DOI: 10.1126/science.1071152. Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality

Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality

10.1126/science.1071152

Crossref journal-article

American Association for the Advancement of Science (AAAS)

Science (221)

Abstract

Recent advances in statistical mechanical theory can be used to solve a fundamental problem in experimental thermodynamics. In 1997, Jarzynski proved an equality relating the irreversible work to the equilibrium free energy difference, Δ G . This remarkable theoretical result states that it is possible to obtain equilibrium thermodynamic parameters from processes carried out arbitrarily far from equilibrium. We test Jarzynski's equality by mechanically stretching a single molecule of RNA reversibly and irreversibly between two conformations. Application of this equality to the irreversible work trajectories recovers the Δ G profile of the stretching process to within k B T /2 (half the thermal energy) of its best independent estimate, the mean work of reversible stretching. The implementation and test of Jarzynski's equality provides the first example of its use as a bridge between the statistical mechanics of equilibrium and nonequilibrium systems. This work also extends the thermodynamic analysis of single molecule manipulation data beyond the context of equilibrium experiments.

Bibliography

Liphardt, J., Dumont, S., Smith, S. B., Tinoco, I., & Bustamante, C. (2002). Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynskiâs Equality. Science, 296(5574), 1832â1835.

Authors 5

Jan Liphardt (first)
Sophie Dumont (additional)
Steven B. Smith (additional)
Ignacio Tinoco (additional)
Carlos Bustamante (additional)

References 31 Referenced 1,023

10.1103/PhysRev.83.34
10.1021/j100176a002
10.1063/1.1353552
10.1021/j100170a054
10.1103/PhysRevLett.78.2690
10.1103/PhysRevE.56.5018
10.1103/PhysRevE.60.2721
10.1023/A:1023208217925
Jarzynski's equality is frequently written in terms of the Helmholtz free energy for systems at constant temperature and volume; for our isothermal-isobaric system the appropriate thermodynamic variable is the Gibbs free energy.
Supplementary material is available on Science Online.
10.1073/pnas.081074598
10.1073/pnas.071034098
10.1103/PhysRevE.61.2361
10.1143/PTP.38.1031
10.1103/PhysRevLett.71.2401
10.1103/PhysRevLett.74.2694
10.1103/PhysRevE.50.1645
10.1088/0305-4470/31/16/003
10.1023/A:1004589714161
10.1023/A:1004541830999
The relation between the three Δ G estimates W A W FD and W JE can be understood by rewriting Jarzynski's equality as a cumulant expansion ΔG=∑n=1∞ (−β)n−1ωn/n!where ω n is the n th cumulant of work values (3). Keeping only the first term of the expansion yields Δ G = W A and is valid at equilibrium. Keeping the first two terms of the expansion yields Δ G = W A – βσ 2 /2 which corresponds to W FD and is valid when all higher ( n > 2) cumulants vanish. This is the case when the work distribution is Gaussian as expected in the near-equilibrium regime from the central limit theorem (3). For large enough N WJE=∑n=1∞ (−β)n−1ωn/n!and this is valid arbitrarily far from equilibrium. In the near-equilibrium regime W FD = W JE and the dotted and dashed lines in Fig. 3A are consequently equal.
10.1126/science.273.5282.1678
10.1126/science.1058498
10.1126/science.271.5250.795
Although this scheme reduces the effect of drift on the estimates of the work it does not eliminate it. Systematic errors can still become large especially for slow switching.
The observation that early stretching is reversible allows us to retain the lower integration limit of Eq. 2 z = 0 at 341 nm and thus neglect the switching history of the system up to this extension even at the two fast switching rates.
J. Liphardt S. Dumont S. B. Smith I. Tinoco C. Bustamante data not shown.
Because the values in the left tail of the work distribution contribute most to W JE widening of the standard deviation of work values leads to underestimation of Δ G.
Here the quoted error is the standard error of the mean σ m /( m − 1) 1/2 . A distribution of W JE estimates of Δ G is not necessarily Gaussian.
O. Mazonka C. Jarzynski arXiv:cond-mat/9912121 (1999).
We thank C. Jarzynski G. E. Crooks and D. Chandler for valuable discussions and suggestions and an anonymous referee for suggesting the analytic sampling efficiency estimation. Supported by the Program in Mathematics and Molecular Biology through a Burroughs Wellcome Fund Fellowship (J.T.L.); the Natural Sciences and Engineering Research Council of Canada and Fonds pour la Formation de Chercheurs et l'Aide à la Recherche du Québec (S.D.); NIH grants GM-10840 and GM-32543; U.S. Department of Energy grants DE-FG03-86ER60406 and DE-AC03-76SF00098; and NSF grants MBC-9118482 and DBI-9732140. Supporting Online Material www.sciencemag.org/cgi/content/full/296/5574/1832/DC1 Materials and Methods

Dates

Type	When
Created	23 years, 1 month ago (July 27, 2002, 5:54 a.m.)
Deposited	1 year, 7 months ago (Jan. 9, 2024, 10:50 p.m.)
Indexed	1 week ago (Aug. 20, 2025, 8:35 a.m.)
Issued	23 years, 2 months ago (June 7, 2002)
Published	23 years, 2 months ago (June 7, 2002)
Published Print	23 years, 2 months ago (June 7, 2002)

Funders 0

None

BibTeX

@article{Liphardt_2002, title={Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski’s Equality}, volume={296}, ISSN={1095-9203}, url={http://dx.doi.org/10.1126/science.1071152}, DOI={10.1126/science.1071152}, number={5574}, journal={Science}, publisher={American Association for the Advancement of Science (AAAS)}, author={Liphardt, Jan and Dumont, Sophie and Smith, Steven B. and Tinoco, Ignacio and Bustamante, Carlos}, year={2002}, month=jun, pages={1832–1835} }

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