050601 (2002)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  8. Ritort, F., Bustamante, C. & Tinoco, I. Jr A two-state kinetic model for the unfolding of single molecules by mechanical force. Proc. Natl Acad. Sci. USA 99, 13544–13548 (2002)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  10. Smith, S. B., Cui, Y. & Bustamante, C. An optical-trap force transducer that operates by direct measurement of light momentum. Methods Enzymol. 361, 134–162 (2003)

    Article  CAS  Google Scholar 

  11. McManus, M. T., Petersen, C. P., Haines, B. B., Chen, J. & Sharp, A. P. Gene silencing using micro-RNA designed hairpins. RNA 8, 842–850 (2002)

    Article  CAS  Google Scholar 

  12. Serganov, A. et al. Role of conserved nucleotides in building the 16S rRNA binding site for ribosomal protein S15. J. Mol. Biol. 305, 785–803 (2002)

    Article  Google Scholar 

  13. Ritort, F. Work fluctuations, transient violations of the second law and free-energy recovery methods. Semin. Poincaré 2, 193–226 (2003)

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  15. Park, S. & Schulten, K. Calculating potentials of mean force from steered molecular dynamics simulations. J. Chem. Phys. 120, 5946–5961 (2004)

    Article  ADS  CAS  Google Scholar 

  16. Liphardt, J., Onoa, B., Smith, S. B., Tinoco, I. Jr & Bustamante, C. Reversible unfolding of single RNA molecules by mechanical force. Science 292, 733–737 (2001)

    Article  ADS  CAS  Google Scholar 

  17. Zuckerman, D. M. & Woolf, T. B. Theory of systematic computational error in free energy differences. Phys. Rev. Lett. 89, 180602 (2002)

    Article  ADS  Google Scholar 

  18. 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)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  19. Shirts, R., Bair, E., Hooker, G. & Pande, V. S. Equilibrium free energies from nonequilibrium measurements using maximum likelihood methods. Phys. Rev. Lett. 91, 140601 (2003)

    Article  ADS  Google Scholar 

  20. Bennett, C. H. Efficient estimates of free energy differences from Monte Carlo data. J. Comp. Phys. 22, 245–268 (1976)

    Article  ADS  Google Scholar 

  21. SantaLucia, J. Jr & Hicks, D. The thermodynamics of DNA structural motifs. Annu. Rev. Biophys. Biomol. Struct. 33, 415–440 (2004)

    Article  CAS  Google Scholar 

  22. Robertus, D. J. et al. Structure of yeast phenylalanine tRNA at 3A resolution. Nature 250, 546–551 (1974)

    Article  ADS  CAS  Google Scholar 

  23. Long, D. M. & Uhlenbeck, O. C. Self-cleaving catalytic RNA. FASEB J. 7, 25–30 (1993)

    Article  CAS  Google Scholar 

  24. Cate, J. H. & Doudna, J. A. Metal binding sites in the major groove of a large ribozyme domain. Structure 4, 1221–1229 (1996)

    Article  CAS  Google Scholar 

  25. Zuker, M. Mfold web server for nucleic acid folding and hybridization predictions. Nucleic Acids Res. 31, 3406–3415 (2003)

    Article  CAS  Google Scholar 

  26. Turner, D. H. in Nucleic Acids: Structures, Properties and Functions (eds Bloomfield, D. A., Crothers, D. M. & Tinoco, I. Jr) Ch. 7 (Univ. Press, New York, 2000)

    Google Scholar 

  27. Carrion-Vazquez, M., et al. Protein nanomechanics studied by AFM single-molecule force spectroscopy. In Emerging Techniques in Biophysics (eds Arrondo, J. L. R. & Alonso, A.) (Biophys. Monogr. Ser., Springer, Heidelberg, in the press)

  28. Milligan, J. F., Groebe, D. R., Witherell, G. W. & Uhlenbeck, O. C. Oligoribonucleotide synthesis using T7 RNA polymerase and synthetic DNA templates. Nucleic Acids Res. 15, 8783–8798 (1987)

    Article  CAS  Google Scholar 

  29. Manosas, M. & Ritort, F. Thermodynamic and kinetic aspects of RNA pulling experiments. Biophys. J. 88, 3224–3242 (2005)

    Article  ADS  CAS  Google Scholar 

  30. Hummer, G. Fast-growth thermodynamics integration: Error and efficiency analysis. J. Chem. Phys. 114, 7330–7337 (2001)

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We thank G. Hummer and A. Szabo for many discussions and G. E. Crooks, D. Chandler and J. Liphardt for a careful reading of the manuscript. F.R. was supported by the Spanish Research council and the Catalan Government (Distinció de la Generalitat). C.J. was supported by an NIH grant and the US Department of Energy. I.T. was supported by an NIH grant. C.B. was supported by the Howard Hughes Medical Institute and the David and Lucile Packard Foundation.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to F. Ritort or C. Bustamante.

Ethics declarations

Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Supplementary information

Supplementary Notes

This contains the Supplementary Discussion with Supplementary Figures and accompanying legends embedded in the text. This file also contains additional references. (PDF 505 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Collin, D., Ritort, F., Jarzynski, C. et al. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437, 231–234 (2005). https://doi.org/10.1038/nature04061

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature04061

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing